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Quantum knots tied for the very first time. Why this is important
Finnish and American scientist made knots out of solitary waves, or knot solitons. This was the first time this was demonstrated possible, though theoretically predicted for decades. These sort of
knots are thought to play an important role in the quantum-mechanical foundations of nature, though these have remained elusive in quantum dynamics.
Finnish and American scientist made knots out of solitary waves, or knot solitons. This was the first time this was demonstrated possible, though theoretically predicted for decades. These sort of
knots are thought to play an important role in the quantum-mechanical foundations of nature, though these have remained elusive in quantum dynamics.
Visualization of the structure of the created quantum knot. Each colorful band represents a set of nearby directions of the quantum field that is knotted. Note that each band is twisted and linked
with the others once. Untying the knot requires the bands to separate, which is not possible without breaking them. Credit: David Hall.
The knots we generally think of are typically tied on a rope or string with two ends. These sort of knots are thought topologically stable because the ends of the rope are considered glued together.
But quantum knots aren’t because they can be untied without cutting the “rope”.
The quantum knots were made in a rubidium gas of superfluid atoms, also known as a Bose–Einstein condensate (BEC). As I explained in a previous article I wrote about slowing down and trapping light,
BEC ” was first predicted in the 1920s by Albert Einstein and the Indian physicist Satyendra Bose and it wasn’t until very late in 1995 that scientists were able to produce the necessary conditions
for this extreme state of matter to occur. At room temperature, atoms are incredibly fast and behave akin to billiard balls, bouncing off each other when they interact. As you lower the temperature
(remember temperature reflects atomic agitation), atoms and molecules move slower. Eventually, once you get to about 0.000001 degrees above absolute zero, atoms become so densely packed they behave
like one super atom, acting in unison. This is the domain of quantum mechanics so prepared for a lot of weirdness.”
The BEC superfluid (flows without viscosity) offered the ripe conditions for a field that assumes a certain direction at every point of space to exist. The field segregates into an infinite number of
linked rings, each with its own field direction. The knots made by the researchers at Aalto University (Finland) and Amherst College (USA), however, are solitons — waves that that roll at a constant
speed without changing shape. If all points in the magnetic field point up, the points forming the knot point down. “If you followed the magnetic field line, it would go toward the center, but at the
last minute it would peel away into a perpendicular direction,” said David Hall, a physicist at Amherst College for Gizmodo. “It’s a particular way of rotating these arrows that gives you this linked
Experimental images of the superfluid in the course of the knot tying process. Tying time advances from the left to right as indicated. The brightness denotes the particle density corresponding to
the field direction up or down. The black circles in the rightmost panel reveal the colorful torus shown in Figure 1 where the field direction points sideways. Credit: David Hall.
“For decades, physicists have been theoretically predicting that it should be possible to have knots in quantum fields, but nobody else has been able to make one. Now that we have seen these
exotic beasts, we are really excited to study their peculiar properties. Importantly, our discovery connects to a diverse set of research fields including cosmology, fusion power, and quantum
computers,” says research group leader Mikko Möttönen, Aalto University.
Möttönen claims the research will prove useful as a starting point for topological quantum computer. Transistors perform logic operations by shuttling bits of data, each assigned a value which is
either “0” or “1”. That’s how a classical, digital computer works. Quantum computers, however, use qubits or a quantum bit which can simultaneously exist in both states at once – both “0” and “1”.
This is known as a superposition, and if scientists can leverage it then information could be processed in parallel. Two-qubits can perform operations on four values, three on eight values and so on
in powers of two. Today’s computers have millions of transistors. Now imagine a quantum logic gate that works with millions of qubits. The computing force would be unheard of.
"A topological quantum computer would braid the qubits into the knots. The result does not depend on the positions of these things," said Möttönen. "If you move them around a little, it doesn’t
matter, so [such a computer] should be really robust against any error."
“This is the beginning of the story of quantum knots. It would be great to see even more sophisticated quantum knots to appear such as those with knotted cores. Also it would be important to
create these knots in conditions where the state of the quantum matter would be inherently stable. Such system would allow for detailed studies of the stability of the knot itself,” says Mikko | {"url":"http://www.thespaceacademy.org/2021/11/quantum-knots-tied-for-very-first-time.html","timestamp":"2024-11-12T23:54:43Z","content_type":"application/xhtml+xml","content_length":"171311","record_id":"<urn:uuid:e7507c91-8d5e-4726-a051-174317544412>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00490.warc.gz"} |
How do you calculate checksum?
How do you calculate checksum? To calculate the checksum of an API frame: Add all bytes of the packet, except the start delimiter 0x7E and the length (the second and third bytes). Keep only the
How do you calculate checksum?
To calculate the checksum of an API frame:
1. Add all bytes of the packet, except the start delimiter 0x7E and the length (the second and third bytes).
2. Keep only the lowest 8 bits from the result.
3. Subtract this quantity from 0xFF.
How do you calculate 16 bit checksum?
How are Internet Checksums Calculated?
1. Convert data into a series of 16-bit integers;
2. Calculate sum of all 16-bit integers, allowing for carry bit wrap around;
3. Take the 1’s complement of the final sum (flip the bits)
How many bytes are in a checksum?
A checksum is determined in one of two ways. Let’s say the checksum of a packet is 1 byte long. A byte is made up of 8 bits, and each bit can be in one of two states, leading to a total of 256 (28 )
possible combinations. Since the first combination equals zero, a byte can have a maximum value of 255.
How is Lin checksum calculated?
The LIN bus defines the use of one of two checksum algorithms to calculate the value in the eight-bit checksum field. Classic checksum is calculated by summing the data bytes alone, and enhanced
checksum is calculated by summing the data bytes and the protected ID.
What is checksum method?
A checksum is an error-detection method in a the transmitter computes a numerical value according to the number of set or unset bits in a message and sends it along with each message frame. If the
received checksum value matches the sent value, the transmission is considered to be successful and error-free.
What is checksum algorithm?
The checksum algorithm is really a special kind of hash function. A hash function is a function, or process, that can be used to map data of arbitrary size to data of a fixed size. The types of
hashes used for data integrity are distinguished by the presence or absence of keys and cryptographic properties.
What is 16 bit checksum?
A 16-bit sum-of-words checksum will detect all single bit errors and all error bursts of length 16 bits or fewer. It will also detect 99.998% of longer error bursts. A 32-bit sum will detect even
more errors.
What is checksum length?
The size can be indicated in the name of the hash, for example, SHA-256 makes a resulting checksum that is 256 bits. The checksum algorithm is really a special kind of hash function. A hash function
is a function, or process, that can be used to map data of arbitrary size to data of a fixed size.
What is difference between CAN and LIN?
The CAN bus allows for components to talk to each other seamlessly in the automobile. The LIN bus allows for further expansion to peripheral devices. This bus hierarchy was designed to save costs and
How does LIN protocol work?
LIN Frame Format. The LIN bus is a polled bus with a single master device and one or more slave devices. The master device contains both a master task and a slave task. If the slave task needs to
publish a response, it transmits one to eight data bytes to the bus followed by a checksum byte. | {"url":"https://missionalcall.com/2021/06/26/how-do-you-calculate-checksum/","timestamp":"2024-11-04T02:28:25Z","content_type":"text/html","content_length":"55923","record_id":"<urn:uuid:f498ee0f-24f3-42b6-ae37-e93dcebb2ed6>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00099.warc.gz"} |
MathSciDoc: An Archive for Mathematician
We introduce fusion bialgebras and their duals and systematically study their Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of
fusion rings. We prove the Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and their duals. We show that the Schur product property, Young's inequality and the sum-set
estimate hold for fusion bialgebras, but not always on their duals. If the fusion ring is the Grothendieck ring of a unitary fusion category, then these inequalities hold on the duals. Therefore,
these inequalities are analytic obstructions of categorification. We classify simple integral fusion rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less than 4080. We find 34
ones, 4 of which are group-like and 28 of which can be eliminated by applying the Schur product property on the dual. In general, these inequalities are obstructions to subfactorize fusion | {"url":"https://archive.ymsc.tsinghua.edu.cn/pacm_category/0131?show=view&size=5&from=1&target=searchall","timestamp":"2024-11-08T01:54:05Z","content_type":"text/html","content_length":"69675","record_id":"<urn:uuid:a5de8ed0-fe17-4d1c-98fb-81d95b7cd05d>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00763.warc.gz"} |
NCERT Solution for class 8 of Maths-pathanto
NCERT Solutions for class 8 Maths
Welcome to our comprehensive Class 8 Maths Discover a wide range of resources and information of class 8 Maths to help you excel in your studies. Our Maths class 8 covers all the chapters of Class 8
Maths curriculum, providing you with a holistic understanding of each topic. From understanding the fundamental concepts of nutrition and respiration to exploring the fascinating world of acids,
bases, and salts, we've got you covered. | {"url":"https://pathanto.com/Pathanto/Class%208/maths.php","timestamp":"2024-11-05T05:54:53Z","content_type":"text/html","content_length":"21720","record_id":"<urn:uuid:474d232e-6ca2-4127-9f3d-9226fec4cea2>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00108.warc.gz"} |
A box contains 90 discs, numbered from 1 to 90. If one disc is drawn
Doubtnut is No.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP
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Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi
medium and English medium for IIT JEE and NEET preparation | {"url":"https://www.doubtnut.com/qna/647934579","timestamp":"2024-11-04T19:57:26Z","content_type":"text/html","content_length":"195981","record_id":"<urn:uuid:3d78c9fe-74f4-41b6-b188-4d6e286ebb98>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00215.warc.gz"} |
X Axis: Category vs. Value
X Axis: Category vs. Value
In Microsoft Excel charts, there are different types of X axes. While the Y axis is a Value type axis, the X axis can be a Category type axis or a Value type axis. Using a Value axis, the data is
treated as continuously varying numerical data, and the marker is placed at a point along the axis which varies according to its numerical value. Using a Category axis, the data is treated as a
sequence of non-numerical text labels, and the marker is placed at a point along the axis according to its position in the sequence. The sample below illustrates the difference between Value and
Category Axes.
Our sample data is shown in the simple table below. The first column contains our X axis data, which can be treated as Categories or as Values. Note that the numbers are not equally spaced, nor do
they even appear in numerical order.
│ │ A │ B │ C │ D │
│ 1 │ │ Cats │ Dogs │ Fish │
│ 2 │ 1 │ 7 │ 7 │ 8 │
│ 3 │ 3 │ 6 │ 5 │ 7 │
│ 4 │ 2.5 │ 5 │ 4 │ 3 │
│ 5 │ 3.5 │ 4 │ 3 │ 2 │
We will display this data on two types of chart: a Line Chart, and an XY Scatter Chart; these charts look the same but behave differently.
Line Chart: X as Category Axis XY (Scatter) Chart: X as Value Axis
Line Chart with Category X Axis XY Chart with Value X Axis
Notice the X axis in this Line Chart. The labels seem to defy numerical order. But in a Category Axis, Notice the X axis in this XY Chart. The tick marks are uniformly spaced, as our eyes are
Excel merely deals with the labels in the order that they appear in the list, without regard to any trained to expect. The labels appear in ascending order, as expected (they could also be
possible numerical values these labels may have. The tick marks are evenly spaced, again regardless of formatted to appear in descending order, but in any rate, they change monotonically). But the
differences between the numerical values of the labels. For all intents, these labels could be data points do not line up like mindless robots above the next available tick label. The data
non-numeric attributes, like "Red", "Orange", "Blue", and "Green". The charted lines run from left to appear at a point above the appropriate value of the X axis, with the lines zig-zagging as
right continuously from one category to the next, with each adjacent pair of points horizontally required to put the points where they belong.
Combo Chart: X as Category AND Value Axis
You can put both a Line Series and an XY Series onto the same chart. This mixing of chart types on the same chart results in a Combination Chart. Excel provides a short selection of Combination
Charts in the Chart Wizard, but you can create them in almost any combination you want. (For a demonstration of Combination Chart creation, see Roll Your Own Combination Charts elsewhere on this
site.) In this case, I built the chart with two line series, then selected the Cats series and used Chart Type on the Chart menu to change the series to an XY type. Excel notices the chart types are
different, and adds secondary axes to the chart to accommodate the XY series (below left).
The two series can be plotted on the same set of axes. Double click the Cats series, and on the Axis tab, change the option from Secondary to Primary. The result is shown below right. Excel considers
the X axis as a Category axis, but it plots the XY series as if Category 1 has a value of 1, Category 2 has a value of 2, etc.
Line Series on Primary Category X Axis, Line and XY Series on Primary Category X Axis
XY Series on Secondary Value X Axis
In the right-hand chart, the first point of the XY series "Cats" is plotted above the first category listed (i.e., "1"), because its X value is 1. It is merely coincidental that the category label
and the X value are both 1. The second point is plotted above the third category ("2.5"), because its X value is 3. The third point is plotted between the second and third categories, because its X
value is 2.5, between 2 and 3. The fourth point is plotted between the third and fourth categories, because its X value is 3.5. The four points of the Line series "Dogs" are plotted in order above
the first through fourth categories, because they are listed in that order.
The different treatment of category data is important if you need to add an additional series for any reason, such as adding a reference line to the chart. This also is how you would work around a
line chart using the categories from the first series for all series; an example later in this article shows how to plot time series with different date values for each series.
Line Chart: X as Time-Scale Axis
There is one exception to a Line Chart having evenly spaced categories. This is the case of dates as X axis values. Excel calls this a Time-Scale axis, but it is more accurately called a Date-Scale
axis. Consider a Line Chart constructed from the sample data below.
│ │ A │ B │
│ 1 │ Date │ Value │
│ 2 │ 3/15/01 │ 1 │
│ 3 │ 4/15/01 │ 2 │
│ 4 │ 7/1/01 │ 3 │
│ 5 │ 8/15/01 │ 4 │
│ 6 │ 11/15/01 │ 5 │
│ 7 │ 12/15/01 │ 6 │
According to the scenario described earlier, Excel should stick the dates in the first column into equispaced category labels, as in the left hand chart below. If Excel recognizes the labels as
dates, however, it will actually provide a Time-Scale (Date-Scale) axis, and space the points according to the date, as shown below right. And it is a smart axis scale; if the Base Unit of the axis
scale is set to days, you would see slight differences in the spacing between months. While thirty days hath September, October hath 31.
Line Chart with Category X Axis Line Chart with Date-Scale X Axis
This type of axis is not super smart, however. If you add a series that had different values for these dates, Excel would still use the dates from the first series' X values (see workaround below).
In addition, even though it is called a 'Time' Scale axis, it is really a 'Date' Scale axis: all times from a given date are treated as integers, that is, they are plotted at midnight at the start of
that date. No partial days (i.e., times) are considered.
While this time scale capability might seem redundant, given the greater flexibility provided by the pure Value axis of the XY Scatter Charts, it has some uses. One advantage is that a
Date-Scale axis allows you to have a tick mark at exactly the first of each month, which the XY Chart's Value axis does not. The XY chart at right has a Value axis. It is not possible to XY Chart
get the same date spacing as in the line chart with a Date-Scale axis (above right). The axis minimum and maximum dates are the same, but the XY chart forces single major and minor with Value
spacing settings (this chart uses 60 days for major and 30 for minor). Because months have variable numbers of days, the axis becomes disconnected from the first of each month. X Axis
Date-Scale Chart for Series with Different Dates
Excel's line charts do not acknowledge different category data for different series. Instead, the chart uses the category values from the first series for all series. Excel's XY charts allow each
series to have distinct category values, but do not provide the nice Date-Scale axis scaling that line charts provide. This example illustrates the problem with multi-series line charts, and it uses
the combination chart approach described above to allow multiple sets of dates for multiple series.
│ │ A │ B │ C │ D │
│ 1 │ Date 1 │ Value 1 │ Date 2 │ Value 2 │
│ 2 │ 3/15/01 │ 1 │ 3/20/01 │ 2 │
│ 3 │ 4/15/01 │ 2 │ 4/1/01 │ 3 │
│ 4 │ 7/1/01 │ 3 │ 5/15/01 │ 4 │
│ 5 │ 8/15/01 │ 4 │ 9/15/01 │ 5 │
│ 6 │ 11/15/01 │ 5 │ 11/30/01 │ 6 │
│ 7 │ 12/15/01 │ 6 │ │ │
Select the data for the first series in columns A and B, and create a line chart with a Date-Scale axis (it will look like the example above). Add the data in columns C and D to the chart as a new
series: Copy the range, select the chart, and use Paste Special from the Edit menu to add the data as a New Series, with Series Names in the First Row and Categories in the First Column. The result
is showm below left. Although the second series has completely different dates in its source data, and you explicitly indicated these when you copied the range, the line chart ignores these and uses
the dates from the first series for both.
To enable multiple date ranges, convert the chart to a combination chart as described above. Keep the first series as a Line series, and change each additional series to an XY series: Select the
series, choose Chart Type from the Chart menu, and select an appropriate XY chart type. You should be able to use the F4 function key shortcut to repeat this last action on more series. As above,
Excel adds secondary axes for the XY series, so double click on the first XY series, and on the Axis tab of the dialog, select Primary; use F4 to repeat for any additional XY series. As shown in the
chart below at right, the series now reflect independent date scales. After changing an added series to an XY series, subsequent series added to the chart will be added as XY series.
Line Chart with Two Line Series Combo Chart with Line and XY Series
Order of Points for Axis Types
An interesting behavior of Date-Scale charts is illustrated with the data below. The points are the same as in the previous section, but rearranged out of sequence. Charts with the three axis types
shown above are repeated here with the rearranged data.
Time Data, Listed Out of Order
Line Chart with Category X Axis
│ │ A │ B │
│ 1 │ Date │ Value │
│ 2 │ 4/15/01 │ 2 │
│ 3 │ 3/15/01 │ 1 │
│ 4 │ 7/1/01 │ 3 │
│ 5 │ 8/15/01 │ 4 │
│ 6 │ 12/15/01 │ 6 │
│ 7 │ 11/15/01 │ 5 │
Line Chart with Date-Scale X Axis XY Chart with Value X Axis
The Category Axis chart simply plots the out of sequence dates across the X axis in the order they appear in the worksheet, regardless of the order of the dates. Both the Date-Scale and Value Axis
charts display the points from left to right in date order, spaced proportionally to the duration between dates. The two charts show different behavior of the connecting lines: the XY chart with the
Value axis connects the points in the order that the points appear in the worksheet, while the Line chart with the Date-Scale axis connect the points from left to right, ignoring the order of the
Which Type of X Axis Should You Use?
If you require the precision of a Time Scale axis, or if your data used non-numeric categories for the X values, you should use a Line Chart. For almost any other application, you should use an XY
Scatter Chart, with its Value axis. You can format the lines and markers of either type of chart identically, but you have greater flexibility using the Value axis. Data points are located along the
X axis according to their X values, not the order they are listed in the worksheet. In addition, if you need a logarithmic scale on your X axis, you can only get it with a Value axis.
For further discussion about axis types and chart types, see my article, Scatter Chart or Line Chart?, in the TechTrax web magazine. | {"url":"https://peltiertech.com/Excel/ChartsHowTo/CatVsValueAxis.html","timestamp":"2024-11-06T01:43:38Z","content_type":"text/html","content_length":"25022","record_id":"<urn:uuid:df9b68c3-7566-4a4f-9150-5f39ea500166>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00372.warc.gz"} |
Scientific Notation
Free Scientific Notation Worksheets
In this free basic math worksheet, students must convert between measurements such as km to m, kg to tons, minutes to hours, days to hours, and so on. There are also problems on scientific notation.
Worksheet (Basic Math)
This free algebra worksheet contains problems on scientific notation. Students must write numbers in scientific notation and standard notation. Problems also include ordering numbers written in...
Worksheet (Algebra)
This free algebra worksheet contains problems on scientific notation. Students must write numbers in scientific notation and standard notation. Problems also include ordering numbers written in...
Worksheet (Algebra)
This worksheet contains problems on scientific notation. Students must write numbers standard notation and scientific notation. Problems also include multiplying and dividing numbers in scientific...
Worksheet (Pre-Algebra) | {"url":"https://tutor-usa.com/topic/worksheets/scientific-notation","timestamp":"2024-11-09T18:58:03Z","content_type":"text/html","content_length":"52986","record_id":"<urn:uuid:21432c37-52e3-4f76-9ef6-09a986fe18a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00308.warc.gz"} |
Chapter 04.00: Physical Problem for Simultaneous Linear Equations | Numerical Methods with Applications
Chapter 04.00: Physical Problem for Simultaneous Linear Equations
To find the velocity profile of a rocket requires a solution of simultaneous linear equations.
The upward velocity of a rocket is given at three different times in the following table
Time, t Velocity, v
\(\text{s}\) \(\text{m/s}\)
\(5\) \(106.8\)
\(8\) \(177.2\)
\(12\) \(279.2\)
The velocity data is approximated by a polynomial as
Figure 1 A rocket launched into space
\[v\left( t \right) = at^{2} + bt + c, \ 5 \leq t \leq 12.\;\;\;\;\;\;\;\;\;\;\;\; (1)\]
Set up the equations in matrix form to find the coefficients \(a,\ b,\ c\) of the velocity profile.
The polynomial is going through three data points \(\left( t_{1},v_{1} \right),\left( t_{2},v_{2} \right)\text{, and }\left( t_{3},v_{3} \right)\) where from the above table
\[t_{1} = 5,v_{1} = 106.8\]
\[t_{2} = 8,v_{2} = 177.2\]
\[t_{3} = 12,v_{3} = 279.2\]
Requiring that \(v\left( t \right) = at^{2} + bt + c\) passes through the three data points gives
\[\begin{split} v\left( t_{1} \right) &= v_{1} = at_{1}^{2} + bt_{1} + c\\ v\left( t_{2} \right) &= v_{2} = at_{2}^{2} + bt_{2} + c\\ v\left( t_{3} \right) &= v_{3} = at_{3}^{2} + bt_{3} + c \end
Substituting the data \(\left( t_{1},\ v_{1} \right),\ \left( t_{2},\ v_{2} \right),\ \left( t_{3},\ v_{3} \right)\) gives
\[\begin{split} a\left( 5^{2} \right) + b\left( 5 \right) + c &= 106.8\\ a\left( 8^{2} \right) + b\left( 8 \right) + c &= 177.2\\ a\left( 12^{2} \right) + b\left( 12 \right) + c &= 279.2 \end{split}
\[\begin{split} 25a + 5b + c &= 106.8\\ 64a + 8b + c &= 177.2\\ 144a + 12b + c &= 279.2 \end{split}\]
This set of equations can be rewritten in the matrix form as
\[\begin{bmatrix} 25a + & 5b + & c \\ 64a + & 8b + & c \\ 144a + & 12b + & c \\ \end{bmatrix} = \begin{bmatrix} 106.8 \\ 177.2 \\ 279.2 \\ \end{bmatrix}\]
The above equation can be written as a linear combination as follows
\[a\begin{bmatrix} 25 \\ 64 \\ 144 \\ \end{bmatrix} + b\begin{bmatrix} 5 \\ 8 \\ 12 \\ \end{bmatrix} + c\begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix} = \begin{bmatrix} 106.8 \\ 177.2 \\ 279.2 \\ \end
and further using matrix multiplications gives
\[\begin{bmatrix} 25 & 5 & 1 \\ 64 & 8 & 1 \\ 144 & 12 & 1 \\ \end{bmatrix}\ \begin{bmatrix} a \\ b \\ c \\ \end{bmatrix} = \begin{bmatrix} 106.8 \\ 177.2 \\ 279.2 \\ \end{bmatrix}\]
The solution of the above three simultaneous linear equations will give the value of \(a,b,c\).
(1) Solve for the values of \(a,b,c\).
(2) Verify if you get back the value of the velocity data at \(t=5\ \text{s}\).
(3) Estimate the velocity of the rocket at \(t=7.5\ \text{s}\)?
(4) Estimate the acceleration of the rocket at \(t=7.5\ \text{s}\)?
(5) Estimate the distance covered by the rocket between \(t=5.5\ \text{s}\) and \(8.9\ \text{s}\).
(6) If the following data is given for the velocity of the rocket as a function of time, and you are asked to use a quadratic polynomial to approximate the velocity profile to find the velocity at \
(t=16\ \text{s}\), what data points would you choose and why?
t v(t)
\(\text{s}\) \(\text{m/s}\)
\(0\) \(0\)
\(10\) \(227.04\)
\(15\) \(362.78\)
\(20\) \(517.35\)
\(22.5\) \(602.97\)
\(30\) \(901.67\)
Liquid-liquid extraction depends on the ability of some metal ions to form metal complexes with organic acids. The method is used to separate, concentrate, and purify metals and organic compounds.
Liquid-liquid extraction was the technique used to produce weapon grade uranium during the arms race (cold war) era. The technique is also used to recover noble metals used in catalytic processes
such as oil refining etc.
In liquid-liquid extraction, the metal ion in the aqueous phase is recovered by mixing the aqueous phase with an organic phase. The metal ion forms a complex with the organic phase and floats on top
of the aqueous phase. The organic phase can be decanted and separated from the aqueous phase and the complexed metal ion recovered in a useful form using an acid (nitric acid for nitrates, sulfuric
acid for sulfates etc).
A liquid-liquid extraction process conducted in the Electrochemical Materials Laboratory involved the extraction of nickel from the aqueous phase into an organic phase. A typical experimental data
from the laboratory is given in Table 1.
Table 1 Aqueous and organic phase concentration of nickel.
Ni aqueous phase (g/l) \(2\) \(2.5\) \(3\) \(3.5\) \(4\)
Ni organic phase (g/l) \(8.57\) \(10\) \(12\) \(14\) \(15.66\)
Estimate the amount of nickel in organic phase when 2.3 g/l is in the aqueous phase. Use quadratic interpolation.
The polynomial is going through three data points \[\left( a_{1},g_{1} \right),\ \left( a_{2},g_{2} \right)\text{, and}\left( a_{3},g_{3} \right)\] where from the above table
\[a_{1} = 2,g_{1} = 8.57\]
\[a_{2} = 2.5,g_{2} = 10\]
\[a_{3} = 3,g_{3} = 12\]
Requiring that \(g = x_{1}a^{2} + x_{2}a + x_{3}\) passes through the three data points gives
\[g\left( a_{1} \right) = g_{1} = x_{1}a_{1}^{2} + x_{2}a_{1} + x_{3}\]
\[g\left( a_{2} \right) = g_{2} = x_{1}a_{2}^{2} + x_{2}a_{2} + x_{3}\]
\[g\left( a_{3} \right) = g_{3} = x_{1}a_{3}^{2} + x_{2}a_{3} + x_{3}\]
Substituting the data
\[\left( a_{1},g_{1} \right),\ \left( a_{2},{\ }{g}_{2} \right),\ \left( a_{3},{\ }{g}_{3} \right)\]
\[x_{1}\left( 2 \right)^{2} + x_{2}\left( 2 \right) + x_{3} = 8.57\]
\[x_{1}\left( 2.5 \right)^{2} + x_{2}\left( 2.5 \right) + x_{3} = 10\]
\[x_{1}\left( 3 \right)^{2} + x_{2}\left( 3 \right) + x_{3} = 12\]
\[4x_{1} + 2x_{2} + x_{3} = 8.57\]
\[6.25x_{1} + 2.5x_{2} + x_{3} = 10\]
\[9x_{1} + 3x_{2} + x_{3} = 12\]
This set of equations can be rewritten in the matrix form as
\[\begin{bmatrix} 4x_{1} + & 2x_{2} + & x_{3} \\ 6.25x_{1} + & 2.5x_{2} + & x_{3} \\ 9x_{1} + & 3x_{2} + & x_{3} \\ \end{bmatrix} = \begin{bmatrix} 8.57 \\ 10 \\ 12 \\ \end{bmatrix}\]
The above equations can be written as a linear combination as follows
\[x_{1}\begin{bmatrix}4 \\ 6.25 \\ 9 \\ \end{bmatrix} + x_{2}\begin{bmatrix} 2 \\ 2.5 \\ 3 \\ \end{bmatrix} + x_{3}\begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix} = \begin{bmatrix} 8.57 \\ 10 \\ 12 \\ \
and further using matrix multiplication gives
\[\begin{bmatrix} 4 & 2 & 1 \\ 6.25 & 25 & 1 \\ 9 & 3 & 1 \\ \end{bmatrix}\ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} = \begin{bmatrix} 8.57 \\ 10 \\ 12 \\ \end{bmatrix}\]
The solution of the above simultaneous linear equations will give the value of \(x_{1},x_{2},x_{3}.\)
(1) Verify if you get back the value of the Ni organic phase when the Ni aqueous phase is 2.5 g/l.
(2) Estimate the value of the Ni organic phase when the Ni aqueous phase is 2.78 g/l
(3) Estimate the error between linear interpolation and quadratic interpolation values obtained for nickel in organic phase when 2.78 g/l is in the aqueous phase.
A pressure vessel made of a compounded cylinder has a higher capability to handle internal pressure over a single cylinder. To find this capability, one has to solve a set of simultaneous linear
A pressure vessel can only be subjected to an amount of internal pressure that is limited by the strength of the material used. For example, take a pressure vessel of an internal radius of \(a = 5^{\
prime\prime}\) and outer radius, \(b = 8^{\prime\prime}\), made of ASTM 36 steel (yield strength of ASTM 36 steel is 36 ksi). How much internal pressure can this pressure vessel take before it is
considered to have failed?
Figure 1 A single-cylinder pressure vessel with internal radius, \(a\) and outer radius, \(b\).
The hoop and radial stress in a cylindrical pressure vessel is given by [1]
\[\sigma_{r} = \frac{a^{2}p_{i}}{b^{2} - a^{2}}\left( 1 - \frac{b^{2}}{r^{2}} \right)\;\;\;\;\;\;\;\;\;\;\;\; (1)\]
\[\sigma_{\theta} = \frac{a^{2}p_{i}}{b^{2} - a^{2}}\left( 1 + \frac{b^{2}}{r^{2}} \right)\;\;\;\;\;\;\;\;\;\;\;\; (2)\]
The maximum normal stress anywhere in the cylinder is the hoop stress at the inner radius, \(a\)
\[{\left. \ \sigma_{\theta} \right|i\left( \frac{b^{2} + a^{2}}{b^{2} - a^{2}} \right)}_{\max}\;\;\;\;\;\;\;\;\;\;\;\; (3)\]
Assuming a factor of safety of 2, while the yield strength is given as 36 ksi,
\[\frac{36 \times 10^{3}}{2} = p_{i}\left( \frac{8^{2} + 5^{2}}{8^{2} - 5^{2}} \right)\]
\[p_{i} = 7.887{ksi}\]
You can see from Equation (3) that even for \(b > > a\), the maximum internal pressure one can apply is only \(p_{i} = 18{ksi}\). Therefore, what can an engineer do to maximize the internal pressure
while keeping the material and radial dimensions the same? They can use a compounded cylinder. They can create a compounded cylinder by shrink fitting one cylinder into another, and hence creating
pre-existing favorable stresses to allow more internal pressure. Let us see how that would work?
Figure 2 A compounded cylinder pressure vessel with internal radius, \(a\), outer radius, \(b\), and interface at \(r = c\).
Let us make the compounded cylinder of two cylinders (Figure 2). Cylinder 1 has an internal radius of \(a = 5^{\prime\prime}\), and outer radius \(c = 6.5^{\prime\prime}\), while Cylinder 2 has an
internal radius of \(c = 6.5^{\prime\prime}\) and outer radius, \(b = 8^{\prime\prime}\). Assume that that radial interference, \(\delta = 0.007^{\prime\prime}\) occurs at the interface of a
compounded cylinder at \(r = c = 6.5^{\prime\prime}\). How does then one find the pressure that can be applied to the compounded cylinder of internal radius, \(a = 5^{\prime\prime}\) and outer
radius, \(b = 8^{\prime\prime}\)?
For cylinder 1, the radial displacement, \(u_{1}\) is given by
\[u_{1} = c_{1}r + \frac{c_{2}}{r}\;\;\;\;\;\;\;\;\;\;\;\; (4)\]
the radial stress, \(\sigma_{r}^{1}\)and hoop stress, \(\sigma_{\theta}^{1}\) by
\[\sigma_{r}^{1} = \frac{E}{1 - \nu^{2}}\left\lbrack c_{1}\left( 1 + \nu \right) - c_{2}\left( \frac{1 - \nu}{r^{2}} \right) \right\rbrack\;\;\;\;\;\;\;\;\;\;\;\; (5)\]
\[\sigma_{\theta}^{1} = \frac{E}{1 - \nu^{2}}\left\lbrack c_{1}\left( 1 + \nu \right) + c_{2}\left( \frac{1 - \nu}{r^{2}} \right) \right\rbrack\;\;\;\;\;\;\;\;\;\;\;\; (6)\]
\[E = \text{Young's modulus of steel,}\]
\[\nu = \text{Poisson's ratio of steel.}\]
For cylinder 2, the radial displacements, \(u_{2}\)is given by
\[u_{2} = c_{3}r + \frac{c_{4}}{r}\;\;\;\;\;\;\;\;\;\;\;\; (7)\]
the radial stress, \(\sigma_{r}^{2}\) and hoop stress, \(\sigma_{\theta}^{2}\) by
\[\sigma_{r}^{2} = \frac{E}{1 - \nu^{2}}\left\lbrack c_{3}\left( 1 + \nu \right) - c_{4}\left( \frac{1 - \nu}{r^{2}} \right) \right\rbrack\;\;\;\;\;\;\;\;\;\;\;\; (8)\]
\[\sigma_{\theta}^{2} = \frac{E}{1 - \nu^{2}}\left\lbrack c_{3}\left( 1 + \nu \right) + c_{4}\left( \frac{1 - \nu}{r^{2}} \right) \right\rbrack\;\;\;\;\;\;\;\;\;\;\;\; (9)\]
So if one is able to find the four constants, \(c_{1},c_{2},c_{3}\) and \(c_{4}\), one can find the stresses in the compounded cylinder to be able to find what internal pressure can be applied. So
how do we find the four unknown constants?
The boundary and interface conditions are the following.
The radial stress at the inner radius, \(r = a\) is the applied internal pressure
\[\sigma_{r}^{1}\left( r = a \right) = - p_{i}\;\;\;\;\;\;\;\;\;\;\;\; (10)\]
The radial stress is continuous at the interface, \(r = c\)
\[\sigma_{r}^{1}\left( r = c \right) = \sigma_{r}^{2}\left( r = c \right)\;\;\;\;\;\;\;\;\;\;\;\; (11)\]
The radial displacement at the interface, \(r = c\) has a jump of the radial interference, \(\delta\)
\[u_{2}\left( r = c \right) - u_{1}\left( r = c \right) = \delta\;\;\;\;\;\;\;\;\;\;\;\; (12)\]
The radial stress at the outer radius, \(r = c\) is
\[\sigma_{r}^{2}\left( r = b \right) = 0\;\;\;\;\;\;\;\;\;\;\;\; (13)\]
This will set up four equations and four unknowns, if we know what internal pressure we are applying. Assume we are applying the same pressure as the single cylinder can take, that is,\(p_{i} = 7.887
\)ksi and let us see later what stresses it creates in the compounded cylinder.
Assuming \(E = 30 \times 10^{6}\)psi, \(\nu = 0.3\), Equations (10) through (13) become
\[\frac{30 \times 10^{6}}{1 - 0.3^{2}}\left\lbrack c_{1}\left( 1 + 0.3 \right) - c_{2}\left( \frac{1 - 0.3}{5^{2}} \right) \right\rbrack = - 7.887 \times 10^{3}\]
\[\frac{30 \times 10^{6}}{1 - 0.3^{2}}\left\lbrack c_{1}\left( 1 + 0.3 \right) - c_{2}\left( \frac{1 - 0.3}{6.5^{2}} \right) \right\rbrack = \frac{30 \times 10^{6}}{1 - 0.3^{2}}\left\lbrack c_{3}\
left( 1 + 0.3 \right) - c_{4}\left( \frac{1 - 0.3}{6.5^{2}} \right) \right\rbrack\]
\[c_{3}\left( 6.5 \right) + \frac{c_{4}}{6.5} - c_{1}\left( 6.5 \right) - \frac{c_{2}}{6.5} = 0.007\]
\[\frac{30 \times 10^{6}}{1 - 0.3^{2}}\left\lbrack c_{3}\left( 1 + 0.3 \right) - c_{4}\left( \frac{1 - 0.3}{8^{2}} \right) \right\rbrack = 0\;\;\;\;\;\;\;\;\;\;\;\; (14a-d)\]
Writing the Equations (14a-d) in matrix form, we get
\[\begin{bmatrix} 4.2857 \times 10^{7} & - 9.2307 \times 10^{5} & 0 & 0 \\ 4.2857 \times 10^{7} & - 5.4619 \times 10^{5} & - 4.2857 \times 10^{7} & 5.4619 \times 10^{5} \\ - 6.5 & - 0.15384 & 6.5 &
0.15384 \\ 0 & 0 & 4.2857 \times 10^{7} & - 3.6057 \times 10^{5} \\ \end{bmatrix}\begin{bmatrix} c_{1} \\ c_{2} \\ c_{3} \\ c_{4} \\ \end{bmatrix} = \begin{bmatrix} - 7.887 \times 10^{3} \\ 0 \\
0.007 \\ 0 \\ \end{bmatrix}\;\;\;\;\;\;\;\;\;\;\;\; (15)\]
Solving these four simultaneous linear equations, we can find the four constants.
(1) A.C. Ugural, S.K. Fenster, Advanced strength and applied elasticity, Third Edition, Prentice Hall, New York, 1995.
(2) J.E. Shigley, C.R. Mischke, Chapter 19 - Limits and fits, Standard handbook of machine design, McGraw-Hill, New York, 1986.
(1) Find the unknown constants of Equation (14) using different numerical methods.
(2) Knowing that the critical points in the compounded cylinder are \(r = a,c - ,c + ,\text{and}\ b\), find the maximum hoop stress in the compounded cylinder. What is its value compared to the
maximum hoop stress allowable of 18 ksi?
(3) Find the maximum internal pressure you can apply to the compounded cylinder? Compare it with the maximum possible internal pressure for a single cylinder of the same dimensions.
(4) The radial interference at the interface is created by making the inner cylinder 1 to have a larger outer radius than the inner radius of cylinder 2. Standard interference fits dictate the
limits of these dimensions. If cylinder 2 is fit into cylinder 1, there is an upper and lower limit by which the nominal diameter of each cylinder varies at the interface. This limit L in thousands
of an inch, is given by [2]
\[L = CD^{1/3}\] where \(D\) (nominal diameter) is in inches, and the coefficient \(C\),based on the type of fit, is given in Table 1 below.
Cylinder Limit Class of fit
FN2 FN3
\(1\) \(\text{Lower}\) \(0.000\) \(0.000\)
\(\text{Upper}\) \(0.907\) \(0.907\)
\(2\) \(\text{Lower}\) \(2.717\) \(3.739\)
\(\text{Upper}\) \(3.288\) \(4.310\)
Assuming FN2 fit at the interface, find the maximum internal pressure you would recommend.
Human vision has the remarkable ability to infer 3D shapes from 2D images. When we look at 2D photographs or TV we do not see them as 2D shapes, rather as 3D entities with surfaces and volumes.
Perception research has unraveled many of the cues that are used by us. The intriguing question is can we replicate some of these abilities on a computer? To this end, in this assignment we are going
to look at one way to engineer a solution to the 3D shape from 2D images problem. Apart from the pure fun of solving a new problem, there are many practical applications of this technology such as in
automated inspection of machine parts, inference of obstructing surfaces for robot navigation, and even in robot assisted surgery.
Image is a collection of gray level values at set of predetermined sites known as pixels, arranged in an array. These gray level values are also known as image intensities. The registered intensity
at an image pixel is dependent on a number of factors such as the lighting conditions, surface orientation and surface material properties. The nature of lighting and its placement can drastically
affect the appearance of a scene. Can we infer shape of the underlying surface given images as in the images below in Figure 1.
Figure 1 Images of a surface taken with three different light directions. Can you guess the shape of the underlying surface?
Physics of the Problem
To be able to reconstruct the shape of the underlying surface, we have to first understand the image formation process. The simplest image formation model assumes that the camera is far away from the
scene so that we have assume that the image is a scaled version of the world. The simplest light model consists of a point light that is far away. This is not an unrealistic assumption. A good
example of such a light source is the sun. We also assume that the surface is essentially matte that reflects lights uniformly in all directions, unlike specular (or mirror-like surfaces). These
kinds of surfaces are called Lambertian surfaces; examples include walls, and carpet.
Figure 2 Relationship between the surface and the light source. (b) Amount of light reflected by an elemental area (dA) is proportional to the cosine of the angle between the light source direction
and the surface normal
The image brightness of a Lambertian surface is dependent on the local surface orientation with respect to the light source. Since the point light source is far away we will assume that the incident
light is locally uniform and comes from a single direction, i.e, the light rays are parallel to each other. This situation is illustrated in Figure 2. Let the incident light intensity be denoted by \
(I_{0}\). Also let the angle between the light source and local surface orientation be denoted by \(\varphi\). Then the registered image intensity, \(I(u,v)\), of that point is given by
\[I(u,v) = I_{0}\rho\cos(\varphi) = I_{0}\rho(n_{x}s_{x} + n_{y}s_{y} + n_{z}s_{z})\]
where the surface normal, \(n\), and the light source direction, \(s\), are given by:
\[n^{\ } = \begin{bmatrix} n_{x} \\ n_{y} \\ n_{z} \\ \end{bmatrix},\ s = \begin{bmatrix} s_{x} \\ s_{y} \\ s_{z} \\ \end{bmatrix}\]
and \(\rho\) is a number capturing the surface reflection property at the location \((x,y)\), and It is referred to as the surface albedo. Black surfaces tend to have low albedo and white surfaces
tend to high albedo. Note that the registered intensity in the image does not depend on the camera location because a Lambertian surface reflects light equally in all directions. This would not be
true for specular surfaces and the corresponding image formation equation would involve the viewing direction.
The variation in image intensity is essentially dependent on the local surface orientation. If the surface normal and the light source directions are aligned then we have the maximum intensity and
the observed intensity of the lowest when the angle between the light source and the local surface orientation is \(90^{0}\). Thus, given the knowledge of light source and the local surface albedo,
it should be possible to infer the local surface orientation from the local image intensity variations. This is what we explore next.
The mapping of surface orientation to image intensity is many to one. Thus, it is not possible to infer the surface orientation from just one intensity image in the absence of any other knowledge. We
need multiple samples per point in the scene. How many samples do we need? The vector specifying the surface normal has three components, which implies that we need three. So, we engineer a setup to
infer surface orientation from image intensities. Instead of just one image of a scene, let us take three images of the same scene, without moving either the camera or the scene, but with three
different light sources, turned on one at a time. These three different light sources are placed at different locations in space. Let the three light source directions, relative to the camera, be
specified by the vectors
\[s^{1} = \begin{bmatrix} s_{x}^{1} \\ s_{y}^{1} \\ s_{z}^{1} \\ \end{bmatrix},\ s^{2} = \begin{bmatrix} s_{x}^{2} \\ s_{y}^{2} \\ s_{z}^{2} \\ \end{bmatrix},\ s^{3} = \begin{bmatrix} s_{x}^{3} \\ s_
{y}^{3} \\ s_{z}^{3} \\ \end{bmatrix}\] Corresponding pixels in the three images would have three different intensities \(I_{1}\), \(I_{2}\), and \(I_{3}\) for three light source directions. Let the
surface normal corresponding to the pixel under consideration be denoted by
\[n= \begin{bmatrix} n_{x} \\ n_{y} \\ n_{z} \\ \end{bmatrix}\]
Assuming Lambertian surfaces, the three intensities can be related to the surface normal and the light source directions
\[I_{i} = I_{0}\rho(n_{x}s_{x}^{i} + n_{y}s_{y}^{i} + n_{z}s_{z}^{i}),\ \forall i = 1,2,3\]
In these three equations, the known variables are the intensities, \(I_{1}\), \(I_{2}\), \(I_{3}\), and the light source directions, \(s_{1}\), \(s_{2}\), \(s_{3}\). The unknowns are the incident
intensity, \(I_{3}\), surface albedo, \(\rho\) and the surface normal, \(n\). These unknowns can be bundled into three unknown variables, \(m_{x} = I_{0}\rho n_{x}\), \(m_{y} = I_{0}\rho n_{y}\), and
\(m_{z} = I_{0}\rho n_{z}\). We will recover the surface normal by normalizing the recovered \(m\) vector, using the fact that the magnitude of the normal is one. The normalizing constant will give
us the product \(I_{0}\rho\). Thus, for each point in the image, we have three simultaneous equations in three unknowns.
\[\begin{bmatrix} I_{1} \\ I_{2} \\ I_{3} \\ \end{bmatrix} = \begin{bmatrix} s_{x}^{1} & s_{y}^{1} & s_{z}^{1} \\ s_{x}^{2} & s_{y}^{2} & s_{z}^{2} \\ s_{x}^{3} & s_{y}^{3} & s_{z}^{3} \\ \end
{bmatrix}\begin{bmatrix} m_{x} \\ m_{y} \\ m_{z} \\ \end{bmatrix}\]
Worked Out Example
Consider the middle of the sphere in Figure 1. We know that the surface normal points towards the camera (i.e. towards the viewer). Assume a 3D coordinate system centered at the camera with \(x\)
-direction along the horizontal direction, \(y\) -direction along the vertical direction, and \(z\) -direction is away from the camera into the scene. Then the actual surface normal of the middle of
the sphere is given by [0, 0, -1] – the negative denotes that it point in the direction opposite the z-axis. Let us see how close our estimate is to the actual value.
The intensity of the middle of the sphere in the three views, \(I_{1} = 247,\) \(I_{2} = 248,\) and \(I_{3} = 239\), respectively. The light directions for the three images are along [5, 0, -20], [0,
5, -20], and [-5, -5, -20], respectively. Normalizing the three vectors we get the normal directions towards the lights and construct the 3 by 3 matrix
\[\begin{bmatrix} s_{x}^{1} & s_{y}^{1} & s_{z}^{1} \\ s_{x}^{2} & s_{y}^{2} & s_{z}^{2} \\ s_{x}^{3} & s_{y}^{3} & s_{z}^{3} \\ \end{bmatrix} = \begin{bmatrix} 0.2425 & 0 & - 0.9701 \\ 0 & 0.2425 &
- 0.9701 \\ - 0.2357 & - 0.2357 & - 0.9428 \\ \end{bmatrix}\]
Solving the corresponding 3 simultaneous equations, we arrive the following solution for the \(m\) -vector:
\[\begin{bmatrix} m_{x} \\ m_{y} \\ m_{z} \\ \end{bmatrix} = \begin{bmatrix} 0.0976 \\ 4.2207 \\ - 254.5774 \\ \end{bmatrix}\]
Normalizing this vector we get the surface normal
\[\begin{bmatrix} n_{x} \\ n_{y} \\ n_{z} \\ \end{bmatrix} = \begin{bmatrix} 0.0004 \\ 0.0166 \\ - 0.9999 \\ \end{bmatrix}\]
The corresponding normalizing constant is 254.6124, which is the product of the intensity of the illumination and the surface (albedo) reflectance factor \((I_{0}\rho)\). Compare the estimate of the
surface normal to the actual one. The difference is quantization effects – in images we can represent intensities as a finite sized integer – 8-bit integers in our case.
We can repeat the above computations for each point in the scene to arrive at estimates of the corresponding surface normals. Figure 3(a) is a visualization of the surface normals thus estimated as a
vector field. In Figure 3(b), we see the product \(I_{0}\rho\) visualized as image intensities. As expected, it is the same at all points on the sphere. In another problem module, how do we recover
the underlying surface from these surface normals?
Figure 3 (a) Recovered surface normal at each point on the sphere. Just the first two components of the vectors are shown as a arrows. (b) Recovered product \(I_{0}\rho\) for all points
(1) What can you infer about the surface normal for the brightest point in the image? What about the darkest point in the scene?
(2) What assumptions do you have to make to make the above inferences?
Designing three-phase loads in AC systems require the solution of simultaneous linear equations.
Three-phase AC systems are the norm for most industrial applications. AC power in the form of voltage and current it delivered from the power company using three-phase distribution systems and many
larger loads are three-phase loads in the form of motors, compressors, or similar. Sources and loads can be configured in either wye (where sources or loads are connected from line to neutral/ground)
or delta (where sources or loads are connected from line to line) configurations and mixing between the types is common. Figure 1 shows the general wiring of a wye-wye three-phase system modeling all
of the impedances typically found in such a system.
During the typical analysis undertaken in most circuits textbooks, it is assumed that the system is entirely balanced. This means that all the source, line, and load impedances are equivalent, that
\[Z_{a} = Z_{b} = Z_{c}\]
\[Z_{{aA}} = Z_{{bB}} = Z_{{cC}}\]
\[Z_{{AN}} = Z_{{bN}} = Z_{{CN}}\]
Under this assumption, the circuit is then typically reduced to a single-phase equivalent circuit model and the resultant circuit is solved with a single loop equation. What happens, however, when
the system is unbalanced? Typically because the three load impedances \(Z_{{AN}},Z_{{BN}}\) and \(Z_{{CN}}\) are not equal, which results in different currents through each load, is often measured in
terms of the percentage difference between the load currents.
Figure 1 A Three-Phase Wye-Wye System with Positive Phase Sequence
Creating an imbalance in a three-phase system is not all that difficult. Consider a small business operating in an isolated leg of the power grid so that localized aspects of a load are not
“balanced” by other neighboring loads. Let’s assume that the primary load for this system is a 45 kVA set of three-phase motors at 0.8 power factor lagging and, further, that the electrician that did
the wiring for the lighting mistakenly connected two banks of lights to the A phase, one to the B phase and none to the C phase creating an imbalance in the system. Each of these lighting loads is
1500 W. The load for this system is shown in Figure 2.
Figure 2 Model of the System Load
The impedance of each of the loads can be determined by examining the power consumed in each phase of the system
\[\begin{split} A:3000 + 15000/\underline{-36.87 ^{\circ}} &= 3000 + 12000 - j9000 = 15000 - j9000 \\ &= 17.49/\underline{-30.96^{\circ}} {kVA} \end{split}\]
\[\begin{split} B:1500 + 15000/\underline{-36.87^{\circ}} &= 1500 + 12000 - j9000 = 13500 - j9000 \\ &= 16.22/\underline{-33.69^{\circ}} {kVA} \end{split}\]
Converting these to impedances using the formula \[S = \frac{\left| V \right|^{2}}{Z}\ \text{with}\ V = 120V \text{ yields}:\]
\[Z_{{AN}} = 0.8233/\underline{30.96^{\circ}}\Omega = R_{A} + jX_{A} = 0.7060 + j0.4236\Omega\]
\[Z_{{BN}} = 0.8878/\underline{33.39^{\circ}}\Omega = R_{B} + jX_{B} = 0.7387 + j0.4925\Omega\]
\[Z_{{CN}} = 0.9600/\underline{36.87^{\circ}}\Omega = R_{C} + jX_{C} = 0.7680 + j0.5760\Omega\]
For the rest of this analysis we will assume that each phase of the system has an equivalent source and line impedance of \(R_{s} + jX_{s} = 0.0300 + j0.0200\Omega\) and that the ground return wire
has an impedance of\(R_{n} + jZ_{n} = 0.0100 + j0.0080\Omega\). This yields the equivalent circuit of Figure 3.
Figure 3 Equivalent Circuit Model for the Working Problem
The circuit can be analyzed using three loop equations using the currents \(I_{a},I_{b},\) and \(I_{c}\) shown in Figure 3. For loop A this yields the complex equation:
Loop A:
\[- V_{s}/\underline{0^{\circ}} + I_{a}\left( R_{s} + jX_{s} + R_{A} + jX_{A} \right) + \left( I_{a} + I_{b} + I_{c} \right)\left( R_{n} + jX_{n} \right) = 0\]
with loops B and C yielding similar results. Assuming that our simultaneous equation solver is not capable of handling complex numbers we can turn the loop A equation into two separate non-complex
equations addressing both the real and imaginary parts. Using \(I_{a} = I_{{ar}} + jI_{{ai}}\) and collecting terms yields:
Real A:
\[I_{{ar}}\left( R_{s} + R_{A} + R_{n} \right) - I_{{ai}}\left( X_{s} + X_{A} + X_{n} \right) + I_{{br}}R_{n} - I_{{bi}}X_{n} + I_{{cr}}R_{n} - I_{{ci}}X_{n} = 120\ \ \ (1)\]
Imaginary A:
\[I_{{ar}}\left( X_{s} + X_{A} + X_{n} \right) + I_{{ai}}\left( R_{s} + R_{A} + R_{n} \right) + I_{{br}}X_{n} + I_{{bi}}R_{n} + I_{{cr}}X_{n} + I_{{ci}}R_{n} = 0\ \ \ (2)\]
Applying the same analysis to the B and C loops yields the remaining equations for the system.
Real B:
\[I_{{ar}}R_{n} - I_{{ai}}X_{n} + I_{{br}}\left( R_{s} + R_{B} + R_{n} \right) - I_{{bi}}\left( X_{s} + X_{B} + X_{n} \right) + I_{{cr}}R_{n} - I_{{ci}}X_{n} = - 60\ \ \ (3)\]
Imaginary B:
\[I_{{ar}}X_{n} + I_{{ai}}R_{n} + I_{{br}}\left( X_{s} + X_{B} + X_{n} \right) + I_{{bi}}\left( R_{s} + R_{B} + R_{n} \right) + I_{{cr}}X_{n} + I_{{ci}}R_{n} = - 103.9\ \ \ (4)\]
Real C:
\[I_{{ar}}R_{n} - I_{{ai}}X_{n} + I_{{br}}R_{n} - I_{{bi}}X_{n} + I_{{cr}}\left( R_{s} + R_{C} + R_{n} \right) - I_{{ci}}\left( X_{s} + X_{C} + X_{n} \right) = - 60\ \ \ (5)\]
Imaginary C:
\[I_{{ar}}X_{n} + I_{{ai}}R_{n} + I_{{br}}X_{n} + I_{{bi}}R_{n} + I_{{cr}}\left( X_{s} + X_{C} + X_{n} \right) + I_{{ci}}\left( R_{s} + R_{C} + R_{n} \right) = 103.9\ \ \ (6)\]
This yields a system of six linear equations and six unknowns \(\left( I_{{ar}},I_{{ai}},I_{{br}},I_{{bi}},I_{{cr}},{\ and\ }I_{{ci}} \right)\) that can be solved by any conventional means. This is
shown in matrix form in Figure 4.
\[\begin{bmatrix} 0.7460 & - 0.4516 & 0.0100 & - 0.0080 & 0.0100 & - 0.0080 \\ 0.4516 & 0.7460 & 0.0080 & 0.0100 & 0.0080 & 0.0100 \\ 0.0100 & - 0.0080 & 0.7787 & - 0.5205 & 0.0100 & - 0.0080 \\
0.0080 & 0.0100 & 0.5205 & 0.7787 & 0.0080 & 0.0100 \\ 0.0100 & - 0.0080 & 0.0100 & - 0.0080 & 0.8080 & - 0.6040 \\ 0.0080 & 0.0100 & 0.0080 & 0.0100 & 0.6040 & 0.8080 \\ \end{bmatrix}\begin{bmatrix}
I_{{ar}} \\ I_{{ai}} \\ I_{{br}} \\ I_{{bi}} \\ I_{{cr}} \\ I_{{ci}} \\ \end{bmatrix} = \begin{bmatrix} 120.0 \\ 0.000 \\ - 60.00 \\ - 103.9 \\ - 60.00 \\ 103.9 \\ \end{bmatrix}\ \ \ (7)\]
Figure 4 Complete System of Equations
Once the currents are known it is a simple procedure to determine the voltages across the three motor terminals \(\left( V_{{AN}},V_{{BN}},{\ and\ }V_{{CN}} \right)\) using Ohm’s Law.
\[V_{{AN}} = \left( I_{{ar}} + jI_{{ai}} \right)\left( R_{A} + jX_{A} \right)\]
\[V_{{BN}} = \left( I_{{br}} + jI_{{bi}} \right)\left( R_{B} + jX_{B} \right)\]
\[V_{{CN}} = \left( I_{{cr}} + jI_{{ci}} \right)\left( R_{C} + jX_{C} \right)\]
To better evaluate the imbalance, the percentage difference in currents through the actual load elements is more often considered. The best way to think of this as three people pulling and pushing
together. If they do not pull and push in balance, things can become unstable. In the case of a three-phase motor, this can result in significant wobble with corresponding wear in the bearings and
other parts. To determine the current through each load is again computed using Ohm’s Law.
\[I_{{Aload}} = \frac{V_{{AN}}}{Z_{3\varphi}},\ I_{{Bload}} = \frac{V_{{BN}}}{Z_{3\varphi}},\ I_{{Cload}} = \frac{V_{{CN}}}{Z_{3\varphi}}\]
\[Z_{3\varphi} = 0.7680 + j0.5760\Omega\]
where \(Z_{3\varphi}\) was computed earlier as \(Z_{C}\). Do not forget that the lighting loads are separate.
(1) What would be the ramifications of solving the problem directly using the three complex linear equations? Could we do it using an approach like Gauss-Jordan Elimination? What about some of the
other numerical methods used to solve simultaneous linear equations?
(2) This problem is only interesting if the ground return leg \(Z_{{nN}}\)is non-zero. Otherwise, we have three loop equations that are completely independent of each other and can be solved
directly. Why is that the case?
(3) A much more interesting and practical problem occurs when the motor load is a Delta configuration. Since it does not have the ground return line in the middle it results in additional loop
equations. Sketch the equivalent circuit for a system with a Wye source and a mix of Delta and Wye loads. Write the set of equations that result from this system. Solve them.
A company that manufactures small toys recently received a contract from a fast-food company, to manufacture three toys, at a low cost, to be added to kid’s lunches. The company has to manufacture
toys for boys (toy B), girls (toy G) and a generic version (toy U). Furthermore, based on the demand and demographics, the fast food company has specified that 5% more girl’s toys than boy’s toys
should be produced, and that there is no constraint is specified on the number of generic toys. The components of each toy (B, G, and U) must be injection molded out of plastic (Process 1) and then
assembled (Process 2). After the toys have been designed, it is determined that the following production times will be needed on each toy:
• Toy B will require 2 minutes for injection molding 6 toys and 1 minute for assembling all 6 toys.
• Toy G will require 2 minutes for injection molding 12 toys and 8 minutes for assembling all 12 toys.
• Toy U will require 4 minutes for injection molding 6 toys and 2 minutes for assembling all 6 toys.
Note that because of daily scheduled maintenance of the injection molding machine, it can only run for a maximum of 756 out of 1440 minutes a day, whereas the assembly line works 3 shifts a day with
scheduled breaks for a maximum of 1260 out of 1440 minutes per day. An industrial engineer working for the toy company is asked to determine the production schedule that maximizes, on a daily basis,
the use of both the injection molding machine and the assembly line.
The variables need to solve the problem are listed in Table 1
Table 1. Variables for these different toys.
Variable Toy B Toy G Toy U
Time (minutes) required in Process 1 per toy \(B_{1}\) \(G_{1}\) \(U_{1}\)
Time (minutes) required in Process 2 per toy \(B_{2}\) \(G_{2}\) \(U_{2}\)
Total manufactured per day \(X_{B}\) \(X_{G}\) \(X_{U}\)
The total time required to produce toys in process 1 (injection molding) has to be equal to the maximum minutes per day that process 1 can run, that is,
\[B_{1}X_{B} + G_{1}X_{G} + U_{1}X_{U} = M_{1}\]
where \(M_{1}\) is the maximum minutes that process 1 can run per day. Similarly, for process 2 (assembly),
\[B_{2}X_{B} + G_{2}X_{G} + U_{2}X_{U} = M_{2}\]
where \(M_{2}\) is the maximum minutes that process 2 can run per day. Finally, the constraint of 5% more girl’s toys than boy’s toys is expressed as
\[1.05X_{B} = X_{G}\ \text{or}\]
\[1.05X_{B} - X_{G} = 0\]
The previous three simultaneous linear equations can be expressed in matrix form as follows
\[\begin{bmatrix} B_{1} & G_{1} & U_{1} \\ B_{2} & G_{2} & U_{2} \\ 1.05 & - 1 & 0 \\ \end{bmatrix}\begin{Bmatrix} X_{B} \\ X_{G} \\ X_{U} \\ \end{Bmatrix} = \begin{Bmatrix} M_{1} \\ M_{2} \\ 0 \\ \
The input variables to the preceding simultaneous linear equations are
\[B_{1} = \frac{2}{6} = \frac{1}{3}\ \text{toy per minute}\]
\[B_{2} = \frac{1}{6}\text{ toy per minute}\]
\[G_{1} = \frac{2}{12} = \frac{1}{6}\text{ toy per minute}\]
\[G_{2} = \frac{8}{12} = \frac{2}{3}\text{ toy per minute}\]
\[U_{1} = \frac{4}{6} = \frac{2}{3}\text{ toy per minute}\]
\[U_{2} = \frac{2}{6} = \frac{1}{3}\text{ toy per minute}\]
\[M_{1} = 756\ \text{minutes per day}\]
\[M_{2} = 1260\ \text{minutes per day}\]
Substituting onto the matrix representation of the simultaneous linear equations yields
\[\frac{1}{6}\begin{bmatrix} 2 & 1 & 4 \\ 1 & 4 & 2 \\ 6.3 & - 6 & 0 \\ \end{bmatrix}\begin{Bmatrix} X_{B} \\ X_{G} \\ X_{U} \\ \end{Bmatrix} = \begin{Bmatrix} 756 \\ 1260 \\ 0 \\ \end{Bmatrix}\]
One needs to solve these simultaneous linear equations to find the number of boys, girl and unisex toys for maximizing the manufacturing facility yield.
To find the diametrical contraction during shrink-fitting in a trunnion requires one to first find how the coefficient of linear thermal expansion of steel is related to temperature. The relation is
given through a second-order polynomial, and calculating the coefficients of the polynomial requires solving a set of simultaneous linear equations.
To make the fulcrum (Figure 1) of a bascule bridge, a long hollow steel shaft called the trunnion is shrink fit into a steel hub. The resulting steel trunnion-hub assembly is then shrunk-fit into the
girder of the bridge.
Figure 1 Trunnion-Hub-Girder (THG) assembly.
This is done by first immersing the trunnion in a cold medium such as dry-ice/alcohol mixture. After the trunnion reaches the steady-state temperature, that is, the temperature of the cold medium,
the trunnion outer diameter contracts. The trunnion is taken out of the medium and slid through the hole of the hub (Figure 2).
Figure 2 Trunnion slid through the hub after contracting
When the trunnion heats up, it expands and creates an interference fit with the hub. In 1995, on one of the bridges in Florida, this assembly procedure did not work as designed. Before the trunnion
could be inserted fully into the hub, the trunnion got stuck. Luckily the trunnion was taken out before it got stuck permanently. Otherwise, a new trunnion and hub would be needed to be ordered at a
cost of $50,000. Coupled with construction delays, the total loss could have been more than a hundred thousand dollars.
Why did the trunnion get stuck? This was because the trunnion had not contracted enough to slide through the hole. Can you find out why?
A hollow trunnion of an outside diameter \(12.363^{\prime\prime}\) is to be fitted in a hub of an inner diameter \(12.358^{\prime\prime}\). The trunnion was put in dry-ice/alcohol mixture
(temperature of the fluid - dry ice/alcohol mixture is \(- 108{^\circ}\text{F}\)) to contract the trunnion so that it can be slided through the hole of the hub. To slide the trunnion without
sticking, a diametrical clearance of at least \(0.01^{\prime\prime}\) is required between the trunnion and the hub. Assuming the room temperature is \(80{^\circ}\text{F}\), is immersing it in dry-ice
/alcohol mixture a correct decision?
To calculate the contraction in the diameter of the trunnion, the coefficient of linear thermal expansion at room temperature is used. In that case, the reduction, \({\Delta D}\) in the outer
diameter of the trunnion is
\[\Delta D = D\alpha\Delta T\ \ \ (1)\]
\[D = \text{outer diameter of the trunnion,}\]
\[\alpha = \text{coefficient of linear thermal expansion at room temperature, and}\]
\[\Delta T = \text{change in temperature,}\]
\[D = 12.363^{\prime\prime}\]
\[\alpha = 6.817 \times 10^{- 6}{in/in/}{^\circ}\text{F}\ \text{at}\ 80{^\circ}\text{F}\]
\[\begin{split} \Delta T&= T_{{fluid}} - T_{{room}}\\ &= - 108 - 80\\ &= - 188{^\circ}\text{F} \end{split}\] where
\[T_{{fluid}}= \text{temperature of dry-ice/alcohol mixture}\]
\[T_{{room}}= \text{room temperature}\]
the reduction in the trunnion outer diameter is given by
\[\begin{split} \Delta D &= (12.363)\left( 6.47 \times 10^{- 6} \right)\left( - 188 \right)\\ &=- 0.01504^{\prime\prime} \end{split}\]
So, the trunnion is predicted to reduce in diameter by \(0.01504^{\prime\prime}\). But is this enough reduction in diameter? As per specifications, he needs the trunnion to contract by
\[\begin{split} &= \text{trunnion outside diameter} - \text{hub inner diameter} + \text{diametral clearance} \\ &= 12.363^{\prime\prime} - 12.358^{\prime\prime} + 0.01^{\prime\prime}\\ &= 0.015^{\
prime\prime} \end{split}\]
So, according to his calculations, immersing the steel trunnion in dry-ice/alcohol mixture gives the desired contraction of \(0.015^{\prime\prime}\) as we predict a contraction of \(0.01504^{\prime\
But as shown in Figure 3, the coefficient of linear thermal expansion of steel decreases with temperature and is not constant over the range of temperature the trunnion goes through. Hence, Equation
1 would overestimate the thermal contraction.
Figure 3 Varying coefficient of linear thermal expansion as a function of temperature for cast steel.
The contraction in the diameter for the trunnion for which the coefficient of linear thermal expansion varies as a function of temperature is given by
\[\Delta D = D\int_{T_{{room}}}^{T_{{fluid}}}{\alpha dT}\ \ \ (2)\]
So one needs to find the curve to find the coefficient of linear thermal expansion as a function of temperature. This curve is found by regression, where we best fit a polynomial through the data
given in Table 1.
Table 1 Coefficient of linear thermal expansion as a function of temperature.
Temperature Coefficient of linear thermal expansion
\({^\circ}\text{F}\) \({\mu }\text{in/in}/{^\circ}\text{F}\)
\(80\) \(6.47\)
\(60\) \(6.36\)
\(40\) \(6.24\)
\(20\) \(6.12\)
\(0\) \(6.00\)
\(-20\) \(5.86\)
\(-40\) \(5.72\)
\(-60\) \(5.58\)
\(-100\) \(5.28\)
\(-120\) \(5.09\)
\(-140\) \(4.91\)
\(-160\) \(4.72\)
\(-180\) \(4.52\)
\(-200\) \(4.30\)
\(-220\) \(4.08\)
\(-240\) \(3.83\)
\(-260\) \(3.58\)
\(-280\) \(3.33\)
\(-300\) \(3.07\)
\(-320\) \(2.76\)
\(-340\) \(2.45\)
Assuming that the coefficient of linear thermal expansion is related to temperature by a second-order polynomial,
\[\alpha = a_{0} + a_{1}T + a_{2}T^{2}\ \ \ (3)\]
Given the data points \(\left( \alpha_{1},T_{1} \right)\), \(\left( \alpha_{2},T_{2} \right)\), …..,\(\left( \alpha_{n},T_{n} \right)\) as in Figure 3 and Table 1, the sum of the square of the
residuals (sum of the square of the differences between the observed and predicted values) is \[\begin{split} S_{r} &= \sum_{i = 1}^{n}\left( \alpha_{i} - \{ a_{0} + a_{1}T_{i} + a_{2}T_{i}^{2}\} \
right)^{2}\\ &= \sum_{i = 1}^{n}\left( \alpha_{i} - a_{0} - a_{1}T_{i} - a_{2}T_{i}^{2} \right)^{2} \ \ \ (4) \end{split}\]
To minimize the value of the sum of the square of the residuals, we take the derivative with respect to each of the three unknown coefficients, \(a_0,\) \(a_1,\) and \(a_2\) to give
\[\begin{split} \frac{\partial S_{r}}{\partial a_{0}} &= \sum_{i = 1}^{n}{2\left( \alpha_{i} - a_{0} - a_{1}T_{i} - a_{2}T_{i}^{2} \right)}\left( - 1 \right)\\ &= 2\left\lbrack - \sum_{i = 1}^{n}\
alpha_{i} + na_{0} + a_{1}\sum_{i = 1}^{n}T_{i} + a_{2}\sum_{i = 1}^{n}T_{i}^{2} \right\rbrack \end{split}\]
\[\begin{split} \frac{\partial S_{r}}{\partial a_{1}} &= \sum_{i = 1}^{n}{2\left( \alpha_{i} - a_{0} - a_{1}T_{i} - a_{2}T_{i}^{2} \right)}\left( - T_{i} \right)\\ &= 2\left\lbrack - \sum_{i = 1}^{n}
{\alpha_{i}T_{i}} + a_{0}\sum_{i = 1}^{n}T_{i} + a_{1}\sum_{i = 1}^{n}{T_{i}}^{2} + a_{2}\sum_{i = 1}^{n}T_{i}^{3} \right\rbrack \end{split}\]
\[\begin{split} \frac{\partial S_{r}}{\partial a_{2}} &= \sum_{i = 1}^{n}{2\left( \alpha_{i} - a_{0} - a_{1}T_{i} - a_{2}T_{i}^{2} \right)}\left( - T_{i}^{2} \right)\\ &= 2\left\lbrack - \sum_{i = 1}
^{n}{\alpha_{i}{T_{i}}^{2}} + a_{0}\sum_{i = 1}^{n}{T_{i}}^{2} + a_{1}\sum_{i = 1}^{n}T_{i}^{3} + a_{2}\sum_{i = 1}^{n}T_{i}^{4} \right\rbrack\ \ \ (5) \end{split}\]
Setting three partial derivatives in Equation (5) equal to zero gives,
\[na_{0} + a_{1}\sum_{i = 1}^{n}{T_{i} + a_{2}\sum_{i = 1}^{n}T_{i}^{2} = \sum_{i = 1}^{n}\alpha_{i}}\]
\[a_{0}\sum_{i = 1}^{n}T_{i} + a_{1}\sum_{i = 1}^{n}T_{i}^{2} + a_{2}\sum_{i = 1}^{n}T_{i}^{3} = \sum_{i = 1}^{n}{\alpha_{i}T_{i}}\]
\[a_{0}\sum_{i = 1}^{n}T_{i}^{2} + a_{1}\sum_{i = 1}^{n}T_{i}^{3} + a_{2}\sum_{i = 1}^{n}T_{i}^{4} = \sum_{i = 1}^{n}{\alpha_{i}T_{i}^{2}}\ \ \ (6)\]
The set of equations given by equations(6a), (6b), and (6c) are simultaneous linear equations. The number of data points in Figure (3) is \(24,\) as given in Table 1. Hence
\[n = 24\]
\[\sum_{i = 1}^{24}T_{i} = - 2860\]
\[\sum_{i = 1}^{24}T_{i}^{2} = 7.26 \times 10^{5}\]
\[\sum_{i = 1}^{24}T_{i}^{3} = - 1.86472 \times 10^{8}\]
\[\sum_{i = 1}^{24}T_{i}^{4} = 5.24357 \times 10^{10}\]
\[\sum_{i = 1}^{24}\alpha_{i} = 1.057 \times 10^{- 4}\]
\[\sum_{i = 1}^{24}\alpha_{i}T_{i} = - 1.04162 \times 10^{- 2}\]
\[\sum_{i = 1}^{24}\alpha_{i}T_{i}^{2} = 2.56799\]
\[\begin{split} &24a_{0} - 2860a_{1} + 7.26 \times 10^{5}a_{2} = 1.057 \times 10^{- 4}\\ &- 2860a_{0} + 7.26 \times 10^{5}a_{1} - 1.8647 \times 10^{8}a_{2} = - 1.04162 \times 10^{- 2}\\ &7.26 \times
10^{5}a_{0} - 1.86472 \times 10^{8}a_{1} + 5.24357 \times 10^{10}a_{2} = 2.56799 \end{split}\;\;\;\;\;\;\;\;\;\;\;\; (7)\]
In matrix form, the three simultaneous linear equations can be written as
\[\begin{bmatrix} 24 & - 2860 & 7.26 \times 10^{5} \\ - 2860 & 7.26 \times 10^{5} & - 1.86472 \times 10^{8} \\ 7.26 \times 10^{5} & - 1.86472 \times 10^{8} & 5.24357 \times 10^{10} \\ \end{bmatrix}\
begin{bmatrix} a_{0} \\ a_{1} \\ a_{2} \\ \end{bmatrix} = \begin{bmatrix} 1.057 \times 10^{- 4} \\ - 1.04162 \times 10^{- 2} \\ 2.56799 \\ \end{bmatrix}\;\;\;\;\;\;\;\;\;\;\;\; (8)\]
(1) Can you now find the contraction in the trunnion outer diameter?
(2) Is the magnitude of contraction more than \(0.015^{\prime\prime}\) as required?
(3) If that is not the case, what if the trunnion were immersed in liquid nitrogen (boiling temperature=\(- 321{^\circ}\text{F}\))? Will that give enough contraction in the trunnion?
(4) Redo problem #1 using a third-order polynomial as the regression model. How much different is the estimate of contraction using the third-order polynomial?
(5) Find the optimum polynomial order to use for the regression model.
(6) Find the effect of the number of significant digits used in solving the set of the equations for problem #4 as you must have noticed that the order of the numbers in the coefficient matrix varies
quite a bit. | {"url":"http://nm.mathforcollege.com/NumericalMethodsTextbookUnabridged/chapter-04.00-physical-problem-for-simultaneous-linear-equations.html","timestamp":"2024-11-03T15:42:33Z","content_type":"text/html","content_length":"391800","record_id":"<urn:uuid:662c5af7-6208-47b4-94ac-dca9f4c36d27>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00693.warc.gz"} |
A Constraint Solver Synthesiser
EPSRC EP/H004092/1
Constraints are a natural, powerful means of representing and reasoning about combinatorial problems that impact all of our lives. For example, in the production of a university timetable many
constraints occur, such as: the maths lecture theatre has a capacity of 100 students; art history lectures require a venue with a slide projector; no student can attend two lectures at once.
Constraint solving offers a means by which solutions to such problems can be found automatically. Its simplicity and generality are fundamental to its successful application in a wide variety of
disciplines, such as: scheduling; industrial design; aviation; banking; combinatorial mathematics; and the petrochemical and steel industries.
Currently, applying constraint technology to a large, complex problem requires significant manual tuning by an expert. Such experts are rare. The central aim of this project is to improve
dramatically the scalability of constraint technology, while simultaneously removing its reliance on manual tuning by an expert. We propose to achieve this by developing a constraint solver
synthesiser, which generates a constraint solver specialised to a given problem. Synthesising a solver from scratch has two key benefits. First, it will enable a fine-grained optimisation not
possible for a general solver, allowing the solution of much larger, more difficult problems. Second, it will open up many research possibilities: there are many techniques in the literature that,
although effective in a limited number of cases, are not suitable for general use. Hence, they are omitted from current general solvers and remain relatively undeveloped. The synthesiser will,
however, select such techniques as they are appropriate for an input problem, creating novel combinations to produce powerful new solvers. The result we hope for is a dramatic increase in the number
of practical problems solvable without the input of a constraints expert. | {"url":"https://dominion.cs.st-andrews.ac.uk/index.php","timestamp":"2024-11-14T05:10:01Z","content_type":"text/html","content_length":"3499","record_id":"<urn:uuid:e2432733-270f-4f9e-8129-bc14362ff2c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00892.warc.gz"} |
New forecasting papers 2008-10-21
In this issue we have: Should the SARB Have Stayed Time Inconsistent? ; Forecasting S&P 500 Daily Volatility using a Proxy for Downward Price Pressure ; The Role of Implied Volatility in Forecasting
Future Realized Volatility and Jumps in Foreign Exchange, Stock, and Bond Markets and more.
Should the SARB Have Stayed Time Inconsistent?
Date: 2008-10
By: Rangan Gupta (Department of Economics, University of Pretoria)
Josine Uwilingiye (Department of Economics, University of Pretoria)
URL: http://d.repec.org/n?u=RePEc:pre:wpaper:200833&r=for
This paper derives the econometric restrictions imposed by the Barro and Gordon (1983) model of dynamic time inconsistency on a bivariate time-series model of Consumer Price Index (CPI) inflation and
real Gross Domestic Product (GDP), and tests these restrictions based on quarterly data for South Africa covering the period of 1960:01 through 1999:04, i.e., for the pre-inflation targeting period.
The results show that the data are consistent with the short- and long-run implications of the theory of time-consistent monetary policy. Moreover, when the model is used to forecast one-step-ahead
inflation over the period of 2001:01 to 2008:02, i.e., the period covering the starting point of the inflation targeting regime till date we, on average, obtain lower rates of inflation. The result
tends to suggest that the South African Reserve Bank (SARB), perhaps needs to manage the inflation targeting framework better than i! t has done so far.
Keywords: Dynamic Time Inconsistency; Inflation Targeting; One-Step-Ahead Forecasts
JEL: E31 E52 E61
Date: 2008-10
Sonali Das (CSIR, Pretoria)
By: Rangan Gupta (Department of Economics, University of Pretoria)
Alain Kabundi (Department of Economics and Econometrics, University of Johannesburg)
URL: http://d.repec.org/n?u=RePEc:pre:wpaper:200831&r=for
This paper develops large-scale Bayesian Vector Autoregressive (BVAR) models, based on 268 quarterly series, for forecasting annualized real house price growth rates for large-, medium- and
small-middle-segment housing for the South African economy. Given the in-sample period of 1980:01 to 2000:04, the large-scale BVARs, estimated under alternative hyperparameter values specifying the
priors, are used to forecast real house price growth rates over a 24-quarter out-of-sample horizon of 2001:01 to 2006:04. The forecast performance of the large-scale BVARs are then compared with
classical and Bayesian versions of univariate and multivariate Vector Autoregressive (VAR) models, merely comprising of the real growth rates of the large-, medium- and small-middle-segment houses,
and a large-scale Dynamic Factor Model (DFM), which comprises of the same 268 variables included in the large-scale BVARs. Based on the one- to fo! ur-quarters ahead Root Mean Square Errors (RMSEs)
over the out-of-sample horizon, we find the large-scale BVARs to not only outperform all the other alternative models, but to also predict the recent downturn in the real house price growth rates for
the three categories of the middle-segment-housing over an ex ante period of 2007:01 to 2008:02.
Keywords: Dynamic Factor Model, BVAR, Forecast Accuracy
JEL: C11 C13 C33 C53
Forecasting S&P 500 Daily Volatility using a Proxy for Downward Price Pressure
Date: 2008-10-14
By: Visser, Marcel P.
URL: http://d.repec.org/n?u=RePEc:pra:mprapa:11100&r=for
This paper decomposes volatility proxies according to upward and downward price movements in high-frequency financial data, and uses this decomposition for forecasting volatility. The paper
introduces a simple Garch-type discrete time model that incorporates such high-frequency based statistics into a forecast equation for daily volatility. Analysis of S&P 500 index tick data over the
years 1988-2006 shows that taking into account the downward movements improves forecast accuracy significantly. The R2 statistic for evaluating daily volatility forecasts attains a value of 0.80,
both for in-sample and out-of-sample prediction.
Keywords: volatility proxy; downward absolute power variation; log-Garch; volatility asymmetry; leverage effect; SP500; volatility forecasting; high-frequency data
JEL: C53 C22 G10
The Role of Implied Volatility in Forecasting Future Realized Volatility and Jumps in Foreign Exchange, Stock, and Bond Markets
Date: 2008-10
Thomas Busch (Danske Bank and CREATES)
By: Bent Jesper Christensen (University of Aarhus and CREATES)
Morten Ørregaard Nielsen (Queen's University and CREATES)
URL: http://d.repec.org/n?u=RePEc:qed:wpaper:1181&r=for
We study the forecasting of future realized volatility in the foreign exchange, stock, and bond markets, and of the separate continuous sample path and jump components of this, from variables in the
information set, including implied volatility backed out from option prices. Recent nonparametric statistical techniques of Barndorff-Nielsen & Shephard (2004, 2006) are used to separate realized
volatility into its continuous and jump components, which enhances forecasting performance, as shown by Andersen, Bollerslev & Diebold (2007). The heterogeneous autoregressive (HAR) model of Corsi
(2004) is applied with implied volatility as an additional forecasting variable, and separating the forecasts of the two realized components. A new vector HAR (VecHAR) model for the resulting
simultaneous system is introduced, controlling for possible endogeneity issues. Implied volatility contains incremental information abo! ut future volatility in all three markets, even when
separating the continuous and jump components of past realized volatility in the information set, and it is an unbiased forecast in the foreign exchange and stock markets. In the foreign exchange
market, implied volatility completely subsumes the information content of daily, weekly, and monthly realized volatility measures when forecasting future realized volatility or the continuous or jump
component of this. In out-of-sample forecasting experiments, implied volatility alone is the preferred forecast of future realized volatility in all three markets, as mean absolute forecast error
increases if realized volatility components are included in the forecast. Perhaps surprisingly, the jump component of realized volatility is, to some extent, predictable, and options appear to be
calibrated to incorporate information about future jumps in all three markets.
Keywords: Bipower variation, HAR, Heterogeneous Autoregressive Model, implied volatility, jumps, options, realized volatility, VecHAR, volatility forecasting
JEL: C22 C32 F31 G1
Forecasting macroeconomic variables using a structural state space model
Date: 2008-09-01
By: de Silva, Ashton
URL: http://d.repec.org/n?u=RePEc:pra:mprapa:11060&r=for
This paper has a twofold purpose; the first is to present a small macroeconomic model in state space form, the second is to demonstrate that it produces accurate forecasts. The first of these
objectives is achieved by fitting two forms of a structural state space macroeconomic model to Australian data. Both forms model short and long run relationships. Forecasts from these models are
subsequently compared to a structural vector autoregressive specification. This comparison fulfills the second objective demonstrating that the state space formulation produces more accurate
forecasts for a selection of macroeconomic variables.
Keywords: State space; multivariate time series; macroeconomic model; forecast; SVAR
JEL: C32 C51 C53
Modelling and Forecasting Multivariate Realized Volatility
Date: 2008-09-01
By: Roxana Chiriac (Universität Konstanz)
Valeri Voev
URL: http://d.repec.org/n?u=RePEc:knz:cofedp:0806&r=for
This paper proposes a methodology for modelling time series of realized covariance matrices in order to forecast multivariate risks. The approach allows for flexible dynamic dependence patterns and
guarantees positive definiteness of the resulting forecasts without imposing parameter restrictions. We provide an empirical application of the model, in which we show by means of stochastic
dominance tests that the returns from an optimal portfolio based on the model's forecasts second-order dominate returns of portfolios optimized on the basis of traditional MGARCH models. This result
implies that any risk-averse investor, regardless of the type of utility function, would be better-off using our model.
A Critical Note on the Forecast Error Variance Decomposition
Date: 2008
By: Seymen, Atilim
URL: http://d.repec.org/n?u=RePEc:zbw:zewdip:7388&r=for
The paper questions the reasonability of using forecast error variance decompositions for assessing the role of different structural shocks in business cycle fluctuations. It is shown that the
forecast error variance decomposition is related to a dubious definition of the business cycle. A historical variance decomposition approach is proposed to overcome the problems related to the
forecast error variance decomposition.
Keywords: Business Cycles, Structural Vector Autoregression Models, Forecast Error Variance Decomposition, Historical Variance Decomposition
JEL: C32 E32
Taken from the NEP-FOR mailing list edited by Rob Hyndman. | {"url":"https://www.appliedforecasting.com/new-forecasting-papers-2008-10-21/","timestamp":"2024-11-05T22:06:32Z","content_type":"text/html","content_length":"52056","record_id":"<urn:uuid:b05d7c9c-00a0-4530-9674-cfa7cdbebe7e>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00789.warc.gz"} |
Why is significance important in history?
Why is significance important in history?
Historical significance is the process used to evaluate what was significant about selected events, people, and developments in the past. The key to understanding significance is to understand the
distinction between teaching significant history, and asking students to make judgements about significance.”
How do you determine the significant difference between two groups?
The determination of whether there is a statistically significant difference between the two means is reported as a p-value. Typically, if the p-value is below a certain level (usually 0.05), the
conclusion is that there is a difference between the two group means.
Can Anova be used to compare two groups?
For a comparison of more than two group means the one-way analysis of variance (ANOVA) is the appropriate method instead of the t test. As the ANOVA is based on the same assumption with the t test,
the interest of ANOVA is on the locations of the distributions represented by means too.
How do you compare two datasets with different sample sizes?
One way to compare the two different size data sets is to divide the large set into an N number of equal size sets. The comparison can be based on absolute sum of of difference. THis will measure how
many sets from the Nset are in close match with the single 4 sample set.
How do you compare two different means?
Comparison of Means Techniques
1. Independent Samples T-Test. Use the independent samples t-test when you want to compare means for two data sets that are independent from each other.
2. One sample T-Test.
3. Paired Samples T-Test.
4. One way Analysis of Variance (ANOVA).
Why is it useful to compare two samples?
Often we wish to compare two samples, in order to make inferences about possible differences between the two sampled populations, or differences between subjects in different experimental conditions.
It is important to remember that distributions may differ in many respects: location, spread, and shape.
How do you know if two samples are independent?
Independent samples are measurements made on two different sets of items. If the values in one sample affect the values in the other sample, then the samples are dependent. If the values in one
sample reveal no information about those of the other sample, then the samples are independent.
How do you know if two sets of data are statistically different?
The Students T-test (or t-test for short) is the most commonly used test to determine if two sets of data are significantly different from each other.
When two data sets are compared what characteristics will give the most accurate prediction?
Sample Answer: If the trend line of scatterplots for two data sets are compared, the one more likely to provide an accurate prediction is the one with the stronger correlation. | {"url":"https://easierwithpractice.com/why-is-significance-important-in-history/","timestamp":"2024-11-12T07:02:53Z","content_type":"text/html","content_length":"130873","record_id":"<urn:uuid:9063c9bd-d1b8-4b74-a74d-6f99b7eb6873>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00684.warc.gz"} |
Relative dating and absolute dating anthropology
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geologists date rocks both absolute dating essay. | {"url":"https://piyanas.com/relative-dating-and-absolute-dating-anthropology/","timestamp":"2024-11-01T23:03:21Z","content_type":"text/html","content_length":"63162","record_id":"<urn:uuid:b8c29c1d-7cea-4ac6-9142-f28a26de5aaf>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00893.warc.gz"} |
Lesson: Eliminating a variable | Oak National Academy
Switch to our new maths teaching resources
Slide decks, worksheets, quizzes and lesson planning guidance designed for your classroom.
Lesson details
Key learning points
1. In this lesson, we will determine how to eliminate a variable by adding or subtracting one equation to or from the other.
This content is made available by Oak National Academy Limited and its partners and licensed under Oak’s terms & conditions (Collection 1), except where otherwise stated.
5 Questions
If x=3 and 2x+5y=41, what is the value of y?
7x+2y=24 AND 4x+2y=18. Find the value of 3x
7x+2y=24 AND 4x + 2y = 18. Find the value of y
9x +6y=168 AND 9x +4y=160. Solve
4x +8y=116 AND 4x +4y=68. Solve
5 Questions
3x+2y=5 AND 3x-2y=7. If I add these two equations together, which unknown will be eliminated?
3x+2y=5 AND 3x-2y=7. If I subtract the two equations, which unknown will be eliminated?
3x-5y=17 AND 2x-5y=10. Subtract the first equation from the second equation.
2x-5y=10 AND 3x-5y=17. Add the two equations together.
3x-5y=17 AND 7x-5y=20. How can I eliminate y from these simultaneous equations? | {"url":"https://www.thenational.academy/teachers/lessons/eliminating-a-variable-75j3jc","timestamp":"2024-11-07T11:03:15Z","content_type":"text/html","content_length":"257002","record_id":"<urn:uuid:9e0199d7-aefc-4e50-98f9-7ec1d1d75d09>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00290.warc.gz"} |
DAA Lab Manual (Design and Analysis of Algorithm) - Code Revise
kmp string matching algorithm in DAA Here you will learn the program code of kmp string matching algorithm in c programming. The Knuth-Morris-Pratt (KMP) algorithm is a more efficient string […]
c program for Naive String Matching algorithm in DAA
Naive String Matching algorithm in DAA The Naive String Matching algorithm is a simple method used to find occurrences of a pattern within a text. Here’s how it works: 1.
c program for Naive String Matching algorithm in DAA Read More » | {"url":"https://coderevise.com/daa-lab-manual/","timestamp":"2024-11-04T01:53:39Z","content_type":"text/html","content_length":"286504","record_id":"<urn:uuid:4b6ac70a-f2ad-43f2-a2b9-808bd8c15c14>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00245.warc.gz"} |
Problem Model Prompt
P_Combinatorics_9 PaLM 2 Medium Direct C w/o H
Ground Truth Answer: n
Use in Code
import champ_dataset # "pip install champ-dataset" if necessary
dataset = champ_dataset.load('v0')
problem = dataset['P_Combinatorics_9']
messages = problem.conversations['PaLM 2 Medium|Direct C w/o H'].messages
for message in messages:
print(f'Role: {message.role}')
print(f'Text: {message.text}')
print(f'Error: {message.error}')
Full Conversation System Prompt User Input Model Output
You are an expert on mathematics.
Solve the following problem. Make sure to show your work before giving the final answer.
Consider all 2^n−1 nonempty subsets of the set {1, 2,..., n}. For every such subset, we find the product of the reciprocals of each of its elements. Find the sum of all these products, as an expression of n.
Let $S = \\{1, 2, \dots, n\\}$. For any subset $A \subseteq S$, let $r(A) = \prod_{a \in A} \frac{1}{a}$. We want to find the sum of all the products $r(A)$, where $A$ ranges over all nonempty subsets of $S$.
We can write this sum as
$$\sum_{A \subseteq S, |A| > 0} r(A).$$
To evaluate this sum, we will use the following lemma.
Lemma: Let $S$ be a set of $n$ elements. Then the sum of all the products of the elements of $S$ is $n!$.
Proof: The proof is by induction on $n$. The base case is $n = 1$, in which case the sum is just $1$, which is equal to $1!$.
For the inductive step, assume that the lemma holds for all sets of $k$ elements for some $k < n$. We want to show that it holds for sets of $n$ elements.
Let $S$ be a set of $n$ elements. We can partition $S$ into two sets: the set of first $k$ elements of $S$ and the set of last $n - k$ elements of $S$. Let $A$ be the set of all products of the elements of the first $k$ elements of $S$, and let $B$ be the set of all products of the elements of the last $n - k$ elements of $S$. Then $A$ and $B$ are both sets of $k$ elements, so by the induction hypothesis, $A = k!$ and $B = (n - k)!$. Therefore, the sum of all the products of the elements of $S$ is
$$A \cdot B = k! (n - k)! = n!.$$
This completes the proof of the lemma.
Now, let's return to the sum we are trying to evaluate. We can write this sum as
$$\sum_{A \subseteq S, |A| > 0} r(A) = \sum_{A \subseteq S, |A| > 0} \prod_{a \in A} \frac{1}{a} = \sum_{A \subseteq S} \prod_{a \in A} \frac{1}{a} - \sum_{A \subseteq S, |A| = 0} \prod_{a \in A} \frac{1}{a}.$$
The first sum is the sum of all the products of the elements of $S$, which by the lemma is $n!$. The second sum is the sum of all the products of the empty set, which is $1$. Therefore, the sum we are looking for is $n! - 1$.
In conclusion, the sum of all the products of the reciprocals of the elements of all nonempty subsets of $\\{1, 2, \dots, n\\}$ is $n! - 1$.
Now, summarize the answer above in one sentence, without any intermediate steps or explanations.
The sum of all the products of the reciprocals of the elements of all nonempty subsets of $\\{1, 2, \dots, n\\}$ is $n! - 1$. | {"url":"https://yujunmao1.github.io/CHAMP/solutions/p_combinatorics_9_palm2m_directcwoh.html","timestamp":"2024-11-14T07:16:43Z","content_type":"text/html","content_length":"6999","record_id":"<urn:uuid:197af910-3bb2-4f2c-b019-c334ac7c5e1e>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00428.warc.gz"} |
EIRP meaning and calculation
EIRP is the power required when the antenna is replaced by an isotropic antenna, and can be calculated from the transmit power, feed line loss, and gain of the transmit antenna.
This article explains the meaning of EIRP and how to calculate it.
Table of Contents
What is EIRP?
EIRP stands for Equivalent Isotropically Radiated Power or Effective Isotropically Radiated Power. The antenna has a large radio strength in a given direction.
Antennas have the property of increasing radio wave strength in a certain direction (directivity).
However, it is inconvenient to compare the strength of radio waves because the strength of directivity differs from antenna to antenna.
Therefore, we consider isotropic antennas.
An isotropic antenna is a theoretical point that radiates radio waves with the same strength in all directions.
n other words, isotropic antennas are not directional.
The EIRP is the power required to reproduce the field strength in the direction of maximum antenna radiation (main lobe) with an isotropic antenna.
The solid circle represents the antenna pattern of an isotropic antenna.
The absolute gain of the antenna is calculated with respect to the isotropic antenna, so the absolute gain of the isotropic antenna is 0 dBi.
The horizontal extension is the main lobe of the antenna, and the EIRP is the power fed to the isotropic antenna to account for this gain.
ERP (Equivalent Radiated Power or Effective Radiated Power), in which the reference antenna for EIRP is a half-wavelength dipole antenna, is also available.
Since the absolute gain of the half-wavelength dipole antenna is 2.14 dBi, the EIRP can be expressed as
EIRP can be expressed as
How to calculate EIRP?
If EIRP \(P_t[\mathrm{dBW}]\) is the transmitting power, the feeder loss between the transmitter and the transmitting antenna, and the gain of the transmitting antenna, then EIRP is
\[\mathrm{EIRP} = P_t-L_{ft}+G_t\]
Example of EIRP calculation
Assuming that the transmitting power is 200 mW (-7 dBW), the feeder loss between the transmitter and the transmitting antenna is 1 dB, and the gain of the transmitting antenna is 2.14 dBi
(half-wavelength dipole antenna)
\[\mathrm{EIRP} = -7\mathrm{dBW}-1\mathrm{dB}+2.14\mathrm{dBi} = -5.86\mathrm{dBW}\]
The result is as follows. As shown in this result, the EIRP can be negative depending on the unit. | {"url":"https://ensatellite.com/eirp/","timestamp":"2024-11-12T19:42:17Z","content_type":"text/html","content_length":"270871","record_id":"<urn:uuid:0439194c-d01c-42ea-9d88-834f4827fb27>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00307.warc.gz"} |
Comparative Analysis of Stellar Temperatures and Energy Flux: Wien's Law vs. Stefan-Boltzmann Law - Essay Quoll
Comparative Analysis of Stellar Temperatures and Energy Flux: Wien’s Law vs. Stefan-Boltzmann Law
When studying stars, understanding their temperatures and energy emission is crucial. Two fundamental laws, Wien’s law and the Stefan-Boltzmann law, provide insights into these aspects. We explore
how Wien’s law helps identify cooler stars and compare them, and how the Stefan-Boltzmann law determines which star emits more energy flux and by how much.
6. Using Wien’s law to find the cooler star
Wien’s law relates the temperature of a star to the wavelength at which its emission is most intense. The formula for Wien’s law is:
λ_max = (b / T),
• λ_max is the wavelength at which emission is most intense.
• b is Wien’s displacement constant, approximately equal to 2.898 × 10^(-3) m·K.
• T is the temperature of the star in Kelvin.
To find the cooler star, we need to compare the values of T (temperature) for both stars and see which one has the higher temperature. The star with the lower temperature is cooler.
7. Using the Stefan-Boltzmann law to find the star emitting more energy flux
The Stefan-Boltzmann law relates the total energy radiated by a star to its temperature. The formula for the Stefan-Boltzmann law is:
E = σ * A * T^4,
• E is the total energy radiated (energy flux) by the star.
• σ is the Stefan-Boltzmann constant, approximately equal to 5.67 × 10^(-8) W/(m^2·K^4).
• A is the surface area of the star.
• T is the temperature of the star in Kelvin.
To determine which star emits more energy flux, we’ll need to calculate the energy flux for both stars using the formula above. The star with the higher energy flux value emits more energy.
Carroll, B. W., & Ostlie, D. A. (2020). Stellar Radiation and the Stefan-Boltzmann Law. An Introduction to Modern Astrophysics (p. 124). Cambridge University Press.
Schwarzschild, M. (2018). Stellar Temperatures and Wien’s Law. Astrophysical Journal, 867(2), 123.
Smith, J. R. (2019). Understanding Stellar Properties: A Comprehensive Guide. Astronomy & Astrophysics, 548, A1.
1. What is Wien’s law, and how does it help determine the temperature of stars?
Wien’s law is a fundamental principle in astrophysics that relates the temperature of a star to the wavelength at which its emission is most intense. By using Wien’s law, astronomers can estimate the
temperature of stars and classify them based on their spectral characteristics.
2. How can I identify the cooler of two stars using Wien’s law?
To identify the cooler star between two, you can compare their temperatures calculated using Wien’s law. The star with the lower temperature, as determined through Wien’s law calculations, is
considered cooler.
3. What does the Stefan-Boltzmann law reveal about stellar energy flux?
The Stefan-Boltzmann law provides a formula to calculate the total energy radiated by a star, which is often referred to as energy flux. It demonstrates how a star’s energy emission is related to its
temperature. Stars with higher temperatures emit significantly more energy flux than cooler stars.
4. Can you explain the significance of surface area in the Stefan-Boltzmann law and its role in determining energy flux?
The Stefan-Boltzmann law incorporates the surface area of a star (A) in its calculations. This parameter highlights that larger stars, with greater surface areas, emit more energy flux compared to
smaller stars at the same temperature. Surface area plays a crucial role in energy flux determination.
5. How do astronomers apply Wien’s law and the Stefan-Boltzmann law in their study of stars and celestial bodies?
Astronomers use Wien’s law and the Stefan-Boltzmann law extensively to study stars, estimate their temperatures, and analyze their energy emission. These laws are foundational tools in astrophysics
and aid in understanding the properties and behavior of stars in the universe.
Last Completed Projects
topic title academic level Writer delivered | {"url":"https://essayquoll.com/2023/09/07/comparative-analysis-of-stellar-temperatures-and-energy-flux-wiens-law-vs-stefan-boltzmann-law/","timestamp":"2024-11-12T12:44:45Z","content_type":"text/html","content_length":"41680","record_id":"<urn:uuid:9c1d1fb1-a121-4440-8231-1487bbd3961b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00137.warc.gz"} |
Simple Training and Inference recipe
In this tutorial, we will see how to use utilites from Modulus to setup a simple model training pipeline. Once the initial setup is complete, we will look into optimizing the training loop, and also
run it in a distributed fashion. We will finish the tutorial with an inference workflow that will demonstrate how to use Modulus models in inference.
Let’s get started. For the purposes of this tutorial, we will focus more on the Modulus utilities and not the correctness of the problem definition or the results. A typical training workflow
requires data, a trainable model and an optimizer to update the model parameters.
In this example, we will look at different ways one can interact with Models in Modulus. Modulus presents a library of models suitable for Physics-ML applications for you to use directly in your
training workflows. In this tutorial we will see how to use a simple model in Modulus to setup a data-driven training. Using the models from Modulus will enable us to use various other Modulus
features like optimization and quality-of-life functionalites like checkpointing and model entrypoints.
Later we will also see how to customize these models in Modulus.
In this example we will use the FNO model from Modulus. To demonstrate the training using this model, we would need some dataset to train the model. To allow for fast prototyping of models, Modulus
provides a set of benchmark datasets that can be used out of the box without the need to setup data-loading pipelines. In this example, we will use one such datapipe called Darcy2D to get the
training data.
Let’s start with importing a few utils and packages.
import torch
import modulus
from modulus.datapipes.benchmarks.darcy import Darcy2D
from modulus.metrics.general.mse import mse
from modulus.models.fno.fno import FNO
In this example we want to develop a mapping between the permeability and its subsequent pressure field for a given forcing function. Refer Modulus Datapipes for additional details.
Then a simple training loop for this example can be written as follows:
normaliser = {
"permeability": (1.25, 0.75),
"darcy": (4.52e-2, 2.79e-2),
dataloader = Darcy2D(
resolution=256, batch_size=64, nr_permeability_freq=5, normaliser=normaliser
model = FNO(
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)
scheduler = torch.optim.lr_scheduler.LambdaLR(
optimizer, lr_lambda=lambda step: 0.85**step
# run for 20 iterations
for i in range(20):
batch = next(iter(dataloader))
true = batch["darcy"]
pred = model(batch["permeability"])
loss = mse(pred, true)
print(f"Iteration:{i}. Loss:{loss.detach().cpu().numpy()}")
That’s it! This shows how to use a model from Modulus. Most of the models in Modulus are highly configurable allowing you to use them out-of-the-box for different applications. Refer Modulus Models
for a more complete list of available models.
Modulus provides a lot of pre-built optimized models. However, there might be times where the shipped models might not serve your application. In such cases, you can easily write your own models and
have them interact with the other Modulus utilites and features. Modulus uses PyTorch in the backend and most Modulus models are, at the core, PyTorch models. In this section we will see how to go
from a typical PyTorch model to a Modulus model.
Let’s get started with the same application of Darcy problem. Let’s write a simple UNet to solve the problem. A simple PyTorch model for a UNet can be written as shown below:
import torch.nn as nn
import modulus
from modulus.datapipes.benchmarks.darcy import Darcy2D
from modulus.metrics.general.mse import mse
class UNet(nn.Module):
def __init__(self, in_channels=1, out_channels=1):
super(UNet, self).__init__()
self.enc1 = self.conv_block(in_channels, 64)
self.enc2 = self.conv_block(64, 128)
self.dec1 = self.upconv_block(128, 64)
self.dec2 = self.upconv_block(64, 32)
self.final = nn.Conv2d(32, out_channels, kernel_size=1)
def conv_block(self, in_channels, out_channels):
return nn.Sequential(
nn.Conv2d(in_channels, out_channels, 3, padding=1),
def upconv_block(self, in_channels, out_channels):
return nn.Sequential(
nn.ConvTranspose2d(in_channels, out_channels, 2, stride=2),
nn.Conv2d(out_channels, out_channels, 3, padding=1),
def forward(self, x):
x = self.enc1(x)
x = self.enc2(x)
x = self.dec1(x)
x = self.dec2(x)
return self.final(x)
Let’s now convert this to a Modulus Model. Modulus provides Module class that is designed to be a drop-in replacement for the torch.nn.module. Along with that, you need to also pass a MetaData that
captures the optimizations and other features supported by the model. Using the Module subclass allows using these optimizations, and other features like checkpointing etc. from Modulus.
Thus, converting a PyTorch model to a Modulus model is very simple. For the above model, the diff would look something like below:
- import torch.nn as nn
+ from dataclasses import dataclass
+ from modulus.models.meta import ModelMetaData
+ from modulus.models.module import Module
- class UNet(nn.Module):
+ @dataclass
+ class MetaData(ModelMetaData):
+ name: str = "UNet"
+ # Optimization
+ jit: bool = False
+ cuda_graphs: bool = True
+ amp_cpu: bool = True
+ amp_gpu: bool = True
+ class UNet(Module):
def __init__(self, in_channels=1, out_channels=1):
- super(UNet, self).__init__()
+ super(UNet, self).__init__(meta=MetaData())
self.enc1 = self.conv_block(in_channels, 64)
self.enc2 = self.conv_block(64, 128)
With simple changes like this you can convert a PyTorch model to a Modulus Model!
The optimizations are not automatically applied. The user is responsible for writing the model with the optimizations supported. However, if the models supports the optimization and the same is
captured in the MetaData, then the downstream features will work out-of-the-box.
For utilizing the checkpointing functionality of Modulus, the Model instantiation arguments must be json serializable.
You can also use a Modulus model as a standard PyTorch model as they are interoperable.
Let’s say you don’t want to make changes to the code, but you have a PyTorch model already. You can convert it to a Modulus model by using the modulus.Module.from_torch method. This is described in
detail in Converting PyTorch Models to Modulus Models.
from dataclasses import dataclass
import torch.nn as nn
from modulus.models.meta import ModelMetaData
from modulus.models.module import Module
class MdlsUNetMetaData(ModelMetaData):
name: str = "MdlsUNet"
# Optimization
jit: bool = False
cuda_graphs: bool = True
amp_cpu: bool = True
amp_gpu: bool = True
MdlsUNet = Module.from_torch(UNet, meta=MdlsUNetMetaData)
And just like that you can use your existing PyTorch model as a Modulus Model. A very similar process can be followed to convert a Modulus model to a Modulus Sym model so that you can use the
Constraints and other defitions from the Modulus Sym repository. Here you will use the Arch class from Modulus Sym that provides utilites and methods to go from a tensor data to a dict format which
Modulus Sym uses.
from typing import Dict, Optional
from modulus.sym.key import Key
from modulus.sym.models.arch import Arch
class MdlsSymUNet(Arch):
def __init__(
super(MdlsSymUNet, self).__init__(
input_keys=input_keys, output_keys=output_keys
self.mdls_model = MdlsUNet(in_channels, out_channels) # MdlsUNet defined above
def forward(self, dict_tensor: Dict[str, torch.Tensor]):
x = self.concat_input(
out = self.mdls_model(x)
return self.split_output(out, self.output_key_dict, dim=1)
Once we have a model defined in the Modulus style, we can use the optimizations like AMP, CUDA Graphs, and JIT using the modulus.utils.StaticCaptureTraining decorator. This decorator will capture the
training step function and optimize it for the specified optimizations.
The StaticCaptureTraining decorator is still under development and may be refactored in the future.
import time
from modulus.utils import StaticCaptureTraining
normaliser = {
"permeability": (1.25, 0.75),
"darcy": (4.52e-2, 2.79e-2),
dataloader = Darcy2D(
resolution=256, batch_size=8, nr_permeability_freq=5, normaliser=normaliser
model = MdlsUNet().to("cuda")
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)
scheduler = torch.optim.lr_scheduler.LambdaLR(
optimizer, lr_lambda=lambda step: 0.85**step
# Create training step function with optimization wrapper
# StaticCaptureTraining calls `backward` on the loss and
# `optimizer.step()` so you don't have to do that
# explicitly.
def training_step(invar, outvar):
predvar = model(invar)
loss = mse(predvar, outvar)
return loss
# run for 20 iterations
for i in range(20):
batch = next(iter(dataloader))
true = batch["darcy"]
input = batch["permeability"]
loss = training_step(input, true)
Modulus has several Distributed utilites to simplify the implementation of parallel training and make inference scripts easier by providing a unified way to configure and query parameters associated
with distributed environment.
In this example, we will see how to convert our existing workflow to use data-parallelism. For an deep-dive on Modulus Distributed utilities, refer Modulus Distributed.
def main():
# Initialize the DistributedManager. This will automatically
# detect the number of processes the job was launched with and
# set those configuration parameters appropriately.
# Get instance of the DistributedManager
dist = DistributedManager()
normaliser = {
"permeability": (1.25, 0.75),
"darcy": (4.52e-2, 2.79e-2),
dataloader = Darcy2D(
resolution=256, batch_size=64, nr_permeability_freq=5, normaliser=normaliser
model = FNO(
# Set up DistributedDataParallel if using more than a single process.
if dist.distributed:
ddps = torch.cuda.Stream()
with torch.cuda.stream(ddps):
model = DistributedDataParallel(
], # Set the device_id to be the local rank of this process on this node
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)
scheduler = torch.optim.lr_scheduler.LambdaLR(
optimizer, lr_lambda=lambda step: 0.85**step
# Create training step function with optimization wrapper
# StaticCaptureTraining calls `backward` on the loss and
# `optimizer.step()` so you don't have to do that
# explicitly.
def training_step(invar, outvar):
predvar = model(invar)
loss = mse(predvar, outvar)
return loss
# run for 20 iterations
for i in range(20):
batch = next(iter(dataloader))
true = batch["darcy"]
input = batch["permeability"]
loss = training_step(input, true)
if __name__ == "__main__":
Running inference on trained model is simple! This is shown by the code below.
model = FNO(
# Save the checkpoint. For demo, we will just save untrained checkpoint,
# but in typical workflows is saved after model training.
# Inference code
# The parameters to instantitate the model will be loaded from the checkpoint
model_inf = modulus.Module.from_checkpoint("untrained_checkpoint.mdlus").to("cuda")
# put the model in evaluation mode
# run inference
with torch.inference_mode():
input = torch.ones(8, 1, 256, 256).to("cuda")
output = model_inf(input)
The static capture and distributed utilities can also be used during inference for speeding up the inference workflow, but that is out of the scope for this tutorial. | {"url":"https://docs.nvidia.com/deeplearning/modulus/modulus-core/tutorials/simple_training_example.html","timestamp":"2024-11-09T07:22:29Z","content_type":"text/html","content_length":"353305","record_id":"<urn:uuid:b0c7b253-899b-4bfd-9198-c2d0ba24cf83>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00224.warc.gz"} |
The presented approach carries enormous analytical potential. The calculated states and eigenenergies of the presented Hamiltonian allow the prediction of a number of electron and magnetic properties
of compounds. The structure of states obtained by solving the Hamiltonian eigenequation makes it possible to calculate a compound’s magnetic properties at extremely low temperatures, including the
value of the magnetic moment of an ordered state, its direction, and the behavior of magnetic moments after the application of an external magnetic field. It is also possible to calculate changes in
the system's physical properties with increasing temperature, when the values (and possibly directions) of magnetic moments change, similar to the change of the splitting of fine structure states
in a self-aligned molecular field. The presented approach facilitates analysis of the mechanism of the formation of magnetic moments and the formation of the magnetic order.
At a temperature of T = 0 K, only the ground state is occupied. In this situation, the magnetic moment of the ion is exactly equal to the momentum of the ground state. At extremely low temperatures,
it is possible excite the system, for example by magnetic interaction with low-energy neutrons (as is used in Inelastic Neutron Scattering Spectroscopy, INS). However, it should be remembered that
the observed transitions are excitations from the ground state. When the temperature rises, the probability of occupying higher states increases according to Boltzmann statistics. The number of ions
with energy E[i] within a system with a temperature T, is:
In the above expression, N[0] denotes the total number of particles, and Z(T) is a statistical sum of states (separation function). Knowing the statistical sum of the states, we can determine the
Helmholtz free energy F(T):
Therefore, the internal energy of the U(T) system can be estimated:
k[B]= 1,38 x 10 ^-23 J/K (Boltzmann constant)
Based on knowledge of thermodynamic state functions [7], we can specify a number of properties defined based on them. In particular, Helmholtz thermal dependence of free energy F(T) allows for the
analysis of the thermal evolution of the properties of d or f-electron systems. A discussion of the various properties of the compounds calculated based on the presented method at T ≠ 0 is shown
below. The fine structure of states (E[i], Γ[i]) makes it possible to determine the thermodynamic functions for the statistical group of N ions. The most sensible value is N=N[0]≈6.022.10^23 mol^-1
(Avogadro constant).
Calculating the entropy of the localized electron system in an unclosed shell d or f is reduced to calculation of the integral:
where c(T) is the system's calculated molar specific heat taken as a derivative of the total internal energy:
or directly as a second derivative of the Helmholtz free energy:
and S is also defined as:
which in conjunction with the definition of statistical entropy allows the number of occupied states of the fine electron structure at a specific temperature to be specified.
In the applied methodology, calculating the magnetization boils down to summing the identically behaving magnetic moments of individual ions with an unclosed electron shell. Due to the fact that
transition metal ions are in the self-aligned molecular field, their magnetic moment value changes with temperature. Bearing in mind that each eigenstate of the CEF Hamiltonian is related to its
magnetic moment, the total moment of a strongly correlated electron system at a given temperature is the resultant moment of occupied states calculated with the inclusion of the Boltzmann weight:
In the above equation, the α index is a directional component, i - numbers the Hamiltonian eigenstates, while <J^i[α]> represents the expected value of the total angular momentum on the α axis in
the i-th state. It should be emphasized that at T = 0, when only the ground state is occupied, the magnetization of the ion system 4f per 1 ion is equal to the expected value of the magnetic moment
of the ground state. Due to the fact that all ions behave identically in this model, the calculated temperature dependence of the magnetic moment is closely linked to the value of the molecular
field, which is:
The value of the molecular field factor n[mol] determines the temperature at which the magnetic order T[c ]occurs. Calculations of magnetization in the ordered state can determine the
magnetocrystalline anisotropy of the system. In the case of tetragonal symmetry, the expansion of the anisotropy energy can be represented in a known form:
where K[i](T) are coefficients of anisotropy dependent on the temperature, and θ and φ are polar angles describing the direction of the magnetic moment relative to the perpendicular crystallographic
axes. The coefficients Ki(T) are closely related to the parameters of the crystal field. In a field with tetragonal symmetry they have the following value:
where:m[c]=m[L]+m[S]. Knowledge of the expected values of the momentum vector components of each state <J^i[α]> makes it possible to specify the share of the spin and orbital sections of the
magnetic moment. Bearing in mind that: J[z]=L[z]+S[z ]and taking into account that g[l ]= 1 and g[s ]= 2.002324:
we will have reached:
The above relationships indicate that the ratios of each component of the magnetic moment to the total momentum are fixed for a given ion and independent of their environment and temperature:
The issue of the accuracy of the above reasoning is discussed at the beginning of the description of the theoretical bases of application, while comparing the results of calculations performed on the
basis |J,J[z]> in relation to the calculations carried out for the fuller basis |L,S,L[z],S[z]> taking into account the coupling between the states of different multiplets. Determining the value of
the orbital and spin components of the magnetic moment of the paramagnetic ion with an electron structure calculated in the representation |L,L[z],S,S[z]> boils down to a simple interpretation of the
expected values L and S.
An external magnetic field eliminates the degeneracy, changes the energy value of the states, and mixes the wave functions of states (Zeeman effect). Elimination of degeneracy and the
temperature-dependent occupation of states leads to the formation of a temperature-dependent magnetic moment of the ion. The magnetization of M is calculated as the statistical sum of the moments of
individual m ions.
M[μ [B]/ f.u.] ( 1μ [B] = 9.27 ·10^-24 J/T, J/T = A × m^2 )
α= x, y, z.
μ[B]–Bohr magneton,
g[L]= Landé g-factor
k[B]-Boltzmann constant= 1,38 · 10 -23 J/K
The magnetic susceptibility of a paramagnetic state is calculated in accordance with its definition as the ratio of induced magnetization to the applied magnetic field. Taking into account the
relationship between the thermodynamic functions, we can relatively simply calculate the directional components of the temperature dependence of magnetic susceptibility of a strongly correlated
electron system in an unclosed shell in a crystal structure. Within the limit of low external fields, susceptibility is defined as the product of:
Where the -α index is the direction of the local coordinate system associated with the quantization axes in the crystal field with a certain symmetry. Bearing in mind the relationships described in
the section THERMODYNAMICS OF MULTIELECTRON SYSTEMS, the matrix elements between different states are obtained within the limit of small fields:
where j, k numbers the eigenstates of the Hamiltonian. For high temperatures and very weak crystal fields, this expression is reduced to the Curie law susceptibility. At temperatures comparable to
the magnitude of the splitting of the states of crystal fields, these susceptibility curves may differ significantly from the shape of hyperbole. In the case of crystal fields with low symmetry, the
magnetic susceptibility will exhibit a significant anisotropy. In a cubic structure, χ[x](T)= χ[y](T)= χ[z](T). Magnetic susceptibility can be calculated for any external fields as the ratio of the
magnetization taken as the sum of the induced magnetic moments to the applied field. Due to the often observed non-linear increase in the value of the magnetic moment, the susceptibility calculated
for different fields may vary considerably.
Fine electron structure is observed in a number of spectroscopic methods (EPR ESR, IR spectroscopy, Raman scattering, etc.). Currently, the best method for exploring the fine electron structure is
Inelastic Neutron Scattering (INS). For a system of N identical ions, a differential cross section in the INS experiments is assumed as follows:
In the above equation γ[N] is the gyromagnetic factor of the neutron, r[e] is the classical radius of the e electron r[e]=e^2/(m[e]c^2)=2.818·10^-15m, k[i] and k[f] are the wave vector of the
incident and reflected neutron, respectively, and the expression exp(-2W(k[i]-k[f])) represents the Debye-Waller factor, which describes the thermal excitation of vibrations in atoms. The delta
functionδ (ω - E[j]+E[k]) corresponds to infinitely narrow crystal field states and is often replaced in exact calculation with the Lorentz distribution with specified half widths at half maximum
(FWHM) of CEF states. The square of matrix elements from the above equation, symbolically denoted as <Γ[k]|J┴|Γ[j]>^2 in the case of a polycrystalline material for dipole transitions is replaced
In the case of a single crystal, the intensity of neutron scattering for dipole transitions between CEF states (Γ[j] and Γ[k]) is represented by the following equation:
where θ is the angle between the neutron beam and the quantization axis z.
INS is a very powerful tool for experimental verification of the theoretically calculated fine structure of electron states. Note, however, that not all CEF states are visible using this technique;
only the transitions between states, where |<Γ[k]|J[z]|Γ[j]>|, |<Γ[k]|J[-]|Γ[j]>|, |<Γ[k]|J[+]|Γ[j] >| is non-zero.
Schottky specific heat occurs when dealing with a low-energy structure of discrete electron states. All we need to determine whether a given atom will exhibit the Schottky anomaly is knowledge of the
electron structure of the element, i.e. the way in which electrons are arranged in shells, subshells and orbitals of the atom. In particular, the existence of fine electron structure with close lying
energy levels ensures that the contribution c[Shottky] is recognizable in the observed c[Shottky](T) relationships:
The Schottky anomaly curve is characterized by one or more peaks. Knowing the shape of this curve, we are often able to use it to determine the system of energy levels of a chemical compound. The
disadvantage of this method is that we may find a number of different systems of energy levels that correspond to a single curve. Molar specific heat for localized electron system can be also
calcualated difectly, as a second derivative of the Helmholtz free energy:
In order to present a realistic form of the total heat capacity of the crystal defined by paramagnetic ions and the CEF potential, we have introduced the ability to visualize the course of
temperature dependence of the total heat of a crystal lattice combined with the Schottky anomaly. The heat of the lattice is simulated with the best-known and most widely accepted formal model which
is based on the phonons-quantized thermal vibrations of the crystal lattice. Such vibrations can be interpreted as normal modes of harmonic oscillators and their energy determined. The energy of a
single phonon is:
A phonon is a quantum of vibration energy of the crystal lattice. Phonons may differ in frequency; therefore, to calculate the total energy of the lattice, the following must be performed: determine
the number of phonons with a specific frequency, multiply it by the energy of a phonon with such frequency, and then sum the energies obtained for different frequencies.
The average number of phonons with a specific frequency can be determined using Bose-Einstein statistics (the so-called Bose-Einstein distribution), given by the formula:
Knowing what frequency phonons of a solid can have, we can determine its total lattice energy, but for simplicity, the Debye model can be successfully used when considering phonon heat. If the length
of the carbon chain consisting of N atoms in a one-dimensional lattice is denoted as L, it is possible to calculate the wavelengths using the following formula:
The number of vibrations in the lattice are calculated by converting the above formula to the one below:
In three-dimensional space, the number of vibrations with a certain frequency is calculated similarly, although the formula will have additional adjustments for fixed coefficients:
where v is the speed of sound in a vacuum, V is the volume of the crystal and ω is the frequency. The last equation is differentiated from ω.We will have reached the full number of vibrations of N
atoms with three degrees of freedom, each computed by integrating the above expression:
Not all vibrations are allowed in the Debye model, so integration must be performed from zero to ω[D], which is maximum frequency of normal mode that may occur in the crystal.
Let us calculate ω[D] using the last equation [^9]
multiply both sides of the equation by
│Solid Element │Θ [K] │
│ Fe │ 467 │
│ Ni │ 456 │
│ Cu │ 339 │
│ Pb │ 95 │
│ Si │ 658 │
Table Debye temperature for several selected elements in the solid phase
The resulting equation defines the Debye temperature (Θ), which is the temperature at which no new normal vibrations occur in the crystal. The Debye temperature also determines the crystal's behavior
at a given temperature and how to calculate its specific heat (depending on the temperature).
If the temperature of the crystal is lower than Θ, its oscillation amplitude increases and new vibrations arise. At a temperature higher than Θ, the amplitude continues to increase, but new
vibrations no longer arise. Table above shows the values of the Debye temperature for several elements .
To derive the formula for the Debye heat, we must first calculate the internal energy, and then differentiate it from temperature. For the sake of simplicity, let us assume that the phonon velocity
does not depend on its polarization. Moreover, in order to calculate the total heat energy, calculate the internal energy of one polarization and multiply it by three.
Differentiating the last equation from temperature, we shall calculate the heat capacity.
The above equation can be simplified by substituting the variables:
Eventually, the expression for the Debye heat takes the form:
This form of the formula for the specific heat of the crystal lattice is used as a comparative in calculations that appear in the result package in the tab associated with specific heat, where by
defining the value of Θ (in K units), we can observe the total heat of the crystal lattice with the Schottky anomaly (provided, of course, it is within a defined temperature range).
• Elliot, K.W.H. Stevens, Proc. Roy. Soc.A215 (1953) 437.
• Elliot, K.W.H. Stevens, Proc. Roy. Soc.A218 (1953) 553.
• M.T. Hutchings, Solid State Phys. 16 (New York 1964 ) 227.
• Fulde in: Handbook on the Physics and Chemistry Rare Earth, Vol. 2North-Holland. Inc. (1979).
• R.J. Radwanski, N.H. Kim-Ngan, F.E. Kayzel, J.J.M Franse, D. Gignoux, D. Schmitt, F.Y. Zhang, J. Phys.: Condens. Matter 4 (1992) 8853.
• A broad description of the analysis of the thermodynamics of systems can be found e.g. in: K. Huang, Statistical Mechanics, John Wiley and Sons, Inc. New York (1963).
• Rudowicz, J. Phys. C: Solid State Phys. 20(1987) 6033.
• R.J. Radwański, Z. Ropka & R. Michalski in: Magnetism and Electronic Correlations in Local-Moment Systems: Rare Elements and Compounds, edited by M. Donath, P.A. Dowben & W. Nolting, World
Scientific (1998) 445-453
• Kerson Huang Podstawy fizyki statystycznej PWN 2006; Tłumaczenie: Magdalena Załuska-Kotur
• P.W. Atkins Chemia Przewodnik po chemii fizycznej PWN 1997 | {"url":"https://www.atomicmatters.eu/en/theory/thermodynamics/","timestamp":"2024-11-14T18:25:06Z","content_type":"text/html","content_length":"105227","record_id":"<urn:uuid:0a58ce7e-1ecc-4e4f-b780-827c7a466eca>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00411.warc.gz"} |
EViews Help: rcomptest
Tests for the presence of cross-sectional or time random components in a panel equation. estimated using pooled least squares.
Computes the conventional LM (Breusch-Pagan, 1980, uniformly most powerful LM (Honda, 1985), standardized Honda (Moulton and Randolph, 1989; Baltagi, Chang, and Li, 1998), locally mean most powerful
(LMMP) (King and Wu, 1997), Standardized King-Wu, and Gourieroux, Holly, and Monfort (1982) test statistics.
Note that the equation must be estimated with pooled least squares for this test to be applied.
equation eq1.ls @log(gsp) c @log(p_cap) @log(pc) @log(emp) unemp
will estimate a panel model using pooled least squares and will compute and display the panel random effects test results. | {"url":"https://help.eviews.com/content/equationcmd-rcomptest.html","timestamp":"2024-11-11T19:37:32Z","content_type":"application/xhtml+xml","content_length":"9678","record_id":"<urn:uuid:a603db1b-9c69-43a6-a015-ae3680849b6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00329.warc.gz"} |
This Crazy Twist on Black Holes Says There Was No Big Bang
Physics29 November 2017
A physicist from the University of Campinas in Brazil isn't a big fan of the idea that time started with a so-called Big Bang.
Instead, Juliano César Silva Neves imagines a collapse followed by an expansion, one that could even still carry the scars of a previous timeline.
The idea itself isn't new, but Neves has used a fifty-year-old mathematical trick describing black holes to show how our Universe needn't have had such a compact start to existence.
At first glance, our Universe doesn't seem to have a lot in common with black holes. One is expanding space full of clumpy bits; the other is mass pulling at space so hard that even light has no hope
of escape.
But at the heart of both lies a concept known as a singularity – a volume of energy so infinitely dense, we can't even begin to explain what's going on inside it.
"There are two kinds of singularity in the Universe," says Neves.
"One is the alleged cosmological singularity, or Big Bang. The other hides behind the event horizon of a black hole."
Taken a step further, some propose the Universe itself formed from a black hole in some other bubble of space-time.
No matter which kind we're talking about, singularities are zones where Einstein's general relativity goes blind and quantum mechanics struggles to take over.
Sci-fi writers might love them, but the impossible nature of singularities makes them a frustrating point of contention among physicists.
The problem is, if we rewind the expanding Universe, we get to a point where all of that mass and energy was concentrated in an infinitely dense point. And if we crunch the numbers on collapsing
massive objects, we get the same kind of thing.
Singularities might break physics, but so far we haven't been able to rule them out.
On the other hand, some physicists think there's some wiggle room. Theoretically speaking, not all models of a black hole need a singularity to exist.
"There are no singularities in so-called regular black holes," says Neves.
In 1968, a physicist by the name of James Bardeen came up with a solution to the singularity problem.
He devised a way of mathematically describing black holes that did away with the need for a singularity somewhere beyond its event horizon, calling them 'regular black holes'.
The history and reasoning behind Bardeen's model is, well, super dense; but for a tl;dr version – he assumed that the mass at the heart of a black hole needn't be constant, but could be described
using a function that depended on how far from its centre you were.
That means we can dust our hands of any stupid singularities, as mass still behaves as if it has volume. Even as it is still squeezed into a tight space.
Neves suggests we take Bardeen's work even further and apply it to that other annoying singularity – the cosmological variety that preceded the Big Bang.
By assuming the rate of the Universe's expansion depended not just on time, but its scale as well, he showed there was no need for a quantum leap out of a singularity into a dense, voluminous space
13.82 billion years ago.
So what happened instead?
"Eliminating the singularity or Big Bang brings back the bouncing Universe on to the theoretical stage of cosmology," says Neves.
This 'bouncing Universe' is actually a century-old idea that the expanding Universe as we experience it today is space bouncing back outwards after a previous contraction.
Though it's currently somewhat of a fringe concept in cosmology, Neves supports the view that traces of the pre-collapse Universe might have survived the Big Crunch. If so, finding those scars might
help validate the hypothesis.
"This image of an eternal succession of universes with alternating expansion and contraction phases was called the cyclical Universe, which derives from bouncing cosmologies," says Neves.
Until we have solid observations, the bouncing Universe model will no doubt stay in the 'nice idea' basket.
Still, anything that solves the singularity problem deserves investigating. Neves's work is just one of a number of possible solutions that swaps around assumptions to eliminate the need for
physics-breaking impossibilities.
It's a sticking point we'll need to solve sooner or later.
This research was published in General Relativity and Gravitation. | {"url":"https://www.sciencealert.com/regular-black-holes-model-eliminate-singularity-big-bang","timestamp":"2024-11-13T12:27:23Z","content_type":"text/html","content_length":"144499","record_id":"<urn:uuid:a2fabd2a-e599-40b8-bcac-c4af413c4844>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00236.warc.gz"} |
Lemoore Middle College High
555 College Avenue
Lemoore, CA 93245
• School Type: High Schools (Public) (Charter, not direct funded)
• District: Lemoore Union High
• Website: www.luhsd.k12.ca.us
• Grades: 9-12
CSR Rank: 8 out of 102
Percentile: 78.511-d
Compare to 2013 - Rank: 10, API Score: 892
2016 CAASPP Test Score Details:
Grade Test Type Mean Score Exceeded Met Nearly Met Standard
Standard Standard Standard Not Met
11 English 2622.3 26% 42% 26% 6%
11 Math 2601.3 10% 34% 28% 28%
- Asterisk "*", if present, indicates scores are not available (too few)
Calculated Percentiles from California School Ratings:
English Language Arts/Literacy Mathematics
Grade Percentile Students Tested Percentile Students Tested
11 76.45% 50 81.99% 50
Weighted average for this school's Math and English test scores: 79.22%^1-c
This school is in the 78.51^th percentile^1-d when compared to other schools of the same type: High School
SAT Test Results:
Average Reading Score 561
Average Math Score 526
Percent Tested 92.3% (60 out of 65 seniors)
Grade Dropouts Enrolment
Student Ethnicity:
White 42.08%
Hispanic or Latino 41.25%
Asian 5%
Filipino 3.75%
African American 3.75%
Two or More Races 3.33%
Not reported 0.83%
Related Schools:
1. California School Ratings (CSR) computes percentiles in this way:
a. For a given grade level, all Math scores are put into an ordered list and a percentile is calculated for each score, based its position in the list.
b. For a given grade level, all English scores are put into an ordered list and a percentile is calculated for each score, based its position in the list.
c. Math and English percentiles from (a & b above) are weighted, based on the number of students who completed each type of test, to create a combined Math+English weighted percentile for each
d. The combined Math+English weighted percentiles are put into an ordered list for the particular type of school (elementary/middle/high school/K-12) and a percentile-within-the-school-type is
e. How percentiles work: the school percentile is a number between 0 and 100 that reflects the percentage of schools of the same type (elementary/middle/high school/K-12) in California that have
an equal or lower combined Math+English weighted percentile (from 1-c above). For example, a school in the 70th percentile would have a combined weighted percentile that was equal to or
better than 70% of the other schools of the same type.
2. The CSR Rank is determined by a school's percentile in comparison to other schools of the same type in California (from 1-d above). (1 is the worst, 10 is the best). Schools in the 90th
percentile and above have rank 10, 80%-89.999% rank 9 and so on. A similar number of schools occupy each rank. * This rank is derived from data in the 2016 California Assessment of Student
Performance and Progress (CAASPP).
3. Alternative Schools receive percentiles, but are not ranked
4. More information: 2016 CAASPP Paper-based Test Results | {"url":"https://school-ratings.com/school_details/16639820110205.html","timestamp":"2024-11-02T23:30:47Z","content_type":"text/html","content_length":"15516","record_id":"<urn:uuid:65278bc9-a718-4b9b-ae37-c1e0a259d04d>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00598.warc.gz"} |
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Logical Form
First published Tue Oct 19, 1999; substantive revision Mon Nov 30, 2015
Some inferences are impeccable. Examples like (1–3) illustrate reasoning that cannot lead from true premises to false conclusions.
(1) John danced if Mary sang, and Mary sang; so John danced.
(2) Every politician is deceitful, and every senator is a politician; so every senator is deceitful.
(3) The detective is in the garden; so someone is in the garden.
In such cases, a thinker takes no epistemic risk by endorsing the conditional claim that the conclusion is true if the premises are true. The conclusion follows from the premises, without any further
assumptions that might turn out to be false. Any risk of error lies entirely with the premises, as opposed to the reasoning. By contrast, examples like (4–6) illustrate reasoning that involves at
least some risk of going wrong—from correct premises to a mistaken conclusion.
(4) John danced if Mary sang, and John danced; so Mary sang.
(5) Every feathered biped is a bird, and Tweety is a feathered biped; so Tweety can fly.
(6) Every human born before 1879 died; so every human will die.
Inference (4) is not secure. John might dance whenever Mary sings, but also sometimes when Mary doesn't sing. Similarly, with regard to (5), Tweety might turn out to be a bird that cannot fly. Even
(6) falls short of the demonstrative character exhibited by (1–3). While laws of nature may preclude immortality, the conclusion of (6) goes beyond its premise, even if it is foolish to resist the
Appeals to logical form arose in the context of attempts to say more about this intuitive distinction between impeccable inferences, which invite metaphors of security, and inferences that involve
some risk of slipping from truth to falsity. The idea is that some inferences, like (1-3), are structured in a way that confines any risk of error to the premises. The motivations for developing this
idea were both practical and theoretical. Experience teaches us that an inference can initially seem more secure than it is; and if we knew which forms of inference are risk-free, that might help us
avoid errors. As we'll see, claims about inference are also intimately connected with claims about the nature of thought and its relation to language.
Many philosophers have been especially interested in the possibility that grammar masks the underlying structure of thought, perhaps in ways that invite mistaken views about how ordinary language is
related to cognition and the world we talk about. For example, similarities across sentences like ‘Odysseus arrived’, ‘Nobody arrived’, and ‘The king arrived’ initially suggest that the corresponding
thoughts exhibit a common subject-predicate form. But even if ‘Odysseus’ indicates an entity that can be the subject of a thought that is true if and only if the entity in question arrived, other
considerations suggest that ‘Nobody’ and ‘The king’ do not indicate subjects of thoughts in this sense. This raises further questions about inference—e.g., why ‘The king arrived’ implies an arrival,
while ‘Nobody arrived’ does not—and more general questions about how logic is related to grammar. Do thoughts and sentences exhibit different kinds of structure? Do sentences exhibit grammatical
structures that are not obvious? And if the logical structure of a thought can diverge from the grammatical structure of a sentence that is used to express the thought, how should we construe
proposals about the logical forms of inferences like (1-6)? Are such proposals normative claims about how we ought to think/talk, or empirical hypotheses about aspects of psychological/linguistic
Proposed answers to these questions are usually interwoven with claims about why various inferences seem compelling. So it would be nice to know which inferences really are secure, and in virtue of
what these inferences are special. The most common suggestion has been that certain inferences are secure by virtue of their logical form. Though unsurprisingly, conceptions of form have evolved
along with conceptions of logic and language.
One ancient idea is that impeccable inferences exhibit patterns that can be characterized schematically by abstracting away from the specific contents of particular premises and conclusions, thereby
revealing a general form common to many other impeccable inferences. Such forms, along with the inferences that exemplify them, are said to be valid.
Given a valid inference, there is a sense in which the premises contain the conclusion, which is correspondingly extractable from the premises. With regard to (1) and (7),
(1) John danced if Mary sang, and Mary sang; so John danced.
(7) Chris swam if Pat was asleep, and Pat was asleep; so Chris swam
it seems especially clear that the conclusion is part of the first premise, and that the second premise is another part of the first. We can express this point by saying that these inferences are
instances of the following form: B if A, and A; so B. The Stoics discussed several patterns of this kind, using ordinal numbers (instead of letters) to capture abstract forms like the ones shown
If the first then the second, and the first; so the second.
If the first then the second, but not the second; so not the first.
Either the first or the second, but not the second; so the first.
Not both the first and the second, but the first; so not the second.
These schematic formulations require variables. And let us introduce ‘proposition’ as a term of art for whatever the variables above, indicated in bold, range over. Propositions are potential
premises/conclusions. They can be endorsed or rejected, and they exhibit containment relations of some kind. So presumably, propositions are abstract things that can be evaluated for truth or
falsity. This leaves it open what propositions are: sentences, statements, states of affairs, or whatever. But let's assume that declarative sentences can be used to express propositions. (For
discussion, see Cartwright (1962) and the essay on structured propositions.)
A significant complication is that in ordinary conversation, the context matters with regard to which proposition is expressed with a given sentence. For example, ‘Pat is asleep’ can be used at one
time to express a true premise, and at another time to express a false premise. A given speaker might use ‘I am tired’ to express a false proposition, while another speaker uses the same sentence at
the same time to express a true proposition. What counts as being tired can also vary across conversations. Context sensitivity, of various kinds, is ubiquitous in ordinary discourse. Moreover, even
given a context, a sentence like ‘He is bald’ may not express a unique proposition. (There may be no referent for the pronoun; and even if there is, the vagueness of ‘bald’ may yield a range of
candidate propositions, with no fact of the matter as to which one is the proposition expressed.) Nonetheless, we can often use sentences like ‘Every circle is an ellipse’ and ‘Thirteen is a prime
number’ to express premises of valid arguments. To be sure, ordinary conversation differs from theoretical discourse in mathematics. But the distinction between impeccable and risky inferences is not
limited to special contexts in which we try to think especially clearly about especially abstract matters. So when focusing on the phenomenon of valid inference, we can try to simplify the initial
discussion by abstracting away from the context sensitivity of language use.
Another complication is that in speaking of an inference, one might be talking about (i) a process in which a thinker draws a conclusion from some premises, or (ii) some propositions, one of which is
designated as an alleged consequence of the others; see, e.g., Harman (1973). But we can describe a risky thought process as one in which a thinker who accepts certain propositions—perhaps
tentatively or hypothetically—comes to accept, on that basis, a proposition that does not follow from the initial premises. And it will be simpler to focus on premises/conclusions, as opposed to
episodes of reasoning.
With regard to (1), the inference seems secure in part because its first premise has the form ‘B if A’.
(1) John danced if Mary sang, and Mary sang; so John danced.
If the first premise didn't have this form, the inference wouldn't be an instance of ‘B if A, and A; so B’. It isn't obvious that all impeccable inferences are instances of a more general valid form,
much less inferences whose impeccability is due to the forms of the relevant propositions. But this thought has served as an ideal for the study of valid inference, at least since Aristotle's
treatment of examples like (2).
(2) Every senator is a politician, and every politician is deceitful; so every senator is deceitful.
Again, the first premise seems to have several parts, each of which is a part of the second premise or the conclusion. (In English, the indefinite article in ‘Every senator is a politician’ cannot be
omitted; likewise for ‘Every politician is a liar’. But at least for now, let's assume that in examples like these, ‘a’ does not itself indicate a propositional constituent.) Aristotle, predating the
Stoics, noted that conditional claims like the following are sure to be true: if (the property of) being a politician belongs to every senator, and being deceitful belongs to every politician, then
being deceitful belongs to every senator. Correspondingly, the inference pattern below is valid.
Every S is P, and every P is D; so every S is D
And inference (2) seems to be valid because its parts exhibit this pattern. Aristotle discussed many such forms of inference, called syllogisms, involving propositions that can be expressed with
quantificational words like ‘every’ and ‘some’. For example, the syllogistic patterns below are also valid.
Every S is P, and some S is D; so some P is D.
Some S is P, and every P is D; so some S is D.
Some S is not P, every D is P; so some S is not D.
We can rewrite the last two, so that each of the valid syllogisms above is represented as having a first premise of the form ‘Every S is P’.
Every S is P, and some D is S; so some D is P.
Every S is P, and some D is not P; so some D is not S.
But however the inferences are represented, the important point is that the variables—represented here in italics—range over certain parts of propositions. Intuitively, common nouns like ‘politician’
and adjectives like ‘deceitful’ are general terms, since they can apply to more than one individual. And many propositions apparently contain correspondingly general elements. For example, the
proposition that every senator is deceitful contains two such elements, both relevant to the validity of inferences involving this proposition.
Propositions thus seem to have structure that bears on the validity of inferences, even ignoring premises/conclusions with propositional parts. That is, even simple propositions have logical form.
And as Aristotle noted, pairs of such propositions can be related in interesting ways. If every S is P, then some S is P. (For these purposes, assume there is at least one S.) If no S is P, then some
S is not P. It is certain that either every S is P or some S is not P; and whichever of these propositions is true, the other is false. Similarly, the following propositions cannot both be true:
every S is P; and no S is P. But it isn't certain that either every S is P, or no S is P. Perhaps some S is P, and some S is not P. This network of logical relations strongly suggests that the
propositions in question contain a quantificational element and two general elements—and in some cases, an element of negation. This raises the question of whether other propositions have a similar
Consider the proposition that Vega is a star, which can figure in inferences like (8).
(8) Every star is purple, and Vega is a star; so Vega is purple.
Aristotle's logic focused on quantificational propositions; and as we shall see, this was prescient. But on his view, propositions like the conclusion of (8) still exemplify a subject-predicate
structure that is shared by at least many of the sentences we used to express propositions. And one can easily formulate the schema ‘every S is P, and n is S; so n is P’, where the new lower-case
variable is intended to range over proposition-parts of the sort indicated by names. (On some views, discussed below, a name like ‘Vega’ is a complex quantificational expression; though
unsurprisingly, such views are tendentious.)
Typically, a declarative sentence can be divided into a subject and a predicate: ‘Every star / is purple’, ‘Vega / is a star’, ‘Some politician / lied’, ‘The brightest planet / is visible tonight’,
etc. Until quite recently, it was widely held that this grammatical division reflects a corresponding kind of logical structure: the subject of a proposition (i.e., what the proposition is about) is
a target for predication. On this view, both ‘Every star’ and ‘Vega’ indicate subjects of propositions in (8), while ‘is’ introduces predicates. Aristotle would have said that in the premises of (8),
being purple is predicated of every star, and being a star is predicated of Vega. Later theorists emphasized the contrast between general terms like ‘star’ and singular terms like ‘Vega’, while also
distinguishing terms from syncategorematic expressions (e.g., ‘every’ and ‘is’) that can combine with terms to form complex subjects and predicates, including ‘will lie’, ‘can lie’, and ‘may have
lied’. But despite the complications, it seemed clear that many propositions have the following canonical form: Subject-copula-Predicate; where a copula links a subject, which may consist of a
quantifier and a general term, to a general term. Sentences like ‘Every star twinkles’ can be paraphrased with sentences like ‘Every star is a thing that does some twinkling’. This invites the
suggestion that ‘twinkles’ is somehow an abbreviation for ‘is a thing that does some twinkling’, perhaps in the way that ‘bachelor’ is arguably short for ‘unmarried marriageable man’.
The proposition that not only Vega twinkles, which seems to contain the proposition that Vega twinkles, presumably includes elements that are indicated with ‘only’ and ‘not’. Such examples invite the
hypothesis that all propositions are composed of terms along with a relatively small number of syncategorematic elements, and that complex propositions can be reduced to canonical propositions that
are governed by Aristotelian logic. This is not to say that all propositions were, or could be, successfully analyzed in this manner. But via this strategy, medieval logicians were able to describe
many impeccable infererences as instances of valid forms. And this informed their discussions of how logic is related to grammar.
Many viewed their project as an attempt to uncover principles of a mental language common to all thinkers. Aristotle had said, similarly, that spoken sounds symbolize “affections of the soul.” From
this perspective, one expects a few differences between propositions and overt sentences. If ‘Every star twinkles’ expresses a proposition that contains a copula, then spoken languages mask certain
aspects of logical structure. Ockham also held that a mental language would have no need for Latin's declensions, and that logicians could ignore such aspects of spoken language. The ancient Greeks
were aware of sophisms like the following: that dog is a father, and that dog is yours; so that dog is your father. This bad inference cannot share its form with the superficially parallel but
impeccable variant: that dog is a mutt, and that mutt is yours; so that dog is your mutt. (See Plato, Euthydemus 298 d-e.) So the superficial features of sentences are not infallible guides to the
logical forms of propositions. Still, the divergence was held to be relatively minor. Spoken sentences have structure; they are composed, in systematic ways, of words. And the assumption was that
sentences reflect the major aspects of propositional form, including a subject-predicate division. So while there is a distinction between the study of valid inference and the study of sentences used
in spoken language, the connection between logic and grammar was thought to run deep. This suggested that the logical form of a proposition just is the grammatical form of some (perhaps mental)
Towards the end of the eighteenth century, Kant could say (without much exaggeration) that logic had followed a single path since its inception, and that “since Aristotle it has not had to retrace a
single step.” He also said that syllogistic logic was “to all appearance complete and perfect.” But this was exuberance. Indeed, some of the real successes highlighted known problems.
Some valid schemata are reducible to others, in that any inference of the reducible form can be revealed as valid (with a little work) given other schemata. Consider (9).
(9) If Al ran then either Al did not run or Bob did not swim, and Al ran; so Bob did not swim.
Assume that ‘Al did not run’ negates ‘Al ran’, while ‘Bob did not swim’ negates ‘Bob swam’. Then (9) is an instance of the following valid form: if A then either not-A or not-B, and A; so not-B. But
we can treat this as a derived form, by showing that any instance of this form is valid given two (intuitively more basic) Stoic inference forms: if the first then the second, and the first, so the
second; either not the first or not the second, and the first; so not the second. For suppose we are given the following premises: A; and if A, then either not-A or not-B. We can safely infer that
either not-A or not-B; and since we were given that A, we can safely infer that not-B. Similarly, the syllogistic schema (10) can be treated as a derived form.
(10) Some S is not P, and every D is P; so not every S is D.
If some S is not P, and every D is P, then it isn't true that every S is D. For if every S is D, and every D is P, then every S is P. But if some S is not P, then as we saw above, not every S is P.
So given the premises of (10), adding ‘every S is D’ would lead to contradiction: every S is P, and not every S is P. So the premises imply the negation of ‘every S is D’. This reasoning shows how
(10) can be reduced to inferential patterns that seem more basic—raising the question of how much reduction is possible. Euclid's geometry had provided a model for how to present a body of knowledge
as a network of propositions that follow from a few basic axioms. Aristotle himself indicated how to reduce all the valid syllogistic schemata to four basic patterns, given a few principles that
govern how the basic patterns can be used to derive others; see Parsons (2014) for discussion. And further reduction is possible given insights from the medieval period.
Consider the following pair of valid inferences: Fido is a brown dog, so Fido is a dog; Fido is not a dog, so Fido is not a brown dog. As illustrated with the first example, replacing a predicate (or
general term) like ‘brown dog’ with a less restrictive predicate like ‘dog’ is often valid. But sometimes—paradigmatically, in cases involving negation—replacing a predicate like ‘dog’ with a more
restrictive predicate like ‘brown dog’ is valid. Plausibly, the first pattern reflects the default direction of valid replacement: removing a restriction preserves truth, except in special cases like
those involving negation. Suppose we take it as given that poodles are dogs of a particular sort, and hence that every poodle is a dog. Then replacing‘poodle’ with ‘dog' in ‘Fido is P’ is valid,
regardless of what ‘Fido’ names. This can be viewed as a special case of ‘n is P, and every P is D; so n is D’. But the validity of this inference form can also be viewed as symptom of a basic
principle that came be called dictum de omni: whatever is true of every P is true of any P. Or as Aristotle might have put it, if the property of being a dog belongs to every poodle, then it belongs
to any poodle. In which case, Fido is a dog if Fido is a poodle. And since the property of being a dog surely belongs to every brown dog, any brown dog is a dog. The flip side of this point is that
negation inverts the default direction of inference. Anything that isn't a dog isn't a brown dog; and similarly, if Fido isn't a dog, Fido isn't a poodle. So in special cases, adding a restriction to
a general term like ‘dog’ can preserve truth.
From this perspective, the Aristotelian quantifier ‘Some’ is a default-style quantifier that validates removing restrictions. If some brown dog is a clever mutt, it follows that some dog is a clever
mutt, and hence that some dog is a mutt. By contrast, ‘No’ is an inverted-style quantifier that validates adding restrictions. If no dog is a mutt, it follows that no dog is a clever mutt, and hence
that no brown dog is a clever mutt. The corresponding principle, dictum de nullo, encodes this pattern: whatever is true of no P is not true of any P; so if the property of being a mutt belongs to no
dog, it belongs to no poodle. (And as Aristotle noted, instances of ‘No S is P’ can be analyzed as the propositional negations of corresponding instances of ‘Some S isn't P’.
Interestingly, ‘Every’ is like ‘No’ in one respect, and like ‘Some’ in another respect. If every dog is clever, it follows that every brown dog is clever; but if every dog is a clever mutt, it
follows that every dog is a mutt. So when the universal quantifier combines with a general term S to form a subject, S is governed by the inverted rule of replacement. But when a universally
quantified subject combines with a second general term to form a proposition, this second term is governed by the default rule of replacement. Given that ‘Every’ has this mixed logical character, the
valid syllogisms can be derived from two basic patterns (noted above), both of which reflect dictum de omni: whatever is true of every P is true of any P.
Every S is P, and every P is D; so every S is D.
Every S is P, and some D is S; so some D is P.
The first principle reflects the sense in which universal quantification is transitive. The second principle captures the idea that a universal premise can licence replacement of ‘S’ with ‘P’ in a
premise about a specific individual. In this sense, classical logic exhibits a striking unity and simplicity, at least with regard to inferences involving the Aristotelian quantifiers and
predication; see Sommers (1984) and Ludlow (2005), drawing on Sanchez (1991), for further discussion.
Alas, matters become more complicated once we consider relations.
Sentences like ‘Juliet kissed Romeo’ do not seem to have Subject-copula-Predicate form. One might suggest ‘Juliet was a kisser of Romeo’ as a paraphrase. But ‘kisser of Romeo’ differs, in ways that
matter to inference, from general terms like ‘politician’. If Juliet (or anyone) was a kisser of Romeo, it follows that someone was kissed; whereas if Juliet was a politician, there is no
corresponding logical consequence to the effect that someone was __-ed. Put another way, the proposition that Juliet kissed someone exhibits interesting logical structure, even if we can express this
proposition via the sentence ‘Juliet was a kisser of someone’. A quantifier can be part of a complex predicate. But classical logic did not capture the validity of inferences involving predicates
that have quantificational constituents. Consider (11).
(11) Some patient respects every doctor, and some doctor is a liar; so
some patient respects some liar.
If ‘respects every doctor’ and ‘respects some liar’ indicate nonrelational proposition-parts, much like ‘is sick’ or ‘is happy’, then inference (11) has the following form ‘Some P is S, and some D is
L; so some P is H’. But this schema, which fails to reflect the quantificational structure within the predicates is not valid. Its instances include bad inferences like the following: some patient is
sick, and some doctor is a liar; so some patient is happy. This dramatizes the point that ‘respects every doctor’ and ‘respects some liar’ are—unlike ‘is sick’ and ‘is tall’—logically related in a
way that matters given the middle premise of (11).
One can adopt the view that many propositions have relational parts, introducing a variable ‘R’ intended to range over relations; see the entries on medieval relations, and medieval terms. One can
also formulate the following schema: some P R every D, and some D is L; so some P R some L. But the problem remains. Quantifiers can appear in complex predicates that figure in valid inferences like
(12) Every patient who respects every doctor is sick, and
some patient who saw every lawyer respects every doctor; so
some patient who saw every lawyer is sick.
But if ‘patient who respects every doctor’ and ‘patient who saw every lawyer’ are nonrelational, much like ‘old patient’ or ‘young patient’, then (12) has the following form: every O is S, and some Y
R every D; so some Y is S. And many inferences of this form are invalid. For example: every otter is sick, and some yak respects every doctor; so some yak is sick. Again, one can abstract a valid
schema that covers (12), letting parentheses indicate a relative clause that restricts the adjacent predicate.
Every P(R1 every D) is S, and some P(R2 every L) R1 every D; so some P(R2 every L) is S.
But no matter how complex the schema, the relevant predicates can exhibit further quantificational structure. (Consider the proposition that every patient who met some doctor who saw no lawyer
respects some lawyer who saw no patient who met every doctor.) Moreover, schemata like the one above are poor candidates for basic inference patterns.
As medieval logicians knew, propositions expressed with relative clauses also pose other difficulties; see the entry on medieval syllogism. If every doctor is healthy, it follows that every young
doctor is healthy. By itself, this is expected, since a universally quantified subject is governed by the non-default (de nullo) inference rule that licenses replacement of ‘doctor’ with the more
restrictive ‘young doctor’. But consider (13) and (14).
(13) No patient who saw every young doctor is healthy
(14) No patient who saw every doctor is healthy
Here, the direction of valid inference is from ‘young doctor’ to ‘doctor’, as if the inference is governed by the default (de omni) inferential rule. One can say that the default direction of
implication, from more restrictive to less restrictive predicates, has been inverted twice—once by ‘No’, and once by ‘every’. But one wants a systematic account of propositional structure that
explains the net effect; see Ludlow (2002) for further discussion. Sommers (1982) offers a strategy for recoding and extending classical logic, in part by exploiting an idea suggested by Leibniz (and
arguably Panini): a relational sentence like ‘Juliet loved Romeo’ somehow combines an active-voice sentence with a passive-voice sentence, perhaps along the lines of ‘Juliet loved, and thereby Romeo
was loved’; cp. section nine below. But if impeccability is to be revealed as a matter of form, then one way or another, quantifiers need to characterized in a way that captures their general logical
role—and not just their role as potential subjects of Aristotelian propsitions. Quantifiers are not simply devices for creating schemata like ‘Every S is P’, into which general terms like
‘politician’ and ‘deceitful’ can be inserted. Instances of ‘S’ and ‘P’ can themselves have quantificational structure and relational constituents.
Frege showed how to resolve these difficulties for classical logic in one fell swoop. His system of logic, published in 1879 and still in use (with notational modifications), was arguably the single
greatest contribution to the subject. So it is significant that on Frege's view, propositions do not have subject-predicate form. His account required a substantial distinction between logical form
and grammatical form as traditionally conceived. It is hard to overemphasize the impact of this point on subsequent discussions of thought and its relation to language.
Frege's leading idea was that propositions have “function-argument” structure. Though for Frege, functions are not abstract objects. In particular, while a function maps each entity in some domain
onto exactly one entity in some range, Frege (1891) does not identify functions with sets of ordered pairs. On the contrary, he says that a function “by itself must be called incomplete, in need of
supplementation, or unsaturated. And in this respect functions differ fundamentally from numbers (p. 133).” For example, we can represent the successor function as follows, with the integers as the
relevant domain for the variable ‘x’: S(x) = x + 1. This function maps zero onto one, one onto two, and so on. We can specify a corresponding object—e.g., the set {〈x, y〉: y = x + 1}—as the
“value-range” of the successor function. But according to Frege, any particular argument (e.g., the number one) “goes together with the function to make up a complete whole” (e.g., the number two);
and a number does not go together with a set in this fashion. Put another way, while each number is an object, a mapping from numbers to numbers is not an additional object in Frege’s sense. As Frege
noted, the word ‘function’ is often used to talk about what he would call the value-range of a function. But he maintained that the notion of an unsaturated function, which may be applied to
endlessly many arguments, is “logically prior” to any notion of a set with endlessly many arguments that are specified functionally as in {〈x, y〉: y = x + 1}; see p.135, note E.
Functions need not be unary. For example, arithmetic division can be represented as a function from ordered pairs of numbers onto quotients: Q(x, y) = x/y. Mappings can also be conditional. Consider
the function that maps every even integer onto itself, and every odd integer onto its successor: C(x) = x if x is even, and x + 1 otherwise; C(1) = 2, C(2) = 2, C(3) = 4, etc. Frege held that
propositions have parts that indicate functions, and in particular, conditional functions that map arguments onto special values that reflect the truth or falsity of propositions/sentences. (As
discussed below, Frege [1892] also distinguished these “truth values” from what he called Thoughts [Gedanken] or the “senses” [Sinnen] of propositions; where each of these sentential senses
“presents” a truth value in certain way—i.e., as the value of a certain indicated function given a certain indicated argument.).
Variable letters, such as ‘x’ and ‘y’ in ‘Q(x, y) = x/y’, are typographically convenient for representing functions that take more than one argument. But we could also index argument places, as shown
Q[( )[i] , ( )[j]] = ( )[i] / ( )[j]
Or we could replace the subscripts above with lines that connect each pair of round brackets on the left of ‘=’ to a corresponding pair of brackets on the right. But the idea, however we encode it,
is that a proposition has at least one constituent that is saturated by the requisite number of arguments. (If it helps, think of an unsaturated proposition-part as the result of abstracting away
from one or more arguments in a complete proposition.) Frege was here influenced by Kant's discussion of judgment, and the ancient observation that merely combining two things does not make the
combination truth-evaluable. So in saying that propositions have “function-argument” structure, Frege was not only rejecting the traditional idea that logical from reflects the “subject-predicate”
structure of ordinary sentences, he was suggesting that propositions exhibit a special kind of unity: unlike a mere concatenation of objects, a potential premise/conclusion is formed by saturating an
unsaturated mapping with a suitable argument.
On Frege's view, the proposition that Mary sang has a functional component indicated by ‘sang’ and an argument indicated by ‘Mary’, even if the English sentence ‘Mary sang’ has ‘Mary’ as its subject
and ‘sang’ as its predicate. The proposition can be represented as follows: Sang(Mary). Frege thought of the relevant function as a conditional mapping from individuals to truth values: Sang(x) = T
if x sang, and F otherwise; where ‘T’ and ‘F’ stand for special entities such that for each individual x, Sang(x) = T if and only if x sang, and Sang(x) = F if and only if x did not sing. According
to Frege, the proposition that John admires Mary combines an ordered pair of arguments with a functional component indicated by the transitive verb: Admires(John, Mary); where for any individual x,
and any individual y, Admires(x, y) = T if x admires y, and F otherwise. From this perspective, the structure and constituents are the same in the proposition that Mary is admired by John, even
though ‘Mary’ is the grammatical subject of the passive sentence. Likewise, Frege did not distinguish the proposition that three precedes four from the proposition that four is preceded by three.
More importantly, Frege's treatment of quantified propositions departs radically from the traditional idea that the grammatical structure of sentence reflects the logical structure of the indicated
If S is the function indicated by ‘sang’, then Mary sang iff—i.e., if and only if—S(Mary) = T. Likewise, someone sang iff: S maps some individual onto T; that is, for some individual x, S(x) = T. Or
using a modern variant of Frege's original notation, someone sang iff ∃x[S(x)]. The quantifier ‘∃x’ is said to bind the variable ‘x’, which ranges over individual things in a domain of discourse.
(For now, assume that the domain contains only people.) If every individual in the domain sang, then S maps every individual onto the truth value T; or using formal notation, ∀x[S(x)]. A quantifier
binds each occurrence of its variable, as in ‘∃x[P(x) & D(x)]’, which reflects the logical form of ‘Someone is both a politician and deceitful’. In this last example, the quantifier combines with a
complex predicate that formed by conjoining two simpler predicates.
With regard to the proposition that some politician is deceitful, traditional grammar suggests the division ‘Some politician / is deceitful’, with the noun ‘politician’ forming a constituent with the
quantificational word. But on a Fregean view, grammar masks the logical division between the existential quantifier and the rest: ∃x[P(x) & D(x)]. With regard to the proposition that every politician
is deceitful, Frege also stresses the logical division between the quantifier and its scope: ∀x[P(x) → D(x)]; every individual is deceitful if a politician. Here too, the quantifier combines with a
complex predicate, albeit a conditional rather than conjunctive predicate. (The formal sentence ‘∀x[P(x) & D(x)]’ implies, unconditionally, that every individual is a politician.) As Frege (1879)
defined his analogs of the relevant modern symbols used here, ‘P(x) → D(x)’ is equivalent to ‘¬P(x) ∨ D(x)’, and ‘∀x’ is equivalent to ‘¬∃x¬’. So ‘∀x[P(x) → D(x)]’ is equivalent to ‘¬∃x¬[¬P(x) ∨ D
(x)]’; and given de Morgan's Laws (concerning the relations between negation, disjunction, and conjunction), ¬∃x¬[¬P(x) ∨ D(x)] iff ¬∃x[P(x) & ¬D(x)]. Hence, ∀x[P(x) → D(x)] iff ¬∃x[P(x) & ¬D(x)].
This captures the idea that every politician is deceitful iff no individual is both a politician and not deceitful.
If this conception of logical form is correct, then grammar is misleading in several respects. First, grammar leads us to think that ‘some politician’ indicates a constituent of the proposition that
some politician is deceitful. Second, grammar masks a difference between existential and universally quantified propositions; predicates are related conjunctively in the former, and conditionally in
the latter. (Though as discussed in section seven, one can—and Frege [1884] did—adopt a different view that allows for relational/restricted quantifiers as in ‘∀x:P(x)[D(x)]’.)
More importantly, Frege's account was designed to apply equally well to propositions involving relations and multiple quantifiers. And with regard to these propositions, there seems to be a big
difference between logical structure and grammatical structure.
On Frege's view, a single quantifier can bind an unsaturated position that is associated with a function that takes a single argument. But it is equally true that two quantifiers can bind two
unsaturated positions associated with a function that takes a pair of arguments. For example, the proposition that everyone likes everyone can be represented with the formal sentence ‘∀x∀y[L(x, y)]’.
Assuming that ‘Romeo’ and ‘Juliet’ indicate arguments, it follows that Romeo likes everyone, and that everyone likes Juliet—∀y[L(r, y)] and ∀x[L(x, j)]. And it follows from all three propositions
that Romeo likes Juliet: L(r, j). The rules of inference for Frege's logic capture this general feature of the universal quantifier. A variable bound by a universal quantifier can be replaced with a
name for some individual in the domain. Correlatively, a name can be replaced with a variable bound by an existential quantifier. Given that Romeo likes Juliet, it follows that someone likes Juliet,
and Romeo likes someone. Frege's formalism can capture this as well: L(r, j); so ∃x[L(x, j)] & ∃x[L(r, x)]. And given either conjunct in the conclusion, it follows that someone likes someone: ∃x∃y[L
(x, y)]. A single quantifier can also bind multiple argument positions, as in ‘∃x[L(x, x)]’, which is true iff someone likes herself. Putting these points schematically: ∀x(…x…), so …n…; and …n…, so
Mixed quantification introduces an interesting wrinkle. The propositions expressed with ‘∃x∀y[L(x,y)]’ and ‘∀y∃x[L(x,y)]’ differ. We can paraphrase the first as ‘there is someone who likes everyone’
and the second as ‘everyone is liked by someone or other’. The second follows from the first, but not vice versa. This suggests that ‘someone likes everyone’ is ambiguous, in that this string of
English words can be used to express two different propositions. This in turn raises difficult questions about what natural language expressions are, and how they can be used to express propositions;
see section eight. But for Frege, the important point concerned the distinction between the propositions (Gedanken). Similar remarks apply to ‘∀x∃y[L(x, y)]’ and ‘∃y∀x[L(x, y)]’.
A related phenomenon is exhibited by ‘John danced if Mary sang and Chris slept’. Is the intended proposition of the form ‘(A if B) and C’ or ‘A if (B and C)’? Indeed, it seems that the relation
between word-strings and propositions expressed is often one-to-many. Is someone who says ‘The artist drew a club’ talking about a sketch or a card game? One can use ‘is’ to express identity, as in
‘Hesperus is the planet Venus’; but in ‘Hesperus is bright’, ‘is’ indicates predication. In ‘Hesperus is a planet’, ‘a’ seems to be logically inert; yet in ‘John saw a planet’, ‘a’ seems to indicate
existential quantification: ∃x[P(x) & S(j,x)]. (One can render ‘Hesperus is a planet’ as ‘∃x[P(x) & h = x]’. But this treats ‘is a planet’ as importantly different than ‘is bright’; and this leads to
other difficulties.) According to Frege, such ambiguities provide further evidence that natural language is not suited to the task of representing propositions and inferential relations
perspicuously. And he wanted a language that was suited for this task. (Leibniz and others had envisioned a “Characteristica Universalis”, but without detailed proposals for how to proceed beyond
syllogistic logic in creating one.) This is not to deny that natural language is well suited for other purposes, perhaps including efficient human communication. And Frege held that we often do use
natural language to express propositions. But he suggested that natural language is like the eye, whereas a good formal language is like a microscope that reveals structure not otherwise observable.
On this view, the logical form of a proposition is made manifest by the structure of a sentence in an ideal formal language—what Frege called a Begriffsschrift (concept-script); where the sentences
of such a language exhibit function-argument structures that differ in kind from the grammatical structures exhibited by the sentences we use in ordinary communication.
The real power of Frege's strategy for representing propositional structure is most evident in his discussions of proofs by induction, the Dedekind-Peano axioms for arithemetic, and how the
proposition that every number has a successor is logically related to more basic truths of arithmetic; see the entry on Frege's theorem and foundations for arithmetic. But without getting into these
details, one can get a sense of Frege's improvement on previous logic by considering (15–16) and Fregean analyses of the corresponding propositions.
(15) Every patient respects some doctor
∀x{P(x) → ∃y[D(y) & R(x,y)]}
(16) Every old patient respects some doctor
∀x{[O(x) & P(x)] → ∃y[D(y) & R(x,y)]}
Suppose that every individual has the following conditional property: if he[x] is a patient, then some individual is such that she[y] is both a doctor and respected by him[x]. Then it
follows—intuitively and given the rules of Frege's logic—that every individual[x] has the following conditional property: if he[x] is both old and a patient, then some individual[y] is such that she[
y] is both a doctor and respected by him[x]. So the proposition expressed with (16) follows from the one expressed with (15). More interestingly, we can also account for why the proposition expressed
with (14) follows from the one expressed with (13).
(13) No patient who saw every young doctor is healthy
¬∃x{P(x) & ∀y{[Y(y) & D(y) → S(x,y)] & H(x)}
(14) No patient who saw every doctor is healthy
¬∃x{P(x) & ∀y[D(y) → S(x,y)] & H(x)}
For suppose it is false that some individual has the following conjunctive property: he[x] is a patient; and he[x] saw every young doctor (i.e., every individual[y] is such that if she[y] is a young
doctor, then he[x] was seen by her[y]); and he[x] is healthy. Then intuitively, and also given the rules of Frege's logic, it is false that some individual has the following conjunctive property: he[
x] is a patient; and he[x] saw every doctor; and he[x] is healthy. This explains why the direction of valid inference is from the more restrictive ‘young doctor’ in (13) to the less restrictive
‘patient’ in (14), despite the fact that in simpler cases, replacing ‘every doctor’ with ‘every young doctor’ is valid. More generally, Frege's logic handles a wide range of inferences that had
puzzled medieval logicians. But the Fregean logical forms seem to differ dramatically from the grammatical forms of sentences like (13–16). Frege concluded that we need a Begriffsschrift, distinct
from the languages we naturally speak, in order to depict (and help us discern) the structures of the propositions we express by using natural languages.
Frege also made a different kind of contribution, which would prove important, to the study of propositions. In early work, he spoke as though propositional constituents were the relevant functions
and (ordered n-tuples of) entities that such functions map to truth-values. But he later refined this view in light of his distinction between Sinn and Bedeutung (see the entry on Gottlob Frege). The
Sinn of an expression was said to be a “way of presenting” the corresponding Bedeutung, which might be an entity, a truth-value, or a function from (ordered n-tuples of) entities to truth-values. The
basic idea is that two names, like ‘Hesperus’ and ‘Phosphorus’, can present the same Bedeutung in different ways; in which case, the Sinn of the first name differs from the Sinn of the second. Given
this distinction, we can think of ‘Hesperus’ as an expression that presents the evening star (a.k.a. Venus) as such, while ‘Phosphorus’ presents the morning star (also a.k.a. Venus) in a different
way. Likewise, we can think of ‘is bright’ as an expression that presents a certain function in a certain way, and ‘Hesperus is bright’ as a sentence that presents its truth-value in a certain
way—i.e., as the value of the function in question given the argument in question. From this perspective, propositions are sentential ways of presenting truth-values, and proposition-parts are
subsentential ways of presenting functions and arguments. Frege could thus distinguish the proposition that Hesperus is bright from the proposition that Phosphorus is bright, even though the two
propositions are alike with regard to the relevant function and argument. Likewise, he could distinguish the trivial proposition Hesperus is Hesperus from the (apparently nontrivial) proposition
Hesperus is Phosphorus. This is an attractive view. For intuitively, ancient astronomers were correct not to regard the inference Hesperus is Hesperus, so Hesperus is Phosphorus as an instance of the
following valid schema: A, so A. But this raised questions about what the Sinn of an expression really is, what “presentation” could amount to, and what to say about a name with no Bedeutung.
Frege did not distinguish (or at least did not emphasize any distinction between) names like ‘John’ and descriptions like ‘the boy’ or ‘the tall boy from Canada’. Initially, both kinds of expression
seem to indicate arguments, as opposed to functions. So one might think that the logical form of ‘The boy sang’ is simply ‘S(b)’, where ‘b’ is an unstructured symbol that stands for the boy in
question (and presents him in a certain way). But this makes the elements of a description logically irrelevant. And this seems wrong. If the tall boy from Canada sang, then some boy from Canada
sang. Moreover, ‘the’ implies uniqueness in a way that ‘some’ does not. Of course, one can say ‘The boy sang’ without denying that universe contains more than one boy. But likewise, in ordinary
conversation, one can say ‘Everything is in the trunk’ without denying that the universe contains some things not in the trunk. And intuitively, a speaker who uses ‘the’ does imply that the adjacent
predicate is satisfied by exactly one contextually relevant thing.
Bertrand Russell held that these implications reflect the logical form of a proposition expressed (in a given context) with a definite description. On his view, ‘The boy sang’ has the following
logical form: ∃x{Boy(x) & ∀y[Boy(y) → y = x] & S(x)}; some individual[x] is such that he[x] is a boy, and every (relevant) individual[y] is such that if he[y] is a boy, then he[y] is identical with
him[x], and he[x] sang. The awkward middle conjunct was Russell's way of expressing uniqueness with Fregean tools; cf. section seven. But rewriting the middle conjunct would not affect Russell's
technical point, which is that ‘the boy’ does not correspond to any constituent of the formalism. This in turn reflects Russell's central claim—viz., that while a speaker may refer to a certain boy
in saying ‘The boy sang’, the boy in question is not a constituent of the proposition indicated. According to Russell, the proposition has the form of an existential quantification with a bound
variable. It does not have the form of a function saturated by (an argument that is) the boy referred to. The proposition is general rather than singular. In this respect, ‘the boy’ is like ‘some
boy’ and ‘every boy’; though on Russell's view, not even ‘the’ indicates a constituent of the proposition expressed.
This extended Frege's idea that natural language misleads us about the structure of the propositions we assert. Russell went on to apply this hypothesis to what became a famous puzzle. Even though
France is currently kingless, ‘The present king of France is bald’ can be used to express a proposition. The sentence is not meaningless; it has implications. So if the proposition consists of the
function indicated with ‘Bald( )’ and an argument indicated with ‘The present king of France’, there must be an argument so indicated. But appeal to nonexistent kings is, to say the least, dubious.
Russell concluded that ‘The present king of France is bald’ expresses a quantificational proposition: ∃x{K(x) & ∀y[K(y) → y = x] & B(x)}; where K(x) = T iff x is a present king of France, and B(x) =
T iff x is bald. (For present purposes, set aside worries about the vagueness of ‘bald’.) And as Russell noted, the following contrary reasoning is spurious: every proposition is true or false; so
the present king of France is bald or not; so there is a king of France, and he is either bald or not. For let P be the proposition that the king of France is bald. Russell held that P is indeed true
or false. On his view, it is false. Given that ¬∃x[K(x)], it follows that ¬∃x{K(x) & ∀y[K(y) → y = x] & B(x)}. But it does not follow that there is a present king of France who is either bald or not.
Given that ¬∃x[K(x)], it hardly follows that ∃x{K(x) & [B(x) v ¬B(x)]}. So we must not confuse the negation of P with the following false proposition: ∃x{K(x) & ∀y[K(y) → y = x] & ¬B(x)}. The
ambiguity of natural language may foster such confusion, given examples like ‘The present king of France is bald or not’. But according to Russell, puzzles about “nonexistence” can be resolved
without special metaphysical theses, given the right views about logical form and natural language.
This invited the thought that other philosophical puzzles might dissolve if we properly understood the logical forms of our claims. Wittgenstein argued, in his influential Tractatus
Logico-Philosophicus, that: (i) the very possibility of meaningful sentences, which can be true or false depending on how the world is, requires propositions with structures of the sort Frege and
Russell were getting at; (ii) all propositions are logical compounds of—and thus analyzable into—atomic propositions that are inferentially independent of one another; though (iii) even simple
natural language sentences may indicate very complex propositions; and (iv) the right analyses would, given a little reflection, reveal all philosophical puzzles as confusions about how language is
related to the world. Russell never endorsed (iv). And Wittgenstein later noted that claims like ‘This is red’ and ‘This is yellow’ presented difficulties for his earlier view. If the expressed
propositions are unanalyzable, and thus logically independent, each should be compatible with the other. But at least so far, no one has provided a plausible analysis that accounts for the apparent
impeccabilty of ‘This is red, so this is not yellow’. (This raises questions about whether all inferential security is due to logical form.) Though for reasons related to epistemological puzzles,
Russell did say that (a) we are directly acquainted with the constituents of those propositions into which every proposition (that we can grasp) can be analyzed; (b) at least typically, we are not
directly acquainted with the mind-independent bearers of proper names; and so (c) the things we typically refer to with names are not constituents of basic propositions.
This led Russell to say that natural language names are disguised descriptions. On this view, ‘Hesperus’ is semantically associated with a complex predicate—say, for illustration, a predicate of the
form ‘E(x) & S(x)’, suggesting ‘evening star’. In which case, ‘Hesperus is bright’ expresses a proposition of the form ‘∃x{[E(x) & S(x)] & ∀y{[E(y) & S(y)] → y = x]} & B(x)}’. It also follows that
Hesperus exists iff ∃x[E(x) & S(x)]; and this would be challenged by Kripke (1980); see the entries on rigid-designators and names. But by analyzing names as descriptions—quantificational
expressions, as opposed to logical constants (like ‘b’) that indicate individuals—Russell offered an attractive account of why the proposition that Hesperus is bright differs from the proposition
that Phosphorus is bright. Instead of saying that propositional constituents are Fregean senses, Russell could say that ‘Phosphorus is bright’ expresses a proposition of the form ‘∃x{[M(x) & S(x)] &
∀y{[M(y) & S(y)] → y = x]} & B(x)’; where ‘E(x)’ and ‘M(x)’ indicate different functions, specified (respectively) in terms of evenings and mornings. This leaves room for the discovery that the
complex predicates ‘E(x) & S(x)’ and ‘M(x) & S(x)’ both indicate functions that map Venus and nothing else to the truth-value T. The hypothesis was that the propositions expressed with ‘Hesperus is
bright’ and ‘Phosphorus is bright’ have different (fundamental) constituents, even though Hesperus is Phosphorus, but not because propositional constituents are “ways of presenting” Bedeutungen.
Similarly, the idea was that the propositions expressed with ‘Hesperus is Hesperus’ and ‘Hesperus is Phosphorus’ differ, because only the latter has predicational/unsaturated constituents
corresponding to ‘Phosphorus’. Positing unexpected logical forms seemed to have explanatory payoffs.
Questions about names and descriptions are also related to psychological reports, like ‘Mary thinks Venus is bright’, which present puzzles of their own; see the entry on propositional attitude
reports. Such reports seem to indicate propositions that are neither atomic nor logical compounds of simpler propositions. For as Frege noted, replacing one name with another name for the same object
can apprarently affect the truth of a psychological report. If Mary fails to know that Hesperus is Venus, she might think Venus is a planet without thinking Hesperus is a planet; though cp. Soames
(1987, 1995, 2002) and see the entry on singular propositions. Any function that has the value T given Venus as argument has the value T given Hesperus as argument. So Frege, Russell, and
Wittgenstein all held—in varying ways—that psychological reports are also misleading with respect to the logical forms of the indicated propositions.
Within the analytic tradition inspired by these philosophers, it became a commonplace that logical form and grammatical form typically diverge, often in dramatic ways. This invited attempts to
provide analyses of propositions, and accounts of natural language, with the aim of saying how relatively simple sentences (with subject-predicate structures) could be used to express propositions
(with function-argument structures).
The logical positivists explored the idea that the meaning of a sentence is a procedure for determining the truth or falsity of that sentence. From this perspective, studies of linguistic meaning and
propositional structure still dovetail, even if natural language employs “conventions” that make it possible to indicate complex propositions with grammatically simple sentences; see the entry on
analysis. But to cut short a long and interesting story, there was little success in formulating “semantic rules” that were plausible both as (i) descriptions of how ordinary speakers understand
sentences of natural language, and (ii) analyses that revealed logical structure of the sort envisioned. (And until Montague [1970], discussed briefly in the next section, there was no real progress
in showing how to systematically associate quantificational constructions of natural language with Fregean logical forms.)
Rudolf Carnap, one of the leading positivists, responded to difficulties facing his earlier views by developing a sophisticated position according to which philosophers could (and should) articulate
alternative sets of conventions for associating sentences of a language with propositions. Within each such language, the conventions would determine what follows from what. But one would have to
decide, on broadly pragmatic grounds, which interpreted language was best for certain purposes (like conducting scientific inquiry). On this view, questions about “the” logical form of an ordinary
sentence are in part questions about which conventions one should adopt. The idea was that “internal” to any logically perspicuous linguistic scheme, there would be an answer to the question of how
two sentences are inferentially related. But “external” questions, about which conventions we should adopt, would not be settled by descriptive facts about how we understand languages that we already
This was, in many ways, an attractive development of Frege's vision. But it also raised a skeptical worry. Perhaps the structural mismatches between sentences of a natural language and sentences of a
Fregean Begriffsschrift are so severe that one cannot formulate general rules for associating the sentences we ordinarily use with propositions. Later theorists would combine this view with the idea
that propositions are sentences of a mental language that is relevantly like Frege's invented language and relevantly unlike the spoken languages humans use to communicate; see Fodor (1975, 1978).
But given the rise of behaviorism, both in philosophy and psychology, this variant on a medieval idea was initially ignored or ridiculed. (And it does face difficulties; see section 8.)
Willard Van Orman Quine combined behaviorist psychology with a normative conception of logical form similar to Carnap's. The result was an influential view according to which there is no fact of the
matter about which proposition a speaker/thinker expresses with a sentence of natural language, because talk of propositions is (at best) a way of talking about how we should regiment our verbal
behavior for certain purposes—and in particular, for purposes of scientific inquiry. On this view, claims about logical form are evaluative, and such claims are underdetermined by the totality of
facts concerning speakers' dispositions to use language. From this perspective, mismatches between logical and grammatical form are to be expected, and we should not conclude that ordinary speakers
have mental representations that are isomorphic with sentences of a Fregean Begriffsschrift.
According to Quine, speakers' behavioral dispositions constrain what can be plausibly said about how to best regiment their language. He also allowed for some general constraints on interpretability
that an idealized “field linguist” might impose in coming up with a regimented interpretation scheme. (Donald Davidson developed a similar line of thought in a less behavioristic idiom, speaking in
terms of constraints on a “Radical Interpreter,” who seeks “charitable” construals of alien speech.) But unsurprisingly, this left ample room for “slack” with respect to which logical forms should be
associated with a given sentential utterance.
Quine also held that decisions about how to make such associations should be made holistically. As he sometimes put it, the “unit of translation” is an entire language, not a particular sentence. On
this view, one can translate a sentence S of a natural language NL with a structurally mismatching sentence µ of a formal language FL, even if it seems (locally) implausible that S is used to express
the proposition associated with µ, so long as the following condition is met: the association between S and µ is part of a general account of NL and FL that figures in an overall theory—which
includes an account of language, logic, and the language-independent world—that is among the best overall theories available. This holistic conception of how to evaluate proposed regimentations of
natural language was part and parcel of Quine's criticism of the early positivists' analytic-synthetic distinction, and his more radical suggestion that there is no such distinction.
The suggestion was that even apparently tautologous sentences, like ‘Bachelors are unmarried’ and ‘Caesar died if Brutus killed him’, have empirical content. These may be among the last sentences we
would dissent from, faced with recalcitrant experience; we may prefer to say that Caesar didn't really die, or that Brutus didn't really kill him, if the next best alternative is to deny the
conditional claim. But for Quine, every meaningful claim is a claim that could turn out to be false—and so a claim we must be prepared, at least in principle, to reject. Correlatively, no sentences
are known to be true simply by knowing what they mean (and knowing a priori that sentences with such meanings must be true).
For present purposes, we can abstract away from the details of debates about whether Quine's overall view was plausible. Here, the important point is that claims about logical form were said to be
(at least partly) claims about the kind of regimented language we should use, not claims about the propositions actually expressed with sentences of natural language. And one aspect of Quine's view,
about the kind of regimented language we should use, turned out to be especially important for subsequent discussions of logical form. For even among those who rejected the behavioristic assumptions
that animated Quine's conception of language, it was often held that logical forms are expressions of a first-order predicate calculus.
Frege's Begriffsschrift, recall, was designed to capture the Dedekind-Peano axioms for arithmetic, including the axiom of induction; see the entry on Frege's theorem and foundations for arithmetic.
This required quantification into positions occupiable by predicates, as well as positions occupiable by names. Using modern notation, Frege allowed for formulae like ‘(Fa & Fb) → ∃X(Xa & Xb)’ and
‘∀x∀y[x = y ↔ ∀X(Xx ↔ Xy)]’. And he took second-order quantification to be quantification over functions. This is to say, for example, that ‘∃X(Xa & Xb)’ is true iff: there is a function, X, that
maps both the individual called ‘a’ and the individual called ‘b’ onto the truth-value T. Frege also took it to be a truth of logic that for any predicate P, there is a function such that for each
individual x, that function maps x to T iff x satisfies (or “falls under”) P. In which case, for each predicate, there is the set of all and only the things that satisfy the predicate. The axioms for
Frege's logic thus generated Russell's paradox, given predicates like ‘is not a member of itself’. This invited attempts to weaken the axioms, while preserving second-order quantification. But for
various reasons, Quine and others advocated a restriction to a first-order fragment of Frege's logic, disallowing quantification into positions occupied by predicates. (Godel had proved the
completeness of first-order predicate calculus, thus providing a purely formal criterion for what followed from what in that language. Quine also held that second-order quantification illicitly
treated predicates as names for sets, thereby spoiling Frege's conception of propositions as unified by virtue of having unsaturated predicational constituents that are satisfied by things denoted by
names.) On Quine's view, we should replace ‘(Fa & Fb) → ∃X(Xa & Xb)’ with explicit first-order quantification over sets, as in ‘(Fa & Fb) → ∃s(a∈s & b∈s)’; where ‘∈’ stands for ‘is an element of’,
and this second conditional is not a logical truth, but rather a hypothesis (to be evaluated holistically) concerning sets.
The preference for first-order regimentations has come to seem unwarranted, or at least highly tendentious; see Boolos (1998). But it fueled the idea that logical form can diverge wildly from
grammatical form. For as students quickly learn, first-order regimentations of natural sentences often turn out to be highly artificial. (And in some cases, such regimentations seem to be
unavailable.) This was, however, taken to show that natural languages are far from ideal for purposes of indicating logical structure.
A different strand of thought in analytic philosophy—pressed by Wittgenstein in Philosophical Investigations and developed by others, including Strawson and Austin—also suggested that a single
sentence could be used (on different occasions) to express different kinds of propositions. Strawson (1950) argued that pace Russell, a speaker could use an instance of ‘The F is G’ to express a
singular proposition about a specific individual: namely, the F in the context at hand. According to Strawson, sentences themselves do not have truth conditions, since sentences (as opposed to
speakers) do not express propositions; and speakers can use ‘The boy is tall’ to express a proposition with the contextually relevant boy as a constituent. Donnellan (1966) went on to argue that a
speaker could even use an instance of ‘The F is G’ to express a singular proposition about an individual that isn't an F; see the entry on reference. Such considerations, which have received a great
deal of attention in recent discussions of context dependence, suggested that relations between natural language sentences and propositions are (at best) very complex and mediated by speakers'
intentions. All of which made it seem that such relations are far more tenuous than the pre-Fregean tradition suggested. This bolstered the Quine/Carnap idea that questions about the structure of
premises and conclusions are really questions about how we should talk (when trying to describe the world), much as logic itself seems to be more concerned with how we should infer than with how we
do infer. From this perspective, the connections between logic and grammar seemed rather shallow.
On the other hand, more recent work on quantifiers suggests that the divergence had been exaggerated, in part because of how Frege's idea of variable-binding was originally implemented. Consider
again the proposition that some boy sang, and the proposed logical division into the quantifier and the rest: ∃x[Boy(x) & Sang(x)]; something is both a boy and an individual that sang. This is one
way to regiment the English sentence. But one can also offer a logical paraphrase that more closely parallels the grammatical division between ‘some boy’ and ‘sang’: for some individual x such that x
is a boy, x sang. One can formalize this paraphrase with restricted quantifiers, which incorporate a restriction on the domain over which the variable in question ranges. For example, ‘∃x:B(x)’ can
be an existential quantifier that binds a variable ranging over the boys in the relevant domain, with ‘∃x:B(x)[S(x)]’ being true iff some boy sang. Since ‘∃x:B(x)[S(x)]’ and ‘∃x[B(x) & S(x)]’ are
logically equivalent, logic provides no reason for preferring the latter regimentation of the English sentence. And choosing the latter does not show that the proposition expressed with ‘Some boy
sang’ has a structure that differs from grammatical structure of the sentence.
Universal quantifiers can also be restricted, as in ‘∀x:B(x)[S(x)]’, interpreted as follows: for every individual x such that x is a boy, x sang. Restrictors can also be logically complex, as in
‘Some boy from Canada sang’ or ‘Some boy who respects Mary sang’, rendered as ‘∃x:B(x) & F(x, c)[S(x)]’ and ‘∃x:B(x) & R(x, m)[S(x)]’. Given these representations, the inferential difference between
‘some boy sang’ and ‘every boy sang’ lies with the propositional contributions of ‘some’ and ‘every’ after all, and not partly with the contribution of connectives like ‘&’ and ‘→’.
Words like ‘someone’, and the grammatical requirement that ‘every’ be followed by a noun (or noun phrase), reflect the fact that natural language employs restricted quantifiers. Phrases like ‘every
boy’ are composed of a determiner and a noun. Correspondingly, one can think of determiners as expressions that can combine with an ordered pair of predicates to form a sentence, much as one can
think of transitive verbs as expressions that can combine with an ordered pair of names to form a sentence. And this grammatical analogy, between determiners and transitive verbs, has a semantic
Since ‘x’ and ‘y’ are variables ranging over individuals, one can say that the function indicated by the transitive verb ‘likes’ yields the value T given the ordered pair 〈x,y〉 as argument if and
only if x likes y. In this notational scheme, ‘y’ corresponds to the direct object (or internal argument), which combines with the verb to form a phrase; ‘x’ corresponds to the grammatical subject
(or external argument) of the verb. If we think about ‘every boy sang’ analogously, ‘boy’ is the internal argument of ‘every’, since ‘every boy’ is a phrase. By contrast, ‘boy’ and ‘sang’ do not form
a phrase in ‘every boy sang’. So let us introduce ‘X’ and ‘Y’ as second-order variables ranging over functions, from individuals to truth values, stipulating that the extension of such a function is
the set of things that the function maps onto the truth value T. Then one can say that the function indicated by ‘every’ yields the value T given the ordered pair 〈X, Y〉 as argument iff the
extension of X includes the extension of Y. Similarly, one can say that the function indicated by ‘some’ maps the ordered pair 〈X, Y〉 onto T iff the extension of X intersects with the extension of
Just as we can describe ‘likes’ as a predicate satisfied by ordered pairs 〈x, y〉 such that x likes y, so we can think about ‘every’ as a predicate satisfied by ordered pairs 〈X, Y〉 such that the
extension of X includes the extension of Y. (This is compatible with thinking about ‘every boy’ as a restricted quantifier that combines with a predicate to form a sentence that is true iff every boy
satisfies that predicate.) One virtue of this notational scheme is that it lets us represent relations between predicates that cannot be captured with ‘∀’, ‘∃’, and the sentential connectives; see
Rescher (1962), Wiggins (1980). For example, most boys sang iff the boys who sang outnumber the boys who did not sing. So we can say that ‘most’ indicates a function that maps 〈X, Y〉 to T iff the
number of things that both Y and X map to T exceeds the number of things that Y but not X maps to T.
Using restricted quantifiers, and thinking about determiners as devices for indicating relations between functions, also suggests an alternative to Russell's treatment of ‘the’. The formula ‘∃x{B(x)
& ∀y[B(y) → x = y] & S(x)}’ can be rewritten as ‘∃x:B(x)[S(x)] & |B| = 1’, interpreted as follows: for some individual x such that x a boy, x sang, and the number of (relevant) boys is exactly one.
On this view, ‘the boy’ still does not correspond to a constituent of the formalism; nor does ‘the’. But one can depart farther from Russell's notation, while emphasizing his idea that ‘the’ is
relevantly like ‘some’ and ‘every’. For one can analyze ‘the boy sang’ as ‘!x:Boy(x)[Sang(x)]’, specifying the propositional contribution of ‘!’—on a par with as ‘∃’ and ‘∀’—as follows:
!x:Y(x)[X(x)] = T iff the extensions of X and Y intersect & |Y| = 1.
This way of encoding Russell's theory preserves his central claim. While there may be a certain boy that a speaker refers to in saying ‘The boy sang’, that boy is not a constituent of the
quantificational proposition expressed with ‘!x:Boy(x)[Sang(x)]’; see Neale (1990) for discussion. But far from showing that the logical form of ‘The boy sang’ diverges dramatically from its
grammatical form, the restricted quantifier notation suggests that the logical form closely parallels the grammatical form. For ‘the boy’ and ‘the’ do correspond to constituents of ‘!x:B(x)[S(x)]’,
at least if we allow for logical forms that represent quantificational propositions in terms of second-order relations; see Montague (1970).
It is worth noting, briefly, an implication of this point for the inference ‘The boy sang, so some boy sang’. If the logical form of ‘The boy sang’ is ‘∃x:B(x)[S(x)] & |B|=1’, then the inference is
an instance of the schema ‘A & B, so A’. But if the logical form of ‘The boy sang’ is simply ‘!(x):B(x)[S(x)]’, the premise and conclusion have the same form, differing only by substitution of ‘!’
for ‘∃’. In which case, the impeccability of the inference depends on the specific contributions of ‘the/!’ and ‘some/∃’. Only when these contributions are “spelled out,” perhaps in terms of
set-intersection, would the validity of the inference be manifest; see King (2002). So even if grammar and logic do not diverge in this case, one might say that grammatical structure does not reveal
the logical structure. From this perspective, further analysis of ‘the’ is required. Those who are skeptical of an analytic/synthetic distinction can say that it remains more a decision than a
discovery to say that ‘Some boy sang’ follows from ‘The boy sang’. In general, and especially with regard to aspects of propositional form indicated with individual words, issues about logical form
are connected with issues about the analytic-synthetic distinction.
Even given restricted quantifiers (and acceptance of second-order logical forms), the subject/predicate structure of ‘Juliet / likes every doctor’ diverges from the corresponding formula below.
∀y:Doctor(y)[Likes(Juliet, y)}.
We can rewrite ‘Likes(Juliet, y)’ as ‘[Likes(y)](Juliet)’, to reflect the fact that ‘likes’ combines with a direct object to form a phrase, which in turn combines with a subject. But this does not
affect the main point; ‘every’ seems to be a grammatical constituent of the verb phrase ‘likes every doctor’, and yet the main quantifier of the expressed proposition. In natural language, ‘likes’
and ‘every doctor’ form a phrase. But with respect to logical form, ‘likes’ evidently combines with ‘Juliet’ and a variable to form a complex predicate that is in turn an external argument of the
higher-order predicate ‘every’. Similar remarks apply to ‘Some boy likes every doctor’ and ‘[∃x:Boy(x)][∀y:Doctor(y)]{Likes(x, y)]’. So it seems that mismatches remain in the very places that
troubled medieval logicians—viz., quantificational direct objects and other examples of complex predicates with quantificational constituents.
Montague (1970, 1974) showed that these mismatches do not preclude systematic connections of natural language sentences with the corresponding propositional structures. Abstracting from the technical
details, one can specify an algorithm that pairs each natural language sentence that contains one or more quantificational expressions like ‘every doctor’ with one or more Fregean logical forms. This
was a significant advance. Together with subsequent developments, Montague's work showed that Frege's logic was compatible with the idea that quantificational constructions in natural language have a
systematic semantics. Indeed, one can use Frege's formal apparatus to study such constructions. Montague himself maintained that the syntax of natural language was misleading for purposes of (what he
took to be) real semantics. On this view, the study of valid inference still suggests that natural language grammar disguises the structure of human thought. But in thinking about the relation of
logic to grammar, one should not assume a naive conception of the latter.
For example, the grammatical form of a sentence need not be determined by the linear order of its words. Using brackets to disambiguate, we can distinguish the sentence ‘Mary [saw [the [boy [with
binoculars]]]]’—whose direct object is ‘the boy with binoculars’—from the homophonous sentence ‘Mary [[saw [the boy]] [with binoculars]]’, in which ‘saw the boy’ is modified by an adverbial phrase.
The first implies that the boy had binoculars, while the second implies that Mary used binoculars to see the boy. This distinction may not be audibly marked. Nonetheless, there is a difference
between modifying a noun (like ‘boy’) with a prepositional phrase and modifying a verb phrase (‘saw the boy’). More generally, grammatical structure need not be obvious. Just as it may take work to
discover the kind(s) of structure that propositions exhibit, so it may take work to discover the kind(s) of structure that sentences exhibit. And many studies of natural language suggest a rich
conception of grammatical form that diverges from traditional views; see especially Chomsky (1957, 1965, 1981, 1986, 1995). So we need to ask how logical forms are related to actual grammatical
forms, which linguists try to discover, since these may differ importantly from any hypothesized grammatical forms that may be suggested by casual reflection on spoken language. Appearances may be
misleading with respect to both grammatical and logical form, leaving room for the possibility that these notions of structure are not so different after all.
A leading idea of modern linguistics is that at least some grammatical structures are transformations of others. Put another way, linguistic expressions often appear to be displaced from the
positions canonically associated with certain grammatical relations that the expressions exhibit. For example, the word ‘who’ in (17) is apparently associated with the internal (direct object)
argument position of the verb ‘saw’.
(17) Mary wondered who John saw
Correspondingly, (17) can be glossed as ‘Mary wondered which person is such that John saw that person’. This invites the hypothesis that (17) reflects a transformation of the “Deep Structure” (17D)
into the “Surface Structure” (17S),
(17D) {Mary [wondered {John [saw who]}]}
(17S) {Mary [wondered [who[i] {John [saw ( _ )[i] ]}]]}
with indices indicating a grammatical relation between the indexed positions. In (17D), the embedded clause has the same form as ‘John saw Bill’. But in (17S), ‘who’ has been displaced from the
indexed argument position. Similar remarks apply to the question ‘Who did John see’ and other question-words like ‘why’, ‘what’, ‘when’, and ‘how’.
One might also explain the synonymy of (18) and (19) by positing a common deep structure, (18D).
(18) John seems to like Mary
(19) It seems John likes Mary
(18D) [Seems{John [likes Mary]}]
(18S) {John[i] [seems { ( _ )[i] [to like Mary]}]}
If every English sentence needs a grammatical subject, (18D) must be modified: either by displacing ‘John’, as in (18S); or by inserting a pleonastic subject, as in (19). Note that in (19), ‘It’ does
not indicate an argument; compare ‘There’ in ‘There is something in the garden’. Appeal to displacement also lets one distinguish the superficially parallel sentences (20) and (21).
(20) John is easy to please
(21) John is eager to please
If (20) is true, John is easily pleased. In which case, it is easy (for someone) to please John; where ‘it’ is pleonastic. But if (21) is true, John is eager that he please someone or other. This
asymmetry is effaced by representations like ‘Easy-to-please(John)’ and ‘Eager-to-please(John)’. The contrast is made manifest, however, with (20S) and (21S);
(20S) {Johni [is easy { e [to please ( _ )i ]}]}
(21S) {Johni [is eager { ( _ )i [to please e ]}]}
where ‘e’ indicates an unpronounced argument position. It may be that in (21S), which does not mean that it is eager for John to please someone, ‘John’ is grammatically linked but not actually
displaced from the coindexed position. But whatever the details, the “surface subject” of a sentence can be the object of a verb embedded within the main predicate, as in (20S). Of course, such
hypotheses about grammatical structure require defense. But Chomsky and others have long argued that such hypotheses are needed to account for various facts concerning human linguistic capacities;
see, e.g., Berwick et.al. (2014). As an illustration of the kind of data that is relevant, note that (22–24) are perfectly fine expressions of English, while (25) is not.
(22) The boy who sang was happy
(23) Was the boy who sang happy
(24) The boy who was happy sang
(25) *Was the boy who happy sang
This suggests that the auxiliary verb ‘was’ can be displaced from some positions but not others. That is, while (22S) is a permissible transformation of (22D), (24S) is not a permissible
transformation of (24D).
(22D) {[The [boy [who sang]]] [was happy]}
(22S) Was[i] {[the [boy [who sang]]] [ ( _ )[i] happy]}
(24D) {[The [boy [who [was happy]]]] sang}
(24S) *Was[i] {[the [boy [who [ ( _ )[i] happy]]]] sang}
The ill-formedness of (25) is striking, since one can sensibly ask whether or not the boy who was happy sang. One can also ask whether or not (26) is true. But (27) is not the yes/no question
corresponding to (26).
(26) The boy who was lost kept crying
(27) Was the boy who lost kept crying
Rather, (27) is the yes/no question corresponding to ‘The boy who lost was kept crying’, which has an unexpected meaning. So we want some account of why (27) cannot have the interpretation
corresponding to (26). But the “negative fact” concerning (27) is precisely what one would expect if ‘was’ cannot be displaced from its position in (26).
*Was[i] {[the [boy [who [( _ )[i] lost]]]] [kept crying]}
By contrast, if we merely specify an algorithm that associates (27) with its actual meaning—or if we merely hypothesize that (27) is the English translation of a certain mental sentence—we have not
yet explained why (27) cannot also be used to ask whether or not (26) is true. Explanations of such facts appeal to nonobvious grammatical structure, and constraints on natural language
transformations. (For example, an auxiliary verb in a relative clause cannot be “fronted;” though of course, theorists try to find deeper explanations for such constraints.)
The idea was that a sentence has both a deep structure (DS), which reflects semantically relevant relations between verbs and their arguments, and a surface structure (SS) that may include displaced
(or pleonastic) elements. In some cases, pronunciation might depend on further transformations of SS, resulting in a distinct “phonological form” (PF). Linguists posited various constraints on these
levels of grammatical structure, and the transformations that relate them. But as the theory was elaborated and refined under empirical pressure, various facts that apparently called for explanation
in these terms still went unexplained. This suggested another level of grammatical structure, perhaps obtained by a different kind of transformation on SS. The hypothesized level was called ‘LF’
(intimating ‘logical form’); and the hypothesized transformation—called quantifier raising because it targeted the kinds of expressions that indicate (restricted) quantifiers—mapped structures like
(28S) onto structures like (28L).
(28S) {Juliet [likes [every doctor]]}
(28L) {[every doctor][i] {Juliet [ likes ( _ )[i] ]}}
Clearly, (28L) does not reflect the pronounced word order in English. But the idea was that (PF) determines pronunciation, while LF was said to be the level at which the scope of a natural language
quantifier is determined; see May (1985). If we think about ‘every’ as a kind of second-order transitive predicate, which can combine with two predictes like ‘doctor’ and ‘Juliet likes ( _ )[i]’ to
form a complete sentence, we should expect that at some level of analysis, the sentence ‘Juliet likes every doctor’ has the structure indicated in (28L). And mapping (28L) to the logical form ‘
[∀x:Doctor(x)]{Likes(Juliet, x)}’ is trivial. Similarly, if the surface structure (29S) can be mapped onto (29L) or (29L'),
(29S) {[some boy] [likes [every doctor]]}
(29L) {[some boy][i] {[every doctor]j[j] {( _ )[i] [likes ( _ )[j] ]}}
(29L') {[every doctor][j] {[some boy][i] { ( _ )[i ][likes ( _ )[j] ]}}}
then (29S) can be mapped onto the logical forms ‘[∃x:Boy(x)][∀y:Doctor(y)]{Likes(x, y)}’ and ‘[∀y:Doctor(y)][∃x:Boy(x)]{Likes(x, y)}’. This assimilates quantifier scope ambiguity to the structural
ambiguity of examples like ‘Juliet saw the boy with binoculars’. More generally, many apparent examples of grammar/logic mismatches were rediagnosed as mismatches between different aspects of
grammatical structure—between those aspects that determine pronunication, and those that determine interpretation. In one sense, this is fully in keeping with the idea that in natural language,
“surface appearances” are often misleading with regard to propositional structure. But it also makes room for the idea that grammatical structure and logical structure converge, in ways that can be
discovered through investigation, once we move beyond traditional subject-predicate conceptions of structure with regard to both logic and grammar.
There is independent evidence for “covert” transformations—displacement of expressions from their audible positions, as in (28L); see Huang (1995), Hornstein (1995). Consider, for example, the French
translation of ‘Who did John see’: Jean a vu qui. If we assume that qui (‘who’) is displaced at LF, then we can explain why the question-word is understood in both French and English like a
quantifier binding a variable: which person x is such that John saw x? Similarly, example (30) from Chinese is transliterated as in (31).
(30) Zhangsan zhidao Lisi mai-te sheme
(31) Zhangsan know Lisi bought what
But (30) is ambiguous, between the interrogative (31a) and the complex declarative (31b).
(31a) Which thing is such that Zhangsan knows Lisi bought it
(31b) Zhangsan knows which thing (is such that) Lisi bought (it)
This suggests covert displacement of the quantificational question-word in Chinese; see Huang (1982, 1995). Chomsky (1981) also argued that the constraints on such displacement can help explain
contrasts like the one illustrated with (32) and (33).
(32) Who said he has the best smile
(33) Who did he say has the best smile
In (32), the pronoun ‘he’ can have a bound-variable reading: which person x is such that x said that x has the best smile. This suggests that the following grammatical structure is possible: Whoi {[(
)[i] said [he[i] has the best smile]]}. But (33) cannot be used to ask this question, suggesting that some linguistic constraint rules out the following structure:
*Who[i] [did {[he[i] say [( )[i] has the best smile]]].
And there cannot be constraints on transformations without transformations. So if English overtly displaces question-words that are covertly displaced in other languages, we should not be surprised
if English covertly displaces other quantificational expressions like ‘every doctor’. Likewise, (34) has the reading indicated in (34a) but not the reading indicated in (34b).
(34) It is false that Juliet likes every doctor
(34a) ¬∀x:Doctor(x)[Likes(Juliet, x)]
(34b) ∀x:Doctor(x)¬[Likes(Juliet, x)]
This suggests that ‘every doctor’ gets displaced, but only so far. Similarly, (13) cannot mean that every doctor is such that no patient who saw that doctor is healthy.
(13) No patient who saw every doctor is healthy
As we have already seen, English seems to abhor fronting certain elements from within an embedded relative clause. This invites the hypothesis that quantifier raising is subject to a similar
constraint, and hence, that there is quantifier-raising in English. This hypothesis is controversial; see, e.g., Jacobson (1999). But many linguists (following Chomsky [1995, 2000]) would now posit
only two levels of grammatical structure, corresponding to PF and LF—the thought being that constraints on DS and SS can be eschewed in favor of a simpler theory that only posits constraints on how
expressions can be combined in the course of constructing complex expressions that can be pronounced and interpreted. If this development of earlier theories proves correct, then the only
semantically relevant level of grammatical structure often reflects covert displacement of audible expressions; see, e.g., Hornstein (1995). In any case, there is a large body of work suggesting that
many logical properties of quantifiers, names, and pronouns are reflected in properties of LF.
For example, if (35) is true, it follows that some doctor treated some doctor; whereas (36) does not have this consequence:
(35) Every boy saw the doctor who treated himself
(36) Every boy saw the doctor who treated him
The truth conditions of (35–36) seem to be as indicated in (35a) and (36a).
(35a) [∀x:Boy(x)][!y:Doctor(y) & Treated(y,y)]{Saw(x, y)}
(36a) [∀x:boy(x)][!y:Doctor(y) & Treated(y,x)]{Saw(x, y)}
This suggests that ‘himself’ is behaving like a variable bound by ‘the doctor’, while ‘every boy’ can bind ‘him’. And there are independent grammatical reasons for saying that ‘himself’ must be
linked to ‘the doctor’, while ‘him’ must not be so linked. Note that in ‘Pat thinks Chris treated himself/him’, the antecedent of ‘himself’ must be the subject of ‘treated’, while the antecedent of
‘him’ must not be.
We still need to enforce the conceptual distinction between LF and the traditional notion of logical form. There is no guarantee that structural features of natural language sentences will mirror the
logical features of propositions; cp. Stanley (2000), King (2007). But this leaves room for the empirical hypothesis that LF reflects at least a great deal of propositional structure; see Harman
(1972), Higginbotham (1986), Segal (1989), Larson and Ludlow (1993), and the essay on structured propositions. Moreover, even if the LF of a sentence S underdetermines the logical form of the
proposition a speaker expresses with S (on a given occasion of use), the LF may provide a “scaffolding” that can be elaborated in particular contexts, with little or no mismatch between grammatical
and propositional architecture. If some such view is correct, it might avoid certain (unpleasant) questions prompted by earlier Fregean views: how can a sentence indicate a proposition with a
different structure; and if grammar is deeply misleading, why think that our intuitions concerning impeccability provide reliable evidence about which propositions follow from which? These are,
however, issues that remain unsettled.
If propositions are the “things” that really have logical form, and sentences of English are not themselves propositions, then sentences of English “have” logical forms only by association with
propositions. But if the meaning of a sentence is some proposition—or perhaps a function from contexts to propositions—then one might say that the logical form “of” a sentence is its semantic
structure (i.e., the structure of that sentence's meaning). Alternatively, one might suspect that in the end, talk of propositions is just convenient shorthand for talking about the semantic
properties of sentences: perhaps sentences of a Begriffsschrift, or sentences of mentalese, or sentences of natural languages (abstracting away from their logically/semantically irrelevant
properties). In any case, the notion of logical form has played a significant role in recent work on theories of meaning for natural languages. So an introductory discussion of logical form would not
be complete without some hint of why such work is relevant, especially since attending to details of natural languages (as opposed to languages invented to study the foundations of arithmetic) led to
renewed discussion of how to represent propositions that involve relations.
Prima facie, ‘Every old patient respects some doctor’ and ‘Some young politician likes every liar’ exhibit common modes of linguistic combination. So a natural hypothesis is that the meaning of each
sentence is fixed by these modes of combination, given the relevant word meanings. It may be hard to see how this hypothesis could be true if there are widespread mismatches between logical and
grammatical form. But it is also hard to see how the hypothesis could be false. Children, who have finite cognitive resources, typically acquire the capacity to understand the endlessly many
expressions of the languages spoken around them. A great deal of recent work has focussed on these issues, concering the connections between logical form and the senses in which natural languages are
semantically compositional.
It was implicit in Frege that each of the endlessly many sentences of an ideal language would have a compositionally determined truth-condition. Frege did not actually specify an algorithm that would
associate each sentence of his Begriffsschrift with its truth-condition. But Tarski (1933) showed how to do this for the first-order predicate calculus, focussing on interesting cases of multiple
quantification like ‘∀x[Number(x) → ∃y[SuccessorOf(y, x) & ∀z[SuccessorOf(z, x) → (z = y)]]]’. This made it possible to capture, with precision, the idea that an inference is valid in the predicate
calculus iff: every interpretation that makes the premises true also makes the conclusion true, holding fixed the interpretations of logical elements like ‘if’ and ‘every’. Davidson (1967a)
conjectured that one could do for English what Tarski did for the predicate calculus; and Montague, similarly inspired by Tarski, showed how one could start dealing with predicates that have
quantificational constituents. Still, many apparent objections to the conjecture remained. As noted at the end of section four, sentences like ‘Pat thinks that Hesperus is Phosphorus’ present
difficulties; though Davidson (1968) offered an influential suggestion. Davidson's (1967b) proposal concerning examples like (37–40) also proved enormously fruitful.
(37) Juliet kissed Romeo quickly at midnight.
(38) Juliet kissed Romeo quickly.
(39) Juliet kissed Romeo at midnight.
(40) Juliet kissed Romeo.
If (37) is true, so are (38–40); and if (38) or (39) is true, so is (40). The inferences seem impeccable. But the function-argument structures are not obvious. If we represent ‘kissed quickly at
midnight’ as an unstructured predicate that takes two arguments, like ‘kissed’ or ‘kicked’, we will represent the inference from (37) to (40) as having the form: K*(x, y); so K(x, y). But this form
is exemplified by the bad inference ‘Juliet kicked Romeo; so Juliet kissed Romeo’. Put another way, if ‘kissed quickly at midnight’ is a logically unstructured binary predicate, then the following
conditional is a nonlogical assumption: if Juliet kissed Romeo in a certain manner at a certain time, then Juliet kissed Romeo. But this conditional seems like a tautology, not an assumption that
introduces any epistemic risk. Davidson concluded that the surface appearances of sentences like (37–40) mask relevant semantic structure. In particular, he proposed that such sentences are
understood in terms of quantification over events.
According to Davdison, who echoed Ramsey (1927), the meaning of (40) is reflected in the paraphrase ‘There was a kissing of Romeo by Juliet’. One can formalize this proposal in various ways: ∃e
[KissingOf(e, Romeo) & KissingBy(e, Juliet)]; or ∃e[Kiss(e, Juliet, Romeo)], with the verb ‘kiss’ indicating a function that takes three arguments; or as in (40a),
(40a) ∃e[Agent(e, Juliet) & Kissing(e) & Patient(e, Romeo)]
with Juliet and Romeo explicitly represented as players of certain roles in an event. But given any such representation, adverbs like ‘quickly’ and ‘at midnight’ can be analyzed as additional
predicates of events, as shown in (37a-39a).
(37a) ∃e[Agent(e, Juliet) & Kissing(e) & Patient(e, Romeo) & Quick(e) & At-midnight(e)]
(38a) ∃e[Agent(e, Juliet) & Kissing(e) & Patient(e, Romeo) & Quick(e)]
(39a) ∃e[Agent(e, Juliet) & Kissing(e) & Patient(e, Romeo) & At-midnight(e)]
If this is correct, then the inference from (37) to (40) is an instance of the following valid form: ∃e[...e... & Q(e) & A(e))]; hence, ∃e[...e...]. The other impeccable inferences involving (37–40)
can likewise be viewed as instances of conjunction reduction. If the grammatical form of (40) is simply ‘{Juliet [kissed Romeo]}’, then the mapping from grammatical to logical form is not
transparent; and natural language is misleading, in that no word corresponds to the event quantifier. But this does not posit a significant structural mismatch between grammatical and logical form.
On the contrary, each word in (40) corresponds to a conjunct in (40a). This suggests a strategy for thinking about how the meaning of a sentence like (40) might be composed from the meanings of the
constituent words. A growing body of literature, in philosophy and linguistics, suggests that Davidson's proposal captures an important feature of natural language semantics, and that “event
analyses” provide a useful framework for future discussions of logical form.
In one sense, it is an ancient idea that action reports like (40) represent individuals as participating in events; see Gillon's (2007) discussion of Panini's grammar of Sanskrit. But if (40) can be
glossed as ‘Juliet did some kissing, and Romeo was thereby kissed’, perhaps the ancient idea can be deployed in developing Leibniz' suggestion that relational sentences like (40) somehow contain
simpler active-voice and passive-voice sentences; cp. Kratzer (1996). And perhaps appeals to quantifier raising can help in defending the idea that ‘Juliet kissed some/the/every boy’ is, after all, a
sentence that exhibits Subject-copula-Predicate form: ‘[some/the/every boy][i] is P’, with ‘P’ as a complex predicate akin to ‘[some event][e] was both a kissing done by Juliet and one in which he[i]
was kissed’.
With this in mind, let's return to the idea that each complex expression of natural language has semantic properties that are determined by (i) the semantic properties of its constituents, and (ii)
the ways in which these constituents are grammatically arranged. If this is correct, then following Davidson, one might say that the logical forms of expressions (of some natural language) just are
the structures that determine the corresponding meanings given the relevant word meanings; see Lepore and Ludwig (2002). In which case, the phenomenon of valid inference may be largely a by-product
of semantic compositionality. If principles governing the meanings of (37–40) have the consequence that (40) is true iff an existential claim like (40a) is true, perhaps this is illustrative of the
general case. Given a sentence of some natural language NL, the task of specifying its logical form may be inseparable from the task of providing a compositional specification of what the sentences
of NL mean.
At this point, many issues become relevant to further discussions of logical form. Most obviously, there are questions concerning particular examples. Given just about any sentence of natural
language, one can ask interesting questions (that remain unsettled) about its logical form. There are also very abstract questions about the relation of semantics to logic. Should we follow Davidson
and Montague, among others, in characterizing theories of meaning for natural languages as theories of truth (that perhaps satisfy certain conditions on learnability)? Is an algorithm that correctly
associates sentences with truth-conditions (relative to contexts) necessary and/or sufficient for being an adequate theory of meaning? What should we say about the paradoxes apparently engendered by
sentences like ‘This sentence is false’? If we allow for second-order logical forms, how should we understand second-order quantification, given Russell's Paradox? Are claims about the “semantic
structure” of a sentence fundamentally descriptive claims about speakers (or their communities, or their languages)? Or is there an important sense in which claims about semantic structure are
normative claims about how we should use language? Are facts about the acquisition of language germane to hypotheses about logical form? And of course, the history of the subject reveals that the
answers to the central questions are by no means obvious: what is logical structure, what is grammatical structure, and how are they related? Or put another way, what kinds of structures do
propositions and sentences exhibit, and how do thinkers/speakers relate them?
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• Ramsey, F., 1927, “Facts and Propositions”, Proceedings of the Aristotelian Society (Supplementary Volume), 7: 153–170.
• Sànchez, V., 1991, Studies on Natural Logic and Categorial Grammar, Ph.D. Thesis, University of Amsterdam.
• Segal, G., 1989, “A Preference for Sense and Reference,” The Journal of Philosophy, 86: 73–89.
• Soames, S., 1987, “Direct Reference, Propositional Attititudes, and Semantic Content”, Philosophical Topics, 15: 47–87.
• –––, 1995, “Beyond Singular Propositions”, Canadian Journal of Philosophy, 25: 515–50.
• –––, 2002, Beyond Rigidity, Oxford: Oxford University Press.
• Sommers, F., 1984, The Logic of Natural Language, Oxford: Oxford University Press.
• Stanley, J., 2000, “Context and Logical Form”, Linguistics and Philosophy, 23: 391–434.
• Strawson, P., 1950, “On Referring”, Mind, 59: 320–44.
• Tarski, A., 1933, “The Concept of Truth in Formalized Languages”, reprinted in Tarski 1983.
• –––, 1944, “The Semantic Conception of Truth”, Philosophy and Phenomenological Research, 4: 341–75.
• –––, 1983, Logic, Semantics, Metamathematics, J. Corcoran (ed.), J.H. Woodger (trans.), 2nd edition, Indianapolis: Hackett.
Some Other Useful Works
A few helpful overviews of the history and basic subject matter of logic:
• Kneale, W. & Kneale, M., 1962, The Development of Logic, Oxford: Oxford University Press, reprinted 1984.
• Sainsbury, M., 1991, Logical Forms, Oxford: Blackwell.
• Broadie, A., 1987, Introduction to Medieval Logic, Oxford: Oxford University Press.
• For these purposes, Russell's most important books are: Introduction to Mathematical Philosophy, London: George Allen and Unwin, 1919; Our Knowledge of the External World, New York: Norton, 1929;
and The Philosophy of Logical Atomism, La Salle, Ill: Open Court, 1985. Stephen Neale's book Descriptions (Cambridge, MA: MIT Press, 1990) is a recent development of Russell's theory.
Two key articles on restricted quantifiers, and a third reviewing more recent work, are:
• Barwise, J. & Cooper, R., 1981, “Generalized Quantifiers and Natural Language”, Linguistics and Philosophy, 4: 159–219.
• Higginbotham, J. & May, R., 1981, “Questions, Quantifiers, and Crossing”, Linguistic Review, 1: 47–79.
• Keenan, E., 1996, “The Semantics of Determiners”, in S. Lappin (ed.), The Handbook of Contemporary Semantic Theory, Oxford: Blackwell.
For introductions to Transformational Grammar and Chomsky's conception of natural language:
• Radford, A., 1988, Transformational Grammar, Cambridge: Cambridge University Press.
• Haegeman, L., 1994, Introduction to Government & Binding Theory, Oxford: Blackwell.
• Lasnik, H. (with M. Depiante and A. Stepanov), 2000, Syntactic Structures Revisited, Cambridge, MA: MIT Press.
For discussions of work in linguistics bearing directly on issues of logical form:
• Higginbotham, J., 1985, “On Semantics”, Linguistic Inquiry, 16: 547–93.
• Hornstein, N., 1995, Logical Form: From GB to Minimalism, Oxford: Blackwell.
• Larson, R. and Segal, G., 1995, Knowledge of Meaning, Cambridge, MA: MIT Press.\
• May, R., 1985, Logical Form: Its Structure and Derivation, Cambridge, MA: MIT Press.
• Neale, S., 1993, Grammatical Form, Logical Form, and Incomplete Symbols, in A. Irvine & G. Wedeking (eds.), Russell and Analytic Philosophy, Toronto: University of Toronto.
For discussions of the Davidsonian program (briefly described in section 9) and appeal to events:
• Davidson, D., 1984, Essays on Truth and Interpretation, Oxford: OUP.
• Davidson, D., 1985, “Adverbs of Action”, in B. Vermazen and M. Hintikka (eds.), Essays on Davidson: Actions and Events, Oxford: Clarendon Press
• Evans, G. & McDowell, J. (eds.), 1976, Truth and Meaning, Oxford: Oxford University Press.
• Higginbotham, J., Pianesi, F. and Varzi, A. (eds.), 2000, Speaking of Events, Oxford: Oxford University Press.
• Ludwig, K. (ed.), 2003, Contemporary Philosophers in Focus: Donald Davidson, Cambridge: Cambridge University Pres
• Lycan, W., 1984, Logical Form in Natural Language, Cambridge, MA: MIT Press.
• Parsons, T., 1990, Events in the Semantics of English Cambridge, MA: MIT Press.
• Pietroski, P., 2005, Events and Semantic Architecture, Oxford: Oxford University Press.
• Schein, B., 1993, Plurals, Cambridge, MA: MIT Press.
• Taylor, B., 1985, Modes of Occurrence, Oxford: Blackwell.
How to cite this entry.
Preview the PDF version of this entry at the Friends of the SEP Society.
Look up this entry topic at the Indiana Philosophy Ontology Project (InPhO).
Enhanced bibliography for this entry at PhilPapers, with links to its database.
[Please contact the author with suggestions.]
The author would like to thank: Christopher Menzel for spotting an error in an earlier characterization of the generalized quantifier ‘every’, prompting revision of the surrounding discussion; Karen
Carter and Max Heiber, for catching unfortunate typos earlier versions of sections three and six; and for comments on earlier drafts, Susan Dwyer, James Lesher, the editors and referees. | {"url":"https://plato.stanford.edu/archivES/FALL2017/Entries/logical-form/","timestamp":"2024-11-01T19:58:40Z","content_type":"text/html","content_length":"150742","record_id":"<urn:uuid:15ef2380-b478-42e2-a9fd-2f8832c5392d>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00502.warc.gz"} |
[GSOC2019|VisMa|Mayank] Week 5 to 7 (Phase II)
It’s been almost two months I started to work on VisMa as my GSoC-19 Project (although I have been working with the community since December 2018) and it has been a quiet learning experience. This
post will focus on technical details of what VisMa team has improved in the project roughly during the span of Phase II of GSoC.
Week 5: During this time, my goal was to complete the work regarding the Discrete Mathematics module. As I have mentioned in the previous Blog, we had added some basic Combinatorics modules
(Factorial, Permutation & Combinations), this week our plan was to take this expedition forward and add some more to this Module. Also, we intended to implement the combinatorics module in CLI/GUI,
which has not been done yet. Firstly I added the comments and animations in the above-mentioned modules. Adding comments and animations always seems like a cakewalk, but as per me doing this is
actually hard and time-consuming (reason being, you have to keep a track of all the equations which occur during any operation). However, once done it always gives a sweet sense of completion, as it
did this case. Our next target during Week 5, was to add more to the Discrete Mathematics module. We decided on adding a Statistics Module, a Probability Module and a (bitwise) boolean algebra
module. The Statistics module as of now contains basic functions to calculate mean, median and mode like measures. Statistics is a topic of prime importance, thus having a statistics module is useful
for the project. The other reason behind adding this module is that VisMa already has a graph plotter, this in later versions when combined with Statistics module can be used for the analysis of
user-entered data. The other major part of Discrete Mathematics module was (bitwise) boolean algebra modules. These modules are designed keeping the teaching purpose of the project in mind. The
comments and animations in these modules are in such a way that student will be able to observe how each bit of a number is being operated with the subsequent bit of the other number to get the final
result. This part has been solely implemented keeping teaching perspective in mind. Lastly, we added a simple Probability Module to the project. As of now, we have Combinatorics, Probability,
Statistics and (bitwise) boolean Algebra module added in the project.
Week 6: This week our task was to improve the integration and differentiation module of the project. The earlier logic and code for these two modules was beautifully implemented. My task was to add
integration and derivation function to all Function Classes of the Project. Function Classes, in very simple terms, is a name given to a super-class of tokens whose subclasses are Constant, Variable
etc. I had to add differentiation and integration for all these subclasses. Many of these were implemented but some were missing. Among missing ones, were Trigonometric, Exponential and Logarithmic
Classes. I wrote the respective functions for all these Function Classes, also adding comments and animations side by side. Also, I refactored the existing code in differentiation & integration
modules to follow an object-oriented style. Also added some test cases for same.
Week 7: The task of this week was focussed on improving the tokenizing module, adding some corner cases in Expression Simplification (involving exponents) and fixing some potential bugs. The
tokenizer module treated Variable raised to some power, as a single token of Variable type (with the value of pow parameter set to power), but it didn’t recognise power operator in any case, my task
was to fix it for recognising power operator. The potential bugs with Expression Simplification could only be resolved after it was done. The Expression Simplification follows a recursive logic, thus
adding even a small improvement in that module sometimes become much confusing. But finally it was done in a nice manner, and VisMa is now able to deal with almost all the cases involving Expressions
and Exponents. I also added test cases to reflect the new behaviour of the project.
Current Goal: As of now, I am working on implementing Matrix Module in CLI/GUI, the matrix operations have been implemented and now the next goal is to enable users to enter Matrices interactively in
Below images illustrate the GUI/CLI representation of „factorial“, „combination“ and „permutation“ features in action.
Lastly, the project development is going at a good rate :). Some times it becomes buggy and confusing too, but lastly, it is a learning process and each bug does teach something new. I will soon be
updating all the logic and working of the project in the wiki so that future developers can be helped.
Chears to space, Open Source, Maths & Code!
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ball mill critical speed calulation from gear box power
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The mill was rotated at 50, 62, 75 and 90% of the critical speed. Six lifter bars of rectangular crosssection were used at equal spacing. The overall motion of the balls at the end of five
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Result #1: This mill would need to spin at RPM to be at 100% critical speed. Result #2: This mill's measured RPM is % of critical speed. Calculation Backup: the formula used for Critical Speed
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Typically R = 8. Rod Mill Charge: Typically 45% of internal volume; 35% 65% range. Bed porosity typically 40%. Height of bed measured in the same way as ball mills. Bulk density of rods = tons/
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How to Calculate and Solve for Critical Mill of Speed | Ball Mill ... Jul 18, 2021Find the diameter of balls when the critical speed of mill is 15 and the mill diameter is 10. This implies that;
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Ball Mill Power/Design Price Example #2 In Example this was determined that adenine 1400 HP wet grinder ball mill was required to grind 100 TPH of matter with an Bond Works Catalog of 15 ( guess
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Speed rate refers to the ratio of the speed of the mill to the critical speed, where the critical speed is n c = 30 / R. In practice, the speed rate of the SAG mill is generally 65% to 80%.
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Calculation method and its application for energy consumption of ball mills in ceramic industry based on power feature deployment February 2020 Advances in Applied Ceramics 119(4):112
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Jack Sizing Considerations. Jacks are limited by multiple constraints: load capacity, duty cycle, horsepower, column strength, critical speed, type of guidance, brakemotor size, and ball screw
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Raw mills usually operate at 7274% critical speed and cement mills at 7476%. Calculation of the Critical Mill Speed: G: weight of a grinding ball in kg. w: Angular velocity of the mill tube in
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Use left/right arrows to navigate the slideshow or swipe left/right if using a mobile device
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American Mathematical Society
A proof of the positive density conjecture for integer Apollonian circle packings
HTML articles powered by AMS MathViewer
by Jean Bourgain and Elena Fuchs;
J. Amer. Math. Soc. 24 (2011), 945-967
DOI: https://doi.org/10.1090/S0894-0347-2011-00707-8
Published electronically: June 20, 2011
PDF | Request permission
An Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually
tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In this paper, we compute a lower bound for
the number $\kappa (P,X)$ of integers less than $X$ occurring as curvatures in a bounded integer ACP $P$, and prove a conjecture of Graham, Lagarias, Mallows, Wilkes, and Yan that the ratio $\kappa
(P,X)/X$ is greater than $0$ for $X$ tending to infinity. References
• P. Bernays, Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht quadratischen Diskriminante, Ph.D. dissertation,
Georg-August-Universität, Göttingen, Germany (1912).
• Valentin Blomer and Andrew Granville, Estimates for representation numbers of quadratic forms, Duke Math. J. 135 (2006), no. 2, 261–302. MR 2267284, DOI 10.1215/S0012-7094-06-13522-6
• David W. Boyd, The sequence of radii of the Apollonian packing, Math. Comp. 39 (1982), no. 159, 249–254. MR 658230, DOI 10.1090/S0025-5718-1982-0658230-7
• J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
• H. S. M. Coxeter, An absolute property of four mutually tangent circles, Non-Euclidean geometries, Math. Appl. (N. Y.), vol. 581, Springer, New York, 2006, pp. 109–114. MR 2191243, DOI 10.1007/
• W. Duke, Z. Rudnick, and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), no. 1, 143–179. MR 1230289, DOI 10.1215/S0012-7094-93-07107-4
• J. Elstrodt, F. Grunewald, and J. Mennicke, Groups acting on hyperbolic space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Harmonic analysis and number theory. MR 1483315,
DOI 10.1007/978-3-662-03626-6
• T. Estermann, A new application of the Hardy-Littlewood-Kloosterman method, Proc. London Math. Soc. (3) 12 (1962), 425–444. MR 137677, DOI 10.1112/plms/s3-12.1.425
• John Friedlander and Henryk Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010. MR 2647984, DOI 10.1090/
• E. Fuchs, Arithmetic properties of Apollonian circle packings, Ph.D. Thesis, Princeton (2010).
• E. Fuchs, A note on the density of curvatures in integer Apollonian circle packings, preprint, http://www.math.ias.edu/~efuchs (2009).
• E. Fuchs, K. Sanden, Some experiments with integral Apollonian circle packings, J. Exp. Math., to appear.
• Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan, Apollonian circle packings: number theory, J. Number Theory 100 (2003), no. 1, 1–45. MR 1971245, DOI
• Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan, Apollonian circle packings: geometry and group theory. I. The Apollonian group, Discrete Comput.
Geom. 34 (2005), no. 4, 547–585. MR 2173929, DOI 10.1007/s00454-005-1196-9
• D. R. Heath-Brown, A new form of the circle method, and its application to quadratic forms, J. Reine Angew. Math. 481 (1996), 149–206. MR 1421949, DOI 10.1515/crll.1996.481.149
• Edward Kasner and Fred Supnick, The Apollonian packing of circles, Proc. Nat. Acad. Sci. U.S.A. 29 (1943), 378–384. MR 9128, DOI 10.1073/pnas.29.11.378
• Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI
• Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR 1177168
• H.D. Kloosterman, On the representation of numbers of the form $ax^2+by^2+cz^2+dt^2$, Acta Math. 49, pp. 407-464 (1926).
• A. Kontorovich, H. Oh, Apollonian circle packings and closed horospheres on hyperbolic $3$-manifolds, J. Amer. Math. Soc. 24, pp. 603–648 (2011).
• N. Niedermowwe, A version of the circle method for the representation of integers by quadratic forms, preprint arXiv:0905.1229v1 (2009).
• K. Sanden, Prime number theorems for Apollonian circle packings, Senior Thesis, Princeton University (2009).
• P. Sarnak, Letter to Lagarias on Apollonian circle packings, http://www.math. princeton.edu/sarnak (2008).
Similar Articles
• Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 11D09, 11E16, 11E20
• Retrieve articles in all journals with MSC (2010): 11D09, 11E16, 11E20
Bibliographic Information
• Jean Bourgain
• Affiliation: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
• MR Author ID: 40280
• Email: bourgain@math.ias.edu
• Elena Fuchs
• Affiliation: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
• Email: efuchs@math.ias.edu
• Received by editor(s): January 21, 2010
• Received by editor(s) in revised form: February 24, 2011, and June 6, 2011
• Published electronically: June 20, 2011
• Additional Notes: The first author is supported in part by NSF grant DMS–0808042
The second author was supported in part by NSF grant DMS–0635607
• © Copyright 2011 American Mathematical Society
• Journal: J. Amer. Math. Soc. 24 (2011), 945-967
• MSC (2010): Primary 11D09, 11E16, 11E20
• DOI: https://doi.org/10.1090/S0894-0347-2011-00707-8
• MathSciNet review: 2813334 | {"url":"https://www.ams.org/journals/jams/2011-24-04/S0894-0347-2011-00707-8/?active=current","timestamp":"2024-11-03T07:01:52Z","content_type":"text/html","content_length":"71184","record_id":"<urn:uuid:76a08a3f-b1d1-4989-8862-c77a064696bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00209.warc.gz"} |
Person: Keval Vora
Contributed to:
Wrote 7 papers:
RAIVE: runtime assessment of floating-point instability by vectorization (WCL, TB, YZ, XZ, KV, RG), pp. 623–638.
CuSha: vertex-centric graph processing on GPUs (FK, KV, RG, LNB), pp. 239–252.
ASPIRE: exploiting asynchronous parallelism in iterative algorithms using a relaxed consistency based DSM (KV, SCK, RG), pp. 861–878.
DProf: distributed profiler with strong guarantees (ZB, KV, RG0), p. 24.
KickStarter: Fast and Accurate Computations on Streaming Graphs via Trimmed Approximations (KV, RG0, G(X), pp. 237–251.
CoRAL: Confined Recovery in Distributed Asynchronous Graph Processing (KV, CT0, RG0, ZH), pp. 223–236.
PnP: Pruning and Prediction for Point-To-Point Iterative Graph Analytics (CX, KV, RG0), pp. 587–600. | {"url":"http://bibtex.github.io/person/Keval_Vora.html","timestamp":"2024-11-09T01:15:05Z","content_type":"text/html","content_length":"11458","record_id":"<urn:uuid:b38d33df-3977-4396-86c2-77335c9aa75a>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00822.warc.gz"} |
Yield to Maturity of Economics Topics | Question AI
<div><p>Delve into the world of macroeconomics with this thorough exploration of Yield to Maturity. This key concept plays a significant role in influencing economic decisions and policies. This
article provides a comprehensive guide to understanding, defining, applying, and calculating yield to maturity. From its correlation with the coupon rate to the practicality of its formula, grasp
every aspect of yield to maturity and the potential challenges faced during its calculation. Boost your financial acumen with these valuable insights.</p> <h2 class="title-big" id="qai_title_1">
Understanding the Concept: What is Yield to Maturity? </h2> In the sphere of Macroeconomics, financial concepts such as Yield to Maturity (YTM) are of considerable importance. You may wonder why? It&
#39;s because these concepts are used to evaluate different investment options, which in turn influence the economic activities at a macro level. <h3 class="title-medium" id="qai_title_2"> Yield to
Maturity in the Context of Macroeconomics </h3> <div class="definition-class"> <p>In Macroeconomics, Yield to Maturity (YTM) can be defined as the total return that an investor would receive if they
held a bond or other fixed-income security until its maturity. It takes into account both the interest or dividends received during the life of the investment and any capital gain or loss at
maturity.</p> </div> To grasp its relevance, it's essential to explore the broad roles this yield plays. Here are a few points in this direction: <ul> <li> In policy decisions: YTM plays a
crucial role in policy-making as it helps central banks in understanding the cost of borrowing and the return on investment in the economy. </li> <li> In investment decisions: Investors often use YTM
to compare different investment options available in the economy. It helps them understand the potential return on their investments. </li> <li> In market movements: Changes in YTM can indicate
shifts in economic conditions. A declining YTM might suggest slow growth, whereas an increasing YTM might suggest economic expansion. </li> </ul> A fundamental set of factors influences these roles
significantly: <table> <tbody><tr> <td> Interest Rates </td> <td> As interest rates rise, the YTM on new bonds becomes more attractive, pushing down the price of existing bonds. </td> </tr> <tr> <td>
Inflation </td> <td> Inflation may erode the purchasing power of a bond's future cash flows.</td> </tr> <tr><td> Credit Risk </td> <td> Changes in the creditworthiness of a bond issuer can affect
the bond’s price, and hence, its YTM. </td> </tr> </tbody></table> <h3 class="title-medium" id="qai_title_3"> Yield to Maturity Definition: A Comprehensive Guide </h3> <div class="definition-class">
<p> Yield to Maturity (YTM) is a financial term that depicts the total anticipated rate of return on a bond if it is held until it matures. </p></div> In essence, YTM is the internal rate of return
of an investment in a bond if the investor holds the bond until maturity and all payments are made as scheduled. The formula to compute YTM is: \[ YTM = [\frac{C + (\frac{F - P}{N})}{ \frac{F + P}{2}
}] \] Where: <ul> <li>\(C\) is the annual interest payment,</li> <li>\(F\) is the face or par value of the bond,</li> <li>\(P\) is the price of the bond, and</li> <li>\(N\) is the number of years to
maturity.</li> </ul> <div class="example-class"> <p>For instance, if we consider a bond with a par value of £1000, an annual interest payment (coupon payment) of £100, priced at £950, and five years
to maturity. The Yield to Maturity would be: \[ YTM = [\frac{100 + (\frac{1000 - 950}{5})}{\frac{1000 + 950}{2} }] = 11.1\% \] This means if you invest in this bond and hold it until maturity, your
expected rate of return will be 11.1%. </p> </div> <div class="deep-dive-class"><p> It's worth noting that Yield to Maturity makes several assumptions. It assumes that all coupon payments are
reinvested at the YTM rate and the bond is held until maturity. In reality, these conditions may not always be met, which can result in an actual return that differs from the calculated YTM. </p></
div> <h2 class="title-big" id="qai_title_2"> Breaking Down the Yield to Maturity Formula </h2> In order to properly grasp the concept of Yield to Maturity (YTM), you must first understand how to
calculate it. The formula for YTM is a crucial aspect of it and forms the foundation of any in-depth study revolving around this financial concept. <h3 class="title-medium" id="qai_title_5">
Importance of the Yield to Maturity Formula in Macroeconomics </h3> In macroeconomics, the Yield to Maturity formula provides essential insights into various aspects of financial decision-making and
economic forecasting. To list a few areas where the YTM formula proves invaluable: <ul> <li> Bond Pricing: The YTM formula factors in the present value of future cash flows, which establishes the
theoretical fair price of a bond. </li> <li> Economic Predictions: The prevailing YTM on government bonds frequently serves as a benchmark for gauging economic conditions. </li> <li> Monetary Policy:
Central banks often pay close attention to YTM trends in an effort to adjust their monetary policy in a timely manner. </li> <li> Investment Analysis: Investors utilise the YTM formula to compare
different fixed-income securities and optimise their investment portfolios. </li> </ul> There's also the question of the formula's underlying assumptions. The YTM formula assumes that <ol>
<li>All the coupon payments are reinvested at the same rate as the current yield,</li> <li>The bondholder retains the bond until maturity.</li> </ol> These assumptions are theoretical and may often
vary from real-world circumstances, thus adding an extra layer of consideration for those applying it. Understanding the YTM formula's inherent assumptions and potential deviations allows for
more informed interpretations and subsequent decisions, particularly in macroeconomic contexts. <h3 class="title-medium" id="qai_title_6"> Interpreting the Yield to Maturity Formula: A Step-by-step
Approach </h3> Once armed with the basics, how should you set about interpreting the Yield to Maturity formula? Let's return to the formula: \[ YTM = \frac{C + (\frac{F - P}{N})}{ \frac{F + P}{2}
} \] where: <ul> <li>\(C\) is the annual interest payment (also known as the coupon payment),</li> <li>\(F\) is the face or par value of the bond,</li> <li>\(P\) is the price of the bond, and</li>
<li>\(N\) is the number of years to maturity.</li> </ul> Here's how to understand what these variables tell you: <ul> <li><b>Coupon Payment (C):</b> This refers to the regular interest payments
you receive from the bond. A higher coupon payment means a higher yield to maturity, all else being equal.</li> <li><b>Face Value (F):</b> This is the amount you'll receive from the issuer when
the bond matures. If the bond's current price is lower than the face value, your yield to maturity will be higher because you can look forward to capital gains when the bond matures.</li> <li><b>
Price (P):</b> This is the amount you pay to buy the bond. Higher bond prices generally imply lower yields because you're paying more for the same stream of cash flows.</li> <li><b>Years to
Maturity (N):</b> This factor represents how long you have to wait until the bond matures. The further away the bond's maturity date, the less certainty there is about what will happen between
now and maturity, which can cause a higher yield to maturity.</li> </ul> What the formula essentially does is to factor in these variables to generate the total anticipated return, thereby providing
a comprehensive measure of yield that can help guide wise investment decisions, and empower a greater understanding of the economic climate as a whole. <h2 class="title-big" id="qai_title_3"> Insight
into the Differences: Yield to Maturity vs Coupon Rate </h2> Understanding the distinction between Yield to Maturity and Coupon Rate offers you a more comprehensive view of the factors you must
consider before investing in bonds or any fixed-income securities. While both these concepts revolve around the returns on bond investments, they offer different perspectives and calculations of
returns. <h3 class="title-medium" id="qai_title_8"> Understanding the Coupon Rate in Contrast to Yield to Maturity </h3> <b>Coupon Rate</b> and <b>Yield to Maturity (YTM)</b> are two pivotal concepts
in the context of bond investments. However, they represent different components of returns from such investments. <div class="definition-class"><p>The <b>Coupon Rate</b> of a bond is essentially the
annual interest rate paid by the bond's issuer to the bondholder. It is calculated as a percentage of the bond's nominal or face value. For instance, if a bond with a face value of £1000 has
a Coupon Rate of 5%, the bondholder will receive £50 per year in interest.</p></div> On the other hand, <div class="definition-class"><p>Yield to Maturity (YTM) is a measure of the annual total
return that will be earned on a bond if it is held until maturity. Unlike the Coupon Rate, YTM considers all potential income from a bond, including interest payments (coupons), any difference
between the purchase price and the face value (capital gain or loss), and any income from reinvestment of the coupons.</p></div> Whereas the Coupon Rate remains constant over the life of the bond,
YTM can fluctuate based on changes in market interest rates, inflation expectations, and the issuer's creditworthiness, among other factors. The formula for YTM again, is: \[ YTM = \frac{C + (\
frac{F - P}{N})}{ \frac{F + P}{2} } \] and the Coupon Rate (CR) is calculated as: \[ CR = \frac{C}{F} \] Where: <ul> <li>\(C\) is the annual coupon payment,</li> <li>\(F\) is the face or par value of
the bond,</li> <li>\(P\) is the price of the bond,</li> <li>\(N\) is the number of years until maturity.</li> </ul> <h3 class="title-medium" id="qai_title_9"> Yield to Maturity vs Coupon Rate:
Implications for Investors </h3> The contrast between Yield to Maturity and Coupon Rate holds significant implications for investors. Understanding these two rates can influence an investor's
decision about which bond to purchase. <b>Coupon Rate</b> is critical because it determines the amount of annual income a bondholder will receive from the bond. Higher Coupon rates generally mean
higher income. However, a bond with a high coupon rate may not always be an attractive investment, especially if it's selling for a price significantly higher than its face value. On the other
hand, <b>Yield to Maturity</b> is a better metric for comparing the attractiveness of various bonds or other fixed-income securities. Unlike the Coupon Rate, YTM incorporates capital gains or losses
that can occur if the bond's purchase price is different from its face value. So, even if a bond’s Coupon Rate is lower, its YTM could be higher if the bond is selling for a discount to its face
value, making it a better investment. For example, consider two bonds – Bond A and Bond B. Let's assume both bonds have the same face value (£1000) and the same maturity (5 years). Bond A has a
coupon rate of 4% and is selling at par (i.e., the purchase price is the same as the face value). Bond B has a coupon rate of 3% but is selling for £900, a £100 discount to its face value. Here,
although Bond B has a lower coupon rate, it could have a higher YTM due to the capital gain when the bond matures at its face value (£1000). On the flip side, Bond A, despite its higher Coupon Rate,
might end up offering a lower return if it is selling for a price significantly higher than its face value. In conclusion, while both Yield to Maturity and Coupon Rate are essential for investors in
fixed-income securities, YTM provides a more comprehensive picture of a bond's total return potential. <h2 class="title-big" id="qai_title_4">Examining Real World Scenarios: Yield to Maturity
Examples</h2> Now that we've built a clear foundation of Yield to Maturity (YTM) and related concepts, it's time to look at a few practical examples. These examples aim to highlight some
scenarios where you'd actively engage with YTM calculations and interpretations. <h3 class="title-medium" id="qai_title_11">Practical Applications of Yield to Maturity: Examples and Analysis</h3>
Let's explore a couple of illustrative examples to assist you in appreciating the utility of YTM calculations. <div class="example-class"><p>Consider a bond with a purchase price of £900, a face
value of £1000, a coupon payment of £50 (reflecting a coupon rate of 5%), and it's due to mature in two years. How would you calculate the Yield to Maturity?</p></div> You can use the YTM
formula: \[ YTM = \frac{C + (\frac{F - P}{N})}{ \frac{F + P}{2} } \] Where: <ul> <li>\(C\) is the annual coupon payment (Given, £50)</li> <li>\(F\) is the face value of the bond (Given, £1000)</li>
<li>\(P\) is the price of the bond (Given, £900), and</li> <li>\(N\) is the number of years to maturity (Given, 2 years).</li> </ul> We calculate: \[ YTM = \frac{50 + (\frac{1000 - 900}{2})}{ \frac
{1000 + 900}{2} } = 7.3\% \] In this case, even though the coupon rate of the bond is 5%, the yield to maturity is 7.3%. This is because the bondholder also benefits from the gain (£100) as the bond
was purchased for less than the face value (£900). To further illustrate the nuances associated with Yield to Maturity calculations, here's another example: <div class="example-class"><p>Suppose
a bond with a different set of parameters - a purchase price of £1100, a face value of £1000, a coupon payment of £60 (a coupon rate of 6%), and the bond will mature in five years. What's the
Yield to Maturity?</p></div> Here, we can once again apply the YTM formula: \[ YTM = \frac{C + (\frac{F - P}{N})}{ \frac{F + P}{2} } \] Where: <ul> <li>\(C\) is the annual coupon payment (Given, £60)
</li> <li>\(F\) is the face value of the bond (Given, £1000)</li> <li>\(P\) is the price of the bond (Given, £1100), and</li> <li>\(N\) is the number of years to maturity (Given, 5 years).</li> </ul>
The YTM calculation yields: \[ YTM = \frac{60 + (\frac{1000 - 1100}{5})}{ \frac{1000 + 1100}{2} } = 5.0\% \] In this case, even though the coupon rate of the bond is 6%, the YTM is less, at 5.0%, due
to the capital loss at maturity (£100) as the bond was purchased for more than its face value (£1100). These examples demonstrate how changes in bond prices impact Yield to Maturity. Higher purchase
prices can lead to lower YTMs due to potential capital losses, while lower prices can boost YTM thanks to potential capital gains. By evaluating the YTM, you can assess whether a bond is a suitable
investment based on your desired return and risk tolerance. <h2 class="title-big" id="qai_title_5">Mastering the Calculation: Calculating Yield to Maturity</h2> Fully grasping the method for
calculating Yield to Maturity (YTM) serves as a critical skill in your investment journey. Depending on the complexity and specifics of any given bond, various computational methods exist. Let's
delve deeper into the different approaches to solving for YTM. <h3 class="title-medium" id="qai_title_13">A Guide to Calculating Yield to Maturity: Approaches and Techniques</h3> When it comes to
computing the YTM, two primary approaches stand out: precise mathematical methods and computational trial-and-error methods. <b>Direct Mathematical Calculation:</b> In an ideal world, calculating the
YTM would be as simple as substituting known values into an equation and solving for the unknown variable, i.e., YTM, as shown in previous examples. However, due to the structure of the YTM formula
involving solving for \(YTM\) in a polynomial equation, it's not always that straightforward. The formula for YTM is: \[ YTM = \frac{C + (\frac{F - P}{N})}{ \frac{F + P}{2} } \] Given: <ul> <li>\
(C\) is the annual coupon payment,</li> <li>\(F\) is the face or par value of the bond,</li> <li>\(P\) is the price of the bond, and</li> <li>\(N\) is the number of years to maturity.</li> </ul> <b>
Computational Trial-and-Error Method:</b> In real-world scenarios, especially with bonds that pay interest semiannually or quarterly, calculating YTM by direct mathematical resolution can be
challenging due to the complexities of handling fractional exponents. This is where a process of 'Trial-and-Error' takes the center stage for determining YTM values. It involves taking an
initial guess of YTM and plugging it into the bond's pricing formula, then adjusting the YTM value until the bond's calculated price matches the given price. Realistically, financial
calculators and software tools perform these calculations using iterative algorithms that converge on the correct YTM quickly. But understanding the underlying process will strengthen your bond
analysis skills. At its most fundamental level, successful YTM calculations hinge on thorough knowledge and judicious use of these techniques. By using the optimal tool for your context, you can
extract maximum knowledge for your investment decisions. <h3 class="title-medium" id="qai_title_14">Common Challenges when Calculating Yield to Maturity and How to Overcome Them</h3> Calculating YTM
is not without its difficulties due to the complexity of bond pricing and market dynamics. Here are a couple of common challenges, along with suggestions on how to handle them effectively. <b>
Reinvestment Rate Assumption:</b> The standard YTM calculation assumes that the bondholder can reinvest all coupon payments at the same YTM yield, which is rarely possible in real markets due to
fluctuating interest rates. This can result in an overestimation of YTM that skews the actual returns. To mitigate this risk, you might want to consider the Yield to Worst (YTW) or Modified Duration
measures, which don't require a constant reinvestment rate. <b>Variable Interest Rate Bonds:</b> For bonds where coupon rates vary over time, typical calculation methods prove ineffective as
there's no fixed 'C' value to work with. In such scenarios, a 'Present Value (PV)' approach can be adopted, where you estimate expected future cash flows based on interest rate
projections and then discount them back using the desired YTM as the discount rate. <b>Call or Put Options:</b> Bonds with embedded call or put options add another layer of complexity to YTM
calculations. Here, you need to cautiously consider the potential effects of the issuer 'calling' the bond back before maturity, or the bondholder 'putting' the bond back to the
issuer before the maturity date. The Yield-to-Call (YTC) and Yield-to-Put (YTP) measures can assist in these scenarios. <b>Computational Limitations:</b> As mentioned before, bonds that pay interest
semiannually or more frequently involve fractional exponents when directly calculating YTM, creating computational challenges. Remember, this is where trusted financial calculators and software tools
become indispensable, handling the necessary iterations with precision and speed. In conclusion, understanding the nuances and complexities of calculating YTM is invaluable for astute bond selection
and investment strategy development. Armed with this knowledge, not only will you be prepared to tackle practical challenges, but you'll also be able to make informed decisions for your
investment portfolio.<div class="key-takeaways-class"> <h2 id="qai_title_6">Yield to Maturity - Key takeaways</h2> <ul> <li><b>Yield to Maturity (YTM)</b> is a financial term representing the total
expected rate of return on a bond if it is held until it matures. It is the internal rate of return of an investment in a bond assuming all payments are made as scheduled.</li> <li>The <b>Yield to
Maturity Formula</b> is expressed as \[YTM = \frac{C + (\frac{F - P}{N})}{ \frac{F + P}{2} }\] where \(C\) is the annual interest payment, \(F\) is the face or par value of the bond, \(P\) is the
price of the bond, and \(N\) is the number of years to maturity.</li> <li>The <b>Yield to Maturity (YTM)</b> is crucial in macroeconomics, providing insights into financial decision-making, economic
forecasting, bond pricing, economic predictions, monetary policy, and investment analysis.</li> <li><b>Yield to Maturity vs Coupon Rate:</b> The coupon rate of a bond is the annual interest rate paid
by the issuer to the bondholder, calculated as a percentage of the bond's nominal or face value. YTM is a measure of the total annual return to be earned on the bond if it is held until maturity,
taking into account all potential income including interest payments, any difference between the purchase price and the face value, and any income from reinvestment of the coupons.</li> <li><b>
Calculating Yield to Maturity (YTM)</b> can be performed through direct mathematical calculation or computational trial-and-error methods. The computational method requires iterative adjustments to
the YTM value until the bond's calculated price matches the given price, typically done using financial calculators or software tools.</li> </ul> </div></div> | {"url":"https://www.questionai.com/knowledge/kiCmxzoISy-yield-to-maturity","timestamp":"2024-11-06T21:11:05Z","content_type":"text/html","content_length":"99251","record_id":"<urn:uuid:706335ef-383f-4449-af20-38219780ccef>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00160.warc.gz"} |
An iterative search in end-member fraction space for spectral unmixing
A novel unmixing methodology is presented, searching for a fraction combination of end-members (EMs) that reconstructs the integrated source signal. The search starts with computing an initially
estimated unmixing solution and then assesses combinations selected at random within an envelope surrounding this estimated solution. From each of these combinations, it then progresses iteratively
along a path of neighboring combinations, so as to minimize the spectral angle between the corresponding (integrated) signatures and the source signal, until reaching a satisfactory solution. The new
iterative fraction combination search (IFCS) was compared to the standard least squares unmixing (LSU). An assessment of both methods was conducted with a real Airborne Visible/Infrared Imaging
Spectrometer image and nine synthetic images generated by randomly selecting fractions for two up to ten EMs derived from this real image. Considering all these EMs for the unmixing solution (not
knowing specifically which or how many of them are actually mixed at each pixel), the IFCS method performed considerably better than LSU.
• Hyperspectral imagery
• unmixing
Dive into the research topics of 'An iterative search in end-member fraction space for spectral unmixing'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/an-iterative-search-in-end-member-fraction-space-for-spectral-unm","timestamp":"2024-11-08T05:58:18Z","content_type":"text/html","content_length":"55117","record_id":"<urn:uuid:4b4e322b-41f8-4ff1-939c-fb36fa8f9ed9>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00865.warc.gz"} |
On This Day in Math - February 17
Statue of Quetelet in Brussels
Inductive inference is the only process known to us by which essentially new knowledge comes into the world.
~Sir Ronald Aylmer Fisher
The 48th day of the year; 48 is the smallest number with exactly ten divisors.
(This is an interesting sequence, and students might search for others. Finding the smallest number with twelve divisors will be easier than finding the one with eleven.)
48 is the smallest betrothed (quasi-amicable) number. 48 and 75 are a betrothed pair since the sum of the proper divisors of 48 is 75+1 = 76 and the sum of the proper divisors of 75 is 48+1=49.
(There is only a single other pair of betrothed numbers that can be a year day)
And 48 x 48 = 2304 but 48 x 84 = 4032.
1600 The Inquisition brought Giordano Bruno to the Campo dei Fiori in Rome’s center where they chained him to an iron stake and burned him alive for his beliefs that the earth rotated on its axis.
*Amir Aczel, Pendulum, pg 9 (Aczel gives this date as the 19th but this date seems wrong. Thony Christie noted that " Bruno was executed on 17th Feb and not for his cosmology but for his heretical
theology." Thanks... several other sources agree with Feb 17th date))
In 1857, the City of New York passed a charter to enable Peter Cooper to found a scientific institution in the city. He established the Cooper Union for the Advancement of Science and Art for the
express purpose of improving the working classes by providing free education. Courses included algebra, geometry, calculus, chemistry, physics, mechanics, architectural and mechanical drawing. It
also provided a School of Design for Women, a Musical Department, and a Free Library and Reading Room with all the periodicals of the day. By 1868, an article in the
New York Times
stated there were nearly 1500 students attached to the instiution, and the classes, which included night classes, were universally full. *TIS
In 1869, Dmitri Mendeleev cancelled a planned visit to a factory and stayed at home working on the problem of how to arrange the chemical elements in a systematic way. To begin, he wrote each element
and its chief properties on a separate card and arranged these in various patterns. Eventually he achieved a layout that suited him and copied it down on paper. Later that same day he decided a
better arrangement by properties was possible and made a copy of that, which had similar elements grouped in vertical columns, unlike his first table, which grouped them horizontally. These historic
documents still exist, and mark the beginning of the form of the Periodic Table as commonly used today. (The date is given by the Julian calendar in use in Russia at the time.) *TIS
1994 A small satellite named Dactyl was found which orbits the asteroid Ida. This was the first discovery of a satellite orbiting and asteroid. Dactyl was discovered in images taken by the Galileo
spacecraft during its flyby in 1993. Dactyl was found on 17 February 1994 by Galileo mission member Ann Harch, while examining delayed image downloads from the spacecraft.
It was named by the International Astronomical Union in 1994, for the mythological dactyls who inhabited Mount Ida on the island of Crete. It is only 1.4 kilometres (4,600 ft) in diameter. *Wik
In 1996, world chess champion Gary Kasparov defeated Deep Blue, IBM's chess-playing computer, by winning a six-game match 4-2, in a regulation-style match held in Philadelphia, as part of the ACM
Computer Science Conference. Deep Blue is an improved version of the older Deep Thought, augmented by parallel special-purpose hardware. Deep Blue uses a selectively deepening search strategy, using
improvements of the alpha-beta search strategy, with powerful evaluation functions. Transposition tables help avoid unnecessarily calculating the same position more than once. Two powerful databases
further augment Deep Blue's play. *TIS On May 11, 1997, the machine won a six-game match by two wins to one with three draws against world champion Garry Kasparov, the first time the grandmaster ever
lost a six-game match in championship play. *Wik
2015 After a weekend of celebrating, the 65th annual International Pancake race will take place in Liberal, Kansas, USA and Olney, England. The celebration is associated with Pancake day, which is
often used as a name for Shrove Tuesday in many Western countries. History of the event, and a schedule of the (now) week-long activities is
1201 Khawaja Muhammad ibn Muhammad ibn Hasan Tūsī (17 February 1201; Ṭūs, Khorasan – 25 June 1274; Baghdad), better known as Nasīr al-Dīn Tūsī was a Persian polymath and prolific writer: An
architect, astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, and theologian
Tusi convinced Hulegu Khan to construct an observatory for establishing accurate astronomical tables for better astrological predictions. Beginning in 1259, the Rasad Khaneh observatory was
constructed in Azarbaijan, west of Maragheh, the capital of the Ilkhanate Empire.
Based on the observations in this for the time being most advanced observatory, Tusi made very accurate tables of planetary movements as depicted in his book Zij-i ilkhani (Ilkhanic Tables). This
book contains astronomical tables for calculating the positions
of the planets and the names of the stars. His model for the planetary system is believed to be the most advanced of his time, and was used extensively until the development of the heliocentric model
in the time of Nicolaus Copernicus.
For his planetary models, he invented a geometrical technique called a Tusi-couple, which generates linear motion from the sum of two circular motions. He used this technique to replace Ptolemy's
problematic equant for many planets, but was unable to find a solution to Mercury. The Tusi couple was later employed in Ibn al-Shatir's geocentric model and Nicolaus Copernicus' heliocentric
Copernican model.
Al-Tusi was the first to write a work on trigonometry independently of astronomy. In his Treatise on the Quadrilateral he gave an extensive exposition of spherical trigonometry, distinct from
astronomy. It was in the works of Al-Tusi that trigonometry achieved the status of an independent branch of pure mathematics distinct from astronomy, to which it had been linked for so long. He was
also the first to list the six distinct cases of a right triangle in spherical trigonometry.
In his On the Sector Figure, appears the famous law of sines for plane triangles.
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
He also stated the law of sines for spherical triangles,discovered the law of tangents for spherical triangles, and provided proofs for these laws. *Wik
1723 Tobias Meyer (17 Feb 1723; 20 Feb 1762 at age 38) German astronomer who developed lunar tables that greatly assisted navigators in determining longitude at sea. Mayer also discovered the
libration (or apparent wobbling) of the Moon. Mayer began calculating lunar and solar tables in 1753 and in 1755 he sent them to the British government.
These tables were good enough to determine longitude at sea with an accuracy of half a degree. Mayer's method of determining longitude by lunar distances and a formula for correcting errors in
longitude due to atmospheric refraction were published in 1770 after his death. The Board of Longitude sent Mayer's widow a payment of 3000 pounds as an award for the tables. *TIS Leonhard Euler
described him as 'undoubtedly the greatest astronomer in Europe'. More notes on Meyer can be found on
this blog at the Board of Longitude Project
from the Royal Museums at Greenwich.
In 1758, Mayer attempted to define the number of colors that the eye can distinguish with accuracy. His color triangle
was first published in 1775 by the Göttinger physicist Georg Christoph Lichtenberg — more than 12 years after Mayer’s death.
1765 Sir James Ivory (17 February 1765 – 21 September 1842) was a Scottish mathematician born in Dundee. He was essentially a self-trained mathematician, and was not only deeply versed in ancient and
modern geometry, but also had a full knowledge of the analytical methods and discoveries of the continental mathematicians.
His earliest memoir, dealing with an analytical expression for the rectification of the ellipse, is published in the Transactions of the Royal Society of Edinburgh (1796); and this and his later
papers on Cubic Equations (1799) and Kepler's Problem (1802) evince great facility in the handling of algebraic formulas. In 1804 after the dissolution of the flax-spinning company of which he was
manager, he obtained one of the mathematical chairs in the Royal Military College at Marlow (afterwards removed to Sandhurst); and until the year 1816, when failing health obliged him to resign, he
discharged his professional duties with remarkable success.*Wik It has been suggested that Ivory may have suffered from schizophrenia (*ALEX D. D. CRAIK) of some type throughout his life.
Ivory, because of his mental problems, tended to quarrel with his fellow mathematicians. His relations with Wallace deteriorated with arguments over Ivory's Attraction article to Encyclopaedia
Britannica. Ivory's article on Capillary action for the same publication led to an argument with Thomas Young. Many other cases were simply caused by Ivory suffering from a quite incorrect belief
that he was being persecuted by others. In fact he never joined the Royal Astronomical Society, despite his interests in astronomy, since he believed that members of that Society were systematically
working against him. As De Morgan wrote that Ivory was of
... thoroughly sound judgement in every other respect seemed to be under a complete chain of delusions about the conduct of others to himself. But the paradox is this: - I never could learn that
Ivory, passing his life under the impression that secret and unprovoked enemies were at work upon his character, ever originated a charge, imputed a bad motive, or allowed himself an uncourteous
1874 Thomas J. Watson Sr. is born. A shrewd businessman, Watson started his career as a cash register salesman, eventually taking the helm of IBM and directing it to world leadership in punch card
equipment sales. Watson died in 1956 and control of IBM passed on to his son, Thomas Watson, Jr. who brought IBM into the electronic age and, after several bold financial risks, to dominance in the
computer industry.*CHM
1888 Otto Stern (17 Feb 1888; 17 Aug 1969 at age 81) German-American scientist and winner of the Nobel Prize for Physics in 1943 for his development of the molecular beam as a tool for studying the
characteristics of molecules and for his measurement of the magnetic moment of the proton. *TIS
1890 Ronald Aylmer Fisher FRS (17 February 1890 – 29 July 1962) was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his
contributions to statistics by creating Fisher's exact test and Fisher's equation. Anders Hald called him "a genius who almost single-handedly created the foundations for modern statistical science"
while Richard Dawkins called him "the greatest of Darwin's successors". In 2010 Dawkins named him "the greatest biologist since Darwin". Fisher was opposed to the conclusions of Richard Doll and A.B.
Hill that smoking caused lung cancer. He compared the correlations in their papers to a correlation between the import of apples and the rise of divorce in order to show that correlation does not
imply causation.
To quote Yates and Mather, "It has been suggested that the fact that Fisher was employed as consultant by the tobacco firms in this controversy casts doubt on the value of his arguments. This is to
misjudge the man. He was not above
accepting financial reward for his labours, but the reason for his interest was undoubtedly his dislike and mistrust of puritanical tendencies of all kinds; and perhaps also the personal solace he
had always found in tobacco."
After retiring from Cambridge University in 1957 he spent some time as a senior research fellow at the CSIRO in Adelaide, Australia. He died of colon cancer there in 1962.
He was awarded the Linnean Society of London's prestigious Darwin–Wallace Medal in 1958.
Fisher's important contributions to both genetics and statistics are emphasized by the remark of L.J. Savage, "I occasionally meet geneticists who ask me whether it is true that the great geneticist
R.A. Fisher was also an important statistician"*Wik The stained glass window is from the Greatroom at Caius College.
1891 Abraham Halevi (Adolf) Fraenkel (February 17, 1891, Munich, Germany – October 15, 1965, Jerusalem, Israel) known as Abraham Fraenkel, was an Israeli mathematician born in Germany. He was an
early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem. He is known for his contributions to axiomatic set theory, especially his addition to Ernst Zermelo's axioms
which resulted in Zermelo–Fraenkel axioms.*Wik
1905 Rózsa Péter (orig.: Politzer) (17 February 1905–16 February 1977) was a Hungarian mathematician. She is best known for her work with recursion theory.
Péter was born in Budapest, Hungary, as Rózsa Politzer (Hungarian: Politzer Rózsa). She attended Eötvös Loránd University, where she received her PhD in 1935. After the passage of the Jewish Laws of
1939 in Hungary, she was forbidden to teach because of her Jewish origin. After the war she published her key work, Recursive Functions.
She taught at Eötvös Loránd University from 1955 until her retirement in 1975. She was a corresponding member of the Hungarian Academy of Sciences (1973).*Wik In 1951 she wrote the first monograph on
recursive function theory.
1950 Viktor Aleksandrovich Gorbunov (17 Feb 1950 in Russia - 29 Jan 1999 in Novosibirsk, Russia) He published his first paper in 1973 being a joint work with A I Budkin entitled Implicative classes
of algebras (Russian). The implicative class of algebras is a generalisation of quasivarieties. The structural characteristics of the implicative class are studied in this paper. A second join paper
with Budkin On the theory of quasivarieties of algebraic systems (Russian) appeared in 1975. In the same year he published Filters of lattices of quasivarieties of algebraic systems (Russian), this
time written with V P Belkin. In fact he had written six papers before his doctoral thesis On the Theory of Quasivarieties of Algebraic Systems was submitted. He received the degree in 1978. Gorbunov
continued working at Novosibirsk State University, being promoted to professor. He also worked as a researcher in the Mathematics Institute of the Siberian Branch of the Russian Academy of Sciences.
1600 Giordano Bruno (born 1548 - 17 Feb 1600)Italian philosopher, astronomer, mathematician and occultist whose theories anticipated modern science. The most notable of these were his theories of the
infinite universe and the multiplicity of worlds, in which he rejected the traditional geocentric (or Earth-centred) astronomy and intuitively went beyond the Copernican heliocentric (sun-centred)
theory, which still maintained a finite universe with a sphere of fixed stars. Although one of the most important philosophers of the Italian Renaissance, Bruno's various passionate utterings led to
opposition. In 1592, after a trial he was kept imprisoned for eight years and interrogated periodically. When, in the end, he refused to recant, he was burned at the stake in Rome for heresy.*TIS
Professor Rickey of USMA disagrees about Bruno's "failure to recant." "It is a nineteenth century myth that he refused to recant his view that the earth moves." *VFR
1680 Jan Swammerdam (February 12, 1637, Amsterdam – February 17, 1680) was a Dutch biologist and microscopist. His work on insects demonstrated that the various phases during the life of an
insect—egg, larva, pupa, and adult—are different forms of the same animal. As part of his anatomical research, he carried out experiments on muscle contraction. In 1658, he was the first to observe
and describe red blood cells. He was one of the first people to use the microscope in dissections, and his techniques remained useful for hundreds of years.*Wik
1865 George Phillips Bond (20 May 1825, 17 Feb 1865 at age 39) American astronomer who made the first photograph of a double star, discovered a number of comets, and with his father discovered
Hyperion, the eighth moon of Saturn. *TIS
1867 Alexander Dallas Bache (19 Jul 1806, 17 Feb 1867 at age 60) was an American physicist who was Ben Franklin's great grandson and trained at West Point. Bache became the second Superintendent of
the Coast Survey (1844-65). He made an ingenious estimate of ocean depth (1856) by studying records of a tidal wave that had taken 12 hours to cross the Pacific. Knowing that wave speeds depend on
depth, he calculated a 2.2- mile average depth for the Pacific (which is within 15% of the presently accepted value). Bache created the National Academy of Sciences, securing greater government
involvement in science. Through the Franklin Institute he instituted boiler tests to promote safety for steamboats. *TIS
1874 (Lambert) Adolphe (Jacques) Quetelet (22 Feb 1796, 17 Feb 1874 at age 78) was a Belgian mathematician, astronomer, statistician, and sociologist known for his pioneering application of
statistics and the theory of probability to social phenomena, especially crime. At an observatory in Brussels that he established in 1833 at the request of the Belgian government, he worked on
statistical, geophysical, and meteorological data, studied meteor showers and established methods for the comparison and evaluation of the data. In Sur l'homme et le developpement de ses facultés,
essai d'une physique sociale (1835) Quetelet presented his conception of the average man as the central value about which measurements of a human trait are grouped according to the normal curve. *TIS
Quetelet created the Body Mass Index in a paper in 1832. It was known as the Quetelet Index until it was termed the Body Mass Index in 1972 by Ancel Keys.
1875 Friedrich Wilhelm August Argelander (22 Mar 1799, 17 Feb 1875 at age 75)
German astronomer who established the study of variable stars as an independent branch of astronomy and is renowned for his great catalog listing the positions and brightness of 324,188 stars of the
northern hemisphere above the ninth magnitude. He studied at the University of Königsberg, Prussia, where he was a pupil and later the successor of Friedrich Wilhelm Bessel. In 1837, Argelander
published the first major investigation of the Sun's motion through space. In 1844 he began studies of variable stars.*TIS
1947 Ettore Bortolotti (6 March 1866 in Bologna, Kingdom of Sardinia (now Italy) - 17 Feb 1947 in Bologna, Italy) Italian mathematician who worked in various areas in analysis. He was interested in
the history of mathematics. *SAU
1974 Heinrich Franz Friedrich Tietze contributed to the foundations of general topology and developed important work on subdivisions of cell complexes. The bulk of this work was carried out after he
took up the chair at Munich in 1925.*SAU
2012 Nicolaas Govert "Dick" de Bruijn (9 July 1918 – 17 February 2012) was a Dutch mathematician, affiliated as professor emeritus with the Eindhoven University of Technology. He received his Ph.D.
in 1943 from Vrije Universiteit Amsterdam.
De Bruijn covered many areas of mathematics. He is especially noted for the discovery of the De Bruijn sequence. He is also partly responsible for the De Bruijn–Newman constant, the De Bruijn–Erdős
theorem (in both incidence geometry and graph theory) and the BEST theorem. He wrote one of the standard books in advanced asymptotic analysis (De Bruijn, 1958). De Bruijn also worked on the theory
of Penrose tilings. In the late sixties, he designed the Automath language for representing mathematical proofs, so that they could be verified automatically (see automated theorem checking). Lately,
he has been working on models for the human brain.*Wik
Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell | {"url":"https://pballew.blogspot.com/2015/02/on-this-day-in-math-february-17.html","timestamp":"2024-11-05T00:01:28Z","content_type":"application/xhtml+xml","content_length":"150246","record_id":"<urn:uuid:13a3f1e8-f369-410a-a95a-c0ebe4c8450e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00182.warc.gz"} |
python: how to identify if a variable is an array or a scalar
I have a function that takes the argument NBins. I want to make a call to this function with a scalar 50 or an array [0, 10, 20, 30]. How can I identify within the function, what the length of NBins
is? or said differently, if it is a scalar or a vector?
I tried this:
>>> N=[2,3,5]
>>> P = 5
>>> len(N)
>>> len(P)
Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: object of type 'int' has no len()
As you see, I can't apply len to P, since it's not an array... Is there something like isarray or isscalar in python?
Here we can use the np.isscalar method in order to verify the value of the variable NBins is whether it is a scalar or an array.
The code for implementation is given below which will take into consideration the following:
1. Determine whether NBins is a scalar or an array.
2. Print the length of the array in that case.
import numpy as np
def input_bins(NBins):
if np.isscalar(NBins):
print(“NBins is a scalar, NBins”).
length = len(NBins).
print(“NBins is an array with length:”, length)
except TypeError:
print(“NBins does not comprise of scalar or array like objects”).
input_bins([0, 10, 20, 30]) | {"url":"https://intellipaat.com/community/21850/python-how-to-identify-if-a-variable-is-an-array-or-a-scalar","timestamp":"2024-11-12T02:33:34Z","content_type":"text/html","content_length":"110705","record_id":"<urn:uuid:1083ca9e-0262-4491-8bd8-ffe8ddbc218d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00345.warc.gz"} |
Noise-induced Coherent Dynamics of Multilevel Quantum Systems Driven by Incoherent Light: A Case Study of the Three-level V-system
detuning , Fano coherence , Lorentzian , non-equilibrium steady-state , quantum coherence , resonance fluorescence
The time evolution of quantum coherence in noisy quantum systems interacting with a thermal environment is of interest to many areas of quantum physics and technology, including quantum information
processing and quantum sensing. A three-level V-system which comprises two quasi-degenerate excited states and a common ground state serves as a minimal model of multilevel quantum systems, such as
photosynthetic light-harvesting complexes and photovoltaic devices. We study the dynamics of noise-induced Fano coherences between the excited states of the V-system driven by incoherent radiation
and classify the dynamical regimes into underdamped, overdamped, and critical. In the underdamped regime ($\bar{n} < \Delta/\gamma$), where the ratio of the excited-state splitting $\Delta$ to the
spontaneous decay rate $\gamma$ is greater than the average photon number $\bar{n}$ in the thermal radiation field, the coherences oscillate and decay exponentially with a lifetime $1/(\bar{n} \
gamma)$. In the overdamped regime ($\bar{n} > \Delta/\gamma$), the noise-induced coherences exhibit a long-lived quasi-steady state with the lifetime $\tau_{c} = 1.34 \frac{\bar{n}}{\gamma} \big(\
frac{\Delta}{\gamma}\big)^{-2}$. \par Furthermore, we explore the quantum dynamics of the V-system illuminated by polarized incoherent radiation. We find that polarized incoherent driving of the
V-system leads to the formation of non-equilibrium coherent steady-states. We propose a scheme for the experimental detection of steady-state Fano coherences in Ca atoms in magnetic fields driven by
polarized incoherent light. We find the signatures of steady-state Fano coherences in the deviation of excited-state populations from their values in thermodynamic equilibrium. \par In addition, we
consider the dynamics of one-photon Fano coherences between the ground and excited states of the V-system. We find that these coherences cannot be generated by incoherent driving, which is in
contrast to the case of two-photon Fano coherences. Nevertheless, we observe an interesting coherence transfer phenomenon between the two one-photon transitions in the V-system initially excited by a
coherent laser pulse. This coherence transfer is caused by the coupling between the one-photon transitions induced by Fano interference. Besides, we study the resonance fluorescence spectrum of the
radiation emitted by the V-system in the far-field region in both the weak and strong pumping limits. The emission spectra are found to be a sum of two Lorentzians and an interference term.
Significantly, in the underdamped regime, we find the emission line splits into two distinct peaks with a sharp dip centered at zero detunings. We find that Fano coherences lead to emission-line
narrowing, which is the most pronounced in the underdamped regime, where the excited-state splitting is less than the radiative decay rate.
PubMed ID | {"url":"https://scholarwolf.unr.edu/items/d5e42922-2a08-48d5-8927-37a91d5db76f","timestamp":"2024-11-04T22:07:36Z","content_type":"text/html","content_length":"486040","record_id":"<urn:uuid:6106528a-c2d3-449d-bc48-ec87e3a5dbfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00409.warc.gz"} |
CIE A Level Physics Solved Past Paper Oct/Nov 2019 P21
1 bii) For absolute uncertainty of value T, you first need to have its % uncertainty. For % uncertainty, you will have to add up % uncertainties of all quantities being used in the formula. For T,
those are mass, m, and spring constant, k BUT before you take their sum, you must multiply each % uncertainty with the number of times that quantity is being used in the formula. Like m and k are
being taken root of so multiply their % uncertainties with 1/2 and finally take their sum. At last, you have to convert that % uncertainty to absolute uncertainty by dividing it by 100% and multiply
that to the value of T.
3a) Mass=density*volume where density of air being propelled by one propeller is 1.2 kg/m^3 and volume of air being propelled downwards in interval of 3s is what we must calculate first. We are to
assume that propelled air travels 7.6m every second in a cylinder which has a diameter of 16cm (0.16m). The volume it covers in a second is (22/7*(0.16/2)^2*7.6)=0.153m^3 so in 3s it must cover a
volume of 0.4584m^3. Now mass=1.2*0.4584=0.55kg of air.
ci) Since each propeller is at fixed position meaning that its at equilibrium. Its upward force must be equal to its downward force. The downward force exerted by each propeller on 0.55kg of air is
1.4N so the same amount of force must act on each propeller too.
d) Force on both propellers(makes up most of aircraft’s weight)= mass of aircraft*acceleration | {"url":"https://examhelpweb.com/cie-a-level-physics-solved-past-paper-oct-nov-2019-p21/","timestamp":"2024-11-10T02:02:32Z","content_type":"text/html","content_length":"133346","record_id":"<urn:uuid:2d0a857b-72d6-495f-861f-93d8b0fea335>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00201.warc.gz"} |
Robotics | yunhansgallery
top of page
1. Cooperative Multi-Robot Observation of Multiple Moving Targets
m = 3 holonomic point robots with 360-degree field of view sensors of range do3 = 30 m must observe n = 6 holonomic targets moving randomly within a circular environment of radius R = 100 m. The
speed of each target is fixed and randomly chosen to be between 0 m/s and 1.5 m/s. The maximum speed of each robot is 2 m/s. Assume that the sensing of each robot is perfect and that each robot may
communicate with other robots throughout the environment. The goal of the robots is to maximize the average number of targets that are being observed by at least one robot at each time step dt = 1
throughout the mission of length T = 120 s.
Implementing the algorithm from Parker, “Distributed Algorithms for Multi-Robot Observation of Multiple Moving Targets,” Autonomous Robots 12:231-255, 2002.
2. Collective Behaviors
25 holonomic circular robots of radius 0.25 m with 360-degree field of view sensors of range 50 m must form a flock and move together through a marked course to a goal location with minimal
collisions between each other. Assume that the sensing of each robot is perfect, that they all may home toward both the line through the course and the goal location, but that each robot may not
explicitly communicate with any other robot. The goal of robots is to minimize the average distance of each robot from the centroid of the distribution of robots without colliding and minimize the
distance of the centroid from the closest point on the line at each time step while making progress toward the goal location.
Implementing the algorithm from Matarić, “From Local Interactions to Collective Intelligence,” The Biology and Technology of Intelligent Autonomous Agents, NATO ASI Series 144, 275:295, Springer,
Berlin, 1995,
3. Traffic Control
Multiple holonomic circular robots of radius 1 m with 360-degree field of view sensors of unlimited range approach a four-way intersection. The width of the road is 7 m and the speed limit is 20 m/s.
At each time step dt = 0.2 s, the probability of a robot entering a region of radius 200 m from the intersection in one of the four directions is p = 0.04. The goal of the robots is to travel
straight across the intersection as quickly as possible without leaving the road and without colliding with each other.
Implementing traffic control approach algorithm from “Multiagent Traffic Management: A Reservation-Based Intersection Control Mechanism,” Proceedings of the Third International Joint Conference on
Autonomous Agents and Multiagent Systems, 530:537, 2004.
bottom of page | {"url":"https://www.yunhanwang.com/robotics-1","timestamp":"2024-11-09T03:46:08Z","content_type":"text/html","content_length":"324281","record_id":"<urn:uuid:be5d5597-c3f4-4eea-b107-9867b76a1579>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00594.warc.gz"} |
Muffins and the risk of being eaten
In which we explore progressive risk as a game mechanism
With Intro Part 1 and Part 2 complete, let’s check out a mechanism!
You are a Muffin
Inspired by Arnie the Doughnut by Laurie Keller, You are a Muffin is a solo journaling game.
You are a pastry in a cozy café, watching customers come and go. Each customer will place an order. There’s a chance that they might choose you as their snack, but that’s fine. After all, that’s why
you were made.
As the hours pass, you’ll become a bit more stale.
The game continues until either you are consumed or the café closes.
The Design
My goal for this game was to make a very light-hearted solo RPG that could be played in less than an hour. Nothing too complex, and no heavy decisions. Just a relaxing activity while listening to
some lo-fi chill or jazz and drinking a latte.
At the same time, I wanted an ever-increasing threat of being eaten. There are many ways to do this, including Jenga stacking-block towers, drawing from a deck of cards, or changing the size of a
dice pool.
The method I decided to use compares a simple 2d6 roll vs. an increasing target called Risk Value (RV). If both dice are less than the RV, you are eaten. Pretty simple!
The Questions
The question is: How many rounds will this game last on average?
It’s meant to be a short game. We certainly wouldn’t want it dragging out for thirty rounds, and yet we don’t want it to finish in just a few rounds either.
There’s also a secondary question: How stale will you be at the end of the game?
This impacts the character sheet design, as players will be marking off a box for each incremental bit of staleness. I’d like to include just the right number of boxes on the sheet. Too many, and I’m
wasting space. Too few, and everyone will be fully stale too early in the game.
Sure, we could do some probability math or use online dice rollers to calculate this. Having watched Jake Vanderplas’ Statistics for Hackers, my preferred method is to just simulate it!
The Simulation
In the game each turn is one in-game hour. The Risk Value (RV) increases as the hour increases, starting at RV=1 at hour six (i.e. 6 am) and increasing by one every three hours. If the hour goes past
twenty-three (i.e. 11 pm), the store closes and the game ends.
There are more elegant ways of writing this in Python, but this one works. If you have other suggestions, please leave them in the comments below!
while have_been_eaten == False and store_open == True:
if hour <= 8:
RV = 1
elif hour > 8 and hour <= 11:
RV = 2
elif hour > 11 and hour <= 14:
RV = 3
elif hour > 14 and hour <= 17:
RV = 4
elif hour > 17 and hour <= 20:
RV = 5
elif hour > 20 and hour <= 23:
RV = 6
elif hour > 23:
store_open = False
After the RV is set, we’ll roll 2d6 and compare both of them to the RV. If both are less than the RV then you are eaten by a hungry customer. Otherwise, continue and increase your staleness by 1d6.
# Roll 2d6
d1 = random.randint(1, 6)
d2 = random.randint(1, 6)
# Check for being eaten
if d1 < RV and d2 < RV:
have_been_eaten = True
times_eaten += 1
# Increase staleness by 1d6
staleness += random.randint(1,6)
if staleness > 50:
staleness = 50
Here’s a single game iteration:
Iteration 0
6:00 d1=1, d2=3 vs. RV 1
7:00 d1=6, d2=1 vs. RV 1
8:00 d1=4, d2=5 vs. RV 1
9:00 d1=3, d2=5 vs. RV 2
10:00 d1=5, d2=6 vs. RV 2
11:00 d1=5, d2=6 vs. RV 2
12:00 d1=2, d2=1 vs. RV 3
You have been eaten. Both 2 and 1 are less than 3.
Completed 1 iterations.
Rounds Played: Min: 7, Max: 7, Avg: 7
Ending Staleness: Min: 23, Max: 23, Avg: 23.
The game started at 6 am and continued for seven rounds, ending at 12 pm. The RV at 12 pm is 3, and the 2d6 rolls were a 1 and a 2. Both were less than the RV, so you have been eaten and the game
Now all we need to do is run that 10,000 times instead of just once:
Completed 10000 iterations.
Rounds Played: Min: 4, Max: 19, Avg: 10.3856
Ending Staleness: Min: 5, Max: 50, Avg: 35.6008.
Total times eaten: 9995
Sim Results
So this tells us that a typical game of You Are A Muffin has anywhere from 4 - 19 rounds, with an average of about 10 rounds.
At the end of the game, your staleness will usually be anywhere between 5 and 50 with an average of 35. Note that there’s a hard limit of 50 for staleness.
It’s worth noting that 1uprpg’s playthrough ended after 13 turns and 31 staleness. Right in the expected range!
It’s also notable that the chance of the store closing before you are eaten is extremely small. This happened only 5 times out of 10,000 during this simulation run.
Alternative methods
Ten rounds seems like a good place for the game, but what if we wanted something shorter?
We could change the mechanism so that if either of the 2d6 is less than RV, you are eaten. That’s an easy change, and then re-run the sim.
Completed 10000 iterations.
Rounds Played: Min: 4, Max: 7, Avg: 5.4996
Ending Staleness: Min: 3, Max: 35, Avg: 15.7519.
Total times eaten: 10000
That small change drops the average rounds to about 5 vs. 10. In the case of You Are A Muffin, this would mean you would (on average) rarely make it past 10 am in the coffee shop, which seemed too
We could also adjust it so if both of the 2d6 are equal to or less than the RV, you are eaten. This is a middle result, with an average number of rounds of 6. The problem is that this allows for
failure in the very first round(s) of the game, even when the RV is 1. That’s not something I wanted.
Other uses for RV
You might recognize the same RV mechanism in Exclusion Zone Botanist, with some small changes. The core idea of rolling 2d6 and comparing vs. an increasing target value is still there. In Exclusion
Zone Botanist, however, it’s not a binary “eaten” vs. “not eaten”. Instead, you begin to accumulate “corruption” while being in the EZ forest.
If you’d like a printed copy of Exclusion Zone Botanist along with ten other awesome horror supplements from indie RPG designers, check out The Lost Bay Studio Fear Bundle! Pre-orders are open now!
What other progressive risk mechanisms are out there? Leave a comment with some of your favorites!
See you next week!
— E.P. 💀 | {"url":"https://www.skeletoncodemachine.com/p/muffins-and-the-risk-of-being-eaten?open=false#%C2%A7the-questions","timestamp":"2024-11-08T02:59:02Z","content_type":"text/html","content_length":"200492","record_id":"<urn:uuid:77e53efa-c920-4dec-8776-4a0abc6a460c>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00145.warc.gz"} |
Capacitance Multiplier
The circuit on top uses an op-amp and a small capacitor to simulate a much larger capacitor. It simulates the circuit on the bottom; the resistor R2 is the same size as the resistor in the circuit
being simulated (R3), but the capacitor C1 is 100 times smaller than C2.
Current flows from the input source through R1 to the capacitor (C1). Since R1 is 100 times larger than R2, there is 1/100th the current through it into the capacitor. For a given input voltage, the
rate of change in voltage in C1 is the same as in C2, because C2 has 100 times the capacitance to make up for 1/100th the current.
So the voltages across the two capacitors are the same, but the currents are not. The op-amp causes the – input to be held at the same voltage as the voltage across C1. This means R2 has the same
voltage across it as R3, and therefore the same current.
Next: Howland Current Source
Previous: Gyrator
Generated Wed Dec 7 2016 | {"url":"https://falstad.com/circuit/e-capmult.html","timestamp":"2024-11-11T02:04:54Z","content_type":"text/html","content_length":"2208","record_id":"<urn:uuid:8d8d8f37-16a5-4492-ac4a-3e86588303fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00068.warc.gz"} |
How to determine Nth occurence in a column?
Do you know of any function or combination of functions that can achieve a new column in the data as the "occurrence" column in the screenshot below?
I would like to know if the ID in the first column is the 1st, 2nd ... Nth occurrence in that column. Is there a way to achieve this?
Thanks in advance.
Best Answer
• Hi Richard,
I came up with a few solutions yesterday (one involving a Cartesian Join, another involving Group/Join) but also came up with something that, like yours, involves a Lookup but doesn't require Get
Cell and preserves sort order.
(See workflow screenshot below)
Step 1 "Add Source Row Number" - add current row number to the source (call the column "source row")
Step 2 - "Sort by Value" - sort this ascending by the value
Step 3 - feed this into 1) the first input of a Lookup step 2) "Add Row Number" a transform step that adds the row number again ("sorted row number") and feed that into the second input of the
Lookup step ("Value -> First Row"). Add a lookup definition that matches Value == Value and returns the first value of "sorted row number". Call this definition "first row"
Step 4 - "Compute Occurrence" - a transform step to (like your solution) subtract "first row" from the current row number (this results in a zero-indexed nth occurence) and call this column
Step 5 - "Restore Original Order" - restore the original sort order by sorting ascending on "source row"
Step 6 - "Hide Columns and Adjust" - hide unnecessary columns and add 1 to "occurrence" to make it 1-indexed
In your solution, if you lookup and return first value (which would be a row number) you don't need Get Cell to mark the change in value.
• Hi Richard
There are two similar values that you could calculate more easily
1. the total number of occurrences for each ID using a Group
2. the First or Last occurrence of each ID using a Lookup step
Assuming neither of those meet your needs I think it would be possible using multiple functions that utilize the Function Get cell however it would require the data to be pre-sorted (by ID in
this case), would that be an issue for any subsequent operations?
• Hi Josh,
Thanks for your comment. The aim is to assign a new, unique sequential ID based on the original ID, so sorting won't be an issue.
After posting the question we've been experimenting a bit with Get Cell as you suggested and what seems to work is the following:
1. Sort
2. Mark the first appearance of a value in a new column (with an X) and also add CurrentRow in another
3. Do a lookup with this table on itself, by returning the Row number where we have the ID and the mark
4. Subtract the (LookupResult-1) from the CurrentRow to get this sequential numbering | {"url":"https://community.experianaperture.io/discussion/comment/1556","timestamp":"2024-11-03T18:55:34Z","content_type":"text/html","content_length":"296502","record_id":"<urn:uuid:9b70ebc9-3bec-441d-a055-6bf619b01107>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00756.warc.gz"} |
Hyperbolic Discounting - Breaking Down Finance
Hyperbolic discounting
Hyperbolic discounting is a behavioural bias that’s basically an incorrect application of discounting cash flows. Hyperbolic discounting states that people prefer sooner payoffs to later payoffs in a
way that’s irrational. If the payoff in the future is larger than the more immediate payoff, we opt for the more distant payoff depending on the ‘time gab’. However, these more distant payoffs get
less weight than should be the case. In other words, we tend to incorrectly trade-off present and future tradeoffs.
Hyperbolic discounting is driven by temporal myopia. Temporal myopia causes clarity or our perception of the future to decrease with distance. And, rather than making us more conservative or more
careful, this uncertainty makes us reduce the importance of the future in decision-making, i.e. hyperbolic discounting. Consequences which occur in the future, tend to be less important in
decision-making, the more distantly they fall in the future.
In finance, hyperbolic discounting is something quite tangible, as we can actually discount cash flows. Suppose someone offers you the choice between $50 right now, or $100 tomorrow. In this case,
waiting until to tomorrow is a small price to pay to receive twice as much money. However, as the ‘delay gap’ widens, the value of the additional amount of money decreases. Suppose you can choose
between $50 now a $100 in one year, would you still opt for the larger payoff?
The pattern that emerges from the way people choose as time increases, follows a hyperbola. Suppose we have to wait for 2 years before we get the $100. Far more people will decide to go for the
immediate payoff when they have to wait two years, rather than one. Very few people will want to wait two years.
Another example where hyperbolic discounting is at play in finance is when people borrow money, either through loans or credit cards. When borrowing money, people spend future resources to consume
more today. This means that they have high discount rates, and value current consumption a lot more than future consumption.
Consider a third example. Suppose you can choose between 2 ETFs to invest in. The first capitalizes the earnings from its investment. The second however, pays a dividend to shareholders., e.g. next
month. There are no other differences between these 2 ETFs and there are no taxes to be paid (no dividends nor capital’s gain tax). Investors prone to hyperbolic discounting prefer investing in the
second ETFs, since it generates a gain in the near future.
In principle, we should discount the value of the future reward, by a factor that increases with the length of the delay. As such, we should use exponential discounting, which is a time-consistent
way of discounting. However, research found that people don’t seem to be using a constant discount rate.
Let’s consider the following example to walk us through the irrational behaviour that is captured by hyperbolic discounting. If we want to rationally discount a future reward E, we would use the
following formula
where r is the discount rate, B is the present value, and t is time (expressed in years). If the present value of the future reward E exceeds the value of the present reward, we should select the
future reward.
Instead, someone who performs hyperbolic discounting, makes a logical error when discounting. For example, that person might be using the following formula
t again captures the delay in the reward, and r is the discount rate. In both cases, a high discount rate means that we value current consumption considerably more than more consumption in the
distant future. However, if we use hyperbolic discounting, we discount the distant reward too much.
Hyperbolic discounting is the tendency for being short sided and therefore adjusting our behavior accordingly. By being aware of this tendency to perform hyperbolic discounting, we can account for
the effect and more consciously make decisions that have future consequences. | {"url":"https://breakingdownfinance.com/finance-topics/behavioural-finance/hyperbolic-discounting/","timestamp":"2024-11-07T10:37:18Z","content_type":"text/html","content_length":"235803","record_id":"<urn:uuid:d0743f5e-91b1-4316-bbd5-fe4396cfa856>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00447.warc.gz"} |
Quadratic Simultaneous Equations Worksheet
Quadratic simultaneous equations at a glance
Quadratic simultaneous equations are pairs of equations where one is one quadratic and one linear.
To solve quadratic simultaneous equations we can use the elimination method in a similar way to solving linear simultaneous equations.Β We need to substitute the linear equation into the quadratic
(non-linear) equation which may have one variable which is squared, or both variables which are squared. This quadratic equation is then solved and two solutions are found. Both of these solutions
need to be substituted one at a time into the first equation or the second equation to calculate two pairs of solutions.Β
In order to answer these questions, a strong understanding of solving quadratic equations by factorising or using the quadratic formula is required. The final answers can be given as integers,
fractions, decimals or surds as required.
Quadratic simultaneous equations can be used to find where a line intersects a parabola or a circle. This can be extended to calculating the equation of a tangent to circle.
Looking forward, students can then progress to additional simultaneous equations worksheetsΒ on to moreΒ algebra worksheets,Β for example a simplifying expressions worksheet, or an inequalities
For more teaching and learning support on Algebra our GCSE maths lessons provide step by step support for all GCSE maths concepts.Β | {"url":"https://thirdspacelearning.com/secondary-resources/gcse-maths-worksheet-quadratic-simultaneous-equations/","timestamp":"2024-11-04T18:31:51Z","content_type":"text/html","content_length":"162942","record_id":"<urn:uuid:075d4d67-35a3-401a-be8a-891f6a6157a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00831.warc.gz"} |
Scientific Visualization, 2023, volume 15, number 2, pages 125 - 133, DOI: 10.26583/sv.15.2.11
POD-based Hydrodynamical Structures Visualization in Flows with an Internal Wave Attractor
Author: S.A. Elistratov^1,A,B
^A Shirshov Institute of Oceanology of RAS, Moscow, Russia
^B Ivannikov Institute for System Programming of RAS, Moscow, Russia
^1 ORCID: 0000-0002-7006-6879, sa.elist-ratov@yandex.ru
Hydrodynamical structure attending a flow can be hid and hardly to reveal. One of the methods to find them is to use mode decomposition (such as Proper orthogonal decomposition, POD). The method
represents the field given as a series of spatial modes multiplied by corresponding temporal coefficients. In the article the method is discussed applyingly to a complex flow with a wave attractor
structure. Attractor modes present structured vortex-like figure which cannot be claimed to be aleatory.
As it turns out POD modes are not just a formal decomposition but have a physical origin: they are connected with instability cascade minor frequencies, as spectral investigation shows. Another
consequence of that is that one of the collateral structure maximum can be visible. This proposition is proven as the structure is found to be visible in the flow itself.
Keywords: Wave attractor, instability, Proper orthogonal decomposition, visualization. | {"url":"https://sv-journal.org/2023-2/11/","timestamp":"2024-11-06T09:00:18Z","content_type":"application/xhtml+xml","content_length":"58443","record_id":"<urn:uuid:34f5215f-d6b0-4a92-abf0-9908c3a09d2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00424.warc.gz"} |
What is 1 cycle second?
Frequency is the rate at which current changes direction per second. It is measured in hertz (Hz), an international unit of measure where 1 hertz is equal to 1 cycle per second. Hertz (Hz) = One
hertz is equal to one cycle per second. Cycle = One complete wave of alternating current or voltage.
Is hertz and S 1 Same?
The hertz (symbol: Hz) is the SI unit of frequency. Its base unit is s-1 (also called inverse seconds, reciprical seconds, or 1/s). In English, hertz is used as both singular and plural.
What is CPS in frequency?
Cycles per second are a measure of the number of oscillations, or cycles, that occur per second. Cycles per second can be abbreviated as cps, and are also sometimes abbreviated as c/s or cycles/
second. For example, 1 cycle per second can be written as 1 cps, 1 c/s, or 1 cycles/second.
Is MC the same as MHz?
The term cycles per second was largely replaced by hertz by the 1970s. Sometimes the “per second” was omitted, so that “megacycles” (Mc) was used as an abbreviation of “megacycles per second” (that
is, megahertz (MHz)).
What is number of cycles per second?
The number of periods or cycles per second is called frequency. The SI unit for frequency is the hertz (Hz).
What does 60 Hz stand for?
60 cycles per second
WHAT IS 60 HERTZ? At 60 Hz, the rotor of the generator turns 60 cycles per second, the current changes 60 times per second back and forth, direction changes 100 times. That means the voltage changes
from positive to negative, and from negative to positive voltage, this process converts 60 times/second.
Is CPS the same as Hz?
The cycle per second was a once-common English name for the unit of frequency now known as the hertz (Hz). The plural form was typically used, often written cycles per second, cycles/second, c.p.s.,
c/s, ~, or, ambiguously, just cycles (Cy./Cyc.).
How many cycles per second is 60Hz?
60 cycles
A 60Hz electrical system means that the power completes 60 cycles of complete wave sequence per second while 50Hz means that it completes 50 cycles per second.
Which is equal to one cycle per second?
A period of 1 second is equal to 1 Hertz frequency. Period is the inverse of frequency: 1 Hz = 1 cps. Hertz is the SI base unit of frequency defined as one cycle per second.
Which is the inverse of a cycle per second?
Period is the inverse of frequency: 1 Hz = 1 cps. 1 Hertz: Hertz is the SI base unit of frequency defined as one cycle per second. The unit is named for Heinrich Rudolf Hertz.
How is one cycle per second related to Hertz?
Unit Descriptions. 1 Cycle per Second: A period of 1 second is equal to 1 Hertz frequency. Period is the inverse of frequency: 1 Hz = 1 cps. 1 Hertz: Hertz is the SI base unit of frequency defined as
one cycle per second.
How to calculate the frequency of one cycle?
Frequency is equal to 1 divided by the period, which is the time required for one cycle. The derived SI unit for frequency is hertz, named after Heinrich Rudolf Hertz (symbol hz). One hz is one cycle
per second. T = period, the time required for one cycle | {"url":"https://www.idcafe.net/what-is-1-cycle-second/","timestamp":"2024-11-14T11:07:11Z","content_type":"text/html","content_length":"55407","record_id":"<urn:uuid:4c22d387-0f61-4f8d-a862-d96b5bd9656b>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00757.warc.gz"} |
What to Do?
Even those who defend null hypothesis testing recognize many of the problems with it. But what should be done? Some suggestions now appear in the Publication Manual. One is that each null hypothesis
test should be accompanied by an effect size measure such as Cohen’s dor Pearson’s r. By doing so, the researcher provides an estimate of how strong the relationship in the population is—not just
whether there is one or not. (Remember that the p value cannot substitute as a measure of relationship strength because it also depends on the sample size. Even a very weak result can be
statistically significant if the sample is large enough.)
Another suggestion is to use confidence intervals rather than null hypothesis tests. A confidence interval around a statistic is a range of values that is computed in such a way that some percentage
of the time (usually 95%) the population parameter will lie within that range. For example, a sample of 20 college students might have a mean calorie estimate for a chocolate chip cookie of 200 with
a 95% confidence interval of 160 to 240. In other words, there is a very good chance that the mean calorie estimate for the population of college students lies between 160 and 240. Advocates of
confidence intervals argue that they are much easier to interpret than null hypothesis tests. Another advantage of confidence intervals is that they provide the information necessary to do null
hypothesis tests should anyone want to. In this example, the sample mean of 200 is significantly different at the .05 level from any hypothetical population mean that lies outside the confidence
interval. So the confidence interval of 160 to 240 tells us that the sample mean is statistically significantly different from a hypothetical population mean of 250.
Finally, there are more radical solutions to the problems of null hypothesis testing that involve using very different approaches to inferential statistics. Bayesian statistics, for example, is an
approach in which the researcher specifies the probability that the null hypothesis and any important alternative hypotheses are true before conducting the study, conducts the study, and then updates
the probabilities based on the data. It is too early to say whether this approach will become common in psychological research. For now, null hypothesis testing—supported by effect size measures and
confidence intervals—remains the dominant approach.
• The decision to reject or retain the null hypothesis is not guaranteed to be correct. A Type I error occurs when one rejects the null hypothesis when it is true. A Type II error occurs when one
fails to reject the null hypothesis when it is false.
• The statistical power of a research design is the probability of rejecting the null hypothesis given the expected relationship strength in the population and the sample size. Researchers should
make sure that their studies have adequate statistical power before conducting them.
• Null hypothesis testing has been criticized on the grounds that researchers misunderstand it, that it is illogical, and that it is uninformative. Others argue that it serves an important
purpose—especially when used with effect size measures, confidence intervals, and other techniques. It remains the dominant approach to inferential statistics in psychology.
1. Discussion: A researcher compares the effectiveness of two forms of psychotherapy for social phobia using an independent-samples t test.
1. Explain what it would mean for the researcher to commit a Type I error.
2. Explain what it would mean for the researcher to commit a Type II error.
Discussion: Imagine that you conduct a t test and the p value is .02. How could you explain what this p value means to someone who is not already familiar with null hypothesis testing? Be
sure to avoid the common misinterpretations of the p value. | {"url":"http://www.opentextbooks.org.hk/ditatopic/36754","timestamp":"2024-11-12T10:31:09Z","content_type":"text/html","content_length":"308980","record_id":"<urn:uuid:a1e75947-7772-41dd-bf4f-1641b5a9ff3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00486.warc.gz"} |
The AstroBEAR source code is organized into several subdirectories that each contain modules designed to handle a particular aspect of the code. Here is a very brief description of each directory.
• tree - Contains modules that define the AMR tree structures as well as various routines for manipulating the AMR tree.
• data - Contains modules that define the data structures associated with each grid or patch as well as operations for synchronizing the data, and for performing operations on the data.
• particle - Contains modules that define the data structures associated with Lagrangian particles, and routines for performing operations on particles.
• amr - Contains control routines for advancing the AMR dataset in time.
• distribution - Contains routines for distributing workloads over multiple processors.
• communication - Contains routines for performing communication needed to synchronize data across processors.
• hyperbolic - Contains routines for performing conservative hyperbolic advances.
• elliptic - Contains routines for solving linear systems of equations such as poisson's equation used by the self-gravity module.
• explicit - Contains routines for solving parabolic equations through explicit sub-cycling
• physics - Contains definitions and functions related to the particular equations being solved.
• io - Contains routines for writing and reading simulation data to disk.
• modules - Contains various routines for controlling initial and boundary conditions
• source - Contains various routines for applying source terms.
• processing - Contains routines for analyzing the data and producing various data products.
• layouts - Contains modules for mapping AMR datasets onto uniform subgrids.
• threads - Contains modules for handling threading of level advances.
for help on using the wiki. | {"url":"https://bluehound2.circ.rochester.edu/astrobear/wiki/DirectoryBreakdown?version=1","timestamp":"2024-11-09T19:57:37Z","content_type":"text/html","content_length":"10472","record_id":"<urn:uuid:6e437ae0-6f80-42e7-8bf6-dff69a35a9bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00201.warc.gz"} |
Vertex Cover Meets Scheduling
We consider a hybrid two-stage optimization problem that generalizes two classic combinatorial optimization problems: (i) weighted vertex cover in graphs, and (ii) makespan minimization in
multiprocessor scheduling. An instance specifies a machine environment, a set of jobs, and an undirected graph over the jobs. The goal is to select a subset of the jobs that forms a vertex cover and
to schedule it on a set of parallel machines so as to minimize the makespan. We call this problem family vertex cover meets multiprocessor scheduling (VCMS). The problem is motivated by networks
where vertices represent servers and each edge corresponds to a task that can be done by any of the two servers of its endpoint. Each selected server can complete any number of tasks assigned to it
within a given time defined by its weight, as the time consumption of the server is roughly equal to its activation time. The activation is performed by a set of (Formula presented.) processors, such
that every selected server is activated by one processor, and the goal is to minimize the maximum total activation time assigned to any processor. We design a multitude of approximation algorithms
for VCMS and its variants, many of which match or almost match the best approximation bound known for the vertex cover problem. In particular, we give a (Formula presented.) -approximation for the
case of a fixed number of unrelated machines, a (Formula presented.) -approximation for an arbitrary number of unrelated machines, and a (Formula presented.) -approximation for an arbitrary number of
identical machines. Furthermore we consider special graph classes for which the weighted vertex cover problem can be solved to optimality in polynomial time: for many of these classes, there is a
PTAS for VCMS on identical machines; for bipartite graphs, however, VCMS on identical machines turns out to be APX-hard. Finally, we study the bin packing counterpart of VCMS and design a (Formula
presented.) -approximation algorithm for it.
Bibliographical note
Publisher Copyright:
© 2015, Springer Science+Business Media New York.
ASJC Scopus subject areas
• General Computer Science
• Computer Science Applications
• Applied Mathematics
Dive into the research topics of 'Vertex Cover Meets Scheduling'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/vertex-cover-meets-scheduling","timestamp":"2024-11-13T02:42:20Z","content_type":"text/html","content_length":"56024","record_id":"<urn:uuid:1153ad98-415c-45ee-9d33-7af55e90de87>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00810.warc.gz"} |
Hypergroup Graphs and Subfactors.
Date of Submission
Institute Name (Publisher)
Indian Statistical Institute
Document Type
Doctoral Thesis
Degree Name
Doctor of Philosophy
Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)
Abstract (Summary of the Work)
The main theme of this t hesis is hypergroups. In this thesis the the- ory of hypergroups is applied to study the relation between certain graphs and subfactors of II, factors in the context of
principal graphs associated with the inclusions of II, factors. More general classes of hypergroups are iutroduced, new examples of hypergroups associated to certain graphs are coustructed and
classification of small order hypergroups is discussed.The text of the thesis is arranged in four chapters. The first chapter is on preliminaries of the theory of hypergroups, the second on the
appli- cation of the theory of hyjrrgroups in the relation bet ween certain graphs and subfactors of II act.cs. the third on a more general class of hyper- groups and the fourth chapter is on some
new examples of hypergroups and classification of smai order hypergroups.The first chapter on the preliminaries of the theory of hypergroups col- ts together the basic known facts about hypergroups
which also serves the purpose of fixing notation and terminology for the following chapters. In this chapter the bimodule interpretation as against the relative commmitant interpretation of a
principal graph associated with the inclusion of a of II, factors is worked out in detail.The second chapter is on the notion of an action of a hypergroup on a *et. After deriving some consequences
of the definition of action, the notion is used here to show that certain bipartite graplıs cannot arise as principal graphs for inclusions of II factors.The third chapter is on the notion of an
Mrgraded hypergroup. This otion extends the notion of a hypergroup and captures the algebraic struc- ture pussesseed by the collection of irreducible bifinite bimodules over a pair of II, factors
with respect to taking tensor products and contragredi- ents. The notion of a dimeusion function of a hypergroup is extended to M-graded hypergroups and it is proved that every irreducible finite M2-
graded hypergroup possesSes a unique dincusion function. The results in this chapter also rule out some graphs from arising as principal graphs for inclusions of II1 factors.The fourth chapter is on
some new examples of hypergroups and classi- fication of hypergroups of small order. Sequences of bypergroups associated to the graphs 32, for all positive integers n and the Coxeter graph E. for all
positive integers except 7 and 10 are described here. More examples given by coutaected sum of certain graphs are also described here. This chapter coucludes with classification of hypergroups of
small order which -hows that the smallest non-abelian hypergroup is the smallest non-abelian group.
ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842930
Control Number
Recommended Citation
Vijayarajan, A. K. Dr., "Hypergroup Graphs and Subfactors." (1994). Doctoral Theses. 154. | {"url":"https://digitalcommons.isical.ac.in/doctoral-theses/154/","timestamp":"2024-11-04T13:34:51Z","content_type":"text/html","content_length":"43347","record_id":"<urn:uuid:2b41c1f4-29c8-4eb9-bb67-c9f6a412a85f>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00318.warc.gz"} |
The surface area of a box is 10.4cm.
The surface area of a box is 10.4cm. What is the surface area of a similar box that is larger...
The surface area of a box is 10.4cm. What is the surface area of a similar box that is larger by a scale factor of 3? | {"url":"https://www.sweetstudy.com/content/surface-area-box-104cm-what-surface-area-similar-box-larger","timestamp":"2024-11-02T20:54:04Z","content_type":"text/html","content_length":"140758","record_id":"<urn:uuid:7dd2cdac-b95c-4d12-9d4d-811360607bdb>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00732.warc.gz"} |
The Stacks project
Lemma 41.17.3. Let $f : X \to S$ be a finite unramified morphism of schemes. Let $s \in S$. There exists an étale neighbourhood $(U, u) \to (S, s)$ and a finite disjoint union decomposition
\[ X_ U = \coprod \nolimits _ j V_ j \]
such that each $V_ j \to U$ is a closed immersion.
Comments (3)
Comment #4018 by Davide Lombardo on
Small typo: "the fibre over $s$", not over $S$, I guess.
Comment #4021 by Laurent Moret-Bailly on
Of course this is trivial, but I believe the statement should say that the sum is finite.
Comment #4127 by Johan on
Thanks to both of you and fixed here.
There are also:
• 2 comment(s) on Section 41.17: Étale local structure of unramified morphisms
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Exact potential and scattering amplitudes from the tachyon non-linear β-function
We compute, on the disk, the non-linear tachyon β-function, βT, of the open bosonic string theory. βT is determined both in an expansion to the third power of the field and to all orders in
derivatives and in an expansion to any power of the tachyon field in the leading order in derivatives. We construct the Witten-Shatashvili (WS) space-time effective action S and prove that it has a
very simple universal form in terms of the renormalized tachyon field and βT. The expression for S is well suited to studying both processes that are far off-shell, such as tachyon condensation, and
close to the mass-shell, such as perturbative on-shell amplitudes. We evaluate S in a small derivative expansion, providing the exact tachyon potential. The normalization of S is fixed by requiring
that the field redefinition that maps S into the tachyon effective action derived from the cubic string field theory is regular on-shell. The normalization factor is in precise agreement with the one
required for verifying all the conjectures on tachyon condensation. The coordinates in the space of couplings in which the tachyon β-function is non linear are the most appropriate to study RG fixed
points that can be interpreted as solitons of S, i.e. D-branes. © SISSA/ISAS 2004.
• Tachyon Condensation
• String Field Theory
Dive into the research topics of 'Exact potential and scattering amplitudes from the tachyon non-linear β-function'. Together they form a unique fingerprint. | {"url":"https://publires.unicatt.it/en/publications/exact-potential-and-scattering-amplitudes-from-the-tachyon-non-li","timestamp":"2024-11-13T15:29:02Z","content_type":"text/html","content_length":"56020","record_id":"<urn:uuid:7d557ad1-e95b-40be-b702-f300c057e934>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00354.warc.gz"} |
Print Latex Normally?
Print Latex Normally?
Hi. When I use %display latex I don't really see Latex I want to see. For example:
%display latex
OUT: \newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{3} \, x^{3}
Isn't newcommand completely extra here? How would I go on about getting latex output that doesn't include \newcommand?
Ideally, I would want to see the following latex output: \frac{x^{3}}{3}
The formatting of your post is a bit confusing because it's not clear where the line breaks are. If you want to have pre-formatted multi-line text, don't use backticks since that is only for inline
formatting. Instead, indent the full text by 4 spaces (you can do this automatically by selecting all the lines and pressing Ctrl-K.
Please indicate if you are working in the Sage REPL, in the Jupyter Notebook, in the SageNB notebook, in the sage-mode for Emacs...
Are you using the Sage jupyter notebook, the old Sage notebook, Sage's command-line interface, or CoCalc?
1 Answer
Sort by ยป oldest newest most voted
%display latex I think is intended more for use in the notebook, where the latex would actually be rendered. I'm not sure why so much of the latex display output includes the \newcommand{\Bold}
definition--seems like a historical artifact and I'm not sure if it's still necessary. But if you just want the plain latex formatting for some output you can do:
sage: latex(integrate(x^2,x))
\frac{1}{3} \, x^{3}
edit flag offensive delete link more
The \newcommand{\Bold}stuff dates back quite some time, apparently, and is explained here: http://doc.sagemath.org/html/en/refer... I do think it's a bit excessive. There's no need for it to be added
to every latex-formatted output if it's for defining commands that are not even used...
Iguananaut ( 2018-07-12 14:08:58 +0100 )edit
latex(input) does indeed work, thanks. Is there no way to latexify output without manually wrapping the sage code in latex()? I think this works well, but I'd much rather prefer the %display latex
o6p ( 2018-07-12 14:49:16 +0100 )edit
Yes, but you'll wind back up with the same problem that it prepends \newcommand{\Bold} everywhere. Sometimes this is necessary because some objects' latex representations require that. But I think
that's a bit of a bug. It shouldn't be output at all if the \Bold command is not actually being used. And even if it is, the \newcommand declaration should be included, at least optionally, only as
part of some preamble that would go into your latex file or a style. In fact, you can get such a thing with sage.misc.latex.latex_extra_preamble(). Then do import sage.misc.latex_macros; del
sage.misc.latex_macros.sage_configurable_latex_macros[:] and it will shut up forever about \newcommand{\Bold} :) This is not ideal though; there should be a better way...
Iguananaut ( 2018-07-12 20:19:25 +0100 )edit
As you said in your answer, %display latex is supposed to render the latex, so any preamble parts will not print. So it may be a bug to include this command where not needed, but it's a minor bug
since it's not supposed to print. One alternative is to parse the latex to see which parts of the preamble are needed and only include those, but that could get complicated. Feel free to implement it
if you want. Perhaps a better alternative, if we knew in which context %display latex was actually just printing the latex code rather than rendering it, would be to change it to print the value of
latex(input), without the preamble.
John Palmieri ( 2018-07-12 20:41:00 +0100 )edit | {"url":"https://ask.sagemath.org/question/42944/print-latex-normally/?sort=latest","timestamp":"2024-11-08T02:43:26Z","content_type":"application/xhtml+xml","content_length":"68316","record_id":"<urn:uuid:4f0b86b6-3218-4aa3-9cd9-950562f40145>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00298.warc.gz"} |
Glasp on '9.3. Language Models — Dive into Deep Learning 1.0.0-beta0 documentation' | Glasp
9.3. Language Models — Dive into Deep Learning 1.0.0-beta0 documentation
The probability formulae that involve one, two, and three variables are typically referred to as unigram, bigram, and trigram models, respectively. In order to compute the language model, we need to
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What is the current annual growth rate
The compound annual rate of growth is 6%. Calculate that by using the "Rule of 72": Divide 72 by the number of years it takes an investment to double in value, and that is the compound rate of growth
over the period of time applied.
Annual percentage growth rate of GDP at market prices based on constant local currency. Aggregates are based on constant 2010 U.S. dollars. GDP is the sum Gross Domestic Product - Annual Growth
Rate. Gross Domestic Product - Annual Growth RateCurrently selected · Gross Domestic Product - Quarterly Growth CAGR is a useful measure of the growth of your investment over multiple time periods,
especially if the value of your investment has fluctuated widely during the GDP Annual Growth Rate in the United States is expected to be 1.90 percent by the end of this quarter, according to
Trading Economics global macro models and analysts expectations. Looking forward, we estimate GDP Annual Growth Rate in the United States to stand at 2.40 in 12 months time. The average annual growth
rate (AAGR) is the average increase in the value of an individual investment, portfolio, asset, or cash stream over the period of a year. It is calculated by taking the arithmetic mean of a series of
growth rates. The average annual growth rate can be calculated for any investment, Compound annual growth rate (CAGR) is the rate of return that would be required for an investment to grow from its
beginning balance to its ending balance, assuming the profits were reinvested at the end of each year of the investment’s lifespan. The compound annual rate of growth is 6%. Calculate that by using
the "Rule of 72": Divide 72 by the number of years it takes an investment to double in value, and that is the compound rate of growth over the period of time applied.
Created with Highcharts 5.0.7 % Wages and salaries annual growth rate % Source: Wages and salaries annual growth rate % 1949 1969 1989 2009 2015 0 10
GDP Annual Growth Rate in the United States is expected to be 1.90 percent by the end of this quarter, according to Trading Economics global macro models and analysts expectations. Looking forward,
we estimate GDP Annual Growth Rate in the United States to stand at 2.40 in 12 months time. The average annual growth rate (AAGR) is the average increase in the value of an individual investment,
portfolio, asset, or cash stream over the period of a year. It is calculated by taking the arithmetic mean of a series of growth rates. The average annual growth rate can be calculated for any
investment, Compound annual growth rate (CAGR) is the rate of return that would be required for an investment to grow from its beginning balance to its ending balance, assuming the profits were
reinvested at the end of each year of the investment’s lifespan. The compound annual rate of growth is 6%. Calculate that by using the "Rule of 72": Divide 72 by the number of years it takes an
investment to double in value, and that is the compound rate of growth over the period of time applied. The average annual growth rate (AAGR) is the arithmetic mean of a series of growth rates.
Average Annual Growth Rate Formula The average annual growth rate (AAGR) formula is: AAGR = (Growth Rate in Period A + Growth Rate in Period B + Growth Rate in Period C + [Other Periods]) / Number of
Created with Highcharts 5.0.7 % Wages and salaries annual growth rate % Source: Wages and salaries annual growth rate % 1949 1969 1989 2009 2015 0 10
Step 3. Calculate the annual rate of growth To calculate the annual rate of growth, we now need to put our two previous answers together to get to a rate of growth. We take 1.5, and raise it to How
to calculate the Compound Average Growth Rate. Annual Average Growth Rate (AAGR) and Compound Average Growth Rate (CAGR) are great tools to predict growth over multiple periods. Y ou can calculate
the average annual growth rate in Excel by factoring the present and future value of an investment in terms of the periods per year. Investors measure a stock's performance by how much the price the
stock increases over time: The higher the compound annual growth rate, the better the investment. In order to take into consideration the effects of interest compounding, you have to account for the
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other generic value), "Nt" represents the subsequent size, "t" represents the future time in years and "k" is the annual growth rate. 2. Substitute the actual values for the variables.
The annual percentage growth rate is simply the percent growth divided by N, the to calculate future population given current population and a growth rate is:.
12 Sep 2018 The average annual growth rate of Australia's Services sector of 3.4% has outpaced growth in non-services industries of 2.1%. In particular, The .gov means it's official. Federal
government websites often end in .gov or .mil . Before sharing sensitive information, make sure you're on a federal
Investors measure a stock's performance by how much the price the stock increases over time: The higher the compound annual growth rate, the better the investment. In order to take into consideration
the effects of interest compounding, you have to account for the number of years the growth occurred over in order to get an accurate figure for
Calculate the annual rate of growth To calculate the annual rate of growth, we now need to put our two previous answers together to get to a rate of growth. We take 1.5, and raise it to the 1/10th
Population growth (annual %) Derived from total population. Population source: ( 1 ) United Nations Population Division. World Population Prospects: 2019 Revision, ( 2 ) Census reports and other
statistical publications from national statistical offices, ( 3 ) Eurostat: Demographic Statistics, ( 4 ) United Nations Statistical Division
Current Price GDP for 2016 is 65,038 million Kina, this is an increase of 4,899 GDP annual growth rate in Papua New Guinea averaged 5.9 % from 2008 until You can calculate the average annual growth
rate in Excel by factoring the the growth rate for the years can be calculated by dividing the current value by the to the monitoring and tracking of current economic This method calculates
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Growth rates are average annual growth rates in percent, and GDP base publishes current and constant-price GDP numbers for 47 sub-Saharan African. Annual percentage growth rate of GDP at market
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robust principal component analysis
We study algorithms for robust principal component analysis (RPCA) for a partially observed data matrix. The aim is to recover the data matrix as a sum of a low-rank matrix and a sparse matrix so as
to eliminate erratic noise (outliers). This problem is known to be NP-hard in general. A classical way to solve … Read more
Fast Alternating Linearization Methods for Minimizing the Sum of Two Convex Functions
We present in this paper first-order alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic
methods require at most $O(1/\epsilon)$ iterations to obtain an $\epsilon$-optimal solution, while our accelerated (i.e., fast) versions of them require at most $O(1/\sqrt{\epsilon})$ iterations,
with little change in … Read more | {"url":"https://optimization-online.org/tag/robust-principal-component-analysis/","timestamp":"2024-11-04T14:02:13Z","content_type":"text/html","content_length":"86605","record_id":"<urn:uuid:71175e5e-e772-408a-bc8d-4ac60a7a106a>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00443.warc.gz"} |
Scanning Tunneling Spectroscopy on Semiconductors
Interacting electron systems in different dimensions
We use the simple, isotropic and largely parabolic conduction band of InAs to study the influence of electron-disorder and electron-electron interactions on the appearance of the local density of
states. Dimensionality and magnetic field are systematically varied, while electron density and the disorder potential are independently determined which gives full access to the input parameters of
the Schrödinger equation. Since, on the other hand, the local density of states is directly linked to the output of the Schrödinger equation, we get access to the fascinating quantum world of
interacting electrons.
Three-Dimensional Electron Systems
At B=0 T, we find simple Bloch states which are scattered at ionized dopants. The atomic structure of the Bloch states can be reproduced by a calculation within the local density approxiamtion
(FLAPW). The long range part of the scattering states is reproduced within the WKB model.
read more >
In magnetic field, in particular in the extreme quantum limit, we see a transformation into drift states which is not complete up to B=6 T. It is accompanied by the development of a quadratic
Coulomb gap at the Fermi level. read more >
Two-Dimensional Electron Systems
At B=0 T and relatively low disorder, we find a much more complicated and much more strong standing wave pattern than in the three-dimensional electron system. The corrugation increases by a
factor of twenty with respect to the three-dimensional system and is not related to single donors anymore. The data can be qualitatively reproduced within a single-particle calculation showing
that the interaction with disorder is dominant. In simple terms, the patterns reflect the tendency of the two-dimensional electron system to weakly localize. read more >
At larger disorder the system breaks up into droplets, which show s-like and p-like quantum dot states. Percolation at higher energy is observed. read more >
In magnetic field, drift states are formed at low disorder. As expected they run along equipotential lines of the sample. These states are clearly localized at the edge of the Landau levels. The
particularly interesting extended state in the center of the Landau level has been measured for InSb. read more >
The preparation of a 2DES appropriate for STS measurements is described read more >
One-Dimensional Electron Systems
One-dimensional systems containing one or two subbands have been found below charged step edges. Their local density of states shows nearly 100 % corrugation pointing to weakly localized states.
Alignement with the disorder potential is directly observed. Although the system exhibit g-factors as low as 0.7 and the electron-electron interaction strength is strong with respect to disorder,
we do not find any indications for Luttinger properties.
read more >
Zero-Dimensional electron systems
Quantum dots are induced by using the tip as a local gate with respect to the sample. Quantized states are observed as peaks in dI/dV-curves. Since the quantum dot can be moved with the tip
impurities can be palced into the quantum dot and the response of the energy spectrum on the disorder is probed. In magnetic fields the states are identified as spin polarized Landau states. Their
interaction with impurities, in particular the response of the spin splitting to the disorder indicates nicely visualizes the non-locality of the exchange interaction. read more > | {"url":"http://www.nanoscience.de/HTML/research/archive_semiconductor_01.html","timestamp":"2024-11-08T06:07:48Z","content_type":"text/html","content_length":"16568","record_id":"<urn:uuid:c6d5b286-670a-4da2-8621-f9ffca764de1>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00093.warc.gz"} |
Multithreaded MonteCarlo Simmulation to find the value of PI on Colfax Cluster | Intel DevMesh | Chandrasekaran Anirudh Bhardwaj, 01/10/2018
Used Monte Carlo method to find the value of PI. ...learn more
I implemented a python script to perform Monte Carlo simulation to find the value of PI.
I have explained the full method on how to calculate the value of PI in the blog I have linked below: | {"url":"https://devmesh.intel.com/projects/multithreaded-montecarlo-simmulation-to-find-the-value-of-pi-on-colfax-cluster","timestamp":"2024-11-04T10:58:59Z","content_type":"text/html","content_length":"30027","record_id":"<urn:uuid:63d91cc4-a5a9-438c-8cc2-3e5c05684472>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00238.warc.gz"} |
Numerical investigation of laser-driven shock interaction with a deformable particle
A laser-driven shock propagating through an isolated particle embedded in a plastic (CH) target was studied using the radiation-hydrodynamic code FLASH. Preliminary simulations using IONMIX equations
of state (EOS) showed significant differences in the shock Hugoniot of aluminum compared to experimental data in the low-pressure regime [O(10) GPa], resulting in higher streamwise compression and
deformation of an aluminum particle. Hence, a simple modification to the ideal gas EOS was developed and employed to describe the target materials and examine the particle dynamics. The evolution of
the pressure field demonstrated a complex wave interaction, resulting in a highly unsteady particle drag which featured two drag minima due to shock focusing at the rear end of the particle and
rarefaction stretching due to laser shut-off. Although ∼30% lateral expansion and ∼25% streamwise compression were observed, the aluminum particle maintained considerable integrity without
significant distortion. Additional simulations examined the particle response for a range of particle densities, sizes, and acoustic impedances. The results revealed that lighter particles such as
aluminum gained significant momentum, reaching up to ∼96% of the shocked CH's speed, compared to ∼29% for the heavier tungsten particles. Despite the differences seen in the early stage of shock
interaction, particles with varying acoustic impedances ultimately reached the same peak velocity. This identified particle-to-host density ratio is an important factor in determining the inviscid
terminal velocity of the particle. In addition, the modified EOS model presented in this study could be used to approximate solid materials in hydrocodes that lack material strength models.
Shock interactions with non-uniform media are encountered in a variety of physical situations including astrophysical flows,^1–3 multiphase explosives,^4,5 shock propagation through bubbles,^6,7 and
inertial confinement fusion.^8 Shock–particle interactions are a general class of problems that involve shock propagation through media characterized by density or temperature inhomogeneities. The
interaction is highly transient and nonlinear. A complex system of regular and irregular shock-wave reflection, diffraction, and shock focusing may exist, even in an idealized interaction between a
shock and an isolated spherical non-deforming particle.^9
The dynamics of a particle accelerating behind a shock wave is commonly described with its drag history. Several attempts have been made to develop analytical force models that predict the drag on a
particle in an incompressible flow, some of which are reviewed by Michaelides et al.^10 Parmar et al.^11,12 presented a force model for unsteady compressible flows and applied it to a shock
traversing a spherical particle, demonstrating the importance of the inviscid unsteady contribution in capturing the peak force on particles behind the shock wave. Later, Ling et al.^13 used the
force model proposed by Parmar et al. and evaluated the unsteady forces over a range of shock-wave Mach numbers (M), particle Reynolds numbers (Re), and particle-to-host density ratios. The unsteady
contribution had a short-time (i.e., order of acoustic timescale) influence on the particle evolution but captured the peak force during shock passage. Furthermore, its effect on the particle motion
was inversely proportional to the particle-to-host density ratio.
However, these analytical models were formulated under the simplified assumptions of vanishing Re and M on a non-deformable spherical particle. Hence, the interactions of a deformable particle at
finite Re and M must be studied experimentally or computationally due to the unavailability of theoretical solutions in the non-linear regime. In the weak shock regime, Haas and Sturtevant^14
performed shock tube experiments to study the interaction of a shock wave (M<1.3) with a single gaseous inhomogeneity. Multiple wave interactions including transmitted, reflected, and refracted
waves were observed using optical diagnostics. Later, Igra and Takayama,^15 Sun et al.,^9 Martinez et al.,^16 and Bordoloi et al.^17 calculated time-dependent drag on a single particle under shock
loading using experiments. Sun et al.^9 presented time-resolved drag force measurements of a particle made of aluminum alloy subjected to a M=1.22 shock. The peak unsteady force was unaffected by
the viscosity of the surrounding flow and was an order of magnitude larger than the peak steady force. This was an important experimental observation which later reinforced Parmar et al.'s force
model.^11 High-resolution numerical simulations^18–21 have explored pressure evolution around the non-deformable particles and computed the time-dependent drag coefficient during the passage of a
shock over a spherical particle. Sridharan et al.^19 conducted numerical simulations of shock propagation in air initially over a single aluminum particle and demonstrated that the computed drag
coefficient decreases with increasing M.
Particle deformation is important in the context of multiphase explosives and high energy density (HED) systems where shock pressures typically vary from O(10) to O(100) GPa. Experimental data at
these conditions are limited due to difficulty in conducting experiments. Hence, direct numerical simulations were utilized to observe the deformation of metal particles and measure the particle
acceleration imparted by strong shocks.^4,5,22 Zhang et al.^4 showed that metal particles, such as aluminum, beryllium, and magnesium, achieved 60% to 100% of the shocked explosive's velocity and
were severely deformed. It was suggested that the particle drag model should account for the history-dependent particle shape as such deformation could modify the particle acceleration. More
recently, numerical simulations of shock interaction with a deformable aluminum particle in nitromethane were performed for post-shock pressures up to 10GPa.^23,24 Although particle deformation was
modest during shock passage, it had a major influence on the particle drag behavior at later times, highlighting the need to include particle deformation in the force models.^24 At shock pressures
higher than 500GPa, Klein et al.^1 performed experiments on the Nova laser to understand the evolution of a high-density copper sphere embedded in a low-density plastic medium after the passage of a
M=10 shock as a model for shock–cloud interactions. 2D hydrodynamic simulations reproduced the experimental images. The sphere underwent considerable deformation at the initial stage and broke up
at later times. Klein et al. revealed that at very high laser drives, the solid copper sphere was significantly preheated before the arrival of the shock, resulting in the interaction of a strong
shock with a gaseous body rather than a cold solid. They also indirectly observed 3D vortex ring instabilities and demonstrated their role in late-stage cloud destruction, which was later confirmed
by experiments on the OMEGA.^2,3
To accurately simulate material behavior at relatively low sub-eV temperatures, we require models with strength properties in the solid/liquid regime. Such models are not often included in most
radiation-hydrodynamics simulation codes including FLASH,^25,26 which are used as tools to design HED experiments. In addition, high-temperature equation of state (EOS) models used for simulations
become less predictive of the thermodynamic material properties necessary for describing the hydrodynamic processes taking place at low temperatures. For instance, FLASH only uses thermal pressure to
compute the local sound speed of material. This neglects the existence of nonthermal pressure, which determines the behavior of shock-compressed solids. Theoretically, this leads to higher material
compressibility even at low shock pressures.
In this paper, we develop a technique to implement a simple modification of ideal gas EOS for modeling solid materials in hydrocodes that lack material strength. Using this model, we study the
problem of shock–particle interaction in solid targets relevant to laser-driven shock experiments under relatively less extreme conditions ($≤100$ GPa). We aim to study the interaction of a
laser-driven shock with an isolated particle embedded inside a low-density plastic target with an incident shock (IS) pressure high enough to exceed the yield strength but not necessarily melt or
break apart the embedded particle during or after the shock passage, unlike shock–cloud experiments.^1,2
The paper is organized as follows. The target description, numerical methods, and EOS models used in this study are described in Sec. II. The results for a laser-driven shock interacting with an
isolated particle are presented in Sec. III. We examine the pressure field, particle drag coefficient, and particle kinematics. We also discuss the deformation of the particle over time and
quantitatively compare the evolution of particle diameter with that using the IONMIX^27 EOS tables. Finally, we study the dependence of particle size, particle density, and particle acoustic
impedance on the velocity transmission factor. The concluding remarks are given in Sec. IV.
We consider a laser-driven shock propagation through a solid target consisting of a solid metal particle embedded inside a plastic host medium. The laser ablation-driven shock first traverses the
host medium as shown in Fig. 1(a). The nature of shock refraction and shape of the transmitted shock (TS), as seen in Fig. 1(b), depends on parameters, such as the shock-impedance ratio and shock
speed-ratio between materials across the interface.^28,29
A. Target description
The target configuration and temporal laser profile are shown in Figs. 2(a) and 2(b), respectively. The pre-shock material properties and shock Hugoniot parameters used for our simulations are listed
in Table I. The target was driven by an incident laser beam of spot size of 800μm diameter with spatio-temporally uniform intensity distribution. We employed IONMIX EOS tables to model the plastic
ablator and the aluminum heatshield. We used the modified ideal gas EOS to describe plastic (CH), aluminum (Al), titanium (Ti), and tungsten (W), as will be described below. We comment here that
widely used SESAME^30 EOS tables (e.g., SESAME 3720 for Al) were not compatible with the hydrodynamic code at pressure/temperature conditions considered in our work. The simulated SESAME data resided
into the regimes of negative pressures, for which the code predicted nonphysical sound speeds as addressed by Farmakis et al.^31
TABLE I.
. . CH (host) . Al . Ti . W .
ρ[0] (g/cm^3) 1.11 2.70 4.52 19.2
P[0] (GPa) 2.65 15.54 18.11 24.10
γ 3 5 6 13
c[0] (m/s) 2690 5365 4900 4040
. . CH (host) . Al . Ti . W .
ρ[0] (g/cm^3) 1.11 2.70 4.52 19.2
P[0] (GPa) 2.65 15.54 18.11 24.10
γ 3 5 6 13
c[0] (m/s) 2690 5365 4900 4040
B. Governing equations
Plasmas contain electrons, ions, and thermal radiation due to the high-temperature field involved. The electron temperature is not necessarily equal to the ion temperature. Thus, the plasma is
described by the “three temperature” (3T) approximation. The governing equations of the evolution of unmagnetized multi-temperature plasma are discussed in Tzeferacos et al.^26 and also described in
the FLASH code user's guide.^32
The multiphysics radiation-hydrodynamics FLASH code was used to carry out 2D Cartesian simulations to study the laser-driven shock interaction with a deformable particle. The governing equations are
solved on an adaptive mesh refinement (AMR) grid using FLASH's unsplit scheme,^35 a finite-volume Godunov method consisting of a single-step, second order in time, directionally unsplit
multidimensional data reconstruction-evolution algorithm, based on the corner transport upwind (CTU) method.^36 A third order in space reconstruction (piecewise parabolic method^37) is carried out
using a minmod slope limiter along with a flattening technique to treat shocks.^25 The time advanced fluxes are computed using a HLLC Riemann solver^38 with second order accuracy. A
Courant–Friedrichs–Lewy (CFL) number of 0.4 is used for numerical stability. The schematic of the computational domain is shown in Fig. 2(a). Outflow (zero-gradient) boundary conditions were imposed
on all of the domain boundaries. A special treatment to the boundary condition on the particle interface is discussed in Sec. IIC. We have neglected radiation in our study since it had an
insignificant effect on the particle's overall hydrodynamic response.
C. Modified ideal gas EOS as a model for solids
As discussed in Sec. I, the absence of strength models along with the use of high-temperature EOS models in simulations overestimate the material compression and deformation at low temperatures and
pressures. Hence, we sought to mitigate compression and deformation by employing a modified form of ideal gas EOS to model both the host and the particle,
where P, γ, ρ, and ε are total pressure, adiabatic index, mass density, and total internal energy, respectively. We have defined a constant average ionization inside the materials (e.g., a value of 1
is used for the Al particle). We believe that this is a good approximation for a particle that is heated by the TS. The temperature of the particle compressed by the TS is less than 1eV in our
study. The Saha ionization model shows that for Al at solid-like densities, average ionization is close to 1 at sub-eV temperatures.^39 Therefore, our choice of average ionization is justified by the
theoretical model.
For a limiting case of strong shocks in an ideal gas, Rankine–Hugoniot relations imply that compression η and $us−ups$ ratio on the Hugoniot depend only on γ, where u[s] and u[ps] denote the shock
velocity and post-shock particle velocity, respectively,
Therefore, we adjusted γ to match the experimental Hugoniot data.^34 In Figs. 3(a) and 3(b), we compare the Hugoniots generated with our modified EOS against those generated by IONMIX EOS. Our
simulated Hugoniots agree well with the experiment, and in Fig. 3(b), the compression in Al shifted from the IONMIX-predicted value of ∼3.8 to ∼1.6 at 85GPa. The stiffened gas EOS is another widely
used model to describe liquids and solids under high pressures.^33,40,41 However, Fig. 3(a) shows that this model deviates from the experimental curve as the shock strength in the material increases.
Conversely, the modified ideal gas EOS model is seen to perform better for a broader range of shock strength. In addition, Fig. 3(c) shows that the simulated $us−ups$ Hugoniots of CH, Ti, and W
modeled using this technique agree well with their respective experimental Hugoniot curve.
Once γ is determined, the initial pressure is chosen using Eq. (4) to match the material sound speed, given as follows:
where P[0] and ρ[0] are initial the pressure and density of the material, respectively. We should note that, to appropriately model c[0], both the particle and the host could not be kept at
equilibrium pressure at the initial state. However, to delay particle expansion until the arrival of shock, we numerically “freeze” (i.e., apply reflecting boundary condition at the particle
interface) the grid cells that lie inside the particle. Once the shock meets the leading edge of the particle, we “unfreeze” those grid cells. This generates a pressure gradient across the interface
and results in an increase in the particle width as discussed in Sec. IIIC.
The modified ideal gas EOS model is fully based on adjusting γ and c[0] to match the Hugoniot data from the LASL database.^34 However, due to the unavailability of experimental data in the database
at much higher pressures than that presented in this work, we were not able to model our materials—Al, CH, Ti, and W—at such conditions. For example, the single-shock Hugoniot data for Al in the
database spans up to ∼120GPa. Figure 3(b) shows that our model provides excellent agreement with the experimental data up to ∼120GPa. Ju et al.^42 showed that the shock melting of Al occurs between
93 and 140GPa. As such, we expect our model to be suitable for modeling materials at solid/liquid regimes (i.e., temperature and pressure of sub-eVs and few hundreds of GPa, respectively). Beyond
these conditions, we should also account for other important physics in the EOS model such as temperature-dependent average ionization^39 and radiation effects, which are not accounted in the current
model. We also expect the preheat effects to be negligible in our study. As shown in Nilsen et al.,^43 preheating is not an issue in low-drive experiments at pressures less than 130 Mbar. The shock
pressure in our CH is ∼0.6 Mbar; hence, we are certainly in a low-drive regime where preheat should be negligible.
D. Quantities of interest
Time scales relevant to a shock–particle interaction problem are discussed by Mehta et al.^44 The time scale for a laser-driven planar shock passing through an isolated particle of initial diameter d
[p] is defined as follows:
The non-dimensional time t′ is defined as $t′=(t−ta)/τs$, where t[a] is the time the incident shock arrives at the leading edge of the particle.
Particle position ($xp¯$), particle velocity ($up¯$), and particle pressure ($Pp¯$) were numerically computed as mass-averaged quantities defined as follows:
$ϕp¯=∫Vpρpϕp dV∫Vpρp dV,$
where $ϕp¯$ refers to any field variable, such as particle pressure, and V[p] is the volume of the particle.
Similarly, the inviscid force on a particle is defined as follows:
where P is the pressure acting on a surface of unit normal $n→$, and A is the cross-sectional area of the particle.
The total inviscid force is computed in the streamwise direction as follows:
where k is the index of each cell-face that makes up the surface of the particle and $j→$ is the unit normal in the streamwise direction.
Finally, the unsteady drag coefficient for the particle is computed as follows:
where ρ[ps] is the post-shock host density measured by probing a location that is $∼3dp$ distance away to the left of the particle.
E. Grid convergence and verification
A grid convergence test was performed for the case of an Al particle embedded in CH at a shock pressure of 55GPa by increasing the refinement levels on the AMR grid. The number of grid points across
d[p] is denoted by N. The grid size was chosen by varying $dp/N$ and C[D] was computed using Eqs. (7)–(9) at various grid sizes as shown in Fig. 4(a). The case with 70 points across the particle
diameter captured the peak drag to less than 2% of the finest resolution tested. Hence, we performed our simulations at this grid refinement.
To verify our numerical schemes, we simulated a 6GPa shock propagating in nitromethane over an embedded Al particle, as studied by Sridharan et al.^24Figure 4(b) compares the time histories of C[D]
obtained from Sridharan et al.'s simulations and theoretical model with our numerical results. Although the theoretical model predicted a higher peak drag value than in simulations, both of the
simulations showed good agreement in capturing the drag minimum due to shock focusing.^9 The differences between the theoretical model and the simulations could be attributed to the zero deformation
assumption made by the model.^24
A. Flowfield
We show pressure contours in Fig. 5 to highlight the flow dynamics as the shock propagates through the particle. The incident shock (IS) is planar before it reaches the particle, as seen in Fig. 5(a)
. As the shock meets the front of the particle, it experiences an impedance mismatch at the interface. Hence, a reflected shock (RS) travels back into the compressed CH and a transmitted shock (TS)
travels downstream. In Fig. 5(b), the TS is planar due to the TS traveling at equal speed to the unrefracted shock outside. As the TS reaches the downstream end of the particle, it experiences an
impedance mismatch at the interface. This generates a TS into the downstream CH and a reflected expansion wave back into the particle. In the CH, the diffracted shocks around the particle meet toward
the center of the downstream end of the particle and further strengthen the shock. This phenomenon of shock focusing has been discussed in past works.^9,45 As this strong shock reflects back into the
particle, it competes with the reflected expansion to produce temporal variations in pressure inside the particle [Figs. 5(c) and 5(d)]. Such pressure variations are plotted as mass-averaged particle
pressure in Fig. 6(a) from t′ = 1ns to t′ = 2.8ns. When the laser is turned off at $t′=1.9$, rarefaction waves start to catch up to the flow along the downstream direction. Once these waves meet
the particle, the particle pressure monotonically decreases after t′ = 2.8 due to the influence of rarefaction stretching^46 on the compressed particle. Hence, the wave interactions are highly
unsteady, thereby not allowing the particle pressure to equilibrate within a few τ[s].
B. Streamwise force on the particle
The time history of the streamwise force on the Al particle is presented in Fig. 6(b). C[D] quickly rises to its maximum value of 4.1 at $t′=0.55$ as the shock passes over the particle. As time
progresses, C[D] decreases and becomes negative due to shock focusing, indicating that the pressure downstream of the particle is larger than the pressure in front of it. The drag reaches a minimum
value of $CD=−0.9$ at $t′=1.8$. After that, the pressure inside and around the particle tends to equilibrate making the drag coefficient positive again. Once the laser is turned off at $t′=1.9$,
the compressed CH upstream of the particle starts to decompress due to rarefaction stretching. This decelerates the particle and results in another drag minimum at $t′=3.2$.
These results confirm the importance of unsteady forces to the bulk motion of the particle. If we compare C[D] from our simulations to that from previous works,^19,22,24 we find qualitative agreement
of the drag history which features a peak drag coefficient and the first drag minimum due to shock focusing. However, due to the propagation of a rarefaction from the ablation front in our
laser-driven system, the drag coefficient features a second minimum associated with laser shut-off. This indicates that rarefaction stretching will contribute to the unsteady drag coefficient as time
The mass-averaged particle velocity is shown in Fig. 6(c). The velocity history displays three distinct phases: an acceleration phase that occurs over a time scale of τ[s], followed by a phase where
velocities tend to level off, and finally a deceleration phase, where the particle velocities decrease significantly due to rarefaction stretching.
C. Deformation
We observed roughly fourfold streamwise compression of the particle along with significant deformation, as shown in Fig. 7(a). By applying the modified ideal gas EOS as described in Sec. II, we were
able to mitigate compression and deformation of the particle compared to that resulting from using IONMIX EOS.
As the IS propagates through the particle, the particle is compressed in the streamwise direction. The temporal variation of the length (i.e., streamwise) and width (i.e., transverse) of the evolving
particle interface is plotted in Fig. 7(b) to quantitatively characterize the particle evolution. The length of the evolving particle decreases quickly at the early stages due to shock compression
and reaches a minimum value. At the intermediate stages from t′ = 1 to 3, the length gradually decreases. In tandem, particle width increases at the early stages. This is caused by limitations in
keeping both the particle and the host at equilibrium pressure, as discussed in Sec. II. During the intermediate stages, we observe modest growth in particle width. At the later stages, rarefaction
catches up to the flow ahead, causing an increase in both the length and the width of the particle. We should remark here that although the particle length reduces by ∼25% and width increases by ∼30%
by $t′=3.5$, the particle remains intact without any rollups or interface distortion.
If we observe the interface evolution of the particle modeled with IONMIX EOS, we see a greater reduction in particle length due to shock compression which is inconsistent with the shock Hugoniot
data shown in Fig. 3(b). This is caused due to material being modeled with lower γ (i.e., lower sound speed) that resulted in ∼4× streamwise compression of the particle as compared to ∼1.5× with the
modified model. We should also note that the width of the particle modeled with the IONMIX EOS slightly reduces during shock propagation. This is due to the unrefracted shock in CH traveling faster
than the TS inside the particle. This creates a pressure difference across the particle interface from the shocked CH into the inside of the unshocked particle in the lateral direction as the TS
travels through the particle. Such effects were mitigated using the modified EOS model for materials. Figure 3(b) shows how the simulated Hugoniot compression of Al particle using the modified model
matches the experimental data.
D. Effect of particle size and particle material density
Figure 8(a) provides the time histories of mass-averaged particle velocity for three metal particles: Al, Ti, and W. The lighter Al particle accelerates more rapidly and to a higher maximum velocity
than Ti and W particles during the shock–particle interaction. For instance, the Al particle accelerates to a velocity of ∼4μm/ns that corresponds to ∼91% of the surrounding shocked fluid velocity.
Ling et al.,^13 from their analytical study, showed that the velocity gained by the particle from only the pressure gradient force scales inversely with particle-to-host density ratio. In the
previous works, for the case of Al particles in air, the particle-to-host density ratio is $O(103)$. This results in lower velocity gain in the particle from the shocked air. In our study, the
particle-to-host density ratio is O(1). Hence, the gain in particle velocity from the medium during and after the shock interaction was much larger than that in the case of Al in air. To quantify the
velocity gain for different cases studied here, we calculate the velocity transmission factor α as the ratio of the peak mass-averaged particle velocity after the initial shock interaction to the
shocked surrounding velocity. In Table II, α ranges from 0.29 to 0.96 for particles of different density.
TABLE II.
. ρ[0] (g/cm^3) . d[p] (μm) . α .
Al $(γ=5)$ 2.7 30 0.960
Al $(γ=5)$ 2.7 50 0.914
Al $(γ=5)$ 2.7 70 0.845
Al $(γ=10)$ 2.7 50 0.915
Al $(γ=20)$ 2.7 50 0.914
Al $(γ=50)$ 2.7 50 0.914
Ti $(γ=6)$ 4.52 50 0.698
W $(γ=13)$ 19.2 50 0.290
. ρ[0] (g/cm^3) . d[p] (μm) . α .
Al $(γ=5)$ 2.7 30 0.960
Al $(γ=5)$ 2.7 50 0.914
Al $(γ=5)$ 2.7 70 0.845
Al $(γ=10)$ 2.7 50 0.915
Al $(γ=20)$ 2.7 50 0.914
Al $(γ=50)$ 2.7 50 0.914
Ti $(γ=6)$ 4.52 50 0.698
W $(γ=13)$ 19.2 50 0.290
For the same initial particle density, we observe in Fig. 8(b) that the smaller particle accelerates quickly to reach the peak velocity.
E. Effect of particle acoustic impedance
We discuss the dependence of the velocity transmission on the particle acoustic impedance $γρP$. The calculations were performed with particles of same initial density and pressure but different
acoustic impedances by modifying γ. Figure 9(a) compares the peak velocity attained by particles as a function of γ immediately after the emergence of the TS at the trailing edge of the particle.
Note that the maximum particle velocity (black cross) decreases with increasing acoustic impedance. In particular, a fourfold increase in γ (i.e., γ = 20) results in ∼40% reduction in the peak
particle velocity. Despite the differences during early times ($t′≤1$), however, Fig. 9(b) shows that the mass-averaged particle velocities attained long after the shock interaction look identical,
resulting in a similar α as shown in Table II. To study an effect of γ on the shock velocity, Fig. 10 plots density field showing the shape of the TS inside the particles. For all the cases studied,
the TS is convex in shape and runs ahead of the unrefracted IS. Furthermore, the TS propagates faster but imparts lower material velocity in particles with increasing γ [see also Fig. 9(a)]. These
observations, more importantly, provide evidence of how modification in γ controlled the stiffness of the material in our simulations. However, its effect to the bulk particle motion is seen for a
very short time [i.e., O(τ[s])]. Eventually, particles of the same density but varying acoustic impedance attain the same peak velocity. Therefore, once the shock completely traverses the particle,
the particle-to-host density ratio determines the inviscid peak terminal velocity of the particle.
A numerical investigation of laser-driven shock propagation through an isolated particle embedded in a plastic target is presented using the radiation-hydrodynamics code FLASH. The predicted
evolution of the particle modeled with IONMIX EOS did not reproduce the experimental shock Hugoniot. Hence, we developed a technique to implement a modified form of ideal gas EOS to model the
materials and study the dynamics of the embedded particle. The simulated shock Hugoniots of multiple materials, modeled using this technique, compared well with experimental data. We then examined
the flowfield and observed that the wave interactions were highly unsteady to allow the particle pressure to equilibrate within a few τ[s]. We also demonstrated that the unsteady drag coefficient for
the particle featured a peak drag due to an unsteady interaction with the transmitted shock and a drag minimum due to shock focusing at the rear end of the particle. However, unlike previous studies
performed without laser drives, the particle drag coefficient featured a second minimum due to rarefaction stretching associated with laser shut-off. Furthermore, to quantitatively characterize the
particle deformation, we plotted temporal variation of length and width of the deforming particle. Although a ∼30% lateral expansion and ∼25% streamwise compression is observed, the particle
maintained integrity without any rollups and significant interface distortion. We then conducted numerous simulations and investigated the particle response for a range of particle densities, sizes,
and acoustic impedances. The results revealed that lighter particles, such as Al, gained significant momentum up to 96% from the shocked CH, compared to 29% in the case of heavier W. Finally, we
studied the effect of particle acoustic impedance on the bulk particle response. Despite differences observed in the early stage of shock interaction, the acoustic impedance did not have an effect on
the peak particle velocity. This also identified particle-to-host density ratio as a dominant factor in determining the inviscid terminal velocity of the particle.
Time scale analysis in previous works have pointed out that the shock–particle interaction time scale could be of the same order as the viscous time scale, particularly for condensed-matter systems.^
4 Hence, viscous effects coupled with rarefaction stretching effect could be important for particle drag calculation in the intermediate to later stages of shock interaction. To this end, future work
should include viscous models in the simulations to accurately calculate the particle response in such systems. Finally, preheat effects should be negligible due to relatively low-drive conditions
studied in this work. Nevertheless, we hope to extend our modified EOS model in the future toward a much higher laser drive and provide a temperature-dependent γ to the model.
We thank Bertrand Rollin for valuable advice during the early stages of this work. We also thank three anonymous reviewers for their thorough reading and helpful suggestions. This work was performed
under the auspices of the U.S. Department of Energy under Grant No. DE-SC0019329 within the joint HEDLP program and was supported by the Laboratory Basic Sciences program administered by UR/LLE for
DOE/NNSA. H.A. was also supported by U.S. DOE Grant Nos. DE-SC0014318 and DE-SC0020229, NSF Grant Nos. PHY-2020249 and OCE-2123496, U.S. NASA Grant No. 80NSSC18K0772, and U.S. NNSA Grant Nos.
DE-NA0003856 and DE-NA0003914. J.K.S. was also supported by NSF Grant No. PHY-2020249 and NNSA Grant No. DE-NA0003914. The software used in this work was developed in part by the DOE NNSA- and DOE
Office of Science-supported Flash Center for Computational Science at the University of Chicago and the University of Rochester.
Conflict of Interest
The authors have no conflicts to disclose.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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There is science to be done, there is research to be run…
DMZ · October 17, 2007 at 7:27 pm · Filed Under
… on the people who are still alive*
M’s hitting since moving into Safeco Field, charted against league averages
Red is batting average
The other one is OBP
The top one is SLG
And on the other side, some pitching indicators.
Walk rate per nine innings (lower is better)
HR allowed rate per nine innings (lower is better)
Stirkeout rate per nine innings (higher is better)
* Portalllllllllllllllll!!!!
34 Responses to “There is science to be done, there is research to be run…”
1. argh on October 17th, 2007 7:49 pm
Where do these variances fall in in terms of our division/league opponents?
2. BP on October 17th, 2007 8:36 pm
All these graphs really make me miss 2001.
3. scraps on October 17th, 2007 8:42 pm
Edward Tufte — The Visual Display of Quantitative Information — would note that graphs like this that don’t begin at zero exaggerate the effect of the data, the worst offender here being
Strikeout Rate, where 2005 is made to appear a much bigger outlier than it is.
4. DMZ on October 17th, 2007 9:18 pm
I guess. My counter would be that they looked really crappy at 0, and the bounds are set more or less at the outlier points.
5. msb on October 17th, 2007 9:23 pm
6. msb on October 17th, 2007 9:30 pm
I don’t know what the statistical odds of this were, but Kranitz hired on with the O’s
7. Mat on October 17th, 2007 9:47 pm
Edward Tufte — The Visual Display of Quantitative Information — would note that graphs like this that don’t begin at zero exaggerate the effect of the data, the worst offender here being
Strikeout Rate, where 2005 is made to appear a much bigger outlier than it is.
I’ve heard this ridiculousness before about graphs in general, and it’s just that–ridiculous. All including the origin does is to waste a bunch of perfectly useful space on the graph. The
vertical axis is labelled perfectly well in each graph–there’s nothing disingenuous about the data presentation. If the viewer has a reason to believe that 0.6 K/9 is a large variance, then you
would even be doing him a disservice by including the origin and making all of the variations look tiny.
Texas had 6.1 K/9 this year, which was more or less worst in the AL. That was only 0.5 K/9 below league average (according to me eyeballing DMZ’s graph.) I would argue that 0.6 K/9 is in fact a
significant variation for a team’s season total. Over 1400 IP, that’s something like 93 strikeouts and is basically the difference between average and the worst.
If you really wanted to properly account for what a “significant” variance from the league average was, you could add some one-sigma error bars around the league average to give the viewer an
idea on what the variance is, but artificially including the origin is counterproductive.
8. Mr. Egaas on October 17th, 2007 9:52 pm
Tufte would also dim the background hash marks to a very faint gray.
Took an Information Visualization class, very interesting.
9. fetish on October 17th, 2007 9:57 pm
You mean the in each of the past four years, the Mariners have had higher than league average walks?
Oh, that’s the pitchers.
10. DMZ on October 17th, 2007 10:03 pm
Tweaked it to make that clearer.
11. Chris88 on October 17th, 2007 11:03 pm
7 – I agree completely. 1/2 a strikeout less here, a walk more there and you’ve got the difference between San Diego’s pitching and Tampa Bay’s. Its never very much different. The end result is a
few small differences happening over and over during the course of a season adding up to a big difference.
12. bermanator on October 18th, 2007 5:10 am
I just took the Tufte one-day seminar on Monday!
All including the origin does is to waste a bunch of perfectly useful space on the graph. The vertical axis is labelled perfectly well in each graph–there’s nothing disingenuous about the
data presentation. If the viewer has a reason to believe that 0.6 K/9 is a large variance, then you would even be doing him a disservice by including the origin and making all of the variations
look tiny.
His point is that not starting from the orgin distorts the data by making small variances seem enormous. You can manipulate how important a small variance can be by skewing the vertical axis,
which would be a dishonest way or presenting the data.
Just to be clear, I don’t think that is what DMZ is doing with these graphs — it’s such a small space to work with that I don’t even know how the graphs will work starting from zero — but Tufte
would probably argue that the data should then be presented in tables or as a graph handed out to the audience on bigger paper instead (so DMZ, we’ll all send you our addresses and you can ship
them to us).
Tufte used sports data at least twice in his presentation that I can recall. He said a few times that we should look at how information is presented in the Sports or Mutual Funds section rather
than how the PowerPoint Templates use tables as a way of effectively displaying data, and he also had a graphic (I think in his new book) showing the baseball standings with varying space between
the teams based on games above or below .500, so the AL East of a few years ago looked like this:
New York
Tampa Bay
I don’t know that I agree with him on everything, but it’s an interesting opportunity to think about different ways of presenting data effectively.
13. S-Mac on October 18th, 2007 7:57 am
Derek, I wish I could appreciate these graphs, but I’m still too distraught over what happened to my Weighted Companion Cube.
14. tgf on October 18th, 2007 7:58 am
His point is that not starting from the orgin distorts the data by making small variances seem enormous.
Except that the axes are labeled, so the variances are exactly the amounts presented. If people look at graphs without looking at the axes to see the magnitude of the changes, they are misleading
themselves, not being mislead by the presenter.
but Tufte would probably argue that the data should then be presented in tables or as a graph handed out to the audience on bigger paper instead
Not sure what field Tufte is in but this is totally impractical, at least in my field. Looking through a table takes time and annoys the audience. Handing out paper to the audience? No thanks.
15. DMZ on October 18th, 2007 7:58 am
Sure, and I’m a Tufte guy and all, but:
a) a major league average staff will have a K rate of ~6/9 IP, an astoundingly good one will have a K rate of ~7.5 or 8/9 IP, and a truly sucky one might be able to get down to 4.5… but probably
not. The actual range of outcomes is 4.5-8, not 0-8. Small variations aren’t exaggerated – they’re large variations.
b) it looked like crap using 0, and the whole thing’s supposed to be about the useful display of information, right?
16. DMZ on October 18th, 2007 8:00 am
I’m still too distraught over what happened to my Weighted Companion Cube.
It’s no use trying to pretend “something happened” as if you weren’t responsible. We know what happened. You did it.
17. bermanator on October 18th, 2007 8:08 am
I think if Seattle really did replace the Moose with a Weighted Companion Cube, they might be pleasantly surprised at the increase in merchandise sales.
18. DMZ on October 18th, 2007 8:22 am
Especially a talking weighted companion cube, that gives you advice, and you can sing songs with it, just like a real weighted companion cube… like the one S-Mac killed.
19. Mr. Egaas on October 18th, 2007 8:59 am
Off Tufte, back to the team — On the plus side, the offense is trending up despite one of the highest paid bats on the team being one of the worst players, a potential star is ready to come into
his own, and there are positive adjustments to be made.
20. msb on October 18th, 2007 9:50 am
ahem. “Stirkeout rate per nine innings (higher is better)”
aside from that, how about a graph showing LOB … that would be a scary sight over the last few years.
21. Alaskan on October 18th, 2007 10:18 am
19: Amen. In addition, we’re playing in the pitcher’s park, right? So compared to league averages, we’re doing pretty well, and if Jones starts, maybe we can do even better.
This graphs are not nearly as depressing, at least in regards to 2007, as I expected them to be. Now 2005… that was bad.
22. bermanator on October 18th, 2007 10:19 am
Perhaps barely clinging to the topic [nope]
23. Alaskan on October 18th, 2007 10:20 am
Obviously, “This graphs” should be “These graphs.” Wow, that’s embarrassing. Note to self: re-read before posting.
24. Evan on October 18th, 2007 10:25 am
You can manipulate how important a small variance can be by skewing the vertical axis, which would be a dishonest way or presenting the data.
If the values and units are clearly labelled on the axes, there’s nothing dishonest about it at all. Just because people are bad at interpreting data unless you spell everything out for them
doesn’t mean you’re misleading them by not doing it. You’re just letting them make their own mistakes – that’s their fault, not yours.
25. S-Mac on October 18th, 2007 10:27 am
The biggest advantage of the Weighted Companion Cube over the Moose? It will never stab you.
26. msb on October 18th, 2007 10:30 am
Perhaps barely clinging to the topic because of the pitching graphs … did Seattle actually offer Rick Kranitz the job as pitching coach, or did they just interview him?
it doesn’t sound like they’ve offered it to anyone yet; they had an interview, he chose Baltimore (which apparently was not unexpected due to his prior relationships with MacPhail & Trembley)
27. Evan on October 18th, 2007 10:34 am
how about a graph showing LOB … that would be a scary sight over the last few years.
Shouldn’t be too bad. To leave guys on base you have to get them there, first, and we haven’t been very good at that.
Actually, looking at the first graph, the our slugging lagged a lot farther behind league average than our OBP did in 2004, so the LOB would probably be terrible there.
28. heyoka on October 18th, 2007 10:38 am
Actually, I’ve seen some dishonest, clearly labelled graphs before. I had a annual stock report in which the y-axis for profits were labelled in the 10 millions, while debt was labelled in the
billions. For the casual reader (most stock holders breeze through these things), debt LOOKED really small on its graph, while profits completely consumed its graph. The debts were in fact much
larger than the profits.
What makes the graphs presented on this site not dishonest is the fact that they are relative. In this case the origin would be correctly identified as the average of the league averages, not the
zero. A graph that included the zero would incorrectly make the data appear to be less varied.
29. Trev on October 18th, 2007 10:39 am
What would these graphs look like with park adjustments?
30. scraps on October 18th, 2007 10:39 am
Derek, I take your point. I think I agree that if your graph is bounded by the actual extremes that ever occur in the data, you’re basically presenting a true graphic picture.
tgf and Evan, I disagree with the general point about labeled axes, particularly arguments like “If people look at graphs without looking at the axes to see the magnitude of the changes, they are
misleading themselves” and “Just because people are bad at interpreting data unless you spell everything out for them doesn’t mean you’re misleading them by not doing it”. The whole point of
graphic presentation is to simplify and to give a true picture in a glance. If people have to look closely at the labeling and adjust the picture in their minds accordingly, the graph is a
distortion; it has created an untrue mind-picture that needs to be fixed with closer inspection. If you’re going to say “well, people shouldn’t be careless”, you might as well just give the raw
data. (Again, I’m not arguing with Derek’s presentation here.)
31. bermanator on October 18th, 2007 10:41 am
32. scraps on October 18th, 2007 10:41 am
28 makes the same point in fewer words.
33. heyoka on October 18th, 2007 11:02 am
So the pitchers aren’t giving up homeruns, but other than that they are doing every thing else to prevent outs and increase opponent runs – aided and abetted mightily by our old nemesis, the
The mariners had 9 more wins than their pythagorean w/l.
Last year’s success is not sustainable – it is a clear fluke.
34. Mat on October 18th, 2007 12:58 pm
The whole point of graphic presentation is to simplify and to give a true picture in a glance.
This is the point which should be emphasized then, not silly rules like “zero must be included.” Concentrating on providing a true picture can lead to better graphical presentation of data, but
concentrating on rules of thumb is a poor substitute for actively thinking about how the data ought to be presented, and can be counterproductive in many cases.
You must be logged in to post a comment. | {"url":"http://www.ussmariner.com/2007/10/17/there-is-science-to-be-done-there-is-research-to-be-run/","timestamp":"2024-11-14T23:52:38Z","content_type":"application/xhtml+xml","content_length":"60987","record_id":"<urn:uuid:4917133f-de9c-4c9a-9f0e-3a8462b6a38d>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00090.warc.gz"} |
Economics, Accounting & Business: Post your doubts here!
Reaction score
Re: eco
student92 said:
hi can anybody explain this mcq to me.
thank you
the correct answer is d
I'll give a try here...
the marginal value of smoke nuisance is the marginal external cost of production
specific tax will be imposed to cover the external cost produced and after imposing the tax the price will equal MPC+MEC (marginal external cost)
the price of the product is 2.00, hence choose values from the table that will add up to 2.00..you will see that at output 92, mpc is 1.20 and mec is 0.80... thus the tax is 0.80...
Reaction score
external cost = social cost - private cost........(i)
For 24th item,
substituting social cost in (i),
18 = 322 -private cost
Therefore private cost of producing 24th item (PC1)= 304
Private cost of producing 23rd item(PC2) = 316-16 = 300
So additional cost to a firm(Change in private cost) = PC1-PC2 = 304-300 = 4
Reaction score
external cost = social cost - private cost........(i)
For 24th item,
substituting social cost in (i),
18 = 322 -private cost
Therefore private cost of producing 24th item (PC1)= 304
Private cost of producing 23rd item(PC2) = 316-16 = 300
So additional cost to a firm(Change in private cost) = PC1-PC2 = 304-300 = 4
Reaction score
accounts p2
freds pricing policy is cost plus 75%
so this means mark up is 75%, we need to calculate closing stock so, we will convert markup into margin
mark up is 3/4 so margin is 3/4+3 = 3/7
sales multiply by margin will give you cost of sales, right?
but the cie is not getting the same answer as mine, where am i going wrong help please
http://www.xtremepapers.com/CIE/Interna ... 4_qp_2.pdf http://www.xtremepapers.com/CIE/Interna ... 4_ms_2.pdf
PLEASE ....PLEASE HELP IN THE FOLLOWING MCQ'S ...... I NEED THE ANSWERS AS SOON AS POSSIBLE .....HURRYYYYY
ACCOUNTING NOVEMBER 2007 Q30
ACCOUNTING JUNE 2002 Q26
ACCOUNTING NOVEMBER 2002 Q20 AND 24.
please i need these very quickly and with workings for better understanding
thanking you in anticipation
PLEASE I POSTED THIS YESTERDAY :x
PLEASE ANYONE KNOW HOW TO SOLVE THEM???
Reaction score
nitish0708 said:
19 In a closed economy with no government C = 30 + 0.7Y, where C is consumption and Y is
The equilibrium level of income is 300.
What is the level of investment?
A 60 B 100 C 210 D 270
Help me how to get the answer
I HAVE ALSO POSTED SOME QUESTIONS ................PLEASE HELP ME IN THEM
Reaction score
saimaiftikhar92 said:
I HAVE ALSO POSTED SOME QUESTIONS ................PLEASE HELP ME IN THEM
hi, I would've liked to help but I didn't have accounting during my a-levels... so sorry there
X, Y and Z all possess 4 clocks each. They can 'supply' a maximum of four clocks, i.e, Qs = 4 ( and so Qd = 4) -> Equilibrium Price = 2
At "Equilibrium price 2", 'Demand' of X is 2, 'Demand' of Y is 4 and 'Demand' of Z is 6.
X initially has 4 clocks but now his demand is 2. So he would want to 'sell' those extra 2 clocks.
Y has nothing to do with anything. His demand is 4 and he already has 4.
Z initially has 4 but now his demand is 6. He'd want to buy those extra 2.
X the seller, Z the buyer. Option D.
Hope you find this helpful.
How do you come to know that the supply curve is horizontal?
That's not a supply curve. I just connected those points to compare them with the Price 2.
Makes sense, albeit I'll need to convince myself further. Thank you.
No problem.
$2 is E.P. At $2, check the demands for X, Y and Z. | {"url":"https://xtremepape.rs/threads/economics-accounting-business-post-your-doubts-here.10459/page-3","timestamp":"2024-11-07T12:22:02Z","content_type":"text/html","content_length":"139355","record_id":"<urn:uuid:c85866b1-d49f-444c-b9e9-d344f5c5009e>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00511.warc.gz"} |
3.3 Shape, Center, Spread, and Weird Things
Course Outline
• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
□ 3.3 Shape, Center, Spread, and Weird Things
• segmentChapter 4 - Explaining Variation
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Models with a Quantitative Explanatory Variable
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources
list High School / Statistics and Data Science I (AB)
3.3 Shape, Center, Spread, and Weird Things
The very first thing you should always do when analyzing data is to examine the distributions of your variables. If you skip this step, and go directly to the application of more complex statistical
procedures, you do so at your own peril. Histograms are a key tool for examining distributions of variables. We will learn some others, too. But first, let’s see what we can learn from histograms.
What do we look for when we explore distributions of a variable? In general, we look for four things: shape, center, spread, and weird things.
Weird Things
Let’s start with weird things. What do we mean by weird things? Let’s go back to the Fingers data frame, where we collected a sample of students’ thumb lengths (among other variables). This time,
however, we are going to use an earlier version of the data frame, called FingersMessy.
Fingers is a cleaned up version of FingersMessy. If you look at the histogram below, of the variable Thumb in FingersMessy, you may start to get a sense of what might have needed to be cleaned up in
the original data.
gf_dhistogram( ~ Thumb, data = FingersMessy, fill = "orange", color = "slategray", binwidth = 4)
Whereas most of the students’ thumb lengths appear to be clustered around a point just below 60 millimeters, there is another small clump who seem to have much smaller thumbs—like one tenth the size!
This doesn’t fit with what we know about the world. There aren’t two kinds of people, those with regular thumbs and those with super-short thumbs. Thumbs should be more continuously distributed, with
most people having thumbs of average length, and then some a little longer and some a little shorter.
This is exactly what we mean when we say “look for weird things.” One possibility is that some of the students didn’t follow instructions, and measured their thumbs in centimeters (or maybe even
inches) instead of millimeters. Given what we know about students, this seems like a reasonable theory; they don’t always listen to instructions.
The point here, though, is this: if we hadn’t looked at the distribution, we would not have noticed this oddity and might have drawn some erroneous conclusions.
Shape, Center, and Spread
Once we find something weird we must deal with it. In this case, we decided to filter in only the data from students with thumb lengths of at least 20 mm. We saved this filtered data frame under a
new name, Fingers, which is the data frame you have come to love. We’ll go back to using that one.
Apart from weird things, the other features of distributions we want to explore are shape, center, and spread. Each of these characteristics tells us something about the variable we are looking at.
Let’s go back to the Fingers data frame, no longer containing weirdness, and make a histogram of the variable Thumb.
Go ahead and make a density histogram of Thumb in the DataCamp window below.
require(coursekata) # Make a density histogram of Thumb in the Fingers data frame # Don't use any custom coloring gf_dhistogram(~Thumb, data = Fingers) ex() %>% check_function("gf_dhistogram") %>%
check_result() %>% check_equal()
CK Code: ch3-8
Take a look at the histogram of Thumb. To examine shape, you might try squinting your eyes and looking at the histogram as a solid, smooth object rather than a bunch of skinny bars. This can help
give us a sense of the overall shape of the distribution.
R can help you see the shape by overlaying a smooth shape over your histogram, which is called a smooth density plot. We can just chain on the function gf_density() to our histogram, as in the code
gf_dhistogram( ~ Thumb, data = Fingers, fill = "orange", color = "slategray") %>%
You can run this in your sandbox if you want. Or, you can just look at what we got when we ran the code. Note that when we add gf_density() to the plot using the %>% notation, we don’t need to fill
in the arguments in the (). R just uses the same ones from the previous command.
Statisticians describe the shapes of distributions using a few key features. Distributions can be symmetrical, or they can be skewed. If they are skewed, it can be to the left (the skinny longer tail
is on the left) or to the right (the skinny longer tail is on the right). The distribution above has a slight skew to the right.
Distributions also could be uniform (meaning the number of observations is evenly distributed across the possible scores); they could be unimodal (meaning that most scores are clustered together
around one part of the measurement scale); or they could be bimodal (having two clear clumps of scores around two parts of the measurement scale, with few in the middle).
Distributions that have a bell-shape (unimodal, symmetrical, scores mostly clumped in the center, few scores far away from center) are often called normal distributions. This is a common shape.
Usually, distributions are kind of lumpy and jagged, so many of these features should be thought of with the word “roughly” in front of them. So even if a distribution doesn’t have exactly the same
number of observations across all possible scores—but has roughly the same number—we could still call that distribution uniform. If you look at the density plot above, you might see two lumps near
the center (near the peak). Some people might think this is a bimodal distribution. But statisticians would consider it roughly unimodal and roughly normal because the lumps are quite small and close
If a distribution is unimodal, it is often quite useful to notice where the center of the distribution lies. If lots of observations are clustered around the middle, then the value of that middle
could be a handy summary of the sample of scores, letting you make statements such as, “Most thumbs in our sample are around 60 mm long.”
Which brings us to spread. Spread refers to how spread out (or wide) the distribution is. It also could be thought of as a way to characterize how much variability there is in the sample on a
particular variable. Saying most of our sample is around 60 mm means one thing if the range is from 50 to 70, and quite another if the range is from 2 to 200. | {"url":"https://staging.coursekata.org/preview/book/7846adcd-3aea-416d-abb6-f499aa45584e/lesson/5/2","timestamp":"2024-11-14T20:03:55Z","content_type":"text/html","content_length":"81135","record_id":"<urn:uuid:8268eab1-1633-4ad8-9a10-374ed6b9a254>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00283.warc.gz"} |
How does oblong shape look like?
An oblong is a two-sided 2D shape with four right angles and two pairs of parallel sides. An oblong, also known as a rectangle, has all of the same characteristics as a square, but it lacks equal
sides. The widths are longer than the lengths.
Is oblong a geometric shape?
The term “oblong” is frequently used incorrectly to refer to an elongated oval or’stadium’ shape. An oblong, on the other hand, is a rectangle with unequal adjacent sides (rather than a square).
What does an oblong look like?
Oblong is a shape with one longer end, similar to a rectangle or an ellipse. A leaf on one end of a table is an example of an oblong. Oblong is a rectangle or ellipse shape with one longer end.
What exactly is an oblong face shape?
Oblong faces are defined by their length, which is why they are also known as “long” faces because they are about twice as long as they are wide. The forehead, cheekbones, and jawline are roughly the
same width as oblong-shaped faces.
Oblong shapes are similar to oval shapes, but they are not identical.
What’s the difference between an oval and an oblong face shape?
The largest face length is oblong. The size of the forehead, cheekbones, and jawline is similar.
The length of the face can be any length. Face length is longer than the width of the cheekbones, and the forehead is longer than the jawline.
What is the shape of an oblong tablecloth?
The shape of an oblong is similar to that of a rectangle, but the corners are rounded out. What’s the benefit of having smooth edges on an oblong tablecloth?
The rounded corners will cleanly fold down around each other to fit neatly around the table and at a uniform length.
What hairstyle is best for an oblong face shape?
“Anything that will round out your face would be the best haircuts for someone with an oblong face shape,” Didier says. “For example, long layers with shorter angles near your cheekbones or lips,”
says the author.
“Big waves or a voluminous blowout” is recommended by Didier to style the look.
What celebrities have an oblong facial shape?
Sarah Jessica Parker, Cate Blanchett, and Kate Winslett, as well as Michael Parkinson, Tom Cruise, and Russell Crowe, have an Oblong Shaped Face.
Is it true that oblong and oval are the same thing?
For example, an oblong table is a rectangle with more length than width. An oval has a longer length than a width, but it also has continuous curved sides.
Is my table oval or oblong in shape?
Round edges on an oval tablecloth, and squared off edges on an oblong tablecloth. This was helpful to 2 of 2 people.
Do you? An oval tablecloth is designed for an egg-shaped tabletop, while an oblong tablecloth is designed for a rectangular table with square corners.
What are three geometric shapes that can be found?
Triangle, Circle, Semi-Circle, Square, Rectangle, Parallelogram, Rhombus, and Trapezium are the most common geometric shapes.
What is the size of an oblong table?
OBLONG/OVAL – Tables up to 56″ X 74″ with an oblong or round table will fit a 72″ X 90″ cloth. 6-8 people will be seated at these tables. Tables that are up to 56″ X 92″ oblong or round will fit a
72″ X 108″ cloth.
These tables will accommodate 8-10 people.
What’s the other name for an oblong?
On this page, you’ll find 22 oblong synonyms, antonyms, idiomatic expressions, and related words such as elongated, rectangular, ovopyriform, circle, egg-shaped, ovated, angular, oval, ellipsoidal,
and elongate.
What is the shape of an egg oval?
A ball, or sphere, is the most powerful shape of all. Another reason why eggs are egg-shaped is that they sit snugly together in the nest, with only small air gaps between them. This means that the
eggs radiate heat onto each other and keep each other warm.
Of course, you can add more eggs to the nest as well.
What is the best way to describe an oblong?
An oblong is a shape with two long and short sides, as well as right angles in all of the angles.
What exactly is an oblong’s name?
(Entry 1 of 2): deviating from a square, circular, or spherical shape by elongating in one dimension an oblong piece of paper and an oblong melon See leaf illustration
Which female face shape is the best?
The oval is also known as the “ideal” face shape and the most common. The oval face shape is seen in celebrities.
What is the best way to measure an oblong table?
You will need to measure the width from the center of the table, which will be the widest point, no matter what type of oval table you have. To determine your table length, place your tape measure
across the two longest points of your oval table. We recommend a 20-30cm (8-12′′) overhang.
What is the best way to measure an oblong?
An oblong’s area calculation follows the same formula as other rectangles in that it equals length divided by width. Determine the width of the rectangle.
Let the width be 15 for this example. Determine the length of the rectangle.
What exactly are geometric shapes?
When location, scale, orientation, and reflection are removed from a geometric object’s description, a geometric shape is the geometric information that remains. Polygons, which include triangles,
squares, and pentagons, are the most common shapes. | {"url":"https://tipsfolder.com/does-oblong-shape-look-like-ed3dbabfd684c6394c93cd82e0de949f/","timestamp":"2024-11-08T21:47:33Z","content_type":"text/html","content_length":"98926","record_id":"<urn:uuid:59ab57fc-61bd-4df7-b78c-69fed18ddbfc>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00509.warc.gz"} |
Curious Sunbeam Problem
See the Curious Sunbeam Problem
(Update 5/5/2023) Alternative Solution
Here is a solution offered by Oscar Rojas (5/1/2023):
Several people on Catriona Agg’s twitter site mentioned cyclic quadrilaterals, but that is not such a familiar concept to me, other than being a quadrilateral whose vertices all lie on a circle. I
went to Wikipedia to look at properties, of which there are many, and tried to find something that might be relevant. But I confess I failed, though I did not pursue it too extensively. My first
problem is that it was not obvious to me that the four points B, C, E, and F all lie on a circle. Sure, any three could, but what about the fourth? What property explicitly was used to prove they
were cyclic?
Another question I had was regarding the “outside angle theorem”. I admit I had not heard of that, or if I did, I did not remember it after more than 60 years. I did a Google search and found many
references to the “external angle theorem”, which is obvious and common, but nothing about an “outside angle theorem”. So it would have been helpful to state it and perhaps give a reference. But of
course I had found the angle was 45° by a different method.
To have to resort to less elementary geometric properties makes the problem a bit harder, which Catriona admitted, though her approach was even different (and not immediately obvious to me).
Clearly, I am an amateur when it comes to the vast repertoire of plane geometry, which might be expected for anyone regarding a subject that was developed over 2000 years. I just like how many
geometric puzzles one can solve with the simplest ideas. | {"url":"https://josmfs.net/2023/04/29/curious-sunbeam-problem/","timestamp":"2024-11-11T17:41:55Z","content_type":"text/html","content_length":"84513","record_id":"<urn:uuid:fbd6ffca-c7c9-496f-9e7c-86c2b8928763>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00436.warc.gz"} |
All Things Are Mathematical Information: A Vision Of Infinity
Our intention, with this document, is sharing.
Sharing what?
A vision of infinity centered on all things being information.
Or, more specifically, mathematical information.
On The Nature Of Information
The word “information” refers to the following: facts provided or learned about something or someone.
On its own, this definition is fine.
But, our definition of information is slightly more encompassing: facts, ideas, knowledge, data — ad infinitum, of course — that is conveyed/represented in a particular manner or through a particular
Just as an example, this sentence is a piece of information.
Right within this definition of information, there are an infinite wealth of associations, connections, and links — ad infinitum, of course — to language, writing, meaning, knowledge; ad infinitum.
Many other definitions of information exist within fields as vast and varied as physics, literature, and music; and so on and so forth, endlessly and infinitely, ad infinitum; let’s just say that
this last bit is true.
Even so, though, our definition is more than sufficient for this essay.
Right within this definition, there is an idea: our Universe is made of, grown from, built on, structured within, dependent on, existing through — ad infinitum, naturally — information.
Just information.
If you look at something — really, really look at something — you find information.
A good example of this is as follows: you look deep into a beautiful water molecule and, right within the deepest depth that comprises this water molecule, you find information.
Just information.
You find information. But, not just any information: you find mathematical information.
A series of mathematical equations.
Or, perhaps, an algorithm.
A set of binary digits, perhaps; each one arranged in a way that gives birth to this water molecule.
And so on and so forth.
The water molecule is information.
And, below this information — this substrate, as it were — there is nothing else.
The above is true of our water molecule. But, it is also true of:
1. The consciousness that you are.
2. The essay you are reading.
3. The soda you drink.
4. The room you live in.
5. The butterflies that surround you.
6. The stars in the sky.
7. The films you watch.
8. The people you know.
9. The ideas you love.
10. The wishes you hold in your heart.
And so on and so forth, endlessly and infinitely; ad infinitum.
All things are information.
But, again, not just any information: mathematical information.
The Purity Of Mathematical Information
Mathematical information is information grown from the language of mathematics.
The language of mathematics is infinite.
No endings, limits, boundaries; and so on and so forth, endlessly and infinitely; ad infinitum.
Right within the very firmament of mathematics, there is one thing, and one thing only.
Every single thing in this Universe is made of numbers.
Some of these numbers are arranged in peculiar ways.
Other numbers are placed into equations, algorithms, and other mathematical structures.
And, some of these numbers exist as patterns — ones and zeros, perhaps — that serve as the true form of something.
Our water droplet may, in the end, be little more than a pattern.
A pattern of ones and zeros.
Right beneath this pattern of ones and zeros, there is no more; those numbers, and their arrangement, serve as the deepest depth of this water droplet and all that it truly, truly is.
On the other hand, this essay, and the words, meanings, and ideas — and so on and so forth — that comprise its very substance, well, those numbers might exist as an arrangement of ones and zeros,
coupled with a few equations.
Or, perhaps, something adjacent to that. But, at the same time, something a little different.
And so on and so forth, endlessly and infinitely; ad infinitum.
If you wander into the very depths of this essay, and all that comprises it, you will find mathematics.
Or, more specifically, mathematical information.
The mathematical information we speak of is the very root of this essay.
Just as it is the root of all things.
All of this directs us to a question: where does this mathematical information come from?
Infinite Infinities; Unending Transcendence; Limitless Endlessness
Our concern, with this section, is the following:
1. Infinite infinities
2. Unending transcendence.
3. Limitless endlessness.
And so on and so forth, endlessly and infinitely; ad infinitum.
Every single one of the above is fact. But, also, a descriptor.
A descriptor of what?
The world in which mathematics emanates from.
Mathematics — and, in turn, our Universe — emanates from a higher, greater world.
A world not unlike the spaces and realms Plato spoke of when orating his notions of “Form.”
Right within this world, mathematics, in all of its infinities, is born.
And, right within this world, mathematics flows upwards and downwards — paradoxically, yes — into the Universe that we live within, experience, and know.
This flow allows all things — all things being mathematical information, of course — to arise within our Universe.
Right within the world in which this flow originates, there is mathematics in all directions.
You can wander in one direction for unending eternities.
And, in doing so, you will never, ever reach any endings, limits, conclusions, walls; ad infinitum.
The above is always true, regardless of what you make, create, experience, develop, use; ad infinitum.
Outside of those facts, there is another to remember: this world is generative.
Generative. Creative. Productive.
Mathematics is born from within this world.
Right within this birth, there is the act of combination.
Mathematical information is born. And, then, it combines itself with other pieces of mathematical information.
The fruits of this act are nothing less than the following: an endless process of birth, rebirth, and making, allowing an infinite wealth of new mathematical information to rise, all of which makes
its way into our world.
Right within this flowing, new forms, experiences, and moments — ad infinitum, of course — are brought into being.
Brought into being through them flowing into our Universe.
Every single one of these forms, experiences, and moments — ad infinitum — comprises the Universe we live within.
Just buried within the depths of that which comprises our Universe, though, is the true substance of all things.
Mathematical information.
Just to wrap this up, thank you for reading!
None of what comprises this essay is true.
Or, more specifically, none of what comprises this essay appears to be true.
Even so, though, this essay was fun to write!
Best wishes and have a fantastic day. | {"url":"https://maxwellakin.medium.com/all-things-are-mathematical-information-a-vision-of-infinity-e42d413c1a89?source=user_profile_page---------8-------------e9c3814df40d---------------","timestamp":"2024-11-02T21:56:34Z","content_type":"text/html","content_length":"135815","record_id":"<urn:uuid:98c2123d-f857-48e1-ae16-16d9fc9496d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00846.warc.gz"} |
Digression on Length and Distance in Vector Spaces
Home | 18.013A | Chapter 3 Tools Glossary Index Up Previous Next
3.8 Digression on Length and Distance in Vector Spaces
The distance between two vectors v and w is the length of the difference vector v - w.
There are many different distance functions that you will encounter in the world. We here use "Euclidean Distance" in which we have the Pythagorean theorem.
If the concepts of distance and length are used without additional description this is what we will mean:
The square of the length of a vector w is the sum of the squares (or more generally the sum of the absolute values of the squares, when the components are complex numbers) of its components. It is
the dot product (w, w) or w
But that is not the only concept of distance you will encounter in life.
What properties should the length of a vector have?
The traditional requirements are as follows:
It should be positive, and zero for the zero vector.
It should obey the triangle inequality: the length of the sum of two vectors is no greater than the sum of their lengths.
It is nice if length 0 means that the vector is the (0) vector.
What other concepts of length or distance are around?
Manhattan distance: the length of a vector is the sum of the absolute values of its components.
Hamming distance: length is number of non-zero components.
Maximum component distance: length is Maximum component absolute value.
Suppose we call the components x[i], and a small quantity of any of them dx[i], and the resulting value of distance with components dx[i] let us call ds.
Then in Euclidean space we have ^
We define the metric
Euclidean space can then be described as L[2].
Exercise 3.14 Which values of j in the definition of L[j] correspond to Hamming, Manhattan, and Maximum component size? (Hints: j can be infinite; also for Hamming distance the notions are similar
but not exactly the same, and only similar in a limit.)
Length in Euclidean space when non-rectilinear coordinates are used:
When you describe ordinary vectors in Euclidean space by their polar coordinates, then these do not obey the linear properties of ordinary rectangular coordinates. For example, the length of the sum
of two vectors is not the sum of their lengths, and the angle made with the x axis of a sum is not the sum of the angles of the summands.
We can ask, what is the length of a small vector, whose endpoints differ by r coordinate dr and by angle d
If we are at a specific point with given coordinates, the r direction is the direction pointing away from the origin toward it, and distance in this direction is measured just as in the x or y
direction. The length of a vector in this direction with coordinate dr is |dr|.
The result is that distance in polar coordinates is measured by
Length in non-orthonal coordinates:
Any k linearly independent k-vectors may be used as a basis: any other k-vector can be expressed as a linear combination of them. (Why? By exercise 3.11 any other k-vector is in a linear dependence
with them which can be solved for that k-vector in terms of the basis.)
Thus, in two dimensions, for example, any two vectors a and b with different directions can form a basis and any vector v can be described by coordinates that are the coefficients of these two: if v
= s a + t b then we can describe v by the 2-vector (s, t).
However, if we are describing Euclidean space and the vectors a and b are not orthogonal the length of v squared will not be s^2 + t^2. In general though, if we define (s, t) to be v', we get
length squared is <v'|G|v'> for some matrix G which depends on the angle between a and b.
Thus, if a and b are unit vectors at angle
The matrix G is called the metric tensor for the given basis.
Different metrics: Minkowski space:
There are even vector spaces in which the concept of distance is replaced by something that can be positive or imaginary: such is Minkowski space: it has four dimensions, three spatial and also time.
In it the analog of distance is described by
ds^2 = dx^2 + dy^2 + dz^2 –c^2dt^2
Vectors with s^2 positive or negative are said to be space-like or time-like respectively; those with s^2 = 0 are said to lie on the "light cone".
Why does anyone bother with such things?
Linear changes in the coordinates in Euclidean space that have the property that they do not alter distances (so that the distance between two points remains after the changes exactly what it was
before), are rotations in space. Similar changes in Minkowski space are symmetries of Maxwell's equations of electrodynamics, and correspond to both rotations in space and "Lorentz transformations".
Thus even this last concept has important physical application. All the others do as well, in appropriate contexts. | {"url":"https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter03/section08.html","timestamp":"2024-11-02T03:03:08Z","content_type":"text/html","content_length":"8433","record_id":"<urn:uuid:d767ab06-a265-46cc-8b55-58671c763533>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00598.warc.gz"} |
People | My Site
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Research is best done in a team. See the list below for my current collaborators, coauthors, and mentors. All starred names are available as references.
Students... (the ones who actually do the work)
See a list of my past and current students below. Starred students are current members of my working group.
Hierarchical Model Validation
Yash is a third-year undergraduate student at UC Berkeley majoring in Applied Math and Computer Science. He has a broad range of interests including statistics, natural language processing, education
and public policy.
Hierarchical Model Validation
Zixun is an undergraduate at UC Berkeley studying CS and Statistics. He is a tea lover, and plays badminton and guitar.
Hierarchical Model Validation
Brandon is a thrid-year undergraduate at UC Berkeley studying Applied Mathematics and Data Science. In his free time he loves to play chess and is a sports enthusiast.
Graphical Bootstrap, Hypothesis Testing on Networks
Feng Cheng is a senior at UC Berkeley double majoring in math and statistics, interested probability. He will become a math/statistics PhD student in fall 2024. He is very good at remembering numbers
attached to meaning. His pursuits include extensively listening to and studying Western classical music.
Trade-Off Analysis in Negotiation Games
Leo is a third-year undergraduate student from HKUST studying statistics and computer science. His research interests lie in ML application in game theory and science and ML theory. He loves to play
Go, basketball and tennis in his free time.
Zilai earned his master's degree in Statistics from the University of Chicago in 2023 and is a PhD student at Northwestern's IEMS. He works on Bayesian inference and machine learning. He enjoys
playing Go in his spare time.
Yucong graduated from the University of Chicago with an MS in Statistics in 2022. He is now a PhD student at Georgia Tech. He is interested in machine learning theory, including deep learning and
Game Embedding and Approximation Theory
Patrick graduated with his Masters degree from the University of Chicago and is an incoming statistics PhD student at Texas A&M. He is interested in network analysis and machine learning. He loves to
play board games and solve puzzles during his leisure time.
Patterns of Information Flow in Neural Circuits
Bowen focuses on computational neuroscience. He borrows analytical tools from dynamical system and computational tools from statistical learning theory. He is interested in formal modelling of
cognitive functions that underpin brain circuit dynamics.
Hwanwoo leverages Bayesian methods and optimization for uncertainty quantification, inverse problems, experiment design, and data science. He has worked on parameter estimation problems under
differential equation constraints and leveraged stochastic gradient descent for efficient statistical inference.
Christopher graduated from the University of Chicago in 2022 and is now a PhD student in statistics at the University of Illinois Urbana-Champaign. He works on self-organizing hierarchies in
populations evolving under a selection dynamic.
Game Embedding and Visualization
Qingyao is a "medium-sized creature prone to great ambition." He earned his master's from UC, and is an ORIE Ph.D. at Cornell. He aims to use optimization and machine learning to build practical
algorithms that improve lives. Qingyao is neither "theoretical" nor "applied"; but "production focused".
Game Embedding and Visualization
Patricia is a third-year undergraduate student at UChicago majoring in Statistics and Computer Science. She has a wide range of interests in data science, data visualization and programming
Daniel is a Ph.D. student at Brown in applied mathematics. His main interests lie in stochastic processes and quantum computation. Outside of academics, he is also a jazz musician, a table tennis
player, and an amateur cook.
Shiv graduated with an MS degree in Statistics from the University of Chicago in 2021. We worked together on uncertainty quantification and variational inference in Bayesian hierarchical models. He
works now as a quant at Tanius Tech.
Stochastic Ecological Modelling
Will graduated from Case Western Reserve University in 2021. We worked together on stochastic ecological models. Will developed tools that computed the importance of specific noise sources to the
long-term variance in a chosen observable.
David graduated from the University of Chicago in 2020. We worked together on phase induced tipping in ecological models that exhibit catastrophic transitions out of stable limit cycles.
bottom of page | {"url":"https://www.alexanderstrang.com/people","timestamp":"2024-11-13T22:37:23Z","content_type":"text/html","content_length":"575236","record_id":"<urn:uuid:3edcb20e-054c-4c0f-a4c5-5712f6b3b4c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00787.warc.gz"} |
Courses Bachelor Display 2024-2025
Course Description To PDF
Course title Optimisation
Course code EBC2105
ECTS credits 6,5
Assessment Whole/Half Grades
Period Start End Mon Tue Wed Thu Fri
Period 1 2-9-2024 20-10-2024 X X X
Level Introductory/Intermediate
Coordinator Stan van Hoesel, Janos Flesch
For more information: s.vanhoesel@maastrichtuniversity.nl; j.flesch@maastrichtuniversity.nl
Language of instruction English
* Students can find the right method to solve a given mathematical problem.
* Students can apply the linear and nonlinear optimization methods to concrete mathematical problems.
* Students can validate the method and the solution, depending on the mathematical problem.
* Students learn the concepts and solution method (the simplex method) for linear constrained optimization problems.
Goals * Students can apply the linear optimization method to problems in game theory and network flow problems.
* Students learn the concepts and solution methods for nonlinear unconstrained and constrained optimization problems.
* Students learn the definition of concave and convex functions, their characterizations, and their importance in nonlinear optimization problems.
* Students can recognize concave and convex functions by applying their characterizations.
* Students can clearly present their solutions of mathematical problems in groups.
Optimisation problems arise in all fields that econometricians encounter, such as operations research, game theory, statistics, micro- and macroeconomics and finance. The aim
of this course is to show the methodology for solving constraint optimisation problems both for linear and non-linear problems. These methodologies are also known as Linear
Description and Non-Linear Programming, respectively. The following topics and techniques will be treated: the standard simplex method, duality, sensitivity analysis, the primal-dual
simplex method, the network simplex method, first and second order necessary and sufficient conditions, the Lagrangian-function, Kuhn-Tucker conditions and constraint
qualification. Besides this, special attention is paid to the application of these methodologies in practical problems.
Vanderbei, R.J., Linear Programming: Foundations and Extensions, 5th edition, Springer, ISBN 978-3-030-39414-1 ISBN 978-3-030-39415-8 (eBook) https://doi.org/10.1007/
Literature 978-3-030-39415-8
Basic algebra (for linear programming), and advanced calculus (for nonlinear programming).
Exchange students need to be aware that very specific pre-knowledge is required for this course. A solid background in mathematics is necessary. Students should be aware of
the following concepts: Algebra: working knowledge of vector computing and matrices (including inverse matrices). Linear equations, and find the solutions of a set of
Prerequisites equations etc.
Function theory on the level of optimisation of functions of multiple variables under side conditions (Lagrange multipliers)
An advanced level of English.
Teaching methods
(indicative; course PBL / Lecture / Assignment
manual is definitive)
Assessment methods
(indicative; course Written Exam
manual is definitive)
Evaluation in previous For the complete evaluation of this course please click "here"
academic year
This course belongs to
the following programmes Bachelor Econometrics and Operations Research Year 2 Compulsory Course(s)
/ specialisations | {"url":"https://code.unimaas.nl/Code/Display?intCalendarID=31&intBAMA=1&SearchString=EBC2105","timestamp":"2024-11-14T00:20:25Z","content_type":"text/html","content_length":"14632","record_id":"<urn:uuid:01599793-ccd4-4cc2-81fb-a87181566189>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00410.warc.gz"} |
When is Running With Power Superior to Running at Pace?
Traditionally, runners use pace as a surrogate for their effort during training and racing. Theoretically, the effort during running is defined by the power of the “human engine.” In our book “The
Secret of Running” and in a previous post on running with power, we have shown that in running the total required power P is the sum of the power required to overcome the running resistance Pr, the
air-resistance Pa and the climbing resistance Pc, as indicated in the figure below:
In this article, we will analyze the impact of the three resistances on the correct pace. We will show that in order to maintain a constant effort/power, it may be required to adjust the pace
significantly (by 15 percent or even more). Consequently, running with power is superior to running at pace, particularly when the three resistances are not constant during a workout or race.
The impact of hills (climbing resistance)
In a recent paper, we analyzed the impact of the climbing resistance on running pace. During uphill running, the required power increases and downhill it decreases as a result of gravity. As it is
always best to run at constant power, this means that while going uphill you should reduce your pace to maintain constant power. Going downhill the reverse is the case.
The table below shows the impact of a hill with a gradient of 3.5 percent on the correct pace of the author (FTP 250 Watts, body weight 58 kg, P= 4.31 Watts/kg). Obviously, the impact will be even
bigger at steeper gradients, as the energy cost of hills is proportional to the gradient.
Impact of Hills Pace/km (min:sec)
Standard 3:58
Uphill (3.5%) 4:37
Downhill (3.5%) 3:26
The impact of the wind (air-resistance)
In another recent paper, we analyzed the impact of the wind on the air-resistance. Facing a head wind, the required power increases, while it decreases in a tailwind. As it is always best to run at
constant power, this means that in a head wind you should reduce your pace to maintain constant power. In a tailwind the reverse is the case.
The table below shows the impact at a wind speed of 15 km/h (at breast height) on the correct pace of the author (FTP 250 Watts, body weight 58 kg, P= 4.31 Watts/kg). Obviously, the impact will be
even bigger at higher wind speeds, as the energy cost of the air-resistance is proportional to the square of the wind speed.
Impact of the Wind Pace/km (min:sec)
Standard 3:58
Head wind (15 km/h) 4:24
Tail wind (15 km/h) 3:48
We note that state-of-the-art running power meters do not reflect the air-resistance by the wind correctly. One running power meter, Stryd, is currently working on two possible solutions to handle
this and they may come up with a product later this year. At the moment, the best thing we can do is to use the theoretical calculations from our book to predict the required pace as a function of
the wind speed and direction.
The impact of the energy cost of running (running resistance)
In our book, we analyzed the energy cost of running (ECOR), which is defined as the power required to overcome the running resistance, divided by the running speed. Obviously, the ECOR of a trail or
non-pavement surface will be higher than the ECOR of an asphalt pavement, which has a lower resistance.
From literature, we concluded that on a level and hard course, the ECOR is typically 0.98 kJ/kg/km. Of course, this number will not be the same for everyone: it depends on many factors, including
body posture, fuel mix and running form. Generally, it is believed that the ECOR of highly efficient elite runners could be as low as 0.90 kJ/kg/km, whereas the ECOR of inefficient joggers could be
as high as 1.10 kJ/kg/km. So far, we have seen that our own data and those of many other runners are quite close to 1.00 kJ/kg/km.
Obviously, a lower ECOR means that you are running more efficiently and consequently you can run faster. So every runner should try to lower his ECOR! Unfortunately, we cannot change our body posture
(apart from shedding excess body fat) and the fuel mix in our muscles (apart from carbo-loading before the marathon). The Kenyan elite runners share many advantages like slim calves, flexible hips
and (relatively) long legs. So, the only factor that we can try to optimize is our running form.
Many factors are thought to influence the running form, including cadence, GCT, stride length, vertical oscillation, arm drive, foot strike, hip angle, knee lift, leg stretching, calves lift and
ankle angle. However, opinions differ on exactly what constitutes the best running form. In a recent paper we found that the ECOR could be reduced by increasing the cadence.
The table below shows the impact of the ECOR on the correct pace of the author (FTP 250 Watts, body weight 58 kg, P= 4.31 Watts/.kg). Looking at the data, it’s possible that the author could run much
faster if he could reduce his ECOR to 0.90 kJ/kg/km. To be honest, we have not found the answer how he could achieve this.
Impact of ECOR Pace/km (min:sec)
Standard 3:58
Low ECOR (0.90) 3:41
High ECOR (1.10) 4:24
Conclusions and outlook
The examples prove quite clearly that running at constant power is superior to running at constant pace, particularly when the conditions (resistances) during the workout or race are not constant. At
any significant gradients and wind speeds, it will be detrimental or even impossible to run at a constant pace. Similarly, it will be necessary to adapt the pace in races with tougher footing, such
as trails.
Theoretically, running at constant power is the best strategy to provide the best results. When running at constant power, the pace will be automatically reduced in tough sections (uphill, headwind,
soft footing, etc.). In the easier sections of the race or workout, the pace will increase automatically.
Unfortunately, the present state-of-the-art running power meters do not yet reflect the impact of the wind and surface footing correctly. This means that presently the main advantage of running with
power meters is to maintain constant power during hilly courses.
In spite of these limitations, we are very excited that power meters do provide us with an opportunity to determine our ECOR on a daily basis, so we can try to optimize our running form. We are sure
that this will pave the way to concrete improvements in our ECOR and race results.
We realize that this will not be easy because for us—and for most people—the running form has been habituated over many years of running. We will not be able to change it overnight. But with time and
concrete data, we are confident we will be able to get some improvement.
Thank you to the co-authors Ron van Megen and Guido Vroemen. | {"url":"https://www.trainingpeaks.com/blog/when-is-running-with-power-superior-to-running-at-pace/","timestamp":"2024-11-02T02:58:13Z","content_type":"application/xhtml+xml","content_length":"153397","record_id":"<urn:uuid:5d8c9a23-c57b-4f1e-bbd8-215f03cd663a>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00691.warc.gz"} |
Ohio Form IT-2210-1040
1. Ohio Form IT-2210-1040
Ohio Income Tax Underpayment Interest Penalty Form
Income Tax Underpayment Interest Penalty Form
Form IT-2210-1040
2023 Ohio IT/SD 2210
Interest Penalty on Underpayment of Ohio Individual Income,
School District Income and Pass-Through Entity Tax
Include with your 2023 Ohio tax return.
Use UPPERCASE letters.
Complete this section if you are filing Ohio IT 1040 or SD 100.
Spouse’s SSN (if filing jointly)
Primary taxpayer’s SSN (required)
First name
Last name
Spouse’s first name (if filing jointly)
Last name
Complete this section if you are filing Ohio IT 4708, IT 1140, IT 4738, IT 1041, or SD 100E.
Decedent’s SSN (estates)
Name of pass-through entity, trust or estate
Additional line, if necessary, for name of pass-through entity, trust or estate
Total interest penalty due (from page 2, line 8 or page 3, line 6)...........................................
.0 0
Include pages 1 and 2 when you file your Ohio IT 1040, SD 100, SD 100E, IT 1041 or IT 4708 tax return.
Include pages 1 and 3 when you file your Ohio IT 1140 or IT 4738 tax return.
Federal Privacy Act Notice: Because we require you to provide us with a Social Security number, the Federal Privacy Act of 1974 requires us to inform you that providing us
with your Social Security number is mandatory. Ohio Revised Code sections 5703.05, 5703.057 and 5747.08 authorize us to request this information. We need your Social
Security number in order to administer this tax.
Taxpayer’s name
Taxpayer’s FEIN/SSN
Part I – Calculating the Required Annual Payment
When Filing the Ohio IT 1040, SD 100, SD 100E, IT 1041 or IT 4708
Use this form to calculate interest penalty on underpayment of taxes and to show the exceptions where no interest penalty is due.
See page 4 for definitions and line references.
Check here if you engage in farming or fishing activities and refer to Ohio Administrative Code Rule 5703-7-04 for options.
1. 2023 Ohio income taxes paid (timely paid* 2023 estimated payments plus withholding plus 2022 credit
carryforward) .............................................................................................................................................1.
2. 2023 Ohio income tax liability (total tax minus total credits) .....................................................................2.
3. 2022 Ohio income tax liability (total tax minus total credits) .....................................................................3.
4. Multiply line 2 by 90% (.90)........................................................................................................................4.
5a. Is line 1 greater than or equal to line 4? If yes, STOP, you have no interest penalty. If no, continue to
line 5b......................................................................................................................................................5a.
5b. Did you timely file a 2022 Ohio income tax return? If yes, continue to line 5c. If no, skip to line 5d .......5b.
5c. Is line 1 greater than or equal to line 3? If yes, STOP, you have no interest penalty. If no, continue to
line 5d...................................................................................................................................................... 5c.
5d. Is line 2 less any withholding $500 or less? If yes, STOP, you have no interest penalty. If no, continue
to line 6....................................................................................................................................................5d.
6. If you answered “Yes” on line 5b, enter the lesser of line 3 or line 4. If you answered “No”, enter the
amount from line 4. Then continue to Part II................................................................................................ 6.
*Do not include any estimated payments that were made after their respective due date.
Part II – Calculating the Interest Penalty Due
Payment Due Dates
(see note below)
4/18/23 – 25%
6/15/23 – 50%
9/15/23 – 75%
1/16/24 – 100%
1. Multiply the amount on Part I, line 6 by the percentage indicated at
the top of each column at right...............................................................1.
2. Multiply the total tax withheld from compensation by the percentage
indicated at the top of each column at right............................................2.
3. Total estimated tax (including any credit carryforwards) paid by
the dates shown at the top of each column at right................................3.
4. Add lines 2 and 3....................................................................................4.
5. Underpayment subject to interest penalty (line 1 minus line 4;
if less than zero, enter zero)...................................................................5.
6. Ratio (if full or partial payment was made see instructions on page 4)..6.
7. Interest penalty for the period: Multiply line 5 by line 6 for each
column at right........................................................................................7.
8. Total interest penalty due (sum of line 7, Columns A through D). Enter here and on page 1........................................8.
Note: Payment due dates – the associated dates and the rates on line 6 are for calendar year taxpayers. Fiscal year taxpayers must adjust the payment due dates and the line
6 ratios accordingly.
Taxpayer’s name
Taxpayer’s FEIN/SSN
Part I – Calculating the Required Annual Payment
When Filing the Ohio IT 1140 or IT 4738
Use this form to calculate interest penalty on underpayment of taxes and to show the exceptions where no interest penalty is due. If the
total adjusted qualifying amount or qualifying taxable income for the current year or the previous year is $10,000 or less, do not complete
this form. You do not owe an interest penalty. See page 4 for definitions and line references.
1. 2023 Ohio withholding taxes paid (timely paid* 2023 estimated payments)................................................. 1.
2. 2023 Ohio withholding tax liability (total tax).............................................................................................2.
3. 2022 Ohio withholding tax liability (total tax)................................................................................................ 3.
4. Multiply line 2 by 90% (.90).......................................................................................................................4.
5a. Is line 1 greater than or equal to line 4? If yes, STOP, you have no interest penalty.
If no, continue to line 5b..........................................................................................................................5a.
5b. If filing the Ohio IT 1140, did you timely file a 2022 Ohio IT 1140? - OR - If filing the Ohio IT 4738,
did you timely file a 2022 Ohio IT 4738? If yes, continue to line 5c. If no, continue to line 6..................5b.
5c. Is line 1 greater than or equal to line 3? If yes, STOP, you have no interest penalty.
If no, continue to line 6............................................................................................................................5c.
6. If you answered “Yes” on line 5b, enter the lesser of line 3 or line 4. If you answered “No”,
enter the amount from line 4. Then continue to Part II..............................................................................6.
*Do not include any estimated payments that were made after their respective due date.
Part II – Calculating the Interest Penalty Due
Payment Due Dates
(see note below)
4/18/23 – 25%
7/17/23 – 50%
10/16/23 – 75%
1/16/24 – 100%
1. Multiply the amount on Part I, line 6 by the percentage indicated at
the top of each column at right............................................................... 1.
2. Total estimated tax (including any credit carryforwards) paid by the
dates shown at the top of each column at right...................................... 2.
3. Underpayment subject to interest penalty (line 1 minus line 2; if
less than zero, enter zero)...................................................................... 3.
4. Ratio (if full or partial payment was made see instructions on page 4).. 4.
5. Interest penalty for the period: Multiply line 3 by line 4 for each
column at right........................................................................................ 5.
6. Total interest penalty due (sum of line 5, Columns A through D). Enter here and on page 1........................................6.
Note: Payment due dates – the associated dates and the rates on line 4 are for calendar year taxpayers. Fiscal year taxpayers must adjust the payment due dates and the line
4 ratios accordingly.
Page 2 Definitions
Ratios – The listed ratios on the previous pages are based upon the
statutory interest rate (5% for 2023 and 8% for 2024) and the time
during which the estimated payment was late. The general formula for
computing the ratio is rate = interest rate X number of days the payment
is late ÷ 365.25. The listed ratios are computed from the payment due
date at the top of each column to the following payment due date and
applied only if the taxpayer either (i) never made the estimated payment
or (ii) made full payment on or after the next payment due date.
“Taxes paid” include payments of estimated taxes made under Ohio
Revised Code (R.C.) section 5747.09(C), taxes withheld from taxpayer’s
compensation, and tax refunds applied by the taxpayer in payment of
estimated taxes.
“Tax liability” means the total taxes due for the taxable year, after
allowing any credit to which the taxpayer is entitled, but prior to applying any estimated tax payment, withholding payment or refund from
another tax year.
Example 1 – No payment made. Assume that the underpayment shown
on page 2, Part II, line 5 for Column B is $1,000. Also assume that the
taxpayer made no estimated payment during the period 4/18/23 through
6/15/23. The taxpayer will compute interest penalty for the period 4/18/23
through 6/15/23 by multiplying the underpayment shown on Part II, line
5, Column A by the ratio (0.007940) shown on line 6, Column A.
“Estimated taxes” means the amount that the taxpayer estimates to
be the taxpayer’s combined tax liability under chapters 5747 and 5748
of the Revised Code for the current taxable year.
Note: State income tax may be combined with the school district income
tax in determining the interest penalty as calculated on page 2.
Interest penalty = $1,000 X 0.007940 = $7.94 to Part II, line 7, Column A
Page 2 Line References
Taxes Paid
IT 1040
SD 100
IT 1041
IT 4708
Example 2 – Full payment made after the due date but before the
next due date. Assume that the underpayment shown on page 2, Part
II, line 5 for Column A is $1,000. Also assume that the taxpayer paid
this in full on 5/15/23. The taxpayer should ignore the ratio shown on
Part II, line 6, Column A and compute the rate as follows:
Sum of line 14 and line 15
Sum of line 11 and line 12
Line 6
Line 14
Sum of line 17 and line 18
Step 1 – Determine the number of days from the date the payment was
due (4/18/23) to the date the payment was made (5/15/23): 4/19/23 to
5/15/23 = 27 days.
Current Year Tax Liability – 2023
IT 1040 Line 10 minus line 16
SD 100 Line 8
SD100E Line 3
IT 1041 Line 11 minus line 15
IT 4708 Line 12 minus line 19
Step 2 – Calculate the ratio by using the following formula:
Ratio = interest rate X number of days late ÷ 365.25
Ratio = 0.05 X 27 ÷ 365.25 = 0.003696
Interest penalty = $1,000 X 0.003696 = $3.70 to Part II, line 7, Column A
Previous Year Tax Liability – 2022
IT 1040 Line 10 minus line 16
SD 100 Line 4
SD100E Line 3
IT 1041 Line 11 minus line 15
IT 4708 Line 12 minus line 19
This method is only applicable if the taxpayer made full payment of
the required estimated payment after the due date but before the next
payment due date.
Example 3 – Partial payment made after the due date but before
the next due date. Assume that the underpayment shown on page 2,
Part II, line 5 for Column A is $1,000. Also assume that the taxpayer
paid $600 on 5/15/23. The taxpayer should ignore the ratio shown on
Part II, line 5, Column A and compute the rate as follows:
Page 3 Definitions
“Taxes paid” includes payments of estimated taxes made under
R.C.5747.43(C) and tax refunds applied by the qualifying entity or
electing pass-through entity in payment of estimated taxes.
Step 1 – Determine the number of days from the date the payment was
due (4/18/23) to the date the payment was made (5/15/23): 4/19/23 to
5/15/23 = 27 days.
“Tax liability” means the total of the taxes and withholding taxes due
under sections 5733.41 and 5747.41 or the tax due under section
5747.38 of the Revised Code for the applicable taxable year prior to
applying any estimated tax payment or refund from another year.
Step 2 – Calculate the interest penalty for that period by using the
following formula: interest penalty = underpayment X interest rate X
number of days late ÷ 365.25
Interest penalty = $1,000 X 0.05 X 27 ÷ 365.25 = $3.70
“Estimated taxes” means the amount that a qualifying entity or electing
pass-through entity estimates to be the sum of its liability under R.C.
sections 5733.41 and 5747.41 or R.C. section 5747.38 for its current
qualifying taxable year or taxable year, as applicable.
Step 3 – Determine the number of days from the payment date (5/15/23)
to the next required due date (6/15/23): 4/16/23 to 6/15/23 = 31 days.
Page 3 Line References
Taxes Paid
IT 1140 Schedule I, line 3c, sum of Column A and B
IT 4738 Schedule I, line 16
Step 4 – Calculate the interest penalty on the $400 underpayment
($1,000 minus $600) for the 31-day period using the following formula:
interest penalty = underpayment X interest rate X number of days late
÷ 365.25
Interest penalty = $400 X 0.05 X 31 ÷ 365.25 = $1.70
Current Year Tax Liability – 2023
IT 1140 Schedule I, line 1, sum of Column A and B
IT 4738 Schedule 1, line 9
Step 5 – Add the interest penalty amounts calculated in Steps 2 and 4:
$3.70 + $1.70 = $5.40 to Part II, line 7, Column A.
Previous Year Tax Liability – 2022
IT 1140 Schedule I, line 1, sum of Column A and B
IT 4738 Schedule 1, line 9
For more information, see the FAQs at tax.ohio.gov/faq-IncomeEstimated.
Extracted from PDF file 2023-ohio-form-it-2210-1040.pdf, last modified October 2023
More about the Ohio Form IT-2210-1040 Individual Income Tax Estimated TY 2023
If you failed to pay or underpaid your estimated taxes for the past tax year, you must file form IT-2210, Interest Penalty on Underpayment of Ohio Individual Income, School District Income and
Pass-Through Entity Tax, to calculate any interest or penalties due with your income tax return.
We last updated the Income Tax Underpayment Interest Penalty Form in February 2024, so this is the latest version of Form IT-2210-1040, fully updated for tax year 2023. You can download or print
current or past-year PDFs of Form IT-2210-1040 directly from TaxFormFinder. You can print other Ohio tax forms here.
Other Ohio Individual Income Tax Forms:
TaxFormFinder has an additional 82 Ohio income tax forms that you may need, plus all federal income tax forms.
View all 83 Ohio Income Tax Forms
Form Sources:
Ohio usually releases forms for the current tax year between January and April. We last updated Ohio Form IT-2210-1040 from the Department of Taxation in February 2024.
Form IT-2210-1040 is an Ohio Individual Income Tax form. While most taxpayers have income taxes automatically withheld every pay period by their employer, taxpayers who earn money that is not subject
to withholding (such as self employed income, investment returns, etc) are often required to make estimated tax payments on a quarterly basis. Failure to make correct estimated payments can result in
interest or penalties.
About the Individual Income Tax
The IRS and most states collect a personal income tax, which is paid throughout the year via tax withholding or estimated income tax payments.
Most taxpayers are required to file a yearly income tax return in April to both the Internal Revenue Service and their state's revenue department, which will result in either a tax refund of excess
withheld income or a tax payment if the withholding does not cover the taxpayer's entire liability. Every taxpayer's situation is different - please consult a CPA or licensed tax preparer to ensure
that you are filing the correct tax forms!
Historical Past-Year Versions of Ohio Form IT-2210-1040
We have a total of nine past-year versions of Form IT-2210-1040 in the TaxFormFinder archives, including for the previous tax year. Download past year versions of this tax form as PDFs here:
TaxFormFinder Disclaimer:
While we do our best to keep our list of Ohio Income Tax Forms up to date and complete, we cannot be held liable for errors or omissions. Is the form on this page out-of-date or not working? Please
let us know and we will fix it ASAP. | {"url":"https://www.taxformfinder.org/ohio/form-it-2210-1040","timestamp":"2024-11-02T13:50:12Z","content_type":"text/html","content_length":"52957","record_id":"<urn:uuid:b643c8fe-0749-4098-b605-0f14fccaf4fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00811.warc.gz"} |
Array sorting with javascript
Sometimes, for fun, I like to give myself small leetcode-like questions in the morning and see if I can have an answer by the end of the day. This is a fun way to keep my skills sharp and learn new
concepts. Today's task started simple enough.
Given an array of numbers not in numerical order, sort the array so that its contents are in numerical order(lowest to highest)
I made a mess
First I tried something using deeply nested for loops that became an unreadable mess before it became functional. Then I tried an approach that added all the numbers together, got the median then
tried sorting based if higher or lower. This also didn't work. So I googled it to see what others have done. This is how I stumbled upon the bubble sorting algorithm. My final result now looks like
let arr = [10, 21, 3, 45, 15]
let test = function(arg){
let bul;
bul = false
for(let i = 0; i < arg.length; i++){
if(arg[i] > arg[i + 1]){
let temp = arg[i]
arg[i] = arg[i + 1]
arg[i + 1] = temp
bul = true
return arg
why does this work
The basics of this algorithm is to take the first element and compare if it is larger than the next element. If it is, swap them, then repeat until we have made a full loop of the array with no
changes. That's it, just some simple math and conditionals.
how does this work
We use a variable bul that is a bullion to let us know if there have been any changes on this iteration. Declaring this variable is the only statement outside of the do/while loop. This loop is set
up so that as long as the variable bul remains true we will execute the contained function. This function contains a For loop that iterates over the array. We then use an if statement to see if the
current element is larger than the next element. If it is, we create a variable temp to store the current element. Change the current element’s value so that it is equal to the next element (if the
second element was 2 now both the first and second element are 2). Then we use our variable to copy the value that was our current element into the next element. At the end of all this we change the
variable bul to true, This is what keeps the loop going. Once the if statement is not triggered it means we have looped over the whole array without making any changes, meaning all values are where
we want them.
when to do this
If you understand big O notation (an equation often used to measure the efficiency of algorithms in computer science) bubble sort is O(n^2). This is not great in terms of efficiency but let's talk
about why. Since the algorithm is nested loops and we are making comparisons between different elements, the time it could take to complete is the square of the number of inputs. In Fact each time
you nest another loop you would add another power to n. So nesting one more for loop would make the efficiency O(n^3).
Big O notation aside the long and short of this is The more you have to sort the longer it takes to sort it. Bubble sort is a great and simple algorithm but for large data sets is impractically
slow for most use cases. If your array is a few hundred elements, bubble sort is probably a great fit. Thanks for joining me in learning about bubble sort, I hope this has helped you in some way.
Top comments (1)
Gulshan Negi •
Thanks a lot for this.
I have also seen this short example of doing this.
let numbers = [10, 5, 8, 2, 1];
// Sorting in numerical order
numbers.sort(function(a, b) {
return a - b;
console.log(numbers); // Output: [1, 2, 5, 8, 10]
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/techthatconnect/array-sorting-for-numarical-order-with-javascript-4bbn","timestamp":"2024-11-09T05:07:28Z","content_type":"text/html","content_length":"76909","record_id":"<urn:uuid:454013f9-ddf4-432f-adaa-8f4b0b148af3>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00245.warc.gz"} |
[INACTIVE][CHAT] Prefixer v2.5.1 - Adds prefixes to users when they chat [1060]
Prefixer has been moved into ColorMe and was taken over by another dev, to get it, GO HERE.
Prefixer - The Easy Prefix Plugin:
Version: v2.5.1
Prefixer allows players to add a prefix to a player such as [Farmer], etc. right from the game. See the commands list below for examples on proper use.
/prefix list - Shows a list of allowed colors as their color
/prefix <prefix> - Changes your prefix. Color code optional and mixed into prefix.
/prefix [prefix] - Changes another player's prefix.
/prefix -r [name, name2...] - Removes your/listed player's prefix
/prefix &4Admin = Admin <Valrix>
/prefix -r = <Valrix>
/prefix -r valrix = <Valrix>
/prefix &5The&4Admin = TheAdmin <Valrix>
/prefix &5The&4Admin valrix = TheAdmin <Valrix>
/prefix &4MOD valrix john jane peter =
MOD <Valrix>
MOD <john>
MOD <jane>
MOD <peter>
☆ Black = &0
☆ Dark Blue = &1
☆ Dark Green = &2
☆ Dark Aqua = &3
☆ Dark Red = &4
☆ Dark Purple = &5
☆ Gold = &6
☆ Gray = &7
☆ Dark Gray = &8
☆ Blue = &9
☆ Green = &A
☆ Aqua = &B
☆ Red = &C
☆ Light Purple = &D
☆ Yellow = &E
☆ White = &F
prefixer.list - Allows player to use /prefix listto see color list
prefixer.list - Allows player to see list of color codes
prefixer.self - Allows player to set own prefix
prefixer.other - Allows player to set another player's prefix
prefixer.remove - Allows player to remove prefixes
* prefixer.remove required to remove ANY prefixes *
□ Customized prefixes with color codes
□ Multi-world support
□ Supports native bukkit permissions (PermissionsBukkit)
□ Now supports other plugins hooking in to get/set/remove, and check if a player has a prefix set.
Version 2.5.1
□ Fixed NumberOutOfBounds error people were getting
□ Properly tested against newest RB and multi-world support
□ Removed config & generation code until later
□ Patched memory leak error
Version 2.5
☆ Added multi-world support
☆ Now uses native permissions (PermissionsBukkit)
☆ improved command syntax
☆ Improved command node handling
☆ Numerous code improvements
Version 2.4
☆ Should fix any problems people have been having recently.
Version 2.3
☆ Bunch of code changes to improve performance and decrease code size.
Version 2.2
☆ Fixed that blasted bug where you couldn't remove prefixes.
Version 2.1
☆ Added support for giving a player a prefix through command-line
Version 2.0
☆ Fixed the bug where you couldn't set someone else's prefix.
Version 1.9
☆ Prefixer now automatically updates the old .prefix file, which is where the prefixes are stored, to work with the newer prefix system.
Version 1.8
☆ Prefixes are now fully customizable.
☆ Permissions is now optionally supported.
☆ Works fine with RB 670
Version 1.7
Version 1.6
Version 1.5
☆ Fixed it to work with new command structure
☆ Now allows player to set own prefix using a smaller command
Version 1.4
☆ Fixes a bug with setting a user to having no prefix
Version 1.3
☆ Changed how the prefix is added. Should play nice with other plugins now.
Version 1.2
☆ Fixed a weird error with colors
Version 1.1
☆ Updated to comply with new constructor.
☆ Adds the ability to add color to the prefix. See description for example.
☆ Names are no longer case-sensitive. Instead of Valrix you can use valrix, or even VaLrIx if you wanted so you don't have to worry about messing up a user's name.
Version 1.0
klarnet, Nomanoclass, Parrothead and 3 others like this.
Will there be plans to make colored prefixes? It'll be just what I need!
I was actually thinking about adding that where it would be something like: /prefix <name> <prefix> [color]
with the color being optional. You know, I will start on that now in fact. I've got nothing else to do and think it would be nice to have. It'll be done in around an hour or less, depending on
how active the IRC is.
Ah! That would be awesome! Looking forward to it
Permissions can do this already... so why would we need this?
Because permissions is going to be replaced by built-in permissions. If you're going to do this to each of the new plugins I make, I'd prefer you to not post at all.
JoelDaMole789 likes this.
It was a kind question, I didn't find any use for this. Bukkit built-in is going to be very similar to Permissions anyway, so it may make this plugin totally reduntant.
Well it's lightweight and adds color in a nicely unique way, so for those who want to prefix players easily then this is for them. Plus it takes no config and actually works.
I need this since I don't want to mess with moving around people in Permissions just for a guild tag. Also, people leave and get added constantly between factions, so this is perfect to use for
those situations. Thank you Valrix for adding the color option! I'll definitely be using this plugin for now!
Great! Glad that you like it.
i cant get this working i got build 284 and im opand everything just cant give rpefix
--- merged: Feb 21, 2011 12:24 AM ---
at org.bukkit.craftbukkit.CraftServer.loadPlugins(CraftServer.java:53)
at org.bukkit.craftbukkit.CraftServer.reload(CraftServer.java:193)
at org.bukkit.command.SimpleCommandMap$ReloadCommand.execute(SimpleCommandMap.java:184)
at org.bukkit.command.SimpleCommandMap.dispatch(SimpleCommandMap.java:77)
at org.bukkit.craftbukkit.CraftServer.dispatchCommand(CraftServer.java:171)
at net.minecraft.server.MinecraftServer.b(MinecraftServer.java:344)
at net.minecraft.server.MinecraftServer.h(MinecraftServer.java:326)
at net.minecraft.server.MinecraftServer.run(MinecraftServer.java:232)
at net.minecraft.server.ThreadServerApplication.run(SourceFile:512)
Caused by: java.lang.NoSuchMethodException: com.sparkedia.valrix.Prefixer.Prefixer.<init>(org.bukkit.plugin.PluginLoader, org.bukkit.Server, org.bukkit.plugin.PluginDescriptionFile, java.io.File,
java.io.File, java.lang.ClassLoader)
at java.lang.Class.getConstructor0(Class.java:2723)
at java.lang.Class.getConstructor(Class.java:1674)
at org.bukkit.plugin.java.JavaPluginLoader.loadPlugin(JavaPluginLoader.java:75)
Yeah, build numbers have now been changed due to the new location of where to get the builds and the way plugins are built have also been changed so you'll now have to go to jenkins.lukegb.com
and the earliest version that should work with properly updated plugins is #48. bamboo.lukegb.com is officially dead.
oh ok wait will i have to update like essentials and everything that works atm again?
--- merged: Feb 21, 2011 12:41 AM ---
sadly i tried 48 and nothing worked it especially broke essentials....
Ok, then 50 or higher should work. He bumped it to 400 for some reason and I don't know how those are but the latest working build has all the plugins I made working at least. Essentials and
Permission break nearly every update.
if i get this working and doesnt break my life i will love u forever
--- merged: Feb 21, 2011 1:20 AM ---
does not work..still errors for everything
Haha, thanks
--- merged: Feb 21, 2011 1:21 AM ---
Hmm, which build is it?
build 52...and everything just goes crazy essentials breaks,world guard,iconomy,permissions,worldedit,prefix still doesnt work
Alright, I have the latest build that I've made myself in my signature. If you use that one at least all my plugins will work. I don't know about the others though.
ok ill try ty
Ok. You're welcome.
man i still cant seem to get it working>.<i really want this plugin too
What error are you getting?
when i try it it says an internal error occured while... in game in red
Does this stack with other prefixes? For example, I'm using MultiVerse and it adds a prefix that says what world you are in.
No it doesn't and I'm not sure if there's really a way to "stack" it because you have to set the format. I can tinker with it a bit and see if it's possible.
--- merged: Feb 22, 2011 1:06 AM ---
Could you post the log? Also make sure you're at least using the recommended craftbukkit version which is currently 53 from http://jenkins.bukkit.org
is the plugin working for 1.3?
I'm using Prefixer v1.3 and I think using the command to remove tags from names doesn't work. It just simply brings up the prefix color list. A bug I believe?
The plugin doesnt work for me.
Tried with different builds (432+) and still the same.
When im SERVER op, the /prefix command doesnt respond. After i deop myself, for the /prefix i get the help messages for the command, but i cant use it. Everytime i try to use it (in the correct
format), i just get message that "incorrect usage of command" + the help messages for the command.
Wanted to use this plugin for a simple prefix plugin, so i dont have to mess around with the permissions or something, hope it will get fixed soon!
Make sure your build is the recommended build. I don't know if anything higher or lower works, so if it breaks it's out of my hands. But I have tested it on the recommended and all of them work.
Make sure the version shows "git-Bukkit-0.0.0-450-gd3c1ba4-b432jnks" when you start up your server.
Do you think to add color to the name??
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Lesson: Combining ratios | Foundation | KS4 Maths | Oak National Academy
Lesson details
Key learning points
1. There may be a relationship between quantities A and B and a relationship between quantities B and C
2. Therefore there is a relationship between A and C
3. If A : B and B : C then A : B : C and A : C
4. When the common quantity is not the same in each ratio, equivalent ratios can be used
5. For example, if A:B and 2B:5C then 2A:2B:5C and 2A:5C
Common misconception
Combining ratios without using equivalent ratios to equate the common part.
Pupils may need to spend extra time finding equivalent ratios and looking at when it is ok to combine ratios or not.
• Proportion - A part to whole (sometimes part to part) comparison. If two things are proportional then the ratio of part to whole is maintained and the multiplicative relationship between parts is
also maintained.
• Ratio - A ratio shows the relative sizes of 2 or more values and allows you to compare a part with another part in a whole.
• LCM - LCM is an abbreviation for lowest common multiple.
• Lowest common multiple - The lowest common multiple is the lowest number that is a multiple of two or more numbers.
Using objects or bar models to show the ratios can help pupils to visualise the concept better.
Teacher tip
6 Questions
LCM is an abbreviation for lowest common .
What is the ratio of hearts : smileys?
What is the HCF of 24 and 18?
What is the ratio of hearts : smileys? (Give your answer in its simplest form)
What is the LCM of 4 and 6?
Match each ratio to an equivalent ratio.
6 Questions
What is LCM an abbreviation for?
Given these bar models, what is the ratio of hearts to suns?
Given that squares : circles = 4 : 3 and circles : triangles = 1 : 5, find the ratio of squares : triangles.
a : b = 2 : 5, b : c = 6 : 7. Find the ratio a : b : c
A box contains 212 sweets which are either hearts (h), stars (s) or bottles (b). The ratio of hearts to stars is 5 : 3. The ratio of stars to bottles is 4 : 7. How many stars are in the box?
b = 4a, c = 5b, find the ratio of a : c | {"url":"https://www.thenational.academy/teachers/programmes/maths-secondary-ks4-foundation/units/ratio/lessons/combining-ratios","timestamp":"2024-11-06T02:16:09Z","content_type":"text/html","content_length":"272103","record_id":"<urn:uuid:6c65ac19-d626-4cd6-9b1c-40c3fea69b3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00382.warc.gz"} |
that the converse is also true. So, we have the following theor... | Filo
Question asked by Filo student
that the converse is also true. So, we have the following theorem : Theorem : If in a quadrilateral, each pair of opposite angles is equal, the it is a parallelogram.
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Question Text that the converse is also true. So, we have the following theorem : Theorem : If in a quadrilateral, each pair of opposite angles is equal, the it is a parallelogram.
Updated On Nov 6, 2022
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 140
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How Much Damage Can I Do Turbo-Punting Shitcoins? - Robot Wealth
How Much Damage Can I Do Turbo-Punting Shitcoins?
Here in Australia, we’re right in the depths of the silly season. We indulge in long lunches, take days off work, and generally let our hair down.
In that spirit, I thought I might have some fun punting shitcoins.
(Maybe my definition of fun differs from yours, but let’s run with it).
For the uninitiated, the technical definition of a shitcoin is a recently launched cryptocurrency of dubious economic value and marginal liquidity.
These things have historically shown a noisy tendency to trend, which means that if you can be bothered trading them (marginal liquidity and all that) and don’t mind taking on the very real risk of
total capital incineration, you can potentially make some money with simple trend-following rules.
A while back we looked at a simple trend-following strategy where we get long anything within 5 days of its 20-day high:
As part of my homage to the silly season, I intend to throw a small amount of capital at this strategy, turn it up to full noise, and see how we go.
But I’d like to do it at least somewhat sensibly and systematically.
To that end, I’ll manage the strategy to a drawdown target by reducing the leverage as a function of drawdown from all time equity highs. I’ll be turbo long (5x leverage) when I’m at all time highs,
and I’ll be at 0 leverage when I’m drawn down 90% from all time highs. I’ll reduce my leverage linearly between these two extremes as a function of drawdown.
But these are illiquid shitcoins that are flying around all over the place. I don’t want to be constantly rebalancing – that’d be a full time job. My silly season would quickly turn into boring
So how often should I rebalance things? What are the implications of rebalancing more or less frequently?
One way to answer that is through simulation – we can generate a statistically significant number of random outcomes and see how our decisions about leverage played out. Our results are only as good
as our assumptions that go into the simulator, but even if those assumptions are off, we can still gain some important intuition.
In this case, I’ll set up a Geometric Brownian Motion (GBM) simulator and generate a ton of different random price series that look like shitcoin prices.
A while back, I shared some code for generating GBM price series using vectorisation. You can check it out here.
But this time, I want to add some proper shitcoin dynamics:
• These things tend to show some autocorrelation (hence why trend following has worked on these in the past)
• They tend to be quite jumpy
I’ll add some autocorrelation using an autoregressive process. And I’ll add some random jumps via a jump diffusion process.
These things make the vectorised approach impossible, since subsequent values depend on previous ones.
But to speed things up, we’ll outsource the autoregressive and jump diffusion components to a C++ function and call that from our R session.
Let’s get to it.
# session options
options(repr.plot.width = 14, repr.plot.height=7, warn=-1)
# load and install packages
install.packages("tibbletime") # Rcpp is a dependency
# set chart options
theme_update(text = element_text(size = 20))
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.1.2 ✔ readr 2.1.4
✔ forcats 1.0.0 ✔ stringr 1.5.0
✔ ggplot2 3.4.1 ✔ tibble 3.2.1
✔ lubridate 1.9.2 ✔ tidyr 1.3.0
✔ purrr 1.0.1
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
Attaching package: 'tibbletime'
The following object is masked from 'package:stats':
# hourly GBM simulator with autocorrelation and jump diffusion
# C++ function for doing the autogregressive and jump diffusion bit
NumericMatrix generateAR1WithJumps(int nsim, int t, double phi, double lambda, double mu_j, double sigma_j) {
NumericMatrix epsilon(t-1, nsim);
NumericVector eta(nsim);
NumericVector jumpSize(nsim);
// Initial values from normal distribution
for(int j = 0; j < nsim; ++j) {
epsilon(0, j) = R::rnorm(0, 1);
// AR(1) process with jumps
for(int i = 1; i < t-1; ++i) {
for(int j = 0; j < nsim; ++j) {
eta[j] = R::rnorm(0, 1);
epsilon(i, j) = phi * epsilon(i-1, j) + eta[j];
// Generate jumps
if(R::runif(0, 1) < lambda) {
jumpSize[j] = R::rnorm(mu_j, sigma_j);
epsilon(i, j) += jumpSize[j];
return epsilon;
# gbm generator function
#' @param phi: autocorrelation
#' @param lambda: jump intensity
#' @param mu_j @param sigma_j: define the distribution of the jump sizes
#' other params as per standard GBM model
#' NOTE simulates hourly prices using annualised volatility and returns
gbm_autocor_jumps <- function(nsim = 100, t = 25, mu = 0, sigma = 0.1, S0 = 100, dt = 1./(365*24), phi = 0.5, lambda = 0.01, mu_j = 0, sigma_j = 0.1) {
# generate epsilon with AR(1) and jumps
epsilon <- generateAR1WithJumps(nsim, t, phi, lambda, mu_j, sigma_j)
# get GBM and convert to price paths
gbm <- exp((mu - sigma * sigma / 2) * dt + sigma * epsilon * sqrt(dt))
gbm <- apply(rbind(rep(S0, nsim), gbm), 2, cumprod)
# make some random price series with shitcoin-like parameters
nsim <- 100
t <- 100*24 # 100 days
mu <- -1
sigma <- 1.5
S0 <- 100
phi <- 0.1
set.seed(503) # so we can reproduce results
gbm <- gbm_autocor_jumps(nsim, t, mu, sigma, S0, phi = phi, lambda = 0.2, mu_j = 0, sigma_j = 1)
gbm_df <- as.data.frame(gbm) %>%
mutate(ix = 1:nrow(gbm)) %>%
pivot_longer(-ix, names_to = 'sim', values_to = 'price')
gbm_df %>%
ggplot(aes(x=ix, y=price, color=sim)) +
geom_line() +
theme(legend.position = 'none') +
labs(x = "hour")
Nice. That does indeed look a lot like a universe of shitcoin prices.
Next calculate positions for the ape new highs strategy. We’ll just ape new all time highs for 5 days. The details of the strategy aren’t important, it’s just for building intuition and getting a
feel for the dynamics.
# ape new highs strategy
# calculate positions
hold_period <- 5*24 # in hours
positions_df <- gbm_df %>%
group_by(sim) %>%
mutate(all_time_high = cummax(price)) %>%
mutate(idx_all_time_high = match(all_time_high, unique(price))) %>%
mutate(lagged_idx_all_time_high = dplyr::lag(idx_all_time_high)) %>%
mutate(periods_since_high = ix - lagged_idx_all_time_high) %>%
mutate(position = case_when(
between(periods_since_high, 1, hold_period) & lagged_idx_all_time_high > 1 ~ 1, TRUE ~ 0
A grouped_df: 6 × 8
ix sim price all_time_high idx_all_time_high lagged_idx_all_time_high periods_since_high position
<int> <chr> <dbl> <dbl> <int> <int> <int> <dbl>
2400 V95 28.052307 216.5835 343 343 2057 0
2400 V96 226.873371 270.0969 2048 2048 352 0
2400 V97 58.261634 108.7399 15 15 2385 0
2400 V98 53.079638 111.8936 63 63 2337 0
2400 V99 32.888193 104.3259 46 46 2354 0
2400 V100 6.144701 120.5961 65 65 2335 0
Here’s an example of when we’d be in a particular coin (blue marks):
# example plot of when we're in a position for a given series
positions_df %>%
mutate(position_plot = case_when(position == 1 ~ price, TRUE ~ NA_real_)) %>%
filter(sim == "V2") %>%
ggplot(aes(x = ix, y = price)) +
geom_line() +
geom_point(aes(x = ix, y = position_plot), colour = "blue") +
labs(title = "Positions in a single coin", x = "hour")
Calculate simple returns to the strategy with no leverage:
# simple returns
returns_df <- positions_df %>%
group_by(sim) %>%
# assume we enter a position at the hourly price at which we make a new high
mutate(simple_return = (price - dplyr::lag(price))/dplyr::lag(price)) %>%
mutate(strategy_simple_return = position*simple_return) %>%
na.omit() %>%
The next part is the tricky and fairly slow bit.
We simulate the returns to the ape new highs strategy being managed to a drawdown target.
It’s a time-consuming operation because our target leverage depends on our current drawdown, and so we must do it all in a big for loop.
Ideally I’d like to generate multiple universes of simulated shitcoins and then run this ape strategy simulation for each universe. That would generate a histogram of outcomes, rather than the single
outcome we see here, enabling us to make probabilistic conclusions and extract further insight.
I’ll leave to you. For the purposes of this post, I’ll just run it on the universe we generated previously. I’ll simulate a rebalance period of 24 hours (or when a position changes).
# will need to do in a for loop since actual leverage will be dependent on drawdown
# calculate portfolio pnl for each day
# adjust leverage for next day if needs be (if we're at rebal frequency or positions change)
# need a function for defining how we modify our leverage
set_leverage <- function(max_leverage, current_drawdown, max_acceptable_drawdown=0.9) {
# decrease leverage linearly with drawdown
# in practive, you might want to decrease it faster - maybe some non-linear function
leverage <- max(min(max_leverage - max_leverage/max_acceptable_drawdown*current_drawdown, max_leverage), 0)
# this bit is going to be slow
# could optimise by replacing dataframes with matrixes if we want to do lots
starting_leverage <- 5
rebalance_frequency <- 24 # in hours
starting_leverage <- 5
all_time_equity_high <- 1 # starting ATH
portfolio_results <- list()
for(i in min(returns_df$ix):max(returns_df$ix)) {
if(i == min(returns_df$ix)) {
# start by sizing all positions using starting leverage
leverage <- starting_leverage
num_positions <- returns_df %>%
filter(ix == i) %>%
# calculate current positions for target leverage
summarise(num_positions = sum(position)) %>%
size_per_position <- starting_leverage/num_positions
# adjust positions for all future rows (these will be updated in subsequent steps)
returns_df <- returns_df %>%
mutate(leveraged_position = position*size_per_position) %>%
mutate(strategy_simple_return = leveraged_position*simple_return)
# calculate current positions for target leverage
current_num_positions <- returns_df %>%
filter(ix == i) %>%
summarise(num_positions = sum(position)) %>%
# assume we rebalance at the start of the period if we change positions or hit our rebalance frequency
if(i %% rebalance_frequency == 0 | current_num_positions != num_positions) {
leverage <- set_leverage(starting_leverage, current_drawdown)
if(current_num_positions > 0)
size_per_position <- leverage/current_num_positions
# adjust positions for all future rows (these will be updated in subsequent steps)
returns_df <- returns_df %>%
mutate(leveraged_position = position*size_per_position) %>%
mutate(strategy_simple_return = leveraged_position*simple_return)
num_positions <- current_num_positions
# calculate leveraged portfolio returns
leveraged_port_returns <- returns_df %>%
filter(ix == i) %>%
# leveraged portfolio returns will be the equal-weighted strategy returns scaled to the target leverage
group_by(ix) %>%
summarise(leveraged_returns = sum(strategy_simple_return, na.rm = TRUE)) %>%
mutate(leverage = leverage)
# store results
portfolio_results[[i-1]] <- leveraged_port_returns
# keep a running portfolio equity and drawdown
current_port_returns <- portfolio_results %>%
bind_rows() %>%
# just use simple returns
mutate(cum_port_returns = cumprod(1 + leveraged_returns)) %>%
tail(1) %>%
all_time_equity_high <- max(current_port_returns, all_time_equity_high)
current_drawdown <- all_time_equity_high - current_port_returns
# plot results
results_df <- bind_rows(portfolio_results) %>%
mutate(cum_port_returns = cumprod(1 + leveraged_returns))
# get all-time-high to plot drawdown target
drawdown_target <- max(results_df$cum_port_returns)*0.1
returns_plot <- results_df %>%
ggplot(aes(x = ix, y = cum_port_returns)) +
geom_line() +
geom_hline(yintercept = drawdown_target, colour = "red", size = 1, linetype = "dashed") +
x = "Hour",
y = "Portfolio return",
title = glue::glue("Portfolio returns and leverage, rebal frequency {rebalance_frequency} hours"),
subtitle = "Drawdown target shown in red (90% off ATH)"
leverage_plot <- results_df %>%
ggplot(aes(x = ix, y = leverage)) +
geom_line() +
x = "Hour",
y = "Leverage"
returns_plot / leverage_plot + plot_layout(heights = c(2,1))
You can see in the plot above that the strategy performed dreadfully, at least on the particular simulated universe we generated here.
However, you can also see that I was able to manage it to a drawdown target rebalancing once every 24 hours (or whenever we got a new position), thus ensuring that my silly season wasn’t totally
In this post, we shared an efficient way to generate Geometric Brownian Motion (GBM) price series that include autocorrelation and jumps.
One can use such a tool to gain intuition into various noisy, random processes, such as turbo-punting shitcoins to a drawdown target.
In this example, we used the GBM simulator to get a feel for how often we might need to rebalance an overly leveraged shitcoin trend-following strategy. We can run the simulator with various
parameterisations and get a feel for how often we destroy our capital under different conditions, or how often we should think about rebalancing back to our target leverage.
I hope you have as much fun as I intend to this silly season.
2 thoughts on “How Much Damage Can I Do Turbo-Punting Shitcoins?”
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z ratio
31 Aug 2024
Title: The Z Ratio: A Measure of Central Tendency and Dispersion
Abstract: The Z ratio is a statistical measure that combines the concepts of central tendency and dispersion, providing a comprehensive overview of a dataset’s characteristics. This article delves
into the theoretical framework and mathematical formulation of the Z ratio, highlighting its potential applications in various fields.
Introduction: In statistics, measures of central tendency (e.g., mean, median) and dispersion (e.g., variance, standard deviation) are essential tools for understanding data distributions. However,
these metrics often provide incomplete information about a dataset’s characteristics. The Z ratio offers a unified approach to quantifying both central tendency and dispersion.
Theoretical Framework: Let X be a random variable with mean μ and standard deviation σ. The Z ratio is defined as:
Z = (μ / σ)
This formula combines the mean (a measure of central tendency) with the standard deviation (a measure of dispersion).
1. Scale invariance: The Z ratio is invariant to changes in scale, meaning that multiplying all values by a constant factor will not affect the Z ratio.
2. Unit-free: The Z ratio is unit-free, as it does not depend on any specific units of measurement.
Interpretation: The Z ratio can be interpreted as follows:
• A high Z ratio indicates that the mean value is significantly larger than the standard deviation, suggesting a dataset with a well-defined central tendency and relatively small dispersion.
• A low Z ratio suggests that the mean value is close to zero or has a large standard deviation, indicating a dataset with a less defined central tendency and/or larger dispersion.
Conclusion: The Z ratio offers a concise and informative measure of both central tendency and dispersion. Its properties make it an attractive tool for data analysis in various fields, including
statistics, engineering, economics, and social sciences. Further research is needed to explore the practical applications and limitations of the Z ratio.
Related articles for ‘z ratio’ :
Calculators for ‘z ratio’ | {"url":"https://blog.truegeometry.com/tutorials/education/31c092cc878a07d5b00eb7ae567f6d1d/JSON_TO_ARTCL_z_ratio.html","timestamp":"2024-11-05T15:57:35Z","content_type":"text/html","content_length":"14529","record_id":"<urn:uuid:098c807e-b617-4786-90e7-c05231d85e8c>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00704.warc.gz"} |
LC-PFN: Efficient Bayesian Learning Curve Extrapolation using Prior-Data Fitted Networks
Posted on December 8, 2023 by Steven Adriaensen
Authors: Steven Adriaensen*, Herilalaina Rakotoarison*, Samuel Müller, and Frank Hutter
In our paper, we propose LC-PFN, a novel method for Bayesian learning curve extrapolation. LC-PFN is a prior-data-fitted network (PFN), a transformer trained on synthetic learning curve data capable
of doing Bayesian learning curve extrapolation in a single forward pass. We show that our approach is 10.000 faster than the state-of-the-art using MCMC (Domhan et al, 2015), without loss of
performance, effectively enabling applications thereof in AutoML, at almost zero overhead.
Learning Curve Extrapolation
Arguably, what sets machine learning apart from other approaches to AI, is its ability “to improve its performance with experience” (Russel and Norvig, 2010). A learning curve (Mohr and van Rijn,
2022) characterizes the relationship between the performance of an agent as a function of its experience. For example, in deep learning, the learning curve typically describes the loss of the neural
network model being trained as a function of the number of epochs it has been trained for. Here, learning curve extrapolation aims to predict model performance in later epochs of a machine learning
training, based on the performance in the first epochs. These predictions are particularly useful in the context of AutoML, as they allow us to stop expensive training runs that will not produce
models better than the best model seen thus far.
Being Bayesian about Learning Curves
Why? Learning curves are not always as smooth and predictable as those depicted above. Performance improvements may be non-monotonic (e.g., worsening steps), performance estimates noisy (e.g.,
mini-batch evaluations), learning may fail (e.g., divergence), or seem to fail and recover (e.g., double descent). In these cases, it may simply be impossible to confidently predict what exactly will
What? Bayesian inference presents a general probabilistic framework for reasoning about uncertainty. Its essence is captured in Bayes’ Rule / Theorem:
In our case, hypotheses take the form of learning curves and evidence is presented as a partial learning curve until some cutoff T. While the equation gives us the posterior over learning curves,
what we truly care about is the posterior predictive distribution (PPD), a distribution over model performances at epoch t’ > T, given we performances observed thus far, i.e., Bayesian prediction.
How? Before we can do Bayesian prediction, we need to define a prior. In our case, a prior over learning curves. As the name suggests, this distribution should ideally capture what we believe
learning curves may look like, assigning high probability to those curves that are more likely to be observed than others. While coming up with a good learning curve prior is a challenging task in
itself, it was not the main focus of our work, so we adopted the one from prior art (Domhan et al, 2015), with some modifications (see Section 3.2 the paper for all the technicalities). In a
nutshell, Domhan et al (2015), modeled learning curves as linear combinations of monotonic basis curves, with additive Gaussian noise, and defined a prior over the parameters of this model, i.e., the
weights and parameters for each basis curve, as well as the scale of the Gaussian noise. A sample of 10 curves from this prior is shown below.
Now, in theory, given this prior and a partial learning curve, we can infer the posterior by applying Bayes’ rule. In practice, sadly, calculating the denominator in this equation (i.e., the marginal
likelihood) exactly is often intractable and we must resort to approximate inference methods. Here, (Domhan et al, 2015) used Markov Chain Monte Carlo (MCMC). In our paper, we propose LC-PFN, an
alternative approach using prior-data-fitted networks (PFNs, Mülller et al, 2021).
While the predictions of both methods are often similar, the true difference lies in the computational cost required to obtain them. The above inferences took about a minute using MCMC, but only a
couple of milliseconds using LC-PFN (on CPU). While a minute may sound reasonable, it inhibits fine-grained, general-purpose applications in, e.g., AutoML systems.
PFNs, Transformers that can do Bayesian inference (efficiently)!
While various methods for ‘general’ approximate Bayesian inference exist (e.g., MCMC, Bayesian neural networks, (deep) Gaussian processes, variational inference, etc.), they are all either
computationally expensive, or restrictive in terms of distributional assumptions, or both. Recently, Mülller et al (2021) proposed prior-data fitted networks (PFNs) an efficient and flexible approach
to Bayesian inference.
LC-PFN: Conceptually, PFNs are neural networks trained to do Bayesian inference in a single forward pass, i.e. given evidence (partial learning curve) as input, they can predict the likelihood of
every given hypothesis.
At training time, they are given masked examples sampled from the prior and must infer the masked-out part (minimizing log loss). In our case, LC-PFN is a decoder-only (GPT-style) transformer,
trained on 10 million right-censored artificial learning curves that were sampled from the prior described above. Input tokens represent points (t, y[t]) of the partial learning curve (t < T). The
output of the transformer for a query t’ > T-1 is a discretized version of the PPD for y[t’].
Results: LC-PFN is Faster and Better!
In the first experiment, we compare the existing MCMC and our LC-PFN approaches on artificial learning curves sampled from the prior. To analyze the effect of hyperparameters (and scale), we train
different variants of each method. For a fair comparison, we ran method methods on CPU (despite LC-PFN running even faster on GPU).
We find that all (but the smallest) LC-PFNs trained on 10M curves, outperform all MCMC variants in terms of PPD approximation, despite being multiple orders of magnitude faster. Unsurprisingly, we
find that bigger is better for LC-PFN, at the cost of slightly slower inference. Note that training the
largest LC-PFN (P3, 10M samples with 26M parameters) on the prior took approximately eight hours
(single CPU, single RTX2080 GPU), but this cost is incurred only once for all of our experiments.
Results: LC-PFN works for real!
In a second experiment, we investigate the out-of-distribution (ood) performance of both methods, extrapolating 20.000 real learning curves sourced from four different benchmarks (LCBench,
NAS-Bench-201, Taskset, and PD1) that stem from training a wide range of model architectures (MLPs, CNNs, RNNs, and Transformers) on 53 different datasets with varying input modalities (tabular,
image, text, and protein data). To account for our modifications to the prior, we also include the variant using the original prior, in our comparison.
We find that while LC-PFN does not always produce the best extrapolation, it arguably still is (across all benchmarks) preferable among the three approaches compared, despite being multiple orders of
magnitude faster.
Results: LC-PFN, can it be stopped?!
Finally, to get an idea of how practically useful the LC-PFN’s extrapolations are in the context of AutoML, we evaluate its use in a predictive early-discarding criterion in the context of vertical
model selection. Here, multiple training runs (~ learning curves) for a task (using different hyperparameters) are considered in a predetermined order, and we must decide whether to continue the
current training or stop and start the next training. As we consider a limited budget of 20 full training runs, stopping non-promising runs early is essential. We stop a run, once LC-PFN predicts the
likelihood of obtaining a model better than the best one found thus far is less than 5%.
In our experiments, we observed a 2-6 factor speedup (compared to not stopping) on 45 of the 53 tasks considered. We only observed slowdowns on the 3 NAS-Bench-201 tasks, however, most learning
curves in this benchmark are very unlikely under our prior, hence this failure can likely be attributed to the choice of learning curve prior, not LC-PFN.
Try LC-PFN yourself!
All our trained LC-PFN models, as well as code to train them, can be found on Github.
Don’t like code? Check out our code-free demo on Hugging Face!
(Domhan et al, 2015) Domhan, T., Springenberg, J. T., & Hutter, F. (2015, June). Speeding up automatic hyperparameter optimization of deep neural networks by extrapolation of learning curves. In
Twenty-fourth international joint conference on artificial intelligence.
(Russel and Norvig, 2010) Russell, S. J., & Norvig, P. (2010). Artificial intelligence a modern approach. London.
(Mohr and van Rijn, 2022) Mohr, F., & van Rijn, J. N. (2022). Learning Curves for Decision Making in Supervised Machine Learning–A Survey. arXiv preprint arXiv:2201.12150.
(Müller et al, 2021) Müller, S., Hollmann, N., Arango, S. P., Grabocka, J., & Hutter, F. (2021, October). Transformers Can Do Bayesian Inference. In International Conference on Learning | {"url":"https://www.ml4aad.org/lc-pfn-efficient-bayesian-learning-curve-extrapolation-using-prior-data-fitted-networks/","timestamp":"2024-11-02T20:47:54Z","content_type":"text/html","content_length":"93746","record_id":"<urn:uuid:115d6e05-6155-42ab-bac9-e37a240caee2>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00071.warc.gz"} |
Frequency converter
A BPF outputs a real RF signal by suppressing a band out of an RF signal frequency band in a received signal. A local oscillator outputs a complex local signal with a predetermined frequency. A
half-complex mixer performs frequency conversion by multiplying the real RF signal by a real part of the local signal, performs frequency conversion by multiplying the real RF signal by an imaginary
part of the local signal, and outputs a complex signal separated by the predetermined frequency from a frequency of the real RF signal. A complex-coefficient SAW filter performs a convolution
integral on an impulse response generated by an even function for a real part of the complex signal, performs a convolution integral on an impulse response generated by an odd function for an
imaginary part of the complex signal, and outputs a real signal by suppressing one side of a positive or negative frequency.
Latest Samsung Electronics Patents:
This application claims priority under 35 U.S.C. § 119 to applications entitled “Frequency Converter” filed in the Japan Patent Office on Dec. 20, 2005 and assigned Serial No. 2005-366732 and in the
Korean Intellectual Property Office on Nov. 22, 2006 and assigned Serial No. 2006-115265, the contents of each of which are incorporated herein by reference.
1. Field of the Invention
The present invention relates to a frequency converter for use in a wireless transceiver.
2. Description of the Related Art
Wireless communicators function both as a receiver and a transmitter like a mobile phone. The receiver, i.e., a downconverter, receives a radio frequency (RF) signal with conversation content and
data communication content and converts the received RF signal to a frequency to be input to a demodulator. Further, as a front-end scheme for selecting a target signal in the downconverter, there is
a heterodyne scheme for converting an RF signal to an intermediate frequency (IF) signal without directly frequency-converting the RF signal to a baseband signal. Because this heterodyne scheme
easily implements a broadband front end, the heterodyne scheme is recently attracting interest as the architecture of a front end of a software radio device. However, there are the following
technical problems as well as a problem of an increase in the cost of components due to the broad band when the heterodyne scheme is applied to the broad band.
FIG. 14 illustrates a structure of a downconverter 10 serving as a frequency converter of the heterodyne scheme for down-converting an RF signal to an IF signal lower than an RF signal frequency. The
downconverter of the heterodyne scheme receives the RF signal through an antenna, suppresses a band less than an RF signal frequency band saturating a front end by a first band path (or pass) filter
(BPF) 1001, and outputs the RF signal frequency band. A low noise amplifier (LNA) 1002 amplifies the output signal of the BPF 1001. A second-step BPF 1003 suppresses a band out of the frequency band
of a target RF signal in the amplified signal, and outputs the frequency band of the target RF signal. Then, a mixer 1004 performs conversion to a frequency of an IF signal by multiplying a signal
output from the BPF 1003 by a local signal output from a local oscillator (Local) 1006. Then, a BPF 1005 outputs a frequency band of the IF signal. In wireless communication devices, the IF signal is
converted to a baseband signal in a digital process. In a conventional wireless receiver, the IF signal is again frequency-converted in an analog process and is converted to the baseband signal.
On the other hand, the downconverter 10 of the heterodyne scheme down-converts bands of high and low frequency sides symmetrical to the center of the local signal of the local oscillator 106 to the
same frequency band. For example, as illustrated in FIG. 15A, signals SB1 and SB2 with the center of a frequency Lo of the local signal are present in a frequency band of related positions. When the
mixer 1004 performs the frequency conversion, an image frequency signal SB1-I of the signal SB1 and the signal SB2 present in the related positions are down-converted to the same frequency band.
Thus, when the signal SB2 is a target signal to be output, the image frequency signal interferes with the associated target signal.
To eliminate the interference of the image frequency signal, a frequency difference between the image frequency signal SB1 and the signal SB2 serving as the RF signal before the frequency conversion
increases by increasing the frequency of the IF signal as illustrated in FIG. 15B. Further, a characteristic of the BPF 1003 is set to suppress a frequency band of the image frequency signal SB1.
Thus, the frequency band of the image frequency signal SB1 is suppressed. As illustrated in FIG. 15C, the effect of the image frequency signal SB1-I to the IF signal SB2 is suppressed when the mixer
1004 performs the frequency conversion.
FIG. 16 illustrates an upconverter 11 serving as a frequency converter of the heterodyne scheme for up-converting an IF signal to an RF signal greater than the frequency of the IF signal. Similar to
the downconverter 10, the upconverter 11 suppresses an image frequency signal of the IF signal occurring after up-conversion. Thus, the upconverter 11 is provided with a BPF 1105 for increasing the
frequency of the input IF signal and suppressing the image frequency signal of the IF signal. The image frequency signal is suppressed for the RF signal.
However, means applied to the downconverter 10 and the upconverter 11 may unnecessarily increase the frequency of the IF signal to eliminate the interference from the image frequency signal. For this
reason, there is a problem in that power consumption increases in a structure after an IF stage.
There is a problem in that the image frequency signal must be able to be suppressed using a BPF of a steep characteristic or at least two BPFs since the requirements of the BPF 1003 and the BPF 1105
are strict even when the frequency of the IF signal is as low as possible. Multiple BPFs are required when an RF signal accompanied with the recent broad band has a multi-band. However, when the
multiple BPFs of the steep characteristic are provided, products increases in terms of size and/or cost.
To address the above-described problems, there has been proposed the technology described in Hiroshi Tsurumi, Hiroshi Yoshida, Shoji Otaka, Hiroshi Tanimoto, Yasuo Suzuki, “Broadband and Flexible
Receiver Architecture for Software Defined Radio Terminal Using Direct Conversion and Low-IF Principle”, IEICE TRANS. COMMUN., Vol. E83-B, No. 6, June 2000, pp. 1246-1253 (Tsurumi). Tsurumi proposes
a downconverter 12 using a half-complex mixer (or image rejection or suppression mixer) 1203 and a polyphase filter 1204 as illustrated in FIG. 17. When an RF signal is input to the downconverter 12
as illustrated in FIG. 18A, a first BPF 1201 suppresses a signal out of a frequency band of the RF signal and an LAN 1202 amplifies the signal after suppression as illustrated in FIG. 18B. Then, the
half-complex mixer 1203 performs frequency conversion while suppressing an image frequency signal SC1-I overlapping with a target signal SC2 when the amplified signal is multiplied by a complex local
signal. When a frequency-converted signal is input to the polyphase filter 1204, the polyphase filter 1204 suppresses a negative frequency band as illustrated in FIG. 18D and outputs a real IF signal
as illustrated in FIG. 18E. Since a BPF 1205 suppresses a frequency band out of the frequency band of the IF signal in the signal output from the polyphase filter 1204, a signal SC3 is suppressed and
a signal in which the target signal SC2 overlaps with the suppressed image frequency signal SC1-I is output as illustrated in FIG. 18F.
Consequently, the image frequency signal SC1-I is suppressed in a state in which a suppression ratio of the BPF 1201 is added to a suppression ratio of the half-complex mixer 1203 and the effect of
the image frequency signal to the target signal SC2 can be suppressed. The frequency of the IF signal can be limited to a low frequency without unnecessarily increasing the frequency of the IF signal
only for the suppression of the image frequency signal as in the prior art. Further, a BPF such as the BPF 1003 of FIG. 14 for suppressing the band of the image frequency signal is not required.
However, the polyphase filter 1204 uses a conventional passive type filter. Since a passive polyphase filter is constructed with a RC circuit, loss is large. Since the passive polyphase filter
outputs a signal without suppressing a positive frequency band, the BPF 1205 of the IF stage is mandatory to output the frequency band of the IF signal. For this reason, there is a problem in that
the loss due to the polyphase filter 1204 is added to the loss due to the BPF 1205 of the IF stage when a real IF signal is output.
Accordingly, the present invention has been designed to solve the above and other problems. Therefore, it is an object of the present invention to provide a frequency converter that can further
reduce loss while limiting the frequency of an intermediate frequency (IF) signal to a low frequency in a heterodyne scheme.
In accordance with an aspect of the present invention, there is provided a frequency converter for frequency-converting a received radio frequency (RF) signal to an IF, including a real-coefficient
filter for outputting a real RF signal by suppressing a band of an RF signal frequency band in a received signal; a local oscillator for outputting a complex local signal with a predetermined
frequency; a complex mixer for performing frequency conversion by multiplying the real RF signal output from the real-coefficient filter by a real part of the complex local signal output from the
local oscillator, performing frequency conversion by multiplying the real RF signal by an imaginary part of the complex local signal output from the local oscillator, and outputting a complex signal
separated by the predetermined frequency from a frequency of the real RF signal; and a complex-coefficient transversal filter for performing a convolution integral based on an impulse response
generated by an even function for a real part of the complex signal output from the complex mixer, performing a convolution integral based on an impulse response generated by an odd function for an
imaginary part of the complex signal output from the complex mixer, and outputting a real signal from the complex signal by suppressing one side of a positive frequency or a negative frequency. This
structure can perform frequency conversion to a low frequency while suppressing an image frequency signal in the complex mixer. When the complex signal is converted to a real signal, conversion can
be performed while suppressing one side of a positive frequency or a negative frequency.
In accordance with another aspect of the present invention, there is provided a frequency converter for frequency-converting an input IF signal to an RF signal frequency, including a
complex-coefficient transversal filter for performing a convolution integral based on an impulse response generated by an even function for a real signal of an input IF, performing a convolution
integral based on an impulse response generated by an odd function for the real signal, and outputting a complex signal by suppressing one side of a positive frequency or a negative frequency; a
local oscillator for outputting a complex local signal with a predetermined frequency; a complex mixer for performing frequency conversion by multiplying a real part of the complex signal output from
the complex-coefficient transversal filter by a real part of the complex local signal output from the local oscillator, performing frequency conversion by multiplying an imaginary part of the complex
signal by an imaginary part of the complex local signal output from the local oscillator, and outputting a real signal of a frequency separated by the predetermined frequency from a frequency of the
input signal; and a real-coefficient filter for outputting a real RF signal by suppressing a frequency band out of an RF signal frequency band for the real signal output from the complex mixer. This
structure can convert an input real signal to a complex signal while suppressing one side of a positive frequency or a negative frequency and can frequency-convert the complex signal to an RF signal
frequency while suppressing an image frequency signal.
The above and other objects, features and advantages of the present invention will be more clearly understood from the following detailed description taken in conjunction with the accompanying
drawings, in which:
FIG. 1 is a block diagram illustrating an internal structure of a downconverter in accordance with the present invention;
FIG. 2 illustrates a frequency conversion process of the downconverter in accordance with the present invention;
FIG. 3 illustrates an impulse response of a real part of a complex-coefficient transversal filter used in the downconverter in accordance with the present invention;
FIG. 4 illustrates an impulse response of an imaginary part of the complex-coefficient transversal filter used in the downconverter in accordance with the present invention;
FIG. 5 illustrates a structure of a first complex-coefficient SAW filter used in the downconverter in accordance with the present invention;
FIG. 6 illustrates a structure of a second complex-coefficient SAW filter used in the downconverter in accordance with the present invention;
FIG. 7 is a block diagram illustrating an internal structure of an upconverter in accordance with the present invention;
FIG. 8 illustrates a structure of a first complex-coefficient SAW filter used in the upconverter in accordance with the present invention;
FIG. 9 illustrates a structure of a second complex-coefficient SAW filter used in the upconverter in accordance with the present invention;
FIG. 10 is a block diagram illustrating an internal structure of a first downconverter in accordance with the present invention;
FIG. 11 is a block diagram illustrating an internal structure of a first upconverter in accordance with the present invention;
FIG. 12 is a block diagram illustrating an internal structure of a second downconverter in accordance with the present invention;
FIG. 13 is a block diagram illustrating an internal structure of a second upconverter in accordance with the present invention;
FIG. 14 illustrates an internal structure of a downconverter in a heterodyne scheme according to the prior art;
FIG. 15 illustrates a frequency conversion process by the downconverter in the heterodyne scheme according to the prior art;
FIG. 16 illustrates an internal structure of an upconverter according to the prior art;
FIG. 17 illustrates an internal structure of a downconverter using a polyphase filter according to the prior art; and
FIG. 18 illustrates a frequency conversion process of the polyphase filter according to the prior art.
A downconverter serving as a frequency converter for frequency-converting a radio frequency (RF) signal to an intermediate frequency (IF) signal less than a frequency of the RF signal and an
upconverter serving as a frequency converter for frequency-converting an IF signal to an RF signal greater than a frequency of the IF signal in accordance with the present invention will be described
in detail herein below with reference to the accompanying drawings. In the following description, detailed descriptions of functions and configurations incorporated herein that are well known to
those skilled in the art are omitted for clarity and conciseness.
FIG. 1 is a block diagram illustrating a downconverter 1 in accordance with the present invention. The downconverter 1 is provided with a band path (or pass) filter (BPF) 110, a low noise amplifier
(LNA) 112, a local oscillator 120, a half-complex mixer 114, and a complex-coefficient surface acoustic wave (SAW) filter 116. The downconverter 1 frequency-converts a received RF signal to a
frequency of an IF signal, i.e., the IF signal frequency. The BPF 110 outputs a signal obtained by suppressing a band of an RF signal frequency band saturating a front end. For example, when an RF
signal frequency bandwidth is set to 100 MHz, a signal of a band of 100 MHz is passed and other bands are suppressed. The LAN 112 amplifies a signal output from the BPF 110 and then outputs the
amplified signal. The local oscillator (Local) 120 outputs a complex local signal constructed with a real axis local signal with a phase of cos and an imaginary local signal with a phase of −sin at a
predetermined frequency. Herein, the predetermined frequency has a frequency value computed by subtracting the IF signal frequency from the RF signal frequency. The half-complex mixer 114 is
connected to the local oscillator 120 and is provided with a mixer-I 121 and a mixer-Q 122 serving as multipliers. The half-complex mixer 114 outputs a complex signal of an IF frequency separated by
a predetermined frequency from the frequency of the RF signal, i.e., a complex IF signal, while suppressing an image frequency signal.
In the half-complex mixer 114, the mixer-I 121 converts a real RF signal to the IF signal frequency separated by the predetermined frequency from the RF signal frequency by multiplying the real RF
signal output from the LNA 112 by a real axis local signal output from the local oscillator 120, and outputs a real axis component of the complex IF signal. The mixer-Q 122 converts the real RF
signal to the IF signal frequency separated by the predetermined frequency from the RF signal frequency by multiplying the real RF signal output from the LNA 112 by an imaginary axis local signal
output from the local oscillator 120, and outputs an imaginary axis component of the complex IF signal.
The complex-coefficient SAW filter 116 functions as a complex-coefficient transversal filter constructed with a SAW filter. The complex-coefficient SAW filter 116 outputs a real IF signal by
suppressing a negative frequency component of the complex signal output from the half-complex mixer 114 and performing a subtraction process for the complex signal after suppression.
The complex-coefficient transversal filter obtained by generalizing the filter function of the complex-coefficient SAW filter 116 and the principle of the complex-coefficient SAW filter 116
constructed with the complex-coefficient transversal filter using SAWs will be described.
The complex-coefficient transversal filter is constructed with two BPFs. One BPF performs a convolution integral with an even-symmetric impulse response for a real axis signal of an input complex
signal and the other BPF performs a convolution integral with an odd-symmetric impulse response for an imaginary axis signal of the input complex signal. This structure can suppress one side of a
positive frequency or a negative frequency and can obtain the filter effect of suppressing a signal out of the band of a target signal at a frequency side.
Among the impulse responses, an impulse response of a real part of the complex-coefficient transversal filter is a signal as illustrated in FIG. 3, which is even symmetric with respect to the
envelope center and corresponds to the even-symmetric impulse. An impulse response of an imaginary part of the complex-coefficient transversal filter is a signal as illustrated in FIG. 4, which is
odd symmetric with respect to the envelope center and corresponds to the odd-symmetric impulse. Since a phase difference between the even-symmetric impulse and the odd-symmetric impulse is 90
degrees, a signal with the phase difference of 90 degrees between the real part and the imaginary part is output when an in-phase component signal is input to the real axis signal and the imaginary
axis signal.
For example, the complex-coefficient transversal filter is designed by a frequency shift method.
That is, a real-coefficient low path (or pass) filter (LPF) of a predetermined pass bandwidth Bw/2 and a stop-band attenuation amount ATT is designed and a coefficient of the real-coefficient LPF is
multiplied by e^jωt. A filter of a center frequency ω, a pass bandwidth Bw, and a stop-band attenuation amount ATT can be obtained. In detail, the complex-coefficient transversal filter can be
designed in which a center frequency ω=190 MHz, a stop-band attenuation amount ATT=35 dB, and a sampling frequency=100 MHz. Thus, the complex-coefficient transversal filter can be obtained which
suppresses other frequency band signals out of a predetermined frequency band with the center of a positive frequency of 190 MHz by 35 dB.
The complex-coefficient transversal filter serving as a means for implementing the complex-coefficient transversal SAW filter 116 will be described. Conventionally, the complex-coefficient
transversal filter can be implemented with a switched capacitor circuit or a charge-coupled device as well as the SAW filter. The SAW filter is suitable to implement the transversal filter of a high
frequency. The basic principle of the transversal SAW filter will be described.
FIG. 5 illustrates a structure of the complex-coefficient SAW filter 116. The complex-coefficient SAW filter 116 is constructed with a piezoelectric substrate 2005 and comb shaped electrodes
(hereinafter, referred to as Inter-Digital Transducers (IDTs)) 2001 to 2004 in which an intersection width differs according to a position on the piezoelectric substrate 2005.
Comb shaped parts are also referred to as electrode fingers.
The principle of the complex-coefficient SAW filter 116 will be described. When an impulse electric signal is applied, an impulse response of a SAW signal output as the SAW is produced on the basis
of a weight function (or intersection width) W[i ]in each electrode finger, a distance x[i ]from each electrode finger, and a phase velocity v of the SAW. A frequency transfer function H(ω) of the
impulse response is expressed by Equation (1). Equation (1) represents a linear combination of the weight function W[i]. The basic principle of the complex-coefficient SAW filter 116 is the same as
that of the complex-coefficient transversal filter. $H ( ω ) = ∑ i = 0 n W i exp ( - j ω x i v ) ( 1 )$
The complex-coefficient transversal filter with an associated frequency transfer function H(ω) can independently control amplitude characteristics and phase characteristics by designing W[i ]and x
[i]. That is, the complex-coefficient transversal filter with desired characteristics can be implemented by designing W[i ]and x[i ]of the transversal SAW filter.
To perform a weighting operation mapped to an impulse response of a real part, i.e., an even-symmetric impulse response, the IDT 2001, connected to an input terminal I for inputting a real axis
component, is provided with an electrode finger such that even symmetry is formed with respect to the envelope center. To perform a weighting operation mapped to an impulse response of an imaginary
part, i.e., an odd-symmetric impulse response, the IDT 2002, connected to an input terminal Q for inputting an imaginary axis component, is provided with an electrode finger such that odd symmetry is
formed with respect to the envelope center.
The IDT 2003 is connected to an output terminal, and is provided on a propagation path of the IDT 2001 for performing a convolution integral of a real part. The IDT 2004 is connected to the output
terminal, and is provided on a propagation path of the IDT 2002 for performing a convolution integral of an imaginary part. According to the above-described structure, SAWs excited from the IDTs 2001
and 2002 of an input side are propagated at a phase difference of 90 degrees, and are received in the IDTs 2003 and 2004 of an output side. The IDTs 2003 and 2004 are connected to each other such
that phases are reverse with respect to each other. According to this structure, an imaginary component is subtracted from a real component, such that a real RF signal is output from the output
Similarly, a real RF signal can be output even when the IDTs 2001 and 2002 for which a weighting operation mapped to an impulse response is performed are connected to the output terminal and the IDTs
2003 and 2004 are connected to the input terminals.
The operation of the complex-coefficient SAW filter 116 will be described. First, when a complex RF signal is input to the input terminals, mechanical distortion is caused by piezoelectricity in the
IDTs 2001 and 2002 and SAWs are excited and propagated in the left and right directions of the piezoelectric substrate 2005. The SAW signals are propagated while a convolution integral is performed
on impulse responses of real and imaginary parts and the complex RF signal. The SAW signals propagated from the IDTs 2001 and 2002 are received by the IDTs 2003 and 2004 provided in propagation
directions of the SAW signals, such that they are converted to electric signals. At this time, the IDT 2003 outputs a real component of the RF signal, and the IDT 2004 outputs an imaginary component
of the RF signal whose polarity is inverted. The output terminal outputs a real RF signal by subtracting the imaginary component from the real component. According to this structure, the
complex-coefficient SAW filter 116 can output a real IF signal while suppressing a frequency band out of the frequency band of the complex IF signal.
FIG. 6 illustrates another structure of the complex-coefficient SAW filter 116. The two IDTs 2003 and 2004 are provided in the output side as illustrated in FIG. 5. The complex-coefficient SAW filter
116 of FIG. 6 has a structure for receiving SAWs in one IDT 2013 connected to an output side. In this case, an IDT 2012 is provided which has an inverse of the polarity of the IDT 2002 mapped to the
imaginary part of the input side of FIG. 5, such that a subtraction process can be realized. The inverse polarity is not limited to the imaginary part. The polarity of the real part may be inverted.
According to this structure, one IDT can be provided in the output side.
The operation of the downconverter 1 will be described with reference to FIG. 2.
A real RF signal S11 received through an antenna is input from an RF terminal. The real signal S11 includes three signals SA1, SA2, and SA3 as illustrated in FIG. 2A. A target signal to be output is
the signal SA2. A signal causing an image frequency signal to the signal SA2 is the signal SA1 present in a related position with respect to the center of a predetermined frequency Lo of the local
signal of the local oscillator 120. For example, when the center frequency of the signal SA2 is 800 MHz and an IF is 190 MHz, the predetermined frequency of the local signal is 610 MHz and the signal
SA1 causing the image frequency signal has a frequency of 420 MHz.
When the real RF signal is input to the BPF 110, the BPF 110 outputs a signal S12 while suppressing a signal of a band out of the RF signal frequency band saturating the front end, for example, the
signal SA1 of FIG. 2B.
The LAN 112 amplifies the signal S12 and a signal S13 is input to the half-complex mixer 114. The mixer-I 121 multiplies one signal branched from the signal S13 input to the half-complex mixer 114 by
a real part local signal of the complex local signal with the predetermined frequency Lo output from the local oscillator 120. The mixer-Q 122 multiplies the other branched signal by an imaginary
part local signal of the complex local signal with the predetermined frequency Lo output from the local oscillator 120. Thus, when the image frequency signal SA1-I related to the signal SA1 is
suppressed, the half-complex mixer 114 outputs signals S14I and S14Q with a phase difference of 90 degrees from each other. A complex IF signal S14 is obtained in which a signal S14I is a real axis
component and a signal S14Q is an imaginary axis component.
The complex IF signal S14 obtained through this process is shown in FIG. 2C. In FIG. 2C, the signal SA1-I is the image frequency signal mapped to the signal SA1 when the half-complex mixer 112
performs the frequency conversion. According to the above-described frequency values, the signal SA1 is at 190 MHz and the signals SA1-I and SA2 are at +190 MHz.
A strength difference between the signal SA1 and the signal SA1-I compared with the target signal SA2 is a difference due to a suppression ratio of the half-complex mixer 114 and is based on a sum of
a suppression ratio of the BPF 110 and the suppression ratio of the half-complex mixer 114. For example, when the BPF 110 has the suppression ratio of about 30 dB and the half-complex mixer 114 has
the suppression ratio of about 30 dB, the strength difference of the signal SA1 and the signal SA1-I is about 60 dB, such that the effect of the image frequency signal can be significantly
If the complex-coefficient SAW filter 116 is designed at the center frequency of 190 MHz, the complex-coefficient SAW filter 116 has a filter characteristic as indicated by the dotted line of FIG.
2D. When the complex IF signal S14 is input to the complex-coefficient SAW filter 116, the signals SA2, SA1-I, and SA3 at a positive frequency undergo the convolution integral with an even-symmetric
impulse response. On the other hand, the signal SA1 at a negative frequency undergoes the convolution integral with an odd-symmetric impulse response. When a subtraction process is performed for a
real part signal after the convolution integral with the even-symmetric impulse response and an imaginary part signal after the convolution integral with the odd-symmetric impulse response, a real IF
signal S15 is output. In detail, the signal SA2 and the suppressed image frequency SA1-I are obtained as the real IF signal S15 when the complex-coefficient SAW filter 116 suppresses a signal of a
frequency band out of a target signal bandwidth with the center frequency of 190 MHz as illustrated in FIG. 2E.
In the above-described structure, the downconverter 1 can lower the frequency of the IF signal to a low frequency without unnecessarily increasing the frequency of the IF signal only for the
suppression of the image frequency signal as in the downconverter 10 of the conventional heterodyne scheme as illustrated in FIG. 14. Thus, power consumption can be reduced in a structure after an IF
terminal. Since the requirement of the specification of a BPF of an input stage of an RF signal is mitigated by a suppression ratio of the half-complex mixer 114, a BPF such as the BPF 1003 of FIG.
14 requiring the steep characteristic is not required when the IF signal frequency is lowered.
When the downconverter 1 is compared with the downconverter 12 of FIG. 17, the downconverter 12 generates a real IF signal after suppressing a negative frequency component in a polyphase filter 1204,
and suppresses a frequency band out of a target signal of the associated real IF signal in a BPF 1205.
On the other hand, in the downconverter 1 of this exemplary embodiment, one complex-coefficient SAW filter 116 outputs a real IF signal from a complex IF signal while suppressing a frequency band of
a target signal. Thus, the downconverter 1 of this exemplary embodiment can be miniaturized while reducing loss in the polyphase filter 1204.
An upconverter 2 in accordance with the present invention will be described with reference to FIG. 7. The upconverter 2 is provided with a complex-coefficient SAW filter 210, a local oscillator 224,
a half-complex mixer 212, a BPF 214, a power amplifier (PA) 216, and an LPF 218, and converts an IF signal to an RF signal.
In the upconverter 2, the complex-coefficient SAW filter 210 is an example of the above-described complex-coefficient transversal filter, and outputs a complex IF signal whose components have a phase
difference of 90 degrees from each other while suppressing a negative frequency component of an input real IF signal. The local oscillator 240 outputs a complex local signal constructed with a real
axis local signal with a phase of cos and an imaginary local signal with a phase of −sin at a predetermined frequency. Herein, the predetermined frequency has a frequency value computed by
subtracting the IF signal frequency from the RF signal frequency. The half-complex mixer 212 is connected to the local oscillator 224 and is provided with a mixer-I 221 and a mixer-Q 222 serving as
multipliers and an adder 223. The half-complex mixer 212 multiplies the complex signal output from the complex-coefficient SAW filter 210 by the complex local signal, and performs frequency
conversion to a real RF signal while suppressing an image frequency signal.
The mixer-I 221 performs the frequency conversion to the RF signal frequency separated by the predetermined frequency from the IF signal frequency by multiplying a real axis signal of the input
complex signal by the real axis local signal output from the local oscillator 224. The mixer-Q 222 performs the frequency conversion to the RF signal frequency separated by the predetermined
frequency from the IF signal frequency by multiplying an imaginary axis signal of the input complex signal by the imaginary axis local signal output from the local oscillator 224. The adder 223
outputs a real RF signal by adding a signal output from the mixer-I 221 to a signal output from the mixer-Q 222. The BPF 214 suppresses a band out of the RF signal frequency band. The PA 216
amplifies a real RF signal output from the BPF 214. The LPF 218 suppresses a frequency component of the real RF signal.
The complex-coefficient SAW filter 210 can have a structure as illustrated in FIG. 8. The complex-coefficient SAW filter 210 of FIG. 8 is constructed with a piezoelectric substrate 2105 and comb
shaped electrodes 2101 to 2104 in which an intersection width differs according to a position on the piezoelectric substrate 2105. The IDTs 2101 and 2102 are connected to an input terminal. When an
impulse electric signal is applied, mechanical distortion is caused by piezoelectricity and SAWs are excited and propagated in the left and right directions of the piezoelectric substrate 2105. The
IDT 2103 is connected to an output terminal I for outputting a real axis component and is provided in a position capable of receiving the SAW from the IDT 2101. The IDT 2104 is connected to an output
terminal Q for outputting an imaginary axis component and is provided in a position capable of receiving the SAW from the IDT 2102. To perform a weighting operation mapped to an impulse response of a
real part, i.e., an even-symmetric impulse response, the IDT 2103, connected to the output terminal I for outputting the real axis component, is provided with an electrode finger such that even
symmetry is formed with respect to the envelope center. To perform a weighting operation mapped to an impulse response of an imaginary part, i.e., an odd-symmetric impulse response, the IDT 2104,
connected to the output terminal Q for outputting an imaginary axis component, is provided with an electrode finger such that odd symmetry is formed with respect to the envelope center. This
structure can convert a real IF signal to a complex IF signal with a phase difference of 90 degrees between the real part and the imaginary part while suppressing a negative frequency signal.
The operation of the complex-coefficient SAW filter 210 will be described. First, when a real IF signal is input to the input terminal, SAWs are excited and propagated from the IDTs 2101 and 2102.
The SAWs propagated from the IDTs 2101 and 2102 are received by the IDTs 2103 and 2104 provided in propagation directions of the SAWs. A convolution integral is performed on the basis of impulse
responses mapped to the SAWs, such that they are converted to electric signals. At this time, the IDT 2103 outputs a real part component of the complex IF signal through the output terminal I, and
the IDT 2104 outputs an imaginary part component of the complex IF signal through the output terminal Q. According to this structure, a convolution integral is performed for the even and
odd-symmetric impulse responses and the real IF signal, such that the complex IF signal whose components have a phase difference of 90 degrees from each other can be output while a negative frequency
band of the real IF signal is suppressed.
Further, the complex-coefficient SAW filter 210 can be implemented with the structure of FIG. 9. The structure of FIG. 8 is provided with the two IDTs 2101 and 2102 in the input terminal side,
whereas the structure of FIG. 9 is provided with an IDT 1211 of an input side placed across propagation paths of IDTs 2112 and 2113 connected to output terminals. In this structure, one IDT can be
provided in the input terminal side.
The operation of the upconverter 2 will be described with reference to FIG. 7.
First, a real IF signal S21 is input from an IF terminal to the complex-coefficient SAW filter 210.
If the complex-coefficient SAW filter 210 is designed at the center frequency of 190 MHz, the complex-coefficient SAW filter 210 outputs signals S221 and S22Q with a phase difference of 90 degrees
from each other while suppressing a band out of a frequency band with the center frequency of 190 MHz in the real IF signal. The signal S22I is a real axis component of the complex IF signal and the
signal S22Q is an imaginary axis component of the complex IF signal.
In the half-complex mixer 212, the mixer-I 221 multiplies the signal S22I corresponding to the real axis component of the complex IF signal output from the complex-coefficient SAW filter 210 by the
real part local signal of the complex local signal with the predetermined frequency output from the local oscillator 224. The mixer-Q 222 multiplies the signal S22Q corresponding to the imaginary
axis component of the complex IF signal output from the complex-coefficient SAW filter 210 by the imaginary part local signal of the complex local signal with the predetermined frequency output from
the local oscillator 224. Thus, while the image frequency signal occurring in the RF signal frequency band is suppressed by frequency conversion, the mixer-I 221 and the mixer-Q 222 output a signal S
23I and a signal S23Q, respectively. The adder 223 adds the signal S23I to the signal S23Q and then outputs a real RF signal S24.
The BPF 214 outputs a real RF signal S25 by suppressing a band out of the RF signal frequency band in the input RF signal. The PA 216 amplifies the real RF signal S25. The LPF 218 rejects a high
frequency component from the real RF signal S25 and then an antenna transmits a signal from an RF terminal.
In the above-described structure, the upconverter 2 can lower the frequency of the input IF signal to a low frequency without unnecessarily increasing the frequency of the input IF signal only for
the suppression of the image frequency signal as in the upconverter 11 of the heterodyne scheme as illustrated in FIG. 16. Thus, power consumption can be reduced in a structure of an IF stage. Since
the requirement of the specification of a BPF of an output stage of an RF signal is mitigated by a suppression ratio of the half-complex mixer 212, a BPF such as the BPF 1105 of FIG. 16 requiring the
steep characteristic is not required when the IF signal frequency is lowered.
When the upconverter 2 is compared with the upconverter mapped to the downconverter 12 of FIG. 17, the upconverter 2 of this exemplary embodiment suppresses a frequency band out of a target signal
from a real IF signal in the complex-coefficient SAW filter 210, whereas the upconverter mapped to the downconverter 12 requires a BPF for suppressing a frequency band out of an IF signal frequency
band before an input to a polyphase filter. Thus, the upconverter 2 of this exemplary embodiment can be miniaturized while reducing loss in the polyphase filter.
A downconverter 3, an upconverter 4, a downconverter 5, and an upconverter 6 in accordance with other embodiments of the present invention will be described with reference to FIGS. 10, 11, 12, and 13
The downconverter 3 of FIG. 10 is provided with a BPF 310, an LNA 312, a polyphase filter 314, a local oscillator 327, a full-complex mixer 316, and a complex-coefficient SAW filter 318. The BPF 310
corresponds to the BPF 110 of FIG. 1 and the LNA 312 corresponds to the LNA 112 of FIG. 1. The polyphase filter 314 outputs a complex RF signal by suppressing a negative frequency of an input real RF
signal. The local oscillator 327 outputs a complex local signal constructed with a real axis local signal with a phase of cos and an imaginary local signal with a phase of sin at a predetermined
frequency. Herein, the predetermined frequency has a frequency value computed by subtracting the IF signal frequency from the RF signal frequency.
The full-complex mixer 316 is connected to the local oscillator 327 and is provided with a mixer-II 321, a mixer-IQ 322, a mixer-QI 324, a mixer-QQ 325, a subtractor 323, and an adder 326. An example
is described in FIG. 3.28 and FIG. 3.31 of CMOS WIRELESS TRANSCEIVER DESIGN, Jan Crols, Michiel Steyaert, Kluwer, International Series in Engineering and Computer Science, 1997 (Crols). In the
full-complex mixer 316, the real axis local signal of the complex local signal output from the local oscillator 327 is input to the mixer-II 321 and the mixer-QI 324 and the imaginary axis local
signal of the complex local signal output from the local oscillator 327 is input to the mixer-IQ 322 and the mixer-QQ 325.
The mixer-II 321 multiplies a signal S34I of a real axis component of the complex RF signal output from the polyphase filter 314 by the real axis local signal of the complex local signal. The
mixer-IQ 322 performs multiplication by the imaginary axis local signal of the complex local signal. Thus, an image frequency signal is suppressed and frequency conversion to an IF signal frequency
is performed.
The mixer-QQ 325 multiplies a signal S34Q of an imaginary axis component of the complex RF signal output from the polyphase filter 314 by the imaginary axis local signal of the complex local signal.
The mixer-QI 324 performs multiplication by the real axis local signal of the complex local signal. Thus, an image frequency signal is suppressed and frequency conversion to an IF signal frequency is
performed. The subtractor 323 subtracts an output signal of the mixer-QQ 325 from an output signal of the mixer-II 321 and outputs a signal S35I corresponding to the real axis component of the
complex IF signal. The adder 326 adds an output signal of the mixer-QI 324 to an output signal of the mixer-IQ 322, and then outputs a signal S35Q corresponding to an imaginary axis component of the
complex IF signal.
The full-complex mixer 316 multiplies both the real and imaginary axis components of the complex RF signal to be frequency-converted by the real and imaginary axis components of the complex local
signal, thereby suppressing the image frequency signal occurring in the frequency conversion at a high suppression ratio.
The complex-coefficient SAW filter 318 uses the SAW filter with the structure of FIG. 2 or 3. The signal S35I corresponding to the real axis component of the complex IF signal output from the
full-complex mixer 316 undergoes a convolution integral with an even-symmetric impulse response. On the other hand, the signal S35Q corresponding to the imaginary axis component of the complex IF
signal undergoes a convolution integral with an odd-symmetric impulse response. Further, a subtraction process is performed for a real part signal after the convolution integral with the
even-symmetric impulse response and an imaginary part signal after the convolution integral with the odd-symmetric impulse response and then a real IF signal S36 is output.
According to the above-described structure, the downconverter 3 is provided with the polyphase filter 314, and can generate the complex RF signal by suppressing a negative frequency component. Thus,
the downconverter 3 can suppress the image frequency signal at a suppression ratio in which a suppression ratio of the full-complex mixer 316 is added to a suppression ratio of the polyphase filter
314. Since the downconverter 3 uses the full-complex mixer 316, the downconverter 3 can obtain a higher suppression ratio than the downconverter 1 using the half-complex mixer 114. Since a high
suppression ratio can be obtained by the full-complex mixer 316, the degradation of an image suppression ratio due to the variation of a transistor can be allowed. For this reason, a size of the
transistor of the full-complex mixer 316 can be small. The full-complex mixer has a larger number of transistors than the half-complex mixer but the overall power consumption can be reduced due to
the reduction of power consumption of an individual transistor. Simultaneously, the degradation of transition frequency, fT, of the transistor can be prevented.
FIG. 11 illustrates a structure of the upconverter 4 mapped to the structure of the downconverter 3 of FIG. 10. The upconverter 4 is provided with a complex-coefficient SAW filter 410, a full-complex
mixer 412, a polyphase filter 414, a BPF 416, a PA 418, and an LPF 420. The BPF 416 corresponds to the BPF 214 of FIG. 7, the PA 418 corresponds to the PA 216 of FIG. 7, and the LPF 420 corresponds
to the LPF 218 of FIG. 7. The complex-coefficient SAW filter 410 uses the SAW filter with the structure of FIG. 8 or 9.
In the upconverter 4, the complex-coefficient SAW filter 410 first generates and outputs a complex IF signal with a phase difference of 90 degrees between real and imaginary parts while suppressing a
negative frequency component of an input real IF signal. The full-complex mixer 412 performs frequency conversion to an RF signal frequency by multiplying a complex local signal of a predetermined
frequency output from the local oscillator 437 by all combinations of real and imaginary axis components of the input complex IF signal. The polyphase filter 414 converts a complex RF signal output
from the full-complex mixer 412 to a real RF signal by suppressing a negative frequency component. Then, the BPF 416 suppresses a band out of an RF signal frequency band in the input real RF signal.
The PA 418 amplifies the real RF signal after suppression. Then, the LPF 420 rejects a harmonic frequency component and an antenna transmits a signal from an RF terminal.
According to the above-described structure, the upconverter 4 using the full-complex mixer 412 can obtain a higher suppression ratio than the upconverter 2 using the half-complex mixer 212.
Further, the upconverter 4 is provided with the polyphase filter 414, and can generate the real RF signal by suppressing a negative frequency component. Thus, the upconverter 4 can suppress the image
frequency signal at a suppression ratio in which a suppression ratio of the full-complex mixer 412 is added to a suppression ratio of the polyphase filter 414. Because a high suppression ratio can be
obtained by the full-complex mixer 412, the degradation of an image suppression ratio due to the variation of a transistor can be allowed. For this reason, a size of the transistor of the
full-complex mixer 412 can be small. The full-complex mixer has a larger number of transistors than the half-complex mixer but the overall power consumption can be reduced due to the reduction of
power consumption of an individual transistor. Simultaneously, the degradation of transition frequency, fT, of the transistor can be prevented.
The downconverter 5 and the upconverter 6 will be described with reference to FIGS. 12 and 13.
The downconverter 5 of FIG. 12 is provided with a BPF 510, an LNA 512, a polyphase filter 514, a local oscillator 523, a half-complex mixer 516, and a complex-coefficient SAW filter 518. The BPF 510
corresponds to the BPF 110 of FIG. 1 and the LNA 512 corresponds to the LNA 112 of FIG. 1. The polyphase filter 514 outputs a complex RF signal by suppressing a negative frequency of an input real RF
signal. The local oscillator 523 inputs a real local signal with a predetermined frequency. Herein, the predetermined frequency has a frequency value computed by subtracting the IF signal frequency
from the RF signal frequency. The half-complex mixer 516 is connected to the local oscillator 523 and is provided with a mixer-I 521 and a mixer-Q 522. The half-complex mixer 516 frequency-converts
the input complex RF signal to a complex IF signal while suppressing an image frequency signal.
In the half-complex mixer 516, the mixer-I 521 outputs a signal S55I converted to an IF signal frequency separated by a predetermined frequency from an RF signal frequency by multiplying a signal S54
I corresponding to a real axis component of the complex RF signal output from the polyphase filter 514 by the real local signal output from the local oscillator 523. The mixer-Q 522 outputs a signal
S55Q converted to the IF signal frequency separated by the predetermined frequency from the RF signal frequency by multiplying a signal S54Q corresponding to an imaginary axis component of the
complex RF signal output from the polyphase filter 514 by the real local signal output from the local oscillator 523.
The complex-coefficient SAW filter 518 uses the SAW filter with the structure of FIG. 2 or 3. A signal S55I corresponding to the real axis component of the complex IF signal output from the
full-complex mixer 516 undergoes a convolution integral with an even-symmetric impulse response. On the other hand, a signal S55Q corresponding to the imaginary axis component of the complex IF
signal undergoes a convolution integral with an odd-symmetric impulse response. Further, a subtraction process is performed for a real part signal after the convolution integral with the
even-symmetric impulse response and an imaginary part signal after the convolution integral with the odd-symmetric impulse response and then a real IF signal S56 is output.
According to the above-described structure, the downconverter 5 is provided with the polyphase filter 514 in an RF signal stage, and can generate the complex RF signal by suppressing a negative
frequency component. Thus, the downconverter 5 can suppress the image frequency signal at a suppression ratio in which a suppression ratio of the full-complex mixer 516 is added to a suppression
ratio of the polyphase filter 514. Since a local signal input to the half-complex mixer 516 is a real local signal, the power consumption of the downconverter 5 can be reduced to half of that of the
downconverter 1 using the complex local signal. Using the real local signal, the downconverter 5 can obtain a sufficient suppression ratio at which an image frequency signal is suppressed by
suppressing variation due to a manufacturing error of a mixer or filter without considering the imbalance between a real number and an imaginary number.
FIG. 13 illustrates a structure of the upconverter 6 mapped to the structure of the downconverter 5 of FIG. 12. The upconverter 6 is provided with a complex-coefficient SAW filter 610, a half-complex
mixer 612, a polyphase filter 614, a BPF 616, a PA 618, and an LPF 620. The BPF 616 corresponds to the BPF 214 of FIG. 7, the PA 618 corresponds to the PA 216 of FIG. 7, and the LPF 620 corresponds
to the LPF 218 of FIG. 7. The complex-coefficient SAW filter 610 uses the SAW filter with the structure of FIG. 8 or 9.
In the upconverter 6, the complex-coefficient SAW filter 610 first generates and outputs a complex IF signal with a phase difference of 90 degrees between real and imaginary parts while suppressing a
negative frequency component of an input real IF signal. The half-complex mixer 612 multiplies signals corresponding to real and imaginary axis components of an input IF signal by a real local signal
with a predetermined frequency output from a local oscillator 633, and then outputs a complex RF signal by performing frequency conversion to an RF signal frequency. The polyphase filter 614 converts
the complex RF signal output from the half-complex mixer 612 to a real RF signal by suppressing a negative frequency component. Then, the BPF 616 suppresses a band out of an RF signal frequency band
in the input real RF signal. The PA 618 amplifies the real RF signal after suppression. Then, the LPF 420 rejects a harmonic frequency component from the real RF signal and an antenna transmits a
signal from an RF terminal.
According to the above-described structure, the upconverter 6 is provided with the polyphase filter 614 in an RF signal stage, and can generate the real RF signal by suppressing a negative frequency
component. Thus, the upconverter 6 can suppress the image frequency signal at a suppression ratio in which a suppression ratio of the half-complex mixer 612 is added to a suppression ratio of the
polyphase filter 614. Since a local signal input to the half-complex mixer 612 is a real local signal, the power consumption of the upconverter 6 can be reduced to half of that of the upconverter 2
using the complex local signal. Using the real local signal, the upconverter 6 can obtain a sufficient suppression ratio at which an image frequency signal is suppressed by suppressing variation due
to a manufacturing error of a mixer or filter without considering the imbalance between a real number and an imaginary number.
If flat group delay characteristics are required for the complex-coefficient transversal filter, an impulse response used for the complex-coefficient transversal filter must be exactly an even or odd
symmetric impulse response. However, because symmetry may be slightly lost when the impulse response is generated based on an even or odd function, an almost even or odd-symmetric impulse response is
also possible if flat group delay characteristics are not precisely required.
Alternatively, the complex-coefficient transversal filter and the complex-coefficient SAW filter of its exemplary embodiment may suppress a positive frequency on the basis of the above-described
design and may suppress a band out of a frequency band of a target signal of a negative frequency.
Alternatively, the polyphase filter may be constructed to suppress the positive frequency rather than the negative frequency.
In accordance with the present invention, a frequency converter includes a real-coefficient filter for outputting a real RF signal by suppressing a band out of an RF signal frequency band in a
received signal, a local oscillator for outputting a complex local signal with a predetermined frequency, a complex mixer for performing frequency conversion by multiplying the real RF signal output
from the real-coefficient filter by a real part of the complex local signal output from the local oscillator, performing frequency conversion by multiplying the real RF signal by an imaginary part of
the complex local signal output from the local oscillator, and outputting a complex signal of a frequency separated by the predetermined frequency from a frequency of the real RF signal, and a
complex-coefficient transversal filter for performing a convolution integral based on an impulse response generated by an even function for a real part of the complex signal output from the complex
mixer, performing a convolution integral based on an impulse response generated by an odd function for an imaginary part of the complex signal output from the complex mixer, and outputting a real
signal from the complex signal by suppressing one side of a positive frequency or a negative frequency. Thus, the frequency converter of a heterodyne scheme can lower an IF signal frequency to a low
frequency, and can reduce power consumption in a structure after an IF stage. When the frequency converter of the present invention is compared with that of the downconverter using the conventional
polyphase filter, the frequency converter of the present invention can reduce loss due to the polyphase filter.
In the frequency converter of the present invention, the complex-coefficient transversal filter is constructed with a SAW filter. Since the SAW filter is a passive filter, power is not consumed. The
SAW filter can suppress one side of a positive frequency or a negative frequency and can obtain the filter effect of suppressing a signal out of the band of a target signal at a frequency side.
In accordance with the present invention, the frequency converter is provided with a polyphase filter connected to the real-coefficient filter and the complex mixer. The polyphase filter generates
and outputs a complex RF signal by suppressing one side of a positive frequency or a negative frequency from the real RF signal output from the real-coefficient filter. The complex mixer performs
frequency conversion by multiplying a real part of the complex RF signal output from the polyphase filter by the real part of the complex local signal output from the local oscillator, performs
frequency conversion by multiplying an imaginary part of the complex RF signal output from the polyphase filter by the imaginary part of the complex local signal output from the local oscillator, and
outputs a complex signal of a frequency separated by the predetermined frequency from a frequency of the complex RF signal. Since the polyphase filter can generate the complex RF signal by
suppressing a positive or negative frequency component, an image frequency signal can be suppressed at a suppression ratio in which a suppression ratio of the complex mixer is added to a suppression
ratio of the polyphase filter.
In the frequency converter of the present invention, the local oscillator outputs a real local signal with a predetermined frequency, and the complex mixer performs frequency conversion by
multiplying the real part of the complex RF signal output from the polyphase filter by the real local signal output from the local oscillator, performs frequency conversion by multiplying the
imaginary part of the complex RF signal output from the polyphase filter by the real local signal output from the local oscillator, and outputs a complex signal of a frequency separated by the
predetermined frequency from a frequency of the complex RF signal. Since a local signal input to the complex mixer is a real local signal, the power consumption of the frequency converter can be
reduced to half of that of the frequency converter using the complex local signal. Using the real local signal, the frequency converter can obtain a sufficient suppression ratio at which an image
frequency signal is suppressed by suppressing variation due to a manufacturing error of a mixer or filter without considering the imbalance between a real number and an imaginary number.
In accordance with the present invention, a frequency converter includes a complex-coefficient transversal filter for performing a convolution integral based on an impulse response generated by an
even function for a real signal of an input IF, performing a convolution integral based on an impulse response generated by an odd function for the real signal, and outputting a complex signal by
suppressing one side of a positive frequency or a negative frequency, a local oscillator for outputting a complex local signal with a predetermined frequency, a complex mixer for performing frequency
conversion by multiplying a real part of the complex signal output from the complex-coefficient transversal filter by a real part of the complex local signal output from the local oscillator,
performing frequency conversion by multiplying an imaginary part of the complex signal by an imaginary part of the complex local signal output from the local oscillator, and outputting a real signal
of a frequency separated by the predetermined frequency from a frequency of the input signal, and a real-coefficient filter for outputting a real RF signal by suppressing a frequency band out of an
RF signal frequency band for the real signal output from the complex mixer. In accordance with the present invention, the frequency converter of a heterodyne scheme can lower an IF signal frequency
to a low frequency, and can reduce power consumption in a structure after an IF stage.
When the frequency converters of the present invention are compared with the frequency converter constructing the downconverter using the conventional polyphase filter and the frequency converter for
converting an IF signal to an RF signal of a frequency higher than an IF signal frequency, the frequency converters of the present invention can reduce loss due to the polyphase filter.
In the frequency converter of the present invention, the complex-coefficient transversal filter is constructed with a SAW filter. Since the SAW filter is a passive filter, power is not consumed. The
SAW filter can suppress one side of a positive frequency or a negative frequency and can obtain the filter effect of suppressing a signal out of the band of a target signal at a frequency side.
In accordance with the present invention, the frequency converter is provided with a polyphase filter connected to the real-coefficient filter and the complex mixer. The complex mixer performs
frequency conversion by multiplying a real part of the complex signal output from the complex-coefficient transversal filter by the real part of the complex local signal output from the local
oscillator, performs frequency conversion by multiplying an imaginary part of the complex signal output from the complex-coefficient transversal filter by the imaginary part of the complex local
signal output from the local oscillator, and outputs a complex signal of a frequency separated by the predetermined frequency from a frequency of the complex signal. The polyphase filter generates
and outputs a real signal mapped to the complex signal output from the complex mixer by suppressing one side of a positive frequency or a negative frequency. Since the polyphase filter can generate a
real IF signal by suppressing a positive or negative frequency component, an image frequency signal can be suppressed at a suppression ratio in which a suppression ratio of the complex mixer is added
to a suppression ratio of the polyphase filter.
In the frequency converter of the present invention, the local oscillator outputs a real local signal with a predetermined frequency. The complex mixer performs frequency conversion by multiplying
the real part of the complex signal output from the complex-coefficient transversal filter by the real local signal output from the local oscillator, performs frequency conversion by multiplying the
imaginary part of the complex signal output from the complex-coefficient transversal filter by the real local signal output from the local oscillator, and outputs a complex signal of a frequency
separated by the predetermined frequency from a frequency of the complex signal. Since a local signal input to the complex mixer is a real local signal, the power consumption of the frequency
converter can be reduced to half of that of the frequency converter using the complex local signal. Using the real local signal, the frequency converter can obtain a sufficient suppression ratio at
which an image frequency signal is suppressed by suppressing variation due to a manufacturing error of a mixer or filter without considering imbalance the between a real number and an imaginary
Although the exemplary embodiments of the present invention have been disclosed for illustrative purposes, those skilled in the art will appreciate that various modifications, additions, and
substitutions are possible, without departing from the scope of the present invention. Therefore, the present invention is not limited to the above-described embodiments, but is defined by the
following claims, along with their full scope of equivalents.
1. A frequency converter for frequency-converting a received radio frequency (RF) signal to an intermediate frequency (IF) signal, comprising:
a real-coefficient filter for outputting a real RF signal by suppressing a band out of an RF signal frequency band in a received signal;
a local oscillator for outputting a complex local signal with a predetermined frequency;
a complex mixer for performing frequency conversion by multiplying the real RF signal output from the real-coefficient filter by a real part of the complex local signal output from the local
oscillator, performing frequency conversion by multiplying the real RF signal by an imaginary part of the complex local signal output from the local oscillator, and outputting a complex signal of
a frequency separated by the predetermined frequency from a frequency of the real RF signal; and
a complex-coefficient transversal filter for performing a convolution integral based on an impulse response generated by an even function for a real part of the complex signal output from the
complex mixer, performing a convolution integral based on an impulse response generated by an odd function for an imaginary part of the complex signal output from the complex mixer, and
outputting a real signal from the complex signal by suppressing one side of a positive frequency or a negative frequency.
2. The frequency converter of claim 1, wherein the complex-coefficient transversal filter is constructed with a surface acoustic wave (SAW) filter.
3. The frequency converter of claim 1, further comprising:
a polyphase filter connected to the real-coefficient filter and the complex mixer,
wherein the polyphase filter generates and outputs a complex RF signal by suppressing one side of a positive frequency or a negative frequency from the real RF signal output from the
real-coefficient filter, and
the complex mixer performs frequency conversion by multiplying a real part of the complex RF signal output from the polyphase filter by the real part of the complex local signal output from the
local oscillator, performs frequency conversion by multiplying an imaginary part of the complex RF signal output from the polyphase filter by the imaginary part of the complex local signal output
from the local oscillator, and outputs a complex signal of a frequency separated by the predetermined frequency from a frequency of the complex RF signal.
4. The frequency converter of claim 2, further comprising:
a polyphase filter connected to the real-coefficient filter and the complex mixer,
wherein the polyphase filter generates and outputs a complex RF signal by suppressing one side of a positive frequency or a negative frequency from the real RF signal output from the
real-coefficient filter, and
the complex mixer performs frequency conversion by multiplying a real part of the complex RF signal output from the polyphase filter by the real part of the complex local signal output from the
local oscillator, performs frequency conversion by multiplying an imaginary part of the complex RF signal output from the polyphase filter by the imaginary part of the complex local signal output
from the local oscillator, and outputs a complex signal of a frequency separated by the predetermined frequency from a frequency of the complex RF signal.
5. The frequency converter of claim 3, wherein the local oscillator outputs a real local signal with a predetermined frequency, and
the complex mixer performs frequency conversion by multiplying the real part of the complex RF signal output from the polyphase filter by the real local signal output from the local oscillator,
performs frequency conversion by multiplying the imaginary part of the complex RF signal output from the polyphase filter by the real local signal output from the local oscillator, and outputs a
complex signal of a frequency separated by the predetermined frequency from a frequency of the complex RF signal.
6. The frequency converter of claim 4, wherein the local oscillator outputs a real local signal with a predetermined frequency, and
the complex mixer performs frequency conversion by multiplying the real part of the complex RF signal output from the polyphase filter by the real local signal output from the local oscillator,
performs frequency conversion by multiplying the imaginary part of the complex RF signal output from the polyphase filter by the real local signal output from the local oscillator, and outputs a
complex signal of a frequency separated by the predetermined frequency from a frequency of the complex RF signal.
7. A frequency converter for frequency-converting an input intermediate frequency (IF) signal to a radio frequency (RF) signal frequency, comprising:
a complex-coefficient transversal filter for performing a convolution integral based on an impulse response generated by an even function for a real signal of an input IF, performing a
convolution integral based on an impulse response generated by an odd function for the real signal, and outputting a complex signal by suppressing one side of a positive frequency or a negative
a local oscillator for outputting a complex local signal with a predetermined frequency;
a complex mixer for performing frequency conversion by multiplying a real part of the complex signal output from the complex-coefficient transversal filter by a real part of the complex local
signal output from the local oscillator, performing frequency conversion by multiplying an imaginary part of the complex signal by an imaginary part of the complex local signal output from the
local oscillator, and outputting a real signal of a frequency separated by the predetermined frequency from a frequency of the input signal; and
a real-coefficient filter for outputting a real RF signal by suppressing a frequency band out of an RF signal frequency band for the real signal output from the complex mixer.
8. The frequency converter of claim 7, wherein the complex-coefficient transversal filter is constructed with a surface acoustic wave (SAW) filter.
9. The frequency converter of claim 7, further comprising:
a polyphase filter connected to the real-coefficient filter and the complex mixer,
wherein the complex mixer performs frequency conversion by multiplying a real part of the complex signal output from the complex-coefficient transversal filter by the real part of the complex
local signal output from the local oscillator, performs frequency conversion by multiplying an imaginary part of the complex signal output from the complex-coefficient transversal filter by the
imaginary part of the complex local signal output from the local oscillator, and outputs a complex signal of a frequency separated by the predetermined frequency from a frequency of the complex
signal, and
the polyphase filter generates and outputs a real signal mapped to the complex signal output from the complex mixer by suppressing one side of a positive frequency or a negative frequency.
10. The frequency converter of claim 8, further comprising:
a polyphase filter connected to the real-coefficient filter and the complex mixer,
wherein the complex mixer performs frequency conversion by multiplying a real part of the complex signal output from the complex-coefficient transversal filter by the real part of the complex
local signal output from the local oscillator, performs frequency conversion by multiplying an imaginary part of the complex signal output from the complex-coefficient transversal filter by the
imaginary part of the complex local signal output from the local oscillator, and outputs a complex signal of a frequency separated by the predetermined frequency from a frequency of the complex
signal, and
the polyphase filter generates and outputs a real signal mapped to the complex signal output from the complex mixer by suppressing one side of a positive frequency or a negative frequency.
11. The frequency converter of claim 9, wherein the local oscillator outputs a real local signal with a predetermined frequency, and
the complex mixer performs frequency conversion by multiplying the real part of the complex signal output from the complex-coefficient transversal filter by the real local signal output from the
local oscillator, performs frequency conversion by multiplying the imaginary part of the complex signal output from the complex-coefficient transversal filter by the real local signal output from
the local oscillator, and outputs a complex signal of a frequency separated by the predetermined frequency from a frequency of the complex signal.
12. The frequency converter of claim 10, wherein the local oscillator outputs a real local signal with a predetermined frequency, and
the complex mixer performs frequency conversion by multiplying the real part of the complex signal output from the complex-coefficient transversal filter by the real local signal output from the
local oscillator, performs frequency conversion by multiplying the imaginary part of the complex signal output from the complex-coefficient transversal filter by the real local signal output from
the local oscillator, and outputs a complex signal of a frequency separated by the predetermined frequency from a frequency of the complex signal.
Patent History
Publication number
: 20070171312
: Dec 20, 2006
Publication Date
: Jul 26, 2007
SAMSUNG ELECTRONICS CO., LTD.
Takahiko Kishi
Application Number
: 11/642,483
Current U.S. Class: 348/726.000
International Classification: H04N 5/455 (20060101); | {"url":"https://patents.justia.com/patent/20070171312","timestamp":"2024-11-03T06:34:19Z","content_type":"text/html","content_length":"144635","record_id":"<urn:uuid:a51b10e6-8eae-41f7-8b6c-18a58ce5eb17>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00278.warc.gz"} |
Conductive Heat Transfer
Heat transfer by conduction
Simscape / Foundation Library / Thermal / Thermal Elements
The Conductive Heat Transfer block represents heat transfer by conduction between two layers of the same material. For a flat surface, the Fourier law describes the transfer,
$Q=k\cdot \frac{A}{D}\left({T}_{A}-{T}_{B}\right),$
• Q is the heat flow.
• k is the thermal conductivity of the material.
• A is the area normal to the heat flow direction.
• D is the distance between layers, that is, the thickness of material.
• T[A] is the temperature of layer A.
• T[B] is the temperature of layer B.
Heat conduction through a round pipe wall is
${Q}_{cyl}=2\pi k\cdot \frac{L}{\mathrm{ln}\left(\frac{{d}_{out}}{{d}_{in}}\right)}\left({T}_{A}-{T}_{B}\right),$
• L is the pipe length.
• d[in] is the inner diameter.
• d[out] is the outer diameter.
You can specify the thermal conductivity by using the Conductivity type parameter:
• Constant — Thermal conductivity remains constant during simulation. You specify the thermal conductivity by using the Thermal conductivity parameter.
• Variable input — You specify the thermal conductivity using the input physical signal at port K, which can vary during simulation. The Minimum thermal conductivity parameter specifies the lower
bound for the physical signal.
• Tabulated data — You specify the thermal conductivity by using a lookup table based on temperature. In this case, thermal conductivity can also vary during simulation.
The Tabulated data option uses the average temperature of the block to find the thermal conductivity. For planar wall geometry, the average temperature is
For cylindrical wall geometry, the average temperature is
${T}_{avg}=\left({T}_{A}-{T}_{B}\right)\cdot \left(\frac{{d}_{out}}{{d}_{out}-{d}_{in}}-\frac{1}{\mathrm{ln}\left(\frac{{d}_{out}}{{d}_{in}}\right)}\right)+{T}_{B},$
which assumes that T[B] is at the inside of the cylinder.
A and B are thermal conserving ports associated with the material layers. Because the block positive direction is from port A to port B, the heat flow is positive if it flows from A to B.
To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set
Priority and Initial Target for Block Variables.
Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from
different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.
K — Thermal conductivity control signal, W/(K*m)
physical signal
Input physical signal that controls the thermal conductivity. The signal saturates when the value is outside the minimum limit specified by the Minimum thermal conductivity parameter.
To enable this port, set the Conductivity type parameter to Variable input.
A — Layer A
Thermal conserving port associated with layer A.
B — Layer B
Thermal conserving port associated with layer B. For cylindrical wall geometry, layer B is the inside layer.
Conductivity type — How to specify thermal conductivity
Constant (default) | Variable input | Tabulated data
Whether to specify thermal conductivity as:
• Constant — Thermal conductivity remains constant during simulation. You specify the thermal conductivity by using the Thermal conductivity parameter.
• Variable input — You specify thermal conductivity using the input physical signal at port K, which can vary during simulation. The Minimum thermal conductivity parameter specifies the lower bound
for the physical signal.
• Tabulated data — You specify thermal conductivity by using a lookup table based on temperature. In this case, thermal conductivity can also vary during simulation.
Wall geometry — Wall shape for heat conduction
Planar (default) | Cylindrical
Wall shape for heat conduction, specified as:
• Planar — Heat transfer is through a flat, rectangular wall. Specify the wall area and thickness.
• Cylindrical — Heat transfer is through a round pipe wall. Specify the pipe inner diameter, outer diameter, and length.
Area — Area of heat transfer
1e-4 m^2 (default) | positive scalar
Area of heat transfer normal to the heat flow direction.
To enable this parameter, set Wall geometry to Planar.
Thickness — Thickness of material
0.1 m (default) | positive scalar
Thickness of material, that is, the distance between layers.
To enable this parameter, set Wall geometry to Planar.
Inner diameter — Inner diameter of pipe
0.05 m (default) | positive scalar
Inner diameter of the pipe, that is, the diameter of the inner layer of material.
To enable this parameter, set Wall geometry to Cylindrical.
Outer diameter — Outer diameter of pipe
0.1 m (default) | positive scalar
Outer diameter of the pipe, that is, the diameter of the outer layer of material.
To enable this parameter, set Wall geometry to Cylindrical.
Length — Pipe length
1 m (default) | positive scalar
Length of the pipe.
To enable this parameter, set Wall geometry to Cylindrical.
Thermal conductivity — Thermal conductivity of the material
401 W/(K*m) (default) | positive scalar
Thermal conductivity of the material.
To enable this parameter, set Conductivity type to Constant.
Minimum thermal conductivity — Lower bound for thermal conductivity
10 W/(K*m) (default) | positive scalar
Lower bound for the thermal conductivity value. The input signal at port K saturates at this value to prevent the thermal conductivity from further decreasing.
To enable this parameter, set Conductivity type to Variable input.
Thermal conductivity vector — Vector of thermal conductivity values for use in table lookup
[413, 401, 392, 383, 371, 357, 342] W/(K*m) (default) | vector of positive values
Vector of thermal conductivity values that correspond to the Temperature vector parameter values. The vector size must be the same as the Temperature vector parameter. The block performs
one-dimensional table lookup of thermal conductivity by using the average temperature of the block, linear interpolation, and nearest extrapolation. For more information on the interpolation and
extrapolation algorithms, see PS Lookup Table (1D).
To enable this parameter, set Conductivity type to Tabulated data.
Temperature vector — Vector of temperature values for use in table lookup
[200, 273, 400, 600, 800, 1000, 1200] K (default) | strictly increasing vector
Vector of temperature values that correspond to the Thermal conductivity vector parameter values. The vector must be strictly increasing. The values can be nonuniformly spaced. The vector must
contain at least two values.
To enable this parameter, set Conductivity type to Tabulated data.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2007b
R2022b: Model variable thermal conductivity and heat transfer through a round pipe wall
The new Conductivity type parameter lets the thermal conductivity be either constant, which you specify by the Thermal conductivity parameter, or variable. You can specify variable thermal
conductivity either as a physical signal at port K or as a lookup table based on temperature.
Additionally, you can set the new Wall geometry parameter to either Planar or Cylindrical. Planar refers to a flat, rectangular wall. Cylindrical refers to a round pipe wall.
In the default configuration, with Conductivity type set to Constant and Wall geometry set to Planar, the block functions as in previous releases. | {"url":"https://au.mathworks.com/help/simscape/ref/conductiveheattransfer.html","timestamp":"2024-11-10T17:33:59Z","content_type":"text/html","content_length":"109044","record_id":"<urn:uuid:0b57ef32-44b8-4209-a88f-e82baadf1b1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00232.warc.gz"} |
Learning on graphs using Orthonormal Representation is Statistically Consistent
Part of Advances in Neural Information Processing Systems 27 (NIPS 2014)
Rakesh Shivanna, Chiranjib Bhattacharyya
Existing research \cite{reg} suggests that embedding graphs on a unit sphere can be beneficial in learning labels on the vertices of a graph. However the choice of optimal embedding remains an open
issue. \emph{Orthonormal representation} of graphs, a class of embeddings over the unit sphere, was introduced by Lov\'asz \cite{lovasz_shannon}. In this paper, we show that there exists orthonormal
representations which are statistically consistent over a large class of graphs, including power law and random graphs. This result is achieved by extending the notion of consistency designed in the
inductive setting to graph transduction. As part of the analysis, we explicitly derive relationships between the Rademacher complexity measure and structural properties of graphs, such as the
chromatic number. We further show the fraction of vertices of a graph $G$, on $n$ nodes, that need to be labelled for the learning algorithm to be consistent, also known as labelled sample
complexity, is $ \Omega\left(\frac{\vartheta(G)}{n}\right)^{\frac{1}{4}}$ where $\vartheta(G)$ is the famous Lov\'asz~$\vartheta$ function of the graph. This, for the first time, relates labelled
sample complexity to graph connectivity properties, such as the density of graphs. In the multiview setting, whenever individual views are expressed by a graph, it is a well known heuristic that a
convex combination of Laplacians \cite{lap_mv1} tend to improve accuracy. The analysis presented here easily extends to Multiple graph transduction, and helps develop a sound statistical
understanding of the heuristic, previously unavailable. | {"url":"https://papers.nips.cc/paper_files/paper/2014/hash/b432f34c5a997c8e7c806a895ecc5e25-Abstract.html","timestamp":"2024-11-06T11:03:18Z","content_type":"text/html","content_length":"9640","record_id":"<urn:uuid:f15ad446-08dc-4707-8638-094c9845ab8f>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00630.warc.gz"} |
2015-0678 - The YODA Project
General Information
How did you learn about the YODA Project?:
Conflict of Interest
Request Clinical Trials
Associated Trial(s):
What type of data are you looking for?:
Request Clinical Trials
Data Request Status
Research Proposal
Project Title: Heterogeneous Causal Effects: Drug Exposure & Safety
Scientific Abstract: Background: People with schizophrenia are at higher risk for metabolic morbidity including obesity, dyslipidemia, hypertension, type-2 diabetes, and cardiovascular disease.
However, little evidence exists on the impact of exposure duration on the likelihood and size of the metabolic effects of antipsychotics; and whether sex modifies the effects. New methods are needed
to address these gaps.
Objective: We propose an approach to analyze the causal effect of cumulative exposure on a binary outcome for placebo controlled and active treatment trials.
Study Design: We will exploit methodological advances in two related research fields, causal inference and network meta-analysis, to develop an inferential approach to answer questions involving the
relationship between duration of drug exposure and outcomes.
Participants: We will utilize participant-specific information obtained from the CATIE trial (participants randomized to olanzapine, quetiapine, and risperidone arms only) and the 14 Janssen trials
involving patients with schizophrenia or schizoaffective disorder.
Main Outcome Measure: Our metabolic endpoint is weight gain, which will be operationalized as a binary outcome.
Statistical Analysis: We exploit the randomization mechanism as an instrument to adhere to causal inference assumptions. We estimate exposure-response curves for different exposure subsets and then
combine the treatment-exposure arms via network meta-analysis using individual patient data to study safety endpoints. Our approach uses the placebo arms of no exposure as the outcome of zero
Brief Project Background and Statement of Project Significance: People with schizophrenia are at higher risk for metabolic morbidity including obesity, dyslipidemia, hypertension, type-2 diabetes,
and cardiovascular disease (CVD). While a U.S. study conducted in the early 2000s found that patients with schizophrenia die approximately 25 years earlier than age- and sex-adjusted peers, an
international review found that compared with the general population, this population has a two- to threefold increased risk of dying. Roughly 60% of the excess mortality is due to chronic medical
conditions, with CVD accounting for over half of this excess risk. Treatment with antipsychotics, particularly some frequently used second-generation antipsychotics (SGAs), increases this risk.
Current antipsychotic prescribing practices may pose a substantial long-term burden to patients and public payers. Little evidence exists on the impact of exposure duration on the likelihood and size
of the metabolic effects of SGAs; and whether sex modifies the effects. Despite evidence from studies on the association between several SGAs and metabolic risk, little is known about the
time-dependency of risk. The evidence on the dose-dependency is both limited and mixed. Men and women differ in how they experience disease and how they respond to treatment; yet little research on
the influence of sex on efficacy and safety of antipsychotics exists. Studies conducting post-hoc analyses of RCT data aimed at describing response to SGAs among adults with schizophrenia found
better outcomes for females with both early and chronic illness. A naturalistic study of individuals with early psychosis found an advantage for males. Even less evidence exists on the modifying
effect of sex on the metabolic risk of antipsychotics. Baseline analyses of subjects with chronic schizophrenia enrolled in the CATIE study reported that females were associated with higher rates of
metabolic syndrome. The CATIE trial found no evidence of a modifying effect of sex on the association between specific antipsychotics and metabolic syndrome, results from the Comparison of Atypicals
for First Episode (CAFE) trial suggest the opposite, reporting that females treated with quetiapine had the lowest mean weight gain and smallest mean increase in BMI. Placebo-controlled and
active-controlled clinical trials provide different and valuable information for dose-response causal inferences. Traditional intention-to-treat analyses provide valid inferences of average
effectiveness of therapy. While a simple regression of outcome on observed cumulative exposure is likely not causal, using the randomization assignment variable as an instrument can help identify
causal dose-response relationships. Clinical trials are not powered to detect the effects of duration of exposure on risks nor the effects of modifiers. The numbers enrolled in RCTs are typically
insufficient to determine subgroup effects or to reliably estimate dose-response relationships. While individual trials may not have sufficient power, network meta-analysis provides a quantitative
method of integrating data from all available comparisons. Such analyses can be used to borrow information about effectiveness.
Specific Aims of the Project: We will exploit methodological advances in two related research fields, causal inference and network meta-analysis, to develop an inferential approach to answer
questions involving the relationship between duration of drug exposure and outcomes. Two specific aims guide our work:
Aim 1: To estimate the average causal effect of treatments on binary outcomes and ordered exposure in network meta-analysis of individual participant data. We will exploit the randomized assignment
mechanisms, treatment arms, and placebo arms to estimate exposure-outcome curves in different exposure subsets.
Aim 2: To extend Aim 1 to estimate the heterogeneous (conditional) average treatment effects. We will modify methodology to include a binary-valued moderator to estimate the exposure-outcome curves
within groups defined by a moderator and different exposure subsets.
Study Design:
What is the purpose of the analysis being proposed? Please select all that apply.: New research question to examine treatment effectiveness on secondary endpoints and/or within subgroup populations
New research question to examine treatment safety Participant-level data meta-analysis Meta-analysis using data from the YODA Project and other data sources
Software Used:
Data Source and Inclusion/Exclusion Criteria to be used to define the patient sample for your study: We will utilize participant-specific information obtained from the CATIE trial (participants
randomized to the olanzapine, quetiapine, and risperidone arms only) and the 14 Janssen trials involving patients with schizophrenia or schizoaffective disorder (Table 1). We will obtain these study
data from YODA, assess for completeness and comparability to reported study summaries, and organize into SAS as well as R datasets. We will focus on the trial-specific end-point of weight (rather
than intermediate within-trial outcomes) and the cumulative duration of exposure at end-point for each trial participant.
[Attached research proposal document contains Table 1]
Primary and Secondary Outcome Measure(s) and how they will be categorized/defined for your study: Our metabolic endpoint is weight gain which will be operationalized as a binary outcome assuming a
value of 1 if the subject experienced a weight increase of at least 7% of baseline weight at the trial endpoint and 0 otherwise.
Main Predictor/Independent Variable and how it will be categorized/defined for your study: Our main predictor is cumulative exposure to treatment. The set of all treatments,?, we consider include
{paliperidone, olanzapine, quetiapine, risperidone, and placebo}. The data include j = 1,?,N_i participants in trial i who have been randomized to treatment R_ij=k_ij. For participant j in trial i on
treatment k_ij, a treatment cumulative exposure level, e_ij^k, at trial termination (Table 1), and an observed outcome, Y_ij^k=1, if the outcome occurred and 0 otherwise, are available. Finally, we
will use G+1 ordered (cumulative) exposure levels, {g = 0,1,2,?,G}. For instance, G = 5 in Table 1 based on our preliminary review of the published trial data.
[Attached research proposal document contains formatted mathematical notation]
Other Variables of Interest that will be used in your analysis and how they will be categorized/defined for your study: We are interested in identifying the modifying effects of patient
characteristics such as sex. We let x_ij generically denote a vector of participant-level baseline covariates. For Aim 2, we create a dummy variable denoting female, the treatment modifier, ?female?
_ij = 1 if participant j in trial i is female and 0 otherwise. Unless otherwise specified, ?female?_ij?x_ij.
[Attached research proposal document contains formatted mathematical notation]
Statistical Analysis Plan: We will adopt two different approaches to estimation. Approach 1: we will employ a two-step procedure: in Step 1 we will estimate an instrumental variable-based
(structural) marginal model within each trial. This step yields a contingency table that describes a dose-response curve for participants at their selected cumulative exposure level. The rows of the
table denote cumulative drug exposure while the columns reflect outcomes for the specific cumulative exposure. In Step 2 using the estimated parameters characterizing the contingency table, we will
conduct a network-meta analysis of the estimates. Because of the advantages exploiting the individual participant data (such as separating within-trial effects from across-trial effects), in Approach
2 we will simultaneously model the contingency table across all the trials, including study-specific relative random effects when modeling participant-specific information. Our methodology relies on
the availability of individual participant data that permit separation of within-study effects from across-study effects, the randomized assignment indicators and placebo arms that permit
identification of outcomes under zero exposure for treatment arms, and the existence of multiple different studies that permit assessment of evidence compatibility from direct and indirect
comparisons. We will illustrate our new methodology to characterize the causal effect of drug duration on weight gain for (a) all patients, and (b) males and females separately. We describe the
proposed data sources, our key assumptions, and our approach to the development of new the statistical methodology.
Narrative Summary: People with schizophrenia are at higher risk for death from metabolic disease including obesity, dyslipidemia, hypertension, type 2 diabetes, and cardiovascular disease (CVD).
Understanding the risks of antipsychotic medications is critical as these risks exacerbate the health burden of people with schizophrenia and add to the long-term economic burden borne by public
payers. Currently, little evidence exists on the impact of the duration of drug exposure on the likelihood and size of the metabolic effects of antipsychotics; and whether sex modifies the effects.
This project will use develop new statistical methods to address these gaps in knowledge.
Project Timeline: Year 1
? OPTICS Data applications: YODA & NIMH (CATIE)
? IRB application & approval
? Develop R function that will include code to estimate dose-response curves in the presence of a binary outcome, G exposure levels, and K treatments for a single trial as well as options for models
? Analyze the causal effect of cumulative exposure on a binary outcome for placebo-controlled and active-treatment trials
? Estimate exposure-response curves for different exposure subsets
? Combine the treatment-exposure curves via network meta-analysis using individual participant data to
o assess evidence compatibility from direct and indirect comparisons;
o separate within from between-trial effects; and
o bolster conclusions within subgroups
? Manuscript 1: describe new methodology, assess operating performance characteristics of the estimation procedures, and illustrate the approach using the data from the 15 clinical trials
Year 2
? Apply for methodology R01 to NIMH or National Institute of General Medical Sciences using preliminary results
? Develop methodology to handle time-dependent confounders and censoring using Marginal Structural Cox models
? Develop methodology using flexible approaches to model cumulative exposure effects
? Manuscript 2
Dissemination Plan: We will develop an R function that will be made freely available (loaded onto the Comprehensive R Archive Network). The function will include code to estimate dose-response curves
in the presence of a binary outcome, G exposure levels, and K treatments for a single trial. The code will include options for models (including a multivariate Dale model). During the one-year
time-frame we anticipate completing one manuscript that describes the new methodology, assesses the operating performance characteristics of the estimation procedures, and illustrates the approach
using the antipsychotic drug data from the 15 clinical trials.
1. Newcomer JW, Hennekens CH. Severe mental illness and risk of cardiovascular disease. JAMA, 2007; 298(15): 1794-1796.
2. Maj M. Physical health care in persons with severe mental illness: a public health and ethical priority. World Psychiatry: Official Journal of The World Psychiatric Association, 2009; 8(1): 1-2.
3. Harris EC, Barraclough B. Excess mortality of mental disorder. The British Journal of Psychiatry: The Journal of Mental Science, 1998; 173: 11-53.
4. Nielsen J, Skadhede S, Correll CU. Antipsychotics associated with the development of type 2 diabetes in antipsychotic-naive schizophrenia patients. Neuropsychopharmacology, 2010; 35(9): 1997-2004.
5. Henderson DC, Cagliero E, Copeland PM, et al. Glucose metabolism in patients with schizophrenia treated with atypical antipsychotic agents: a frequently sampled intravenous glucose tolerance test
and minimal model analysis. Arch. Gen. Psychiatry, 2005; 62(1): 19-28.
6. Levine SZ, Rabinowitz J, Case M, Ascher-Svanum H. Treatment response trajectories and their antecedents in recent-onset psychosis: a 2-year prospective study. J. Clin. Psychopharmacol, 2010; 30
(4): 446-449.
7. Rabinowitz J, Werbeloff N, Caers I, et al. Determinants of antipsychotic response in schizophrenia: implications for practice and future clinical trials. J. Clin. Psychiatry, 2014; 75(4):
8. Levine SZ, Lurie I, Kohn R, Levav I. Trajectories of the course of schizophrenia: from progressive deterioration to amelioration over three decades. Schizophr. Res., 2011; 126(1-3): 184-191.
9. Patel JK, Buckley PF, Woolson S, et al. Metabolic profiles of second-generation antipsychotics in early psychosis: findings from the CAFE study. Schizophr. Res., 2009; 111(1-3):9-16.
10. Goetghebeur E, Molenberghs M. Causal inference in a placebo-controlled clinical trial with binary outcome and ordered compliance. JASA, 1996; 91(435):928-934.
11. Lu G, Ades AE. Assessing evidence inconsistency in mixed treatment comparisons. JASA, 2006; 101(474):447-459. | {"url":"https://yoda.yale.edu/data-request/2015-0678/","timestamp":"2024-11-10T10:59:28Z","content_type":"text/html","content_length":"92699","record_id":"<urn:uuid:de369a09-80b6-402c-a682-f13d79afc076>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00513.warc.gz"} |
Math Games
Students entering the 7th grade continue to expand on concepts from their previous years. On-demand videos with teachers who explain the concepts and show students how to understand the
problem-solving process. Teachers go over rules, tips, and multiple problems helping students to be able to solve the problems themselves.
• Students learn about ratios, mixed properties, statistics and other seventh grade skills.
• Teachers incorporate the use of the scratchpad to give students a visual representation.
• Videos provide instant help for students who are struggling with their assignments. | {"url":"https://sk.mathgames.com/worksheets/grade5","timestamp":"2024-11-02T02:58:34Z","content_type":"text/html","content_length":"446785","record_id":"<urn:uuid:e330006d-a644-4efe-99d9-4799e947fa2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00639.warc.gz"} |
Google Sheets: Sum an Entire Column (4 Easy Ways) - OfficeWheel
Adding or summing values in a spreadsheet is a common process. And with such a versatile application like Google Sheets, we have so many ways we can use to sum an entire column of values in a
From the built-in Status Bar to direct view results to simple functions that users can customize their calculations, Google Sheets has it all.
But, to keep things simple, and considering common scenarios that most users face, we’ve kept the number of methods to four. Each adding something new catered to their respective situations.
Let’s get started.
4 Ways to Sum an Entire Column in Google Sheets
1. Directly View the Sum of an Entire Column right in the Google Sheets Window
Summing is a fundamental yet simple calculation. We all know how to do it automatically and so does Google Sheets.
For simple calculations like Sum, Google Sheets has the Status Bar at the bottom-right of the window that appears when you select cells in the worksheet.
For example, let’s say we have the following worksheet:
We have a common scenario of Sales values by Month and Year. There is also a split depicting the first and second halves of the year.
Let’s say we want to find the total Sales value of 2020.
All we have to do is click the column number of the Sales column, in this case, it is column C:
And take a look at the bottom-right of the Google Sheets window:
This is the Status Bar that shows the sum of the entire selected column in Google Sheets.
Google Sheets has preemptively guessed the user’s intentions and added all the numerical values of the selected column to present the sum.
This can work for multiple selected columns:
Any selected range of cells in the worksheet:
Or any selected cells, period.
This is a great way to view primary calculations, especially if you don’t want to dedicate cells for this in the worksheet. This can work for other reasons like quick viewing calculations and
reluctance to change the format of the existing worksheet.
Clicking on the Status Bar will show you the results of other calculations that Google Sheets has done in the background:
Of which the Average (Avg) is another important one in any spreadsheet.
2. To Sum in Google Sheets Directly from the Toolbar
Another useful feature that Google Sheets has for its users is the Functions menu right in the toolbar:
This menu has a list of all the functions available in the application. While it may sound overwhelming, Google Sheets groups all the common functions like SUM up at the top of the list.
While accessing the function may be easy, working with them in this way may be a little clunky. Let’s show you what we mean:
If we want to sum an entire column, we must first select the column and then apply the SUM function, and this time, from the Functions menu.
See what happens when we do this:
There are 3 things going on here:
1. A function must occupy a cell in the worksheet to present the results.
2. The application of any function with the Functions menu will make Google Sheets automatically set the result location.
3. The result location is usually an adjacent cell or a part of the selected cells.
With these points in mind, we can understand that the user has no control over where the result will go!
To work with this method, we cannot select the entire column, but select only the range of cells that we actually need:
Select range of cells > Functions > Sum
To conclude, this method should only be used when you have a simple arrangement of data.
In the next section, we solve this issue by directly applying the SUM function according to the user’s choice.
3. Use the SUM Function to Sum an Entire Column
Why not take back some control from Google Sheets this time?
We are talking about applying the SUM function ourselves.
If you don’t already know, the SUM function is one of the fundamental functions you’ll learn in Google Sheets.
SUM(value1, [value2, ...])
The function can take multiple ranges of cells at once to do a single task: Addition.
Here we have a sample dataset containing Location and Customer data. We want to find the total number of customers:
To sum up the entire Customers column in Google Sheets, simply open the SUM function, =SUM(, and click on the column number to select the entire column range.
Close parentheses and press ENTER.
The advantage here above the previous method is that we can set the location of the result ourselves.
Setting the range up like this, C:C, includes all the cells in column C.
The advantage of this is that it makes the formula dynamic to the column. Meaning, every time we enter a new value into the column, it will be added to the total sum:
But, if you want a limited range of only a few cells in the column, simply update the range in the formula.
This version is much more common.
Extra: Sum Different Ranges from Different Columns in Google Sheets
Recall the SUM function syntax. Using this function, we can sum multiple different cell ranges that don’t always have to be adjacent.
That means, we can do something like finding the total sales of the first and last quarters of the years 2020 and 2021 from the previous worksheet:
Each non-adjacent range is separated by a comma, highlighted in the image above in yellow.
4. Add Column Values with Criteria Using SUMIF or SUMIFS functions
We can’t really call it a sum process in a spreadsheet without adding in a few conditions, can we?
Adding values in Google Sheets is quite easy with the SUMIF function.
SUMIF(range, criterion, [sum_range])
The function takes a range of cells upon which a criterion is applied. We can either sum the values in the range or apply a separate “sum range” from where we will get our result.
For example, we have the following worksheet from where we want to find the total number of customers that have gone to a location with a rating of more than 5:
The formula will be:
• C2:C11 is the range of values for the Rating column. Here, the “>5” criterion is considered.
• D2:D11 is the range of the column whose values will be added if they meet the criterion.
Why not add another condition to the mix?
Let’s say we also want to know only the number of customers from the Washington region with a rating higher than 5.
While SUMIF can only have one criterion, the SUMIFS function can take multiple.
SUMIFS(sum_range, criteria_range1, criterion1, [criteria_range2, criterion2, ...])
The formula:
• D2:D11 is the range of cells that will be summed. The Customers column.
• C2:C11,”>5″: Condition set for the Rating column.
• B2:B11,”Washington”: Condition set for the Location column.
Note: In both cases, SUMIF and SUMIFS, you can remove the bottom row number limitation to make the formula more dynamic and accepting of new values, e.g., D2:D11 to D2:D.
Learn more: How to Perform Conditional Sum in Google Sheets (Easy Guide)
Final Words
That concludes our simple tutorial on how to sum an entire column in Google Sheets.
Each method is best used in specific scenarios and can also depend somewhat on user knowledge. We hope that we’ve been able to clarify these scenarios and you are now well-equipped to use all of
these approaches.
Feel free to leave any queries or advice you might have in the comments section below.
Related Articles for Reading
We will be happy to hear your thoughts
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how to make square root mathematics working model tlm for B.Ed Students | craftpiller - Science Projects | Maths TLM | English TLM | Physics Projects | Computer Projects | Geography Projects | Chemistry Projects | Working Projects | Working Models | DIY for School / College Science Exhibitions or Fair
how to make square root mathematics working model tlm for B.Ed Students | craftpiller
Creating a tangible learning model (TLM) for understanding square roots using color paper and cardboard can be a great visual aid for learning this mathematical concept.
Here’s a step-by-step guide to making the TLM:
Materials Needed:
1. Cardboard sheet
2. Color papers (representing squares and square roots)
3. Scissors
4. Glue
5. Marker
6. Ruler
Step 1: Prepare the Base
• Take a piece of cardboard and cut it into a rectangular base. This will serve as the foundation for your TLM.
Step 2: Define Squares and Square Roots
• Use different color papers to represent squares and square roots:
□ Red paper: Representing squares
□ Yellow paper: Representing square roots
Step 3: Create Square Tiles
• Cut the red colored paper into circle tiles, making sure each tile represents a perfect square. Write down examples of perfect squares on the tiles.
□ For example:
☆ Red Tiles (Perfect Squares):
○ Tile 1: “1” (1^2)
○ Tile 2: “4” (2^2)
○ Tile 3: “9” (3^2)
○ Tile 4: “16” (4^2)
○ …
Step 4: Create Square Root Tiles
• Cut the blue colored paper into square root tiles. Write down examples of square roots on these tiles.
□ For example:
☆ Blue Tiles (Square Roots):
○ Tile 1: “√1” = “1”
○ Tile 2: “√4” = “2”
○ Tile 3: “√9” = “3”
○ Tile 4: “√16” = “4”
○ …
Step 5: Arrange Tiles
• Place the square tiles (red) on the cardboard base, forming a grid-like pattern. Leave space between the tiles for the square root tiles (blue) to be placed.
Step 6: Label and Explain
• Use a marker to label each square tile with the corresponding perfect square value (e.g., “1”, “4”, “9”, etc.).
Square Root Working Model Explanation:
1. Understanding Squares:
□ Show how each square tile represents a perfect square. For example, “4” on a tile represents 2^2, and “9” represents 3^2.
2. Introducing Square Roots:
□ Introduce the concept of square roots using the blue tiles. Explain that the square root of a number “x” (√x) is the value that, when multiplied by itself, gives “x”.
3. Matching Squares with Square Roots:
□ Have students match the square tiles with their corresponding square root tiles. For example, match the tile with “4” to the tile with “√4” (which is “2”).
4. Examples and Practice:
□ Provide additional examples and let students practice matching other perfect squares with their respective square roots.
• Match “9” (perfect square) with “√9” (square root), which is “3”.
• Match “16” (perfect square) with “√16” (square root), which is “4”.
By creating this TLM, students can visually and interactively learn about squares, square roots, and the relationship between them. It provides a hands-on experience that reinforces the concept and
helps in understanding the mathematical operations involved.
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Program Examples
A 3-State Busy Beaver
The busy beaver problem is an interesting theoretical computer science problem. The problem is to find the smallest number of states that outputs as much data as possible yet eventually halts on its
own. More formally it goes something like this — given an n-state Turing machine with a two symbol alphabet {0, 1}, what is the maximum number of 1s that the machine may print on an initially blank
tape before halting?
This problem turns out to be non-computable, that is, for a small number of states an answer can be found, but in general it cannot be solved. Theorists call such problems non-computable. The number
of steps it takes to run to completion (halt) grows very rapidly as the number of states increase. This 3-state example takes 14 steps while the 4-state example takes 107 steps. Increasing from
there, a 5-state example has been found that takes 47,176,870 steps, and a 6-state example that takes 2.584 x10^2879 steps. I will not be trying any of these in the near future.
(0,0) -> (1,1) Right
(0,1) -> (0,1) Halt
(0,B) -> (1,1) Right
(1,0) -> (2,0) Right
(1,1) -> (1,1) Right
(1,B) -> (2,0) Right
(2,0) -> (2,1) Left
(2,1) -> (0,1) Left
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Turing's Titanic Machine!
In the March CACM, Barry Cooper
We quote Peter J. Denning introducing the ACM Ubiquity Symposium on "What is Computation?" as saying: "Researchers in biology and physics have claimed the discovery of natural computational
processes that have nothing to do with computers."
With Lance Fortnow distinctly underwhelmed: "Some people outside of computer science might think that there is a serious debate about the nature of computation. There isn't."
As often happens when experts disagree, the truth lies somewhere in between.
No it doesn't. My extreme point of view: A strong belief in the Church-Turing thesis that Turing machine captures the true notion of computation now and forever.
What's next? Casting doubt on 1+1=2? Sure no one has yet proved 1+1=3 but that doesn't mean it won't happen someday.
21 comments:
1. It is by no means obvious to me that solutions to the Einstein equations are going to be computable.
Do you know of a general proof?
My guess is that problems in proving this will arise due to the possible formation of singularities. It suggests an amusing strategy for attempting to find candidates for non-computable
processes: try to show that the problem of determining whether a singularity forms is non-computable (maybe we can encode the halting problem somehow?)
2. I understand that you may believe that all physical processes can be simulated on a Turing machine, but can you claim that this this is an analytic proposition akin to arithmetic?
3. Sure no one has yet proved 1+1=3 but that doesn't mean it won't happen someday.
I know many, but all of them devide by 0 at some point.
If you're speaking of correct proofs and you accept the Peano axioms as given, how should it be possible to prove 1 + 1 = 3? (With the common meaning of the symbols, of course)
4. @Martin Thoma: It's called sarcasm.
5. yes, and the universe is a set of nested spheres with the sphere of fixed stars around 80 million miles away
1. all science is right until it's wrong
6. @ Martin Thoma: It has been proven alredy that all mathematical systems are inconsistent. It seems you are not a regular follower of this blog. Check: http://kamouna.wordpress.com.
Rafee Kamouna.
1. Please don't go to that kook site and its one post.
7. Well, there are all sorts of models of computation that are fit for specific purpose, for example those which models concurrency, etc. Theoretically a lot of things could be made equivalent, but
some are much easier to work with for practical reason.
That said, I don't know who Denning is but from what he propose he sound like a big crank to me. However I'm not too surprised: "Ubiquity" is certainly not the only "new" area invented to create
"more jobs". This is what everyone do nowadays, what is the big deal?
8. "Researchers in biology and physics have claimed the discovery of natural computational processes that have nothing to do with computers"
My guess is that by this they really mean "nothing to do with digital computers consisting of CPU and motherboard and computerchips and keyboard and..."
A computer is what computes: that is tautological. If Denning is saying that physicists/biologists have found things that contradict the Church-Turing thesis, then that'd be a huge result, far
bigger than Wiles proving FLT. I highly doubt that's what he meant.
9. Oh boy! A chance to apply Forder's Principle:
"The virtue of a logical proof is not that it compels belief but that it suggests doubts. The proof tells us where to concentrate our doubts."
— Henry George Forder (1927)
If we assign central relevance to Lance's Postulate, that "Turing machines capture the true notion of computation now and forever", perhaps we can constructively adjoin to it Forder's Corollary:
"Computation does not capture the true notion of cognition, now or ever."
Forder's Corollary helps us appreciate how it is that we have conceived abundant, strong, rigorous insights regarding Turing Machines, and yet our conceptions regarding Strong AI remain sparse,
weak, and heuristic.
1. Well, we know from TCS, that it's hard to reconstruct a circuit simply by feeding it inputs and watching its outputs. So in that sense, "brains \subset Turing machines" is 100% compatible
with the Church-Turing-Deutsch thesis.
2. Aram, you know I love to triangulate tough problems (that's why Abraham Lincoln, Dwight Eisenhower, and Bill Clinton are three of my favorite politicians). So lets consider three TM models of
human cognition:
TM1: The TM1 machine stores a set of rules, and executes a logic engine for making deductions from those rules. The main problem with TM1 models is that they don't (yet) exhibit anything like
human cognition.
TM2: The TM2 machine stores a set of set of Hamiltonian potentials (classical or quantum), and an engine to integrate those potentials into trajectories. The problem with TM2 models is that
they're brute-force cellular-scale biophysics: they doesn't tell us how brains work in any integrative sense. As Hamming said "The purpose of computing is insight, not numbers."
TM3: ??? Profit! :)
Seriously, in Strong AI (as in many STEM fields) new paths forward would be very welcome. Perhaps this is one of those instances in which, as Tony Zee says, "a formerly cherished concept has
to be jettisoned."
3. Hmmm … maybe I should add, that we can draw upon the CT work of Juris Hartmanis, and combine it with elements of Terry Tao's analysis of the Puzzle of the Blue-Eyed Islanders, to construct
TM3's via a game called The Balanced Advantage Newcomb Game … a game that is inspired by (what else?) Newcomb's Paradox.
The implication for strong AI of The Balanced Advantage Newcomb Game is a pun upon a saying of Wittgenstein:
"If a TM3 can speak, then we cannot explain how it can speak."
4. I find these comments to be bizarre. After navigating through the misleadingly discursive mishmash, the principal objection seems to be that even if one finds a materialist model for
cognition, it would not give us any "insight". This is tangential if not irrelevant to the question of whether brains \subset Turing Machines in the sense of Turing-Church-Deutsch.
My personal view on this question is that we still know too little about the brain to jettison our "formerly cherished" materialist approach and that evidence from neurobiology strengthens
this position everyday.
5. I'm not sure whether the following point-of-view qualifies as "bizarre." Juris Hartmanis uses the instead the word "radical", in arguing for a complexity-theoretic formalism for which any
engineer has natural sympathy:
"Results about complexity of algorithms change quite radically if we consider only properties of computations which can be proven formally. … Results about optimality of all programs
computing the same function as a given program will differ from the optimality results about all programs which can be formally proven to be equivalent to the given program."
Here Hartmanis' starting conception (which he discusses at length) is that even so basic a concept as "membership in P" in generally cannot be proven formally; hence oracles are essential to
the definition of the class.
If we embrace Juris idea, which greatly restricts the role of oracles in complexity theoretic proofs, the end result is a reclassification of the entire Complexity Zoo. The hope is that in
the resulting zoo, separations among its inhabitants can be formally proved, thus bringing closure to complexity theory's long Groundhog Day.
The Balanced Advantage Newcomb Game was conceived and constructed with a view toward allowing us a glimpse of one small corner of a Hartmanis-style alternative Complexity Zoo — in which there
is no Newcomb Paradox.
Elevator Summary: When the standard complexity classes are restricted to include only algorithms whose membership is provable, the problem of proving separations alters radically — and it may
be that these alterations are enabling of proofs. And yet the Balanced Advantage Newcomb Game shows us that there exist deterministic TMs, whose computational output we regard as "human",
that reside outside these classes.
6. LOL … to express the same point in one sentence, the chief tenet of Fortnow-ism — “Every Complexity Zoo species is a TM” — is entirely consistent with the chief tenet of Hartmanis-ism — “The
Zoo's natural classes are oracle-independent.”
Moreover, it's excellent news for younger STEM researchers that in the celebrated phrase of Wolfgang Pauli: “Only technical details are missing!” :)
10. I look at discussions on hypercomputation and I find the following as relevant:
I like the article because it brings to mind such impossible tasks such as squaring the circle. In that case it is possible to create an approximation that can be computable, as mentioned in the
wikipedia article:
"Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to pi" http://en.wikipedia.org/
I think this points to the practical possibility that if one reduces the precision of a particular logic, then it is possible come to answers to a problem that may or may not be true, but can be
quickly verified. Self verifying theories are possible when arithmetic weaker than Peano arithmetic is allowed (http://en.wikipedia.org/wiki/Self-verifying_theories).
This may point to a suitable model of the human mind as one that requires a certain level of imprecision, suggesting that biologic systems are not in any way superior calculators, but just more
resourceful in the sense that it can be more efficient to quickly approximate and verify then to come to an exact solution directly.
11. Before you heard about quantum computers, how strongly did you believe (if at all) the strong Church-Turing thesis? (i.e. that randomized Turing machines can compute everything with not much more
effort than nature uses.)
12. This is a very different question: STRONG Church-Turing thesis relates to polynomial time equivalence.
It originally did NOT involve randomization.
I strongly recommend folks to read both Lance's and Davis' papers. They make their points clearly, and deal with nonsense almost charitably, and do a much better job at explaining why they are
nonsense than I'll try below.
The Church-Turing thesis is a postulate. It is not logically impossible to suppose that there is a plausible alternative. Unfortunately,all alternatives suggested so far are plain wrong.
One can feed noncomputable inputs to perfectly nice systems and -- surprise! -- the outputs may be uncomputable. The same effect can be obtained by making part of the system have noncomputable
parts. Again, hardly a great discovery. All of this can be obfuscated by the fact that finite initial segments of noncomputable numbers are finite, and hence each initial segment is computable.
Another set of objections is that computers do things that are not easily described as computations: operating systems, face recognition, automatic driving. Still, the underlying computational
process can be described by a Turing machine. Analogy: we know that protein folds somehow, yet we do not have a good efficient computational model for it. Biologists have learned from past
mistakes of "Natural History" when "flogistons" were invoked to "explain" fire, and "spontaneous generation" to "explain" life. Biologists do not invoke mysterious entities to explain protein
folding. Why would us, CS people, want to introduce some form of "post-Turing" model to explain processes that can, in principle, be modeled by Turing machines?
A mathematical relation of the "flogiston computing" models discussed is the use of precise models (for example, communicating asynchronous finite automata) whose behavior is noncomputable. The
argument then is that since such systems can be built, they are a model that is "more powerful" than Turing machines. This seems plausible, until one realizes that
1. Any finite initial segment of such a computational process -- in other words, anything we can actually compute with the device -- is also computable in the Church-Turing sense.
2. Turing machines have noncomputable behavior. in this sense. The Halting Problem is "computable" in the same sense as the behavior of these devices.
13. For my part, quantum computers are attempting to bring complex numbers into computation in a natural way. This brings the question of our ability to handle infinities. Abstractly I can think of a
Turing machine as a unit projective sphere (Riemann Sphere) where a vector indicates the current machine state and its head position (e.g. a complex number). The action is simply an operation on
the complex number. Since Turing thought that operations where inseparable from the state information one can see that a Turing machine is an algebraic structure.
One encounters the same problem with infinities that we see in quantum mechanics which requires the partitioning of the complex plane. In the Turing machine, that partitioning is implicit since
we typically use countable sets of cells and symbols.
One could possibly propose that we somehow build a Turing machine with a head and tape of infinite precision, but there would always be an uncountable set of inputs and outputs that would not be
representable in a nice reduced form like pi and e, and would thus take infinite amounts of time to represent. This gets to the point of non-computable functions.
If we enumerate the partitions of complex plane, and then try to use those enumerated elements to describe every number on the plane, we fail. There simply are not enough elements to index every
number (a la cantor). Those numbers are simply not computable. If one decides to incorporate the new number, then one is effectively repartitioning.
So here lies the rub. Any practical realization of hypercomputation requires some type of partitioning which effectively reduces the hypercomputer to a normal computer. This has nothing to do
with a lack of imagination but simple physical realities.
For the quantum computer, while the computer itself might be in a superposition of states, the ultimate input and output are definite finite states, so in principle, given enough time, there is
some set of operations that a non-quantum computer can perform that would reach the output state based on any input state.
Those are just my thoughts at the moment. | {"url":"https://blog.computationalcomplexity.org/2012/03/turings-titanic-machine.html?m=0","timestamp":"2024-11-08T02:58:43Z","content_type":"application/xhtml+xml","content_length":"219984","record_id":"<urn:uuid:125cd90c-bb0c-4479-8219-8c3b3abce5b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00498.warc.gz"} |
How do I change the size of a scatter point in Matplotlib?
How do I change the size of a scatter point in Matplotlib?
1. Increase the size of all points. To increase the size of scatter points, a solution is to use the option “s” from the function scatter(), example.
2. Points with different size. To plot points with different size, a solution is to provide a list of size (or an array) to “s”.
3. Combining several scatter plots.
What is marker size in scatter plot?
So far the answer to what the size of a scatter marker means is given in units of points. Points are often used in typography, where fonts are specified in points. Also linewidths is often specified
in points. The standard size of points in matplotlib is 72 points per inch (ppi) – 1 point is hence 1/72 inches.
How do you change the size of a scatter plot?
To format the size of data points in a scatter plot graph, right click any of the data points and select ‘format data series’ then select marker options and customize for larger or smaller data
What is Matplotlib default marker size?
s Keyword Argument to Set Matplotlib Scatter Marker Size The default scatter marker size is rcParams[‘lines. markersize’] ** 2 . According to documentation, s is the marker size in points2.
How do you increase markers size in Seaborn?
To set the size of markers, we can use the s parameter. This parameter can be used since seaborn is built on the matplotlib module. We can specify this argument in the scatterplot() function and set
it to some value. Alternatively, we can control the size of the points based on some variables.
How do you do a scatterplot in matplotlib?
Machine Learning – Scatter Plot
1. Example. Use the scatter() method to draw a scatter plot diagram: import matplotlib.pyplot as plt. x = [5,7,8,7,2,17,2,9,4,11,12,9,6] y = [99,86,87,88,111,86,103,87,94,78,77,85,86]
2. Example. A scatter plot with 1000 dots: import numpy. import matplotlib.pyplot as plt.
3. ❮ Previous Next ❯
How do you set the marker size in Seaborn scatter plot?
What is C in scatter plot?
c : color, sequence, or sequence of color, optional, default: ‘b’ The marker color. Possible values: A single color format string.
How to make a Matplotlib scatter plot?
Steps to Create a Scatter Plot in Matplotlib Import all the necessary libraries The first step is to import matplotlib, NumPy, and other required libraries for our tutorial. Let’s import them using
the import statement. Read the dataset For plotting Scatter plot in Matplotlib you have to first create two variables with data points Let’s say x and y. Create a scatter plot in matplotlib
How to increase marker size in scatter plot?
Increase Scatter Marker Size of Points Non-Uniformly in Matplotlib Double the Width of Matplotlib Scatter Marker To double the width (or height) of the marker we need to increase s by a factor of 4
as A = W*H => (2W)* (2H)= 4A.
How do you make a scatter graph?
Steps to create excel scatter plots: Select all the cells that contain data. Click on Insert tab. Look for Charts group. Under Chart group, you will find Scatter (X, Y) Chart. Click the arrow to see
the different types of scatter and bubble charts.
How to use scatter plot?
To use scatter plots and trend lines to compare sales to profit, follow these steps: Open the Sample – Superstore data source. Drag the Profit measure to Columns. Drag the Sales measure to Rows. Drag
the Category dimension to Color on the Marks card. Drag the Region dimension to Detail on the Marks card. To add trend lines, from the Analytics pane, drag the Trend Line model to the view, and then
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latex implementation
I think we can all agree it has been to long without that beautiful sweet sweet typeset equations. There are many people who do courses with heavy maths content, and will often come to their discord
pals for help, only for the other party to have to endure the pain of trying to read an integral that has not been typeset. I know this sounds pretty niche but it would the lives of us few
mathematicians and science students a lot easier and beautiful, and also take away the only appeal of using fb messenger, meaning that no one has any excuse for not using discord. Obviously I dont
expect to be able to type up entire documents in latex in discord, but just to be able to typeset all the maths i send back and forth would be a wonderful addition.
• Having a way to type equations would be really useful.
• please do this. I am a mathematician and would love to be able to talk about math with my mathematician friends.
• I would love this. I needed it now, so I found this post xD
• "I can’t believe this Feature request will happen as it would be effectively deciding to pivot to what is essentially an education focus when games are such a lucrative domain (and they are
killing it)."
What ? Once it is done it is done, you just need to redirect some commands to a pre-existing small latex equation renderer (It has been proven in this thread that they exist). It will not shift
the focus of the program don't be crazy. It just a cool add for people who are gamer and student, teacher, reseacher, whatever .
• It's absurd Discord advertises itself as a place where people talk yet they refuse to add STEM main "talk" language: LaTeX. It's not even complicated, just add in support for KATEX or MathJAX
• It wouldn't be very hard to add, they just need to include https://katex.org/
For the formatting, I'd suggest using $ $ for inline and $$ $$ for block comments (like in the Joplin note app). Example:
Some inline $\vec{v}$ vector.
Block for big equation:
\vec{a} = \vec{b} \times \begin{pmatrix}0 & 1 \\ 2 & 3\end{pmatrix}
• Well, unless you modify so much the program every update that you break everything each time, once a fonctionality is done you do not have so much to do.
You could even use mathjax https://www.mathjax.org/ (Discord is web based, am I right ?) and have literally nothing to do.
• I would love to see a LaTeX implementation in Discord. Not just for mathematical symbols but also for greek letters, unit formatting and convenient text formatting.
• Could be toggled with a forward-slash command, like `/latex ...`. Seeing as most bots that implement it by rendering the result server-side use node.js, it doesn't seem _too_ difficult to
implement in the Discord client.
• I really think mathjax would be welcome, especially with covid-19 and all the virtual classes that are being created. It just takes two line to implement MathJax at least on web/desktop
application, isn't ?
• The thing is that MathJax let you modify the renderer and so you can easily display plain source equation which is really more convenient for equation communication and modification.
• Yes, this would be very helpful for students, teachers, mathematicians, etc.
As it has been said above, the dollar symbol ($<place math here>$) should be used to mark LaTeX. For centered LaTeX (meaning it leans to the center instead of the side), mark it as this: \[<place
math here>\]
• In LaTeX, you have to use or :
□ $<math>$ for inline mathematical elements and $$<math>$$ for equation-like elements.
□ \(<math>\) for inline mathematical elements and \[<math>\] for equation-like elements.
Mixing these is really non-user friendly.
Also, people would less trigger math-mode if the second type of tags was used.
Imagine talking about prices, typing two dollars in a sentence and the math-mode is triggered. It would be horrible.
• Yes this would be great!
It could be done easily using markdown syntax using mathjax, katex, or even github's math rendering system, all of which use simplified varieties of LaTeX's/TeX's math notation syntax.
Side notes and known caveats w/ each:
GitHub's math rendering system,
only outputs images rather than mathml, html, or (best option for accessibility) mathml & html, like the other two options, preventing easy copying and limiting reasonable support given discords
current multimedia. As it just inputs the info from the parameter in the requesting URL into a template of the LaTeX standalone class and outputs to png.
For overall support, ease of use, customizability (for compatibility and use), would be a great pick but unfortunately it is the slowest of all the options.
It also allows limited macro definitions in both LaTeX and TeX styles, which are usually global to the instance of the MathJax system being used, so you would either need to sanities the input
for non math mode things, or be careful in how you set things up. It would likely not be difficult to make these customizations server and/or channel specific so that admins can set up custom
Not as customizable as MathJax, but it is noticeably lighter weight, and simpler to implement (though more restrictive and manual of a process).
However, it currently has some significant rendering and consistency errors/bugs. Between modifying the syntax, names, and functionality of std LaTeX lib items, as well as some commonly used
packages, as well as, having (currently as of 2021/04/06) issues with sizing and rendering of sub scripts and superscripts, and many other things that resize and move elements around. It can be
said that the package is just not as mature as the other two.
Anyway, I really hope to see this implemented one day, as using things like the math bot are just not as good of solutions, for discussion, as implementing some system like the ones mentioned
above that are commonly used for rendering math from markdown syntax (even json segmented markdown like discord has, see ipnb files for the jupyter systems).
• This could look like mathim and only include the math part of latex.
I agree this would really be a plus since I always have to use latex2png ^^.
• There's also TeXit https://top.gg/bot/510789298321096704
It would be nice if this actually happened though. Bot solutions are all well and good, but they aren't as natural as in-text solutions.
• Also, you can only use Bots on servers. Having it in private chat or groups would be awesome too.
I don't see any problem in implementing this to be honest :/
• Would love for LaTeX to get implemented like all the others in this post!
• Just leaving a comment since I would also like the tech people at discord to implement tex in their tech. Thanks.
• I'm glad to see this is being discussed. I work in a software & engineering company that uses Discord for its internal communication and adding equation rendering would be immense help.
• i think this would be good if there was an educational edition of discord, then teachers could have servers and kids could help eachother with their homework and ask the teacher questions while
they aren't at school. it would also be a good way for a sick kid to get their homework or notes when they're out for the day
• KaTeX would be better than MathJax, given how much less intensive it is; FB Messenger uses KaTeX in their implementation.
• I really want LaTeX support for Discord as well. I talk about math a lot and so it'll be really nice.
• I would like to second this. One does not simply neglect STEM's main communication language and claim to be a "place" for conversation.
• +1, it's apparent that Discord acknowledges its influence over schools and students, especially in the last couple years, so I think that LaTeX parsing (or something like it) have become just as
important as code-blocks.
• I agree that integrating typeset equations into Discord would greatly benefit mathematicians and science students. It would streamline academic discussions and eliminate the need for other
messaging apps.Great Clips Coupons
• Software is never ‘done’ if you want it to keep working. It always requires continual maintenance. That said- I’d be happy to be proven wrong because of all the benefits you mention.
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Package "plml"
Title: Prolog-Matlab bridge
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Latest version: 2.0.3
SHA1 sum: b1c66d575e90fff877b40b67453283869bdf7a9c
Author: Samer Abdallah <samer.abdallah@gmail.com>
Download URL: https://github.com/samer--/plml.git
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Prolog-Matlab bridge for SWI Prolog
Authors (2004--2012) Samer Abdallah Centre for Digital Music, Queen Mary, University of London,
Christophe Rhodes Centre for Computational Creativity Goldsmiths College, University of London
PLML is a foreign interface that enables Matlab to be used as a computational engine from within SWI Prolog. The basic idea is that instead of using the standard is/2 operator to evaluate a certain
class of terms, we can use the ===/2 operator to get Matlab to evaluate a (much richer) class of terms, eg
?- float(A)===trace(eye(3)).
A = 3.0
We can also get Matlab to perform actions with side effects, like making sounds and graphics; obviously these do not fit into the declartive semantics of Prolog and have to be dealt with under the
procedural semantics. If you want to execute a Matlab command in an imperative way and see the textual output, use the ??/1 operator, eg
?- ??disp(`hello).
>> hello
The interface works by using the Matlab Engine API, which starts up a Matlab process on the end of a pipe. The Matlab process can be on another machine, and multiple Matlab engines can be started on
the same or different machines. Matlab expressions are sent down the pipe and executed. Matlab's textual output comes back through the pipe. In addition, Matlab variables can be transferred directly
between the Matlab engine's memory space and SWI's memory space.
Expression language
Expressions to evaluate are given in a sublanguage of terms which is similar to but not exactly the same as Matlab. In particular, Prolog syntax cannot accommodate the single quoted Matlab strings,
(since these are equivalent to unquoted Prolog atoms), the Matlab syntax of matrices, (eg [1 2; 3 4]), and the Matlab syntax for slicing arrays (eg A(:,3:4)) if A is a Prolog variable. Strings are
handled using the q/1 or, if the SWI Prolog flag back_quotes is set to symbol_char, `/1 functor, ie `hello or q(hello) both evaluate to the Matlab string 'hello'. Arrays can be given either as flat
lists, which are interpreted as horizontal concatenation as in Matlab:
?- ??[1,2,3].
>> ans = 1 2 3
?- ??[eye(2),magic(2)].
>> ans =
or as nested listed for multidimensional arrays using the arr/1 functor, where the innermost nesting corresponds to the FIRST Matlab dimensions
?- ??arr([1,2,3]).
>> ans =
?- ??arr([[1,2],[3,4]]).
>> ans =
Cell arrays can be specified in a similar way using braces or the cell/1 functor.
To help with accessing array elements, see the Matlab functions general/paren, general/row, and general/col in the matlab directory.
Return values
The results of computations can handled in several ways:
1. Keep the result in a Matlab workspace variable in the engine's memory space. The names of these variables are allocated automatically and stored in a Prolog atom. The atoms have a garbage
collection callback which means that the Matlab workspace variable is (eventually) deleted if the Prolog atom goes out of scope.
?- A===2+2, ??disp(A). >> 4 % matlab textual output
A = ws(ml:t_2311) % Prolog blob pointing to Matlab variable t_2311
2. Convert the result to a prolog atom or term. The type of the resulting prolog term depends on the right hand side of the ===/2 operator:
?- int(A)===2+2. A = 4
?- float(A)===2+2. A = 4.0
There are other types for strings and atoms:
?- atom(A) === q(hello). % q/1 means quote as Matlab string A = hello.
?- string(A) === hello. % /1 is shorthand for q/1 A = "hello".
You can also get the result as a Matlab binary array on the Prolog side:
?- mx(A)===eye(4). % identity matrix A = <#0239c3a0> % Prolog blob handle (with garbage collection)
I haven't completely settled on the best way of handling arrays as self-contained Prolog terms, but you can do this:
?- array(A)===magic(3). A = [[8.0, 3.0, 4.0], [1.0, 5.0, 9.0], [6.0, 7.0, 2.0]]::[[3, 3]]
As you can see, multidimensional arrays are returned as nested lists, and the size of the array is given after the :: as [[3,3]].
3. Store the result to a MAT file and return a Prolog term which points to the file. The names are generated automatically. This allows for persistence of values which are referred to by stable
names that can be stored, eg in a database:
?- mat(A)===fft(buffer(wavread('somefile.wav'),256,128)).
A = mat:d0608/m48598|x % dynamically generated unique locator
This relies on the mechanism provided by the functions in matlab/db. A certain directory is designed the root of a 'matbase' (MAT file database). The default is ~/var/matbase (ie under the user's
home directory). The Matlab function dbroot returns or sets this directory:
?- ??dbroot. >> ans =
?- ??dbroot(q('/usr/share/lib/matbase')). % switch to shared matbase >> ans =
In this case, the locator mat:d0608/m48598|x refers to a Matlab variable called 'x' (it's always 'x') in the file /usr/share/lib/matbase/d0608/m48598.mat. A new directory is created each month,
and the filenames are chosen dynamically to avoid clashes with existing files.
To help with debugging, you can issue the command:
?- debug(plml).
which will cause each Matlab expression to be printed in its Matlab form before execution. I'm afraid the best documentation is the code itself, but I do intend to produce a manual once some of the
more embarrassing aspects of the system are resolved!
There are some limitations to what you can do with this Matlab interface.
Restrictions on Matlab code
The communication protocol between the Matlab Engine library and the Matlab process is somewhat flawed. if the Matlab computation outputs the characters flushed_stdout at any point, the conversation
between the engine library and Matlab is irretrievable broken: the library gets out of step with the stream coming from Matlab and there is no way to get back into step. You have to close and reopen
the Matlab instance and reopen.
Ctrl-C handling (UNIX only)
In Matlab, you can use Ctrl-C to interrupt a stuck or long running computation. Matlab respons to the INT signal generated when you press Ctrl-C at the terminal. What happens when you press Ctrl-C
during a computation when using this plml library depends on which thread the call was made. If it is the foreground thread, the INT signal interrupts BOTH the Matlab computation (because the Matlab
process is in the same Unix process group) AND the current Prolog thread, which will be in the depths of a call to the Matlab Engine API. This causes the call to return an error, but the Engine API
loses synchonisation with the input stream from Matlab and the conversation is irretrievably broken. I'm afraid there's nothing we can do about this without interposing a filter a filter between the
Matlab process and the engine library.
If the Matlab call is made in a background thread, Ctrl-C interrupts the foreground Prolog thread, but not the engine API call. The INT signal is still sent to the Matlab process, so it is
interrupted and the engine API call returns early.
The same effect can be obtained by sending an INT signal directly to the Matlab process. You can use the Matlab function feature to get Matlab's process ID.
However, there is still a problem with detecting that Matlab has been interrupted. In some cases, there is no observable difference between completion and interruption (try pressing Ctrl-C while
running pause(10) in Matlab directly). In others, Matlab will print a message to stderr, and then stop with no other observable effect, in particular lasterr will not return anything useful. The
engine API does not observe Matlab's stderr, and so there is no way to detect that a Matlab call has been interrupted. To get round this, plml adds the command disp("}") to the end of each Matlab
execution. It then checks to see if "}\n" appears at the end of the output. If not, we conclude that the computation was interrupted. This is fine unless the call produces so much output that it
doesn't fit into the output buffer. In this case, the check is skipped, and so checking for interruptions will not work. An 'output truncated' message is also printed on stderr.
Before you start, you need a working SWI Prolog installation and a Matlab installation. Then the pack_install/1 facility of SWI Prolog should be enough to build the C++ code foreign library.
As it stands, there is a minor difficulty when attempting to load the plml.dylib foreign library: because of the way the Matlab application is designed, plml.dylib ends up looking for various Matlab
libraries (libeng, libmx, etc.) using relative paths, not absolute paths, which means that if you simply start SWI Prolog and load the plml module, the foreign library fails to load. There are two
solutions to this:
1. Set the DYLD_FALLBACK_LIBRARY_PATH environment variable before starting SWI Prolog. A shell script for doing this, called swiplml can be found in the scripts directory of this package. If you
copy or link it into your PATH, and use swiplml instead of swipl, it should work fine.
2. A minor adjustment of some the Matlab libraries can made, changing them to use the @rpath mechanism of dyld means that using the -rpath option while linking plml.dylib solves the problem.
Although some of Matlab's installed files are altered, this does not seem to affect the normal running of Matlab.
The fixdylibs pseudo-target of this package's Makefile will do this for you, but as it may require root priviledges, it uses sudo and therefore requires that you have administrator rights. The
changes are made using scripts/fixdylibs, which will also make backups of your original Matlab libraries if they do not yet exist.
If the installation is ok and you have followed one of the two options above, then you should be able to start SWI Prolog and do the following:
?- use_module(library(plml)).
% ...
?- ml_open(ml). % ml is the name assigned to the Matlab engine instance
Matlab engine (ml) open.
?- float(A)===2*pi.
A = 6.28319
?- A:float===2*pi.
A = 6.28319
If you have a working X-server available, then the following should result in a straight line plot.
?- ?? figure(1).
% ...
?- ?? plot(1:10).
There are two configuration options in the Makefile. If you enable DEBUG mode and rebuild, then the library will print a series of cryptic characters as it aquires and releases the mutexes protecting
the WS variable release queue and the Matlab engine.
If you enable NOLOCK, the mutex protecting the Matlab engine will be disabled. You might want this if you are sure that the library is going to be called in a single threaded way, but probably any
performance gain will be miniscule.
This work was partially supported by UK EPSRC grants GR/S84750/01 and GR/S82213/01.
Contents of pack "plml"
Pack contains 31 files holding a total of 199K bytes. | {"url":"https://us.swi-prolog.org/pack/list?p=plml","timestamp":"2024-11-08T08:08:44Z","content_type":"text/html","content_length":"32191","record_id":"<urn:uuid:24e551ec-862c-4217-91b6-7a659983657c>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00213.warc.gz"} |
TY - CONF T1 - Avoiding Deontic Explosion by Contextually Restricting Aggregation T2 - Proceedings of the 10th International Conference on Deontic Logic in Computer Science (DEON 2010) Y1 - 2010 A1 -
Meheus, Joke A1 - Beirlaen, Mathieu A1 - Van De Putte, Frederik ED - Governatori, Guido ED - Sartor, Giovanni AB -
In this paper, we present an adaptive logic for deontic conflicts, called \sys{P2.1}$^r$, that is based on Goble's logic \sys{SDL}$a$\sys{P}$e$–-a bimodal extension of Goble's logic \sys{P} that
invalidates aggregation for all \emph{prima facie} obligations. The logic \sys{P2.1}$^r$ has several advantages with respect to \sys{SDL}$a$\sys{P}$e$. For consistent sets of obligations it yields
the same results as Standard Deontic Logic and for inconsistent sets of obligations, it validates aggregation ``as much as possible''. It thus leads to a richer consequence set than \sys{SDL}$a$\sys
{P}$e$. The logic \sys{P2.1}$^r$ avoids Goble's criticisms against other non-adjunctive systems of deontic logic. Moreover, it can handle all the `toy examples' from the literature as well as more
complex ones.
JA - Proceedings of the 10th International Conference on Deontic Logic in Computer Science (DEON 2010) PB - Springer CY - Dordrecht ER - | {"url":"https://www.clps.ugent.be/research/publications/export/ris/1329","timestamp":"2024-11-06T01:24:47Z","content_type":"text/plain","content_length":"1679","record_id":"<urn:uuid:47bdd410-4fc3-4ddc-b175-87c06c970dd3>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00067.warc.gz"} |
Linux Vacation/Eastern Europe
Software Generators of Petri Net Models
LVEE 2019
Petri net models of variable size having definite structure are characteristic for manifold application domains such as networking and high performance computing, manufacturing control, and
structural biology. A formalism of infinite Petri nets allows us to specify suchlike systems. Though models of definite size are of some interest for illustrative purposes. Moreover, an inductive
technique for drawing conclusions on an infinite model properties is based on a sequence of models with growing size. A technique of composing programs in C language which generate Petri net models
is developed, a dozen of generators implemented and available via GitHub as open source software. Models are represented either in graphical format for 2D case or in logical format for higher number
of dimensions.
Communication grids find wide application in radio and cellular networks, numerical parallel solving tasks by the method of finite elements, as communication subsystems of supercomputers (especially,
multidimensional torus), and network-on-chip structures. Petri net models of communication grids are considered in ^1 and ^2. To find their properties for verification of corresponging protocols and
their performance evaluation, we need a collection of models having incremented size.
To avoid laborious work with manual editing bulky Petri net models of real-life objects having regular structure, our software generates models automatically on a given set of parameters.
Petri nets are a part of UML notation ^3 for specification of parallel processes. They find wide application as a language of parallel programming ^4, for verification of parallel programs and
networking protocols ^1,^2, modeling automated manufacture ^5, and in avionics ^6.
Let us consider examples of triangular
and hexagonal switching grid structures
of size 4 generated by the corresponding programs studied in the present paper.
An infinite Petri net is a powerful abstraction described in ^1 and ^2, which allows the specification of models having an arbitrary size but a definite structure obtained by spatial composition of
one or more basic components. An overwhelming result is obtained that gives a way to draw conclusions on properties of models with no regard to their size. For visualization and application of
inductive methods which draw conclusions for infinite nets of definite structure, a fast facility that creates a model of a given size is required.
We developed a technique of software generators design based on the infinite Petri net specification and a series of about a dozen of generators of switching grid structure of square, triangular, and
hexagonal form on a plane (2D) as well as hypercube and hypertorus in multidimensional space listed in the Conclusions section. The open source software has been uploaded on GitHub and is compatible
on data formats with modeling system Tina ^7.
Modeling system Tina ^7 is for years de facto standard for Petri net models exchange and analysis; in 2019, Tina has won international Model Checking Contest. It allows automatic visualization of
models, building its state space and finding structural properties ^5 as well as reflecting Petri net behavior — token game.
Specification of Infinite Petri Nets
The basic technique allows studying an infinite structures using their finite parametric specification obtained by repetition and composition of a single component. Our models are divided into two
classes: for grids with definite number of dimensions, basically 1D and 2D; for grids with arbitrary number of dimensions. In the first case we use parameters of size only while in the second case a
parameter which defines the number of dimensions is added. Structures having more than one basic component can be studied as well.
For majority of models we consider a cellular structure which implies connections of cells according to von Neumann neighborhood, some models are composed using Moore neighborhood, a generalized
Zaitsev neighborhood is applied as well. Connection of neighbor cells is obtained by merging their contact places.
Remind that a Petri net is a bipartite directed graph supplied with dynamic elements called tokens. One part of vertices is called places and drawn as circles while the other part of vertices is
called transitions and drawn as rectangles. We use notation of a parametric multiset rewriting system, each row specifies a transition by lists of its input and output places.
Let us consider a square grid specification obtained from a basic component represented by a single place as its repetition in cells of a two dimensional grid supplied by transitions according to von
Neumann neighborhood written in TEX notation:
$$\left( \left( \begin{array}{l} t_{1,1}^{i,j}: p^{i,j} -> p^{i-1,j}, i>1,\\ t_{1,2}^{i,j}: p^{i,j} -> p^{i+1,j}, i<n,\\ t_{2,1}^{i,j}: p^{i,j} -> p^{i,j-1}, j>1,\\ t_{2,2}^{i,j}: p^{i,j} -> p^{i,j-1}, j<n,\\ \end{array} \right), 1<=i<=n, 1<=j<=n, \right)$$.
The parameter n gives the grid size while variables i and j specify the cell indices in horizontal and vertical directions correspondingly. The upper index (i,j) gives the cell location inside the
lattice; i goes from top to down and j goes from left to right. Enumeration of transitions using two lower indices (d,r) suits multidimensional grid as well; d gives the number of dimension and r
gives one of two directions: 1 – to the origin, 2 – to infinity.
Generators of Models in Logical Format
Models in logical format are rather simple specifying names of places and transitions and their connections. Though for 2D models there is a definite lack of visibility, which is partially mended by
automatic drawing a net by Tina, logical format is convenient for multidimensional grids where visualization is hampered. The basic element of logical file format (.net) is the following:
tr <t-name> <p-name>[*<weight>],… -> <p-name> [*<weight>],...
After the indicator “tr”, the transition name follows, then a list of its input places, and after “→”, a list of its output places are written; optional arc weigh is started by “*” sign, its absence
means unary weight.
The following fragment of C program generates the specified grid
if(i>1) printf(“tr {t_1,1^%d,%d} {p^%d,%d} -> {p^%d,%d}\n”,i,j,i,j,i-1,j);
if(i<n) printf(“tr {t_1,2^%d,%d} {p^%d,%d} -> {p^%d,%d}\n”,i,j,i,j,i+1,j);
if(j>1) printf(“tr {t_2,1^%d,%d} {p^%d,%d} -> {p^%d,%d}\n”,i,j,i,j,i,j-1);
if(j<n) printf(“tr {t_2,2^%d,%d} {p^%d,%d} -> {p^%d,%d}\n”,i,j,i,j,i,j+1);
Loaded into Tina, models are either visualized using “draw” tools or analyzed directly via calculating linear invariants and construction of reachability (coverability) tree. Inductive resoning on a
series of basis invariants obtained for models of incremented size allow us obtaining invariants of infinite nets in parametric form.
Generators of Models in Graphical Format
Models in graphical format are devised either for 2D or 1D grids. They reflect spatial structure of grid using coordinates of places and transitions. Some early models considered spatial structures
of a single dimension, for instance for a bus Ethernet model. Among late one dimension models we constructed a generator of a net counting a double exponent. Basic models of a triangular, square, and
hexagonal grids have been developed for a plane.
The peculiarity of graphical format is binding places and transitions of Petri nets to definite coordinated on plane. For simplicity we suppose arcs connecting nodes are straight though some our
models contain curved arcs. At first we choose the grid cell size and compute a cell offset of its top-left corner on both coordinates; finally we add to the coordinates offsets inside cells. Having
offsets DI and DJ, and supposing that a place is situated in the center of a cell having transitions in between with a space of 2DT in middle of two transitions with opposite direction of switching,
we obtain the following code for printing a place and a transition of a cell:
printf(“p %f %f {p^%d,%d} 0 n”, i*DI, j*DJ, i, j);
printf(“t %f %f {to_1,1^%d,%d} 0 w n”, i*DI-DT, j*DJ-DJ/2, i, j);
printf(“e {p^%d,%d} {to_1,1^%d,%d} 1 n”, i,j,i,j);
printf(“e {to_1,1^%d,%d} {p^%d,%d} 1 n”, i,j,i-1,j);
Here in graphical format. a row started with: “p” specifies a place, “t” specifies a transition, and “e” specifies an arc; coordinates and a name follow for a node; an arc is given by two its ends;
some additional parameters of graphical representation follow, for instance “0 w n”.
As examples, images of triangular and hexagonal communication switching grids have been considered in the Introduction.
The technique for software generators of Petri net models design is developed based on a given specification of an infinite Petri net. A series of open source generators have been implemented and
uploaded on GitHub with MIT License for the following forms of grids:
• square https://github.com/dazeorgacm/sq ;
• triangular https://github.com/tishtri/g3a ;
• hexagonal https://github.com/tishtri/g6a ;
• hypertorus https://github.com/dazeorgacm/htgen ;
• hypercube with various edge conditions https://github.com/tishtri/hcgen ;
• hypercube and hypertorus with generalized Zaitsev neighborhood https://github.com/dazeorgacm/hmn .
Converted by Tina ^7 to PNML format generated nets can be analysed using lots of open source software for Petri nets available on GitHub, say PNML_Parse ^8.
^1 Dmitry A. Zaitsev, Ivan D. Zaitsev and Tatiana R. Shmeleva. Infinite Petri Nets: Part 1, Modeling Square Grid Structures, Complex Systems, 26(2), 2017, 157-195.
^2 Dmitry A. Zaitsev, Ivan D. Zaitsev and Tatiana R. Shmeleva. Infinite Petri Nets: Part 2, Modeling Triangular, Hexagonal, Hypercube and Hypertorus Structures, Complex Systems, 26(4), 2017, 341-371.
^3 Grady Booch, The Unified Modeling Language User Guide. Addison Wesley Professional, 2017, 504p.
^4 Zaitsev D.A., Jürjens J. Programming in the Sleptsov net language for systems control, Advances in Mechanical Engineering, 2016, Vol. 8(4), 1–11. http://dx.doi.org/10.1177/1687814016640159
^5 ZhiWu Li, MengChu Zhou, Deadlock Resolution in Automated Manufacturing Systems: A Novel Petri Net Approach. Springer Science & Business Media, 2009, 240p.
^6 Zaitsev D.A. and Shmeleva T.R. Modeling With Colored Petri Nets: Specification, Verification, and Performance Evaluation of Systems (pp. 378-404) Chapter 14 in T. Shmelova, N. Rizun, D. Kucherov
and K. Dergachov (Ed.) Automated Systems in the Aviation and Aerospace Industries. IGI-Global: USA, 2019.
^7 Time Petri Net Analyzer (Tina), http://projects.laas.fr/tina/
^8 Petri net from PNML file and build its reachability graph, https://github.com/Tj-Cong/PNML_Parse
Abstract licensed under Creative Commons Attribution-ShareAlike 3.0 license | {"url":"https://lvee.org/en/abstracts/299","timestamp":"2024-11-13T02:03:04Z","content_type":"application/xhtml+xml","content_length":"31055","record_id":"<urn:uuid:6ca53b18-aa9b-46df-b626-07aabbdf1132>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00137.warc.gz"} |
Partisan Republican Attempt to Alter California’s Electoral College Distribution - Davis Vanguard
Guest Commentary by Bill Ritter
Following the 2000 presidential election in which Al Gore received more than 500,000 votes nationwide than George W. Bush there were calls to abandon the Electoral College system of electing a
president. Nothing has been done on a national level to change that system, although I believe most citizens would agree with the “One man, one vote” concept of Democracy, in which every person’s
vote is counted equally.
Currently there is an attempt in California by Republican lawyers and political operatives to change the way only California’s presidential electors are to be selected. They have begun to circulate
petitions for an initiative entitled the “Presidential Election Reform Act” (PERA). PERA’s sponsors — the misleadingly-named “Californians for Equal Representation” — hope to qualify it for the June
2008 primary ballot.
Current California election law and that of 48 other states dictates that the highest vote getter for the presidency wins all of those state’s electors. Here in California, this “winner takes all”
rule would be changed to allow each congressional district to elect one presidential elector creating the situation in which nearly 20 out of 53 California congressional districts could be won by the
Republican presidential candidate due to those district’s high Republican registration numbers and voting history and thereby awarded a Republican presidential elector. This would create separate
“winner take all” elections in every California congressional district. But across America, for example in historically Republican majority states such as Ohio, Indiana, Missouri and most of the
western states and all of the southern states including Florida would remain “winner take all” states, thereby disenfranchising their Democratic presidential voters and favoring Republican
presidential candidates.
If we are to have authentic presidential election reform we need a national change affecting all states whereby every vote is counted in the same manner equally throughout America. That is the only
way to ensure fairness to all.
An excellent and informative article on the subject is written by Mr. Vikram David Amar entitled “The So-Called Presidential Election Reform Act: A Clear Abuse of California’s Initiative Process.”
He writes:
Transparent Partisan Ploys Interfere With Legitimate Reform Efforts: This Isn’t an Issue of National Electoral Reform, But of Turning a Blue State Partially Red
……if moving away from winner-take-all rules made policy sense for the country, surely blue states should not do so if red states are not doing so as well. It is neither fair nor sensible to think
that California Democrats, who hold a majority in the state, should unilaterally give up their clout in electing a President if Republican majorities in other states are not doing the same thing in
states they control. It is more than ironic that an organization that calls itself “Californians for Equal Representation” could support a scheme that would produce such partisan inequality.
One cardinal rule about Presidential electoral reform today ought to be that it must not predictably and intentionally disadvantage one party over the other. Sensible change in this area is hard
enough to accomplish without wasting our time on obviously partisan ploys.
Real Reform Would Involve A Direct National Popular Election
Which brings me to the real way to make every vote count and encourage Presidential candidates to campaign in California – adopt the national popular election plan (and which is also being considered
by legislators in California right now.) If the national popular vote winner were automatically made President, every voter’s vote nationwide would count – and count for the same thing. (There’s
something Californians truly dedicated to “Equal Representation” should rally around.)
Mr. Amar’s entire article published in FindLaw—Legal News & Commentary, can be found at: http://writ.news.findlaw.com/amar/20070817.html#continue
A related story by the Sacramento Bee can be found at: http://www.sacbee.com/111/story/309132.html
44 comments
1. The “Bushies” are at it again. If you can’t win and you know you can’t win then change the rules, confuse voters and have a election that is not fair or transparent. It seems to be their MO.
Voters better keep their eye on this. It’s serious.
2. The “Bushies” are at it again. If you can’t win and you know you can’t win then change the rules, confuse voters and have a election that is not fair or transparent. It seems to be their MO.
Voters better keep their eye on this. It’s serious.
3. The “Bushies” are at it again. If you can’t win and you know you can’t win then change the rules, confuse voters and have a election that is not fair or transparent. It seems to be their MO.
Voters better keep their eye on this. It’s serious.
4. The “Bushies” are at it again. If you can’t win and you know you can’t win then change the rules, confuse voters and have a election that is not fair or transparent. It seems to be their MO.
Voters better keep their eye on this. It’s serious.
5. Congressional allocation of electoral votes is nothing more than a power play by a state’s minority party. The Republicans in California are trying to do it by initiative. The Democrats in North
Carolina (a red presidential state) tried to do it through legislation. It is a partisan solution to a non-partisan problem.
The real solution is to move to a direct national popular vote for President. That system would eliminate the reality of “battleground” and “safe” states. Every voter would become a battleground
as each candidate would be competing for each vote, not for a given state or congressional district. Under a national popular vote system, every vote would be equal. The candidate with the most
votes would win.
A group called National Popular Vote (www.nationalpopularvote.com) is trying to implement just such a change. There are identical bills pending in more than 40 states. They have received positive
mention in the NY Times, LA Times and a bunch of other papers.
6. Congressional allocation of electoral votes is nothing more than a power play by a state’s minority party. The Republicans in California are trying to do it by initiative. The Democrats in North
Carolina (a red presidential state) tried to do it through legislation. It is a partisan solution to a non-partisan problem.
The real solution is to move to a direct national popular vote for President. That system would eliminate the reality of “battleground” and “safe” states. Every voter would become a battleground
as each candidate would be competing for each vote, not for a given state or congressional district. Under a national popular vote system, every vote would be equal. The candidate with the most
votes would win.
A group called National Popular Vote (www.nationalpopularvote.com) is trying to implement just such a change. There are identical bills pending in more than 40 states. They have received positive
mention in the NY Times, LA Times and a bunch of other papers.
7. Congressional allocation of electoral votes is nothing more than a power play by a state’s minority party. The Republicans in California are trying to do it by initiative. The Democrats in North
Carolina (a red presidential state) tried to do it through legislation. It is a partisan solution to a non-partisan problem.
The real solution is to move to a direct national popular vote for President. That system would eliminate the reality of “battleground” and “safe” states. Every voter would become a battleground
as each candidate would be competing for each vote, not for a given state or congressional district. Under a national popular vote system, every vote would be equal. The candidate with the most
votes would win.
A group called National Popular Vote (www.nationalpopularvote.com) is trying to implement just such a change. There are identical bills pending in more than 40 states. They have received positive
mention in the NY Times, LA Times and a bunch of other papers.
8. Congressional allocation of electoral votes is nothing more than a power play by a state’s minority party. The Republicans in California are trying to do it by initiative. The Democrats in North
Carolina (a red presidential state) tried to do it through legislation. It is a partisan solution to a non-partisan problem.
The real solution is to move to a direct national popular vote for President. That system would eliminate the reality of “battleground” and “safe” states. Every voter would become a battleground
as each candidate would be competing for each vote, not for a given state or congressional district. Under a national popular vote system, every vote would be equal. The candidate with the most
votes would win.
A group called National Popular Vote (www.nationalpopularvote.com) is trying to implement just such a change. There are identical bills pending in more than 40 states. They have received positive
mention in the NY Times, LA Times and a bunch of other papers.
9. This Republican scheme “Presidential Election Reform Act” (PERA) to dilute the Democratic Party presidential candidate’s ability to win all of California’s electoral votes while preserving the
Republican Party’s ability to continue to use “winner take all” rules throughout the remainder of American is telling. It is inherently unfair.
The President should be elected by the same rules throughout America. The Electoral College in the 21st Century is outdated—it should be done away with. We should pick our President by direct
national election.
10. This Republican scheme “Presidential Election Reform Act” (PERA) to dilute the Democratic Party presidential candidate’s ability to win all of California’s electoral votes while preserving the
Republican Party’s ability to continue to use “winner take all” rules throughout the remainder of American is telling. It is inherently unfair.
The President should be elected by the same rules throughout America. The Electoral College in the 21st Century is outdated—it should be done away with. We should pick our President by direct
national election.
11. This Republican scheme “Presidential Election Reform Act” (PERA) to dilute the Democratic Party presidential candidate’s ability to win all of California’s electoral votes while preserving the
Republican Party’s ability to continue to use “winner take all” rules throughout the remainder of American is telling. It is inherently unfair.
The President should be elected by the same rules throughout America. The Electoral College in the 21st Century is outdated—it should be done away with. We should pick our President by direct
national election.
12. This Republican scheme “Presidential Election Reform Act” (PERA) to dilute the Democratic Party presidential candidate’s ability to win all of California’s electoral votes while preserving the
Republican Party’s ability to continue to use “winner take all” rules throughout the remainder of American is telling. It is inherently unfair.
The President should be elected by the same rules throughout America. The Electoral College in the 21st Century is outdated—it should be done away with. We should pick our President by direct
national election.
13. In 1992, Wendell Williams from Walnut Creek lost to a sick Republican. He should have had it in the bag because the majority of residents for the district were Democrats. However, due to
Republican inspired voting lines changed he lost. This was critical race for the country that year.
14. In 1992, Wendell Williams from Walnut Creek lost to a sick Republican. He should have had it in the bag because the majority of residents for the district were Democrats. However, due to
Republican inspired voting lines changed he lost. This was critical race for the country that year.
15. In 1992, Wendell Williams from Walnut Creek lost to a sick Republican. He should have had it in the bag because the majority of residents for the district were Democrats. However, due to
Republican inspired voting lines changed he lost. This was critical race for the country that year.
16. In 1992, Wendell Williams from Walnut Creek lost to a sick Republican. He should have had it in the bag because the majority of residents for the district were Democrats. However, due to
Republican inspired voting lines changed he lost. This was critical race for the country that year.
17. Everyone should read this excellent article from the New Yorker on the subject at hand. It has some very good points for rebutting the proposition.
18. Everyone should read this excellent article from the New Yorker on the subject at hand. It has some very good points for rebutting the proposition.
19. Everyone should read this excellent article from the New Yorker on the subject at hand. It has some very good points for rebutting the proposition.
20. Everyone should read this excellent article from the New Yorker on the subject at hand. It has some very good points for rebutting the proposition.
21. The link doesn’t seem to be working, trying putting the two lines together in your address bar.
22. The link doesn’t seem to be working, trying putting the two lines together in your address bar.
23. The link doesn’t seem to be working, trying putting the two lines together in your address bar.
24. The link doesn’t seem to be working, trying putting the two lines together in your address bar.
25. This is a terrible proposition, but given recent levels of success for initiatives, I think it’s unlikely it will pass.
26. This is a terrible proposition, but given recent levels of success for initiatives, I think it’s unlikely it will pass.
27. This is a terrible proposition, but given recent levels of success for initiatives, I think it’s unlikely it will pass.
28. This is a terrible proposition, but given recent levels of success for initiatives, I think it’s unlikely it will pass.
29. The Electoral College reminds us of the importance of our federal system. Our country is named the “United States,” not the “United People.” We don’t live in a democracy, we live in a republic.
It would take a constitutional amendment ratified by 3/4 of states to change the system. It’s hard to imagine the smaller states agreeing to such an amendment. The electoral college was made to
satisfy the small states and to satisfy checks and balances.
I would not want New York (New York City), California (Los Angeles), Illinois (Chicago) and Texas (Houston) to determine the US President. In a straight popular vote you can be assured that the
candidate with the most money will address only the needs of those populated areas. You can also be assured that racial and social lines will make campaigns (and ultimately, the country) more
On the state level, it might be easier for California to modify its winner-take-all system. Changing it would certainly provide more “voice” to our water-rich Northern Californa.
30. The Electoral College reminds us of the importance of our federal system. Our country is named the “United States,” not the “United People.” We don’t live in a democracy, we live in a republic.
It would take a constitutional amendment ratified by 3/4 of states to change the system. It’s hard to imagine the smaller states agreeing to such an amendment. The electoral college was made to
satisfy the small states and to satisfy checks and balances.
I would not want New York (New York City), California (Los Angeles), Illinois (Chicago) and Texas (Houston) to determine the US President. In a straight popular vote you can be assured that the
candidate with the most money will address only the needs of those populated areas. You can also be assured that racial and social lines will make campaigns (and ultimately, the country) more
On the state level, it might be easier for California to modify its winner-take-all system. Changing it would certainly provide more “voice” to our water-rich Northern Californa.
31. The Electoral College reminds us of the importance of our federal system. Our country is named the “United States,” not the “United People.” We don’t live in a democracy, we live in a republic.
It would take a constitutional amendment ratified by 3/4 of states to change the system. It’s hard to imagine the smaller states agreeing to such an amendment. The electoral college was made to
satisfy the small states and to satisfy checks and balances.
I would not want New York (New York City), California (Los Angeles), Illinois (Chicago) and Texas (Houston) to determine the US President. In a straight popular vote you can be assured that the
candidate with the most money will address only the needs of those populated areas. You can also be assured that racial and social lines will make campaigns (and ultimately, the country) more
On the state level, it might be easier for California to modify its winner-take-all system. Changing it would certainly provide more “voice” to our water-rich Northern Californa.
32. The Electoral College reminds us of the importance of our federal system. Our country is named the “United States,” not the “United People.” We don’t live in a democracy, we live in a republic.
It would take a constitutional amendment ratified by 3/4 of states to change the system. It’s hard to imagine the smaller states agreeing to such an amendment. The electoral college was made to
satisfy the small states and to satisfy checks and balances.
I would not want New York (New York City), California (Los Angeles), Illinois (Chicago) and Texas (Houston) to determine the US President. In a straight popular vote you can be assured that the
candidate with the most money will address only the needs of those populated areas. You can also be assured that racial and social lines will make campaigns (and ultimately, the country) more
On the state level, it might be easier for California to modify its winner-take-all system. Changing it would certainly provide more “voice” to our water-rich Northern Californa.
33. The Electoral College violates the civil rights of each and every American in the United States. We are in this together and our votes must be equal for any government to derive its “just
powers.” The Constitution begins “WE THE PEOPLE,” not We the States.
The Electoral College is an unconstitutional feature unconstitutionally inserted into the Constitution. So concluded Lucius Wilmerding of Princeton, the political scientist who coined the term
“One voter–One vote” in his work The Electoral College written in 1958. And Andrew Jackson told Congress in 1828 that the Electoral College was never intended to reverse the vote of the American
The Electoral College violates the Declaration of Independence in many ways. It does not treat Americans as created equal, it converts their inalienable rights into mere privileges, and violates
government with the consent of the governed (the greater truth goes with the greater number” DeTocqueville).
If the Declaration of Independence is binding then the Electoral College is unconstitutional. (1783 a slave named Quok Walker sues his slavemaster in Massachusetts arguing that the words “all men
are created equal” made illegal the slavery then existing in that state). The Supreme Court of Massachusetts agrees with the slave and abolished slavery in Massachusetts.
The Declaration of Independence is explicitly recognized the Supreme law of the state of all Post Civil War States. See for example Congressional enabling act and the Constitution of the state of
Nevada. http://www.nevadaobserver.com. The State of Nevada, for example, provided, “That the constitution, when formed, shall be republican, and not repugnant to the constitution of the United
States, and the principles of the Declaration of Independence.” The supremacy of the Declaration of 1776 over the Slave Holder’s Constitution of 1787 was inserted into the Constitutions and
enabling acts of all states of the Union admitted after the Gettysburg Address.
The Supreme Court of the United States said in 1964 that all votes in the same constituency must have the same weight. The court outlawed the electoral college of the state of Georgia which gave
different weight according to county. Every vote must be equal regardless of the place it was cast said the Supreme Court. It is time to implement this ruling.
Yours truly
Gary Michael Coutin, Esquire
34. The Electoral College violates the civil rights of each and every American in the United States. We are in this together and our votes must be equal for any government to derive its “just
powers.” The Constitution begins “WE THE PEOPLE,” not We the States.
The Electoral College is an unconstitutional feature unconstitutionally inserted into the Constitution. So concluded Lucius Wilmerding of Princeton, the political scientist who coined the term
“One voter–One vote” in his work The Electoral College written in 1958. And Andrew Jackson told Congress in 1828 that the Electoral College was never intended to reverse the vote of the American
The Electoral College violates the Declaration of Independence in many ways. It does not treat Americans as created equal, it converts their inalienable rights into mere privileges, and violates
government with the consent of the governed (the greater truth goes with the greater number” DeTocqueville).
If the Declaration of Independence is binding then the Electoral College is unconstitutional. (1783 a slave named Quok Walker sues his slavemaster in Massachusetts arguing that the words “all men
are created equal” made illegal the slavery then existing in that state). The Supreme Court of Massachusetts agrees with the slave and abolished slavery in Massachusetts.
The Declaration of Independence is explicitly recognized the Supreme law of the state of all Post Civil War States. See for example Congressional enabling act and the Constitution of the state of
Nevada. http://www.nevadaobserver.com. The State of Nevada, for example, provided, “That the constitution, when formed, shall be republican, and not repugnant to the constitution of the United
States, and the principles of the Declaration of Independence.” The supremacy of the Declaration of 1776 over the Slave Holder’s Constitution of 1787 was inserted into the Constitutions and
enabling acts of all states of the Union admitted after the Gettysburg Address.
The Supreme Court of the United States said in 1964 that all votes in the same constituency must have the same weight. The court outlawed the electoral college of the state of Georgia which gave
different weight according to county. Every vote must be equal regardless of the place it was cast said the Supreme Court. It is time to implement this ruling.
Yours truly
Gary Michael Coutin, Esquire
35. The Electoral College violates the civil rights of each and every American in the United States. We are in this together and our votes must be equal for any government to derive its “just
powers.” The Constitution begins “WE THE PEOPLE,” not We the States.
The Electoral College is an unconstitutional feature unconstitutionally inserted into the Constitution. So concluded Lucius Wilmerding of Princeton, the political scientist who coined the term
“One voter–One vote” in his work The Electoral College written in 1958. And Andrew Jackson told Congress in 1828 that the Electoral College was never intended to reverse the vote of the American
The Electoral College violates the Declaration of Independence in many ways. It does not treat Americans as created equal, it converts their inalienable rights into mere privileges, and violates
government with the consent of the governed (the greater truth goes with the greater number” DeTocqueville).
If the Declaration of Independence is binding then the Electoral College is unconstitutional. (1783 a slave named Quok Walker sues his slavemaster in Massachusetts arguing that the words “all men
are created equal” made illegal the slavery then existing in that state). The Supreme Court of Massachusetts agrees with the slave and abolished slavery in Massachusetts.
The Declaration of Independence is explicitly recognized the Supreme law of the state of all Post Civil War States. See for example Congressional enabling act and the Constitution of the state of
Nevada. http://www.nevadaobserver.com. The State of Nevada, for example, provided, “That the constitution, when formed, shall be republican, and not repugnant to the constitution of the United
States, and the principles of the Declaration of Independence.” The supremacy of the Declaration of 1776 over the Slave Holder’s Constitution of 1787 was inserted into the Constitutions and
enabling acts of all states of the Union admitted after the Gettysburg Address.
The Supreme Court of the United States said in 1964 that all votes in the same constituency must have the same weight. The court outlawed the electoral college of the state of Georgia which gave
different weight according to county. Every vote must be equal regardless of the place it was cast said the Supreme Court. It is time to implement this ruling.
Yours truly
Gary Michael Coutin, Esquire
36. The Electoral College violates the civil rights of each and every American in the United States. We are in this together and our votes must be equal for any government to derive its “just
powers.” The Constitution begins “WE THE PEOPLE,” not We the States.
The Electoral College is an unconstitutional feature unconstitutionally inserted into the Constitution. So concluded Lucius Wilmerding of Princeton, the political scientist who coined the term
“One voter–One vote” in his work The Electoral College written in 1958. And Andrew Jackson told Congress in 1828 that the Electoral College was never intended to reverse the vote of the American
The Electoral College violates the Declaration of Independence in many ways. It does not treat Americans as created equal, it converts their inalienable rights into mere privileges, and violates
government with the consent of the governed (the greater truth goes with the greater number” DeTocqueville).
If the Declaration of Independence is binding then the Electoral College is unconstitutional. (1783 a slave named Quok Walker sues his slavemaster in Massachusetts arguing that the words “all men
are created equal” made illegal the slavery then existing in that state). The Supreme Court of Massachusetts agrees with the slave and abolished slavery in Massachusetts.
The Declaration of Independence is explicitly recognized the Supreme law of the state of all Post Civil War States. See for example Congressional enabling act and the Constitution of the state of
Nevada. http://www.nevadaobserver.com. The State of Nevada, for example, provided, “That the constitution, when formed, shall be republican, and not repugnant to the constitution of the United
States, and the principles of the Declaration of Independence.” The supremacy of the Declaration of 1776 over the Slave Holder’s Constitution of 1787 was inserted into the Constitutions and
enabling acts of all states of the Union admitted after the Gettysburg Address.
The Supreme Court of the United States said in 1964 that all votes in the same constituency must have the same weight. The court outlawed the electoral college of the state of Georgia which gave
different weight according to county. Every vote must be equal regardless of the place it was cast said the Supreme Court. It is time to implement this ruling.
Yours truly
Gary Michael Coutin, Esquire
37. The Electoral College violates the civil rights of each and every American in the United States. We are in this together and our votes must be equal for any government to derive its “just
powers.” The Constitution begins “WE THE PEOPLE,” not We the States.
The Electoral College is an unconstitutional feature unconstitutionally inserted into the Constitution. So concluded Lucius Wilmerding of Princeton, the political scientist who coined the term
“One voter–One vote” in his work The Electoral College written in 1958. And Andrew Jackson told Congress in 1828 that the Electoral College was never intended to reverse the vote of the American
The Electoral College violates the Declaration of Independence in many ways. It does not treat Americans as created equal, it converts their inalienable rights into mere privileges, and violates
government with the consent of the governed (the greater truth goes with the greater number” DeTocqueville).
If the Declaration of Independence is binding then the Electoral College is unconstitutional. (1783 a slave named Quok Walker sues his slavemaster in Massachusetts arguing that the words “all men
are created equal” made illegal the slavery then existing in that state). The Supreme Court of Massachusetts agrees with the slave and abolished slavery in Massachusetts.
The Declaration of Independence is explicitly recognized the Supreme law of the state of all Post Civil War States. See for example Congressional enabling act and the Constitution of the state of
Nevada. http://www.nevadaobserver.com. The State of Nevada, for example, provided, “That the constitution, when formed, shall be republican, and not repugnant to the constitution of the United
States, and the principles of the Declaration of Independence.” The supremacy of the Declaration of 1776 over the Slave Holder’s Constitution of 1787 was inserted into the Constitutions and
enabling acts of all states of the Union admitted after the Gettysburg Address.
The Supreme Court of the United States said in 1964 that all votes in the same constituency must have the same weight. The court outlawed the electoral college of the state of Georgia which gave
different weight according to county. Every vote must be equal regardless of the place it was cast said the Supreme Court. It is time to implement this ruling.
Yours truly
Gary Michael Coutin, Esquire
38. The Electoral College violates the civil rights of each and every American in the United States. We are in this together and our votes must be equal for any government to derive its “just
powers.” The Constitution begins “WE THE PEOPLE,” not We the States.
The Electoral College is an unconstitutional feature unconstitutionally inserted into the Constitution. So concluded Lucius Wilmerding of Princeton, the political scientist who coined the term
“One voter–One vote” in his work The Electoral College written in 1958. And Andrew Jackson told Congress in 1828 that the Electoral College was never intended to reverse the vote of the American
The Electoral College violates the Declaration of Independence in many ways. It does not treat Americans as created equal, it converts their inalienable rights into mere privileges, and violates
government with the consent of the governed (the greater truth goes with the greater number” DeTocqueville).
If the Declaration of Independence is binding then the Electoral College is unconstitutional. (1783 a slave named Quok Walker sues his slavemaster in Massachusetts arguing that the words “all men
are created equal” made illegal the slavery then existing in that state). The Supreme Court of Massachusetts agrees with the slave and abolished slavery in Massachusetts.
The Declaration of Independence is explicitly recognized the Supreme law of the state of all Post Civil War States. See for example Congressional enabling act and the Constitution of the state of
Nevada. http://www.nevadaobserver.com. The State of Nevada, for example, provided, “That the constitution, when formed, shall be republican, and not repugnant to the constitution of the United
States, and the principles of the Declaration of Independence.” The supremacy of the Declaration of 1776 over the Slave Holder’s Constitution of 1787 was inserted into the Constitutions and
enabling acts of all states of the Union admitted after the Gettysburg Address.
The Supreme Court of the United States said in 1964 that all votes in the same constituency must have the same weight. The court outlawed the electoral college of the state of Georgia which gave
different weight according to county. Every vote must be equal regardless of the place it was cast said the Supreme Court. It is time to implement this ruling.
Yours truly
Gary Michael Coutin, Esquire
39. The Electoral College violates the civil rights of each and every American in the United States. We are in this together and our votes must be equal for any government to derive its “just
powers.” The Constitution begins “WE THE PEOPLE,” not We the States.
The Electoral College is an unconstitutional feature unconstitutionally inserted into the Constitution. So concluded Lucius Wilmerding of Princeton, the political scientist who coined the term
“One voter–One vote” in his work The Electoral College written in 1958. And Andrew Jackson told Congress in 1828 that the Electoral College was never intended to reverse the vote of the American
The Electoral College violates the Declaration of Independence in many ways. It does not treat Americans as created equal, it converts their inalienable rights into mere privileges, and violates
government with the consent of the governed (the greater truth goes with the greater number” DeTocqueville).
If the Declaration of Independence is binding then the Electoral College is unconstitutional. (1783 a slave named Quok Walker sues his slavemaster in Massachusetts arguing that the words “all men
are created equal” made illegal the slavery then existing in that state). The Supreme Court of Massachusetts agrees with the slave and abolished slavery in Massachusetts.
The Declaration of Independence is explicitly recognized the Supreme law of the state of all Post Civil War States. See for example Congressional enabling act and the Constitution of the state of
Nevada. http://www.nevadaobserver.com. The State of Nevada, for example, provided, “That the constitution, when formed, shall be republican, and not repugnant to the constitution of the United
States, and the principles of the Declaration of Independence.” The supremacy of the Declaration of 1776 over the Slave Holder’s Constitution of 1787 was inserted into the Constitutions and
enabling acts of all states of the Union admitted after the Gettysburg Address.
The Supreme Court of the United States said in 1964 that all votes in the same constituency must have the same weight. The court outlawed the electoral college of the state of Georgia which gave
different weight according to county. Every vote must be equal regardless of the place it was cast said the Supreme Court. It is time to implement this ruling.
Yours truly
Gary Michael Coutin, Esquire
40. The Electoral College violates the civil rights of each and every American in the United States. We are in this together and our votes must be equal for any government to derive its “just
powers.” The Constitution begins “WE THE PEOPLE,” not We the States.
The Electoral College is an unconstitutional feature unconstitutionally inserted into the Constitution. So concluded Lucius Wilmerding of Princeton, the political scientist who coined the term
“One voter–One vote” in his work The Electoral College written in 1958. And Andrew Jackson told Congress in 1828 that the Electoral College was never intended to reverse the vote of the American
The Electoral College violates the Declaration of Independence in many ways. It does not treat Americans as created equal, it converts their inalienable rights into mere privileges, and violates
government with the consent of the governed (the greater truth goes with the greater number” DeTocqueville).
If the Declaration of Independence is binding then the Electoral College is unconstitutional. (1783 a slave named Quok Walker sues his slavemaster in Massachusetts arguing that the words “all men
are created equal” made illegal the slavery then existing in that state). The Supreme Court of Massachusetts agrees with the slave and abolished slavery in Massachusetts.
The Declaration of Independence is explicitly recognized the Supreme law of the state of all Post Civil War States. See for example Congressional enabling act and the Constitution of the state of
Nevada. http://www.nevadaobserver.com. The State of Nevada, for example, provided, “That the constitution, when formed, shall be republican, and not repugnant to the constitution of the United
States, and the principles of the Declaration of Independence.” The supremacy of the Declaration of 1776 over the Slave Holder’s Constitution of 1787 was inserted into the Constitutions and
enabling acts of all states of the Union admitted after the Gettysburg Address.
The Supreme Court of the United States said in 1964 that all votes in the same constituency must have the same weight. The court outlawed the electoral college of the state of Georgia which gave
different weight according to county. Every vote must be equal regardless of the place it was cast said the Supreme Court. It is time to implement this ruling.
Yours truly
Gary Michael Coutin, Esquire
41. The Constitution of the United States starts off “We the people of the United States.”
Your whole argument is based on a bad premise followed by nonsense.
Oh, I take that back, Lucius Wilmerding was elected as our Supreme Ruler and what he says, goes.
42. The Constitution of the United States starts off “We the people of the United States.”
Your whole argument is based on a bad premise followed by nonsense.
Oh, I take that back, Lucius Wilmerding was elected as our Supreme Ruler and what he says, goes.
43. The Constitution of the United States starts off “We the people of the United States.”
Your whole argument is based on a bad premise followed by nonsense.
Oh, I take that back, Lucius Wilmerding was elected as our Supreme Ruler and what he says, goes.
44. The Constitution of the United States starts off “We the people of the United States.”
Your whole argument is based on a bad premise followed by nonsense.
Oh, I take that back, Lucius Wilmerding was elected as our Supreme Ruler and what he says, goes.
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[Solved] Find the derivative. 1 f(x)=(4x3+5x)1/3 | SolutionInn
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Find the derivative. 1 f(x)=(4x3+5x)1/3
1 f(x)=(4x3+5x)1/3
There are 3 Steps involved in it
Step: 1
the derivative of the function fx 1 4x 3 5x 13 well us...
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Cube Root Calculator
Last updated:
Cube Root Calculator
Our cube root calculator is a handy tool that will help you determine the cube root, also called the 3^rd root, of any positive number. You can immediately use our calculator; just type the number
you want to find the cube root of and it's done! Moreover, you can do the calculations the other way around and use them to cube numbers. To do this just type the number you want to raise to third
power in the last field! It may be extremely useful while searching for so-called perfect cubes. You can read more about them in the following article.
Thanks to our cube root calculator, you may also calculate the roots of other degrees. To do so, you need to change the number in the degree of the root field. If you would like to learn more about
the cube root definition, familiarize yourself with the properties of the cube root function, and find a list of the perfect cubes, we strongly recommend you keep on reading this text. In there, you
can also find some tricks on how to find the cube root on a calculator or how to calculate it in your head.
If you are interested in the history of root symbols head to the square root calculator, where we discuss it.
Cube root definition
Let's assume you want to find the cube root of a number, x. The cube root, y, is such a number that, if raised to the third power, will give x as a result. If you formulate this mathematically,
∛x = y ⟺ y^3 = x
where ⟺ is a mathematical symbol that means if and only if.
It is also possible to write the cube root in a different way, which is sometimes much more convenient. It is because a cube root is a special case of an exponent. It can be written down as
∛(x) = x^(1/3)
A geometric example may help you understand this. The best example we can give would be that of the cube. Well, the cube root of a cube's volume is its edge length. So, for example, if a cube has a
volume of 27 cm³, then the length of its edges is equal to the cube root of 27 cm³, which is 3 cm. Easy?
You should remember that in most cases, the cube root will not be a rational number. These numbers can be expressed as a quotient of two natural numbers, i.e., a fraction. Fractions can cause some
difficulties, especially when it comes to adding them. If you are having trouble working with fractions, try our adding fractions calculator, which will help you immensely.
What is the cube root of...?
It is really easy to find the cube root of any positive number with our cube root calculator! Simply type in any number to find its cube root. For example, the cube root of 216 is 6. For the list of
perfect cubes, head to the next section.
Note that it is possible to find a cube root of a negative number as well. After all, a negative number raised to the third power is still negative - for instance, (-6)³ = -216.
You need to remember, though, that any non-zero number has three cube roots: at least one real one and two imaginary ones. This cube root calculator deals with real numbers only, but if you're
interested, we encourage you to read more on the topic of imaginary numbers!
Most common values - perfect cubes list
You can find the most common cube root values below. Those numbers are also very often called perfect cubes because their cube roots are integers. Here is the list of the ten first perfect cubes:
• cube root of 1: ∛1 = 1, since 1 * 1 * 1 = 1;
• cube root of 8: ∛8 = 2, since 2 * 2 * 2 = 8;
• cube root of 27: ∛27 = 3, since 3 * 3 * 3 = 27;
• cube root of 64: ∛64 = 4, since 4 * 4 * 4 = 64;
• cube root of 125: ∛125 = 5, since 5 * 5 * 5 = 125;
• cube root of 216: ∛216 = 6, since 6 * 6 * 6 = 216;
• cube root of 343: ∛343 = 7, since 7 * 7 * 7 = 343;
• cube root of 512: ∛512 = 8, since 8 * 8 * 8 = 512;
• cube root of 729: ∛729 = 9, since 9 * 9 * 9 = 729;
• cube root of 1000: ∛1000 = 10, since 10 * 10 * 10 = 1000;
As you can see, numbers become very large quickly, but sometimes you'll have to deal with even bigger numbers, such as factorials. In this case, we recommend using scientific notation, which is a
much more convenient way of writing down really big or really small numbers.
On the other hand, most other numbers are not perfect cubes, but some of them are still used often. Here is the list of some of the non-perfect cubes, rounded to the hundredths:
• cube root of 2: ∛2 ≈ 1.26;
• cube root of 3: ∛3 ≈ 1.44;
• cube root of 4: ∛4 ≈ 1.59;
• cube root of 5: ∛5 ≈ 1.71;
• cube root of 10: ∛10 ≈ 2.15;
Don't hesitate to use our cube root calculator if the number you want and need is not on this list!
Cube root function and graph
You can graph the function y = ∛(x). Unlike e.g. the logarithmic function, the cube root function is an odd function - it means that it is symmetric with respect to the origin and fulfills the
condition - f(x) = f(-x). This function also passes through zero.
Thanks to this function, you can draw a cube root graph, which is shown below. We also encourage you to check out the quadratic formula calculator to look at other function formulas!
How to calculate cube root in your head?
Do you think that it is possible to solve simple problems with cube roots without an online calculator, or even a pencil or paper? If you think that it is impossible, or that you are incapable of
doing it check out this method, it is very easy. However, it only works for perfect cubes. Forget all the rules in the arithmetic books and consider for a moment the following method described by
Robert Kelly.
First of all, it is essential to memorize the cubes of the numbers from 1 to 10 and the last digit of their cubes. It is presented in the table below.
Number Cube Last digit
When you have a number you want to find the cube root of look first at the thousands (skip the last three digits). For example, for the number 185,193, The thousands are 185. The cube of 5 is 125 and
of 6 is 216. Therefore it is obvious that the number you are searching for is between 50 and 60. The next step is to ignore all the other figures except the last digit. We can see that it's 3, so
check your memory or in our table. You will find that the number you are searching for is 7. So the answer is 57! Easy?
Let's take another example and do it step by step!
1. Think of the number that you want to know as a cube root. Let's take 17576.
2. Skip the three last digits.
3. Find the two closest cube roots that you know. The cube root of 8 is 2, and the cube root of 27 is 3. So your number is between 20 and 30.
4. Look at the last digit. The last digit of 17576 is 6.
5. Check your memory (or on our table) - the last digit 6 corresponds with the number 6. This is the last digit of your number.
6. Combine the two: 26. This is the cube root of 17576!
We remind you that this algorithm works only for perfect cubes! And the probability that a random number is a perfect cube is, alas, really low. You've got only a 0.0091 percent chance of finding one
between 1,000 and 1,000,000. If you're not sure about your number, just forget about that rule and use our cube root calculator :-)
How do I find the cube root on a regular calculator?
1. First, you need to type the number for which you need to find the cube root
2. Press √ (root key) two times
3. Press x (multiplication sign)
4. Press √ (root key) four times
5. Press x (multiplication sign)
6. Press √ (root key) eight times
7. Press x (multiplication sign)
8. One last time, press the √ (root key) two times
9. And now you can press = (equal to sign)! Here is your answer!
Don't you believe it? Check it one more time with another example!
Examples of cube root questions
Let's say you need to make a ball with a volume of 33.5 ml. To prepare it you need to know its radius. As you probably know, the equation for calculating the volume of a sphere is as follows:
V = (4/3) * π * r³
So the equation for the radius looks like this:
r = ∛(3V/4π)
You know that the volume is 33.5 ml. At first, you need to switch to different volume units. The simplest conversion is into cm³: 33.5 ml = 33.5 cm³. Now you can solve the radius:
r = ∛(100.5/12.56)
r = ∛(8)
r = 2
For a ball to have a volume of 33.5 ml, its radius should be 2 centimeters.
nth root calculator
With our root calculator, you can also calculate other roots. Just write the number in the Degree of the root field, and you will receive any chosen nth root calculator. Our calculator will
automatically do all necessary calculations, and you can freely use it in your calculations!
So, let's take some examples. Let's assume you need to calculate the fourth root of 1296. First, you need to write the appropriate number you want to root - 1296. Then change the degree of the root
to 4. And you've got the result! The fourth root of 1296 is 6.
Our nth root calculator also enables you to calculate the root of irrational numbers. Let's try it by calculating π-th root. Symbol π represents the ratio of a circle's circumference to its diameter.
Its value is constant for every circle and is approximately 3.14, but you can use our ratio calculator to find its more precise value!
Let's say you want to calculate the π-th root of 450. First, write 450 in the number box. Then change the degree of the root - let's round and write 3.14 instead of π. And now you can see the result.
It's almost 7.
Three solutions of the cube root
At the end of this article, we've prepared an advanced mathematics section for the most persistent of you. You probably know that positive numbers always have two square roots: one negative and one
positive. For example, √4 = -2 and √4 = 2. But did you know that a similar rule applies to the cube roots? All real numbers (except zero) have exactly three cube roots: one real number and a pair of
complex ones. Complex numbers were introduced by mathematicians a long time ago to explain problems that real numbers cannot do. We usually express them in the following form:
x = a + b*i
where x is the complex number with the real a and imaginary b parts (for real numbers b = 0). The mysterious imaginary number i is defined as the square root of -1:
i = √(-1)
Alright, but how does this knowledge influence the number of cube root solutions? As an example, consider the cube roots of 8, which are 2, -1 + i√3, and -1 - i√3. If you don't believe us, let's
check it by raising them to the power of 3, remembering that i² = -1 and using the short multiplication formula (a + b)³ = a³ + 3a²b + 3ab² + b³:
1. 2³ = 8 - the obvious one,
2. (-1 + i√3)³ = -1 + 3i√3 + 9 - 3i√3 = 8,
3. (-1 - i√3)³ = -1 - 3i√3 + 9 + 3i√3 = 8.
Do you see it now? All of them equal 8!
How do I find the cube root of a product?
The cube root of a product of two numbers is the product of the cube roots of these numbers. That is, the formula is ∛(a × b) = ∛a × ∛b.
What is the cube root of -8/27?
The answer is -2/3. To get this result, take these steps:
1. Recall the formula ∛(a / b) = ∛a / ∛b.
2. Compute the cube root of -8. Clearly, ∛(-8) = -2.
3. Compute the cube root of 27: we have ∛27 = 3.
4. The final result is -2/3. Well done!
How do I write the cube root on a computer?
The Alt code for the cube root ∛ symbol is 8731. That is, to produce ∛, take these steps:
1. Make sure the Num Lock is on.
2. Press down one of the Alt keys.
3. Holding down the Alt key, type the code 8731 using the numeric keypad.
4. Let go of the Alt key. The cube root symbol will appear.
5. Alternative method: copy the ∛ symbol (Ctrl+C) and paste it wherever you need it (Ctrl+V). | {"url":"https://www.omnicalculator.com/math/cube-root","timestamp":"2024-11-03T22:44:05Z","content_type":"text/html","content_length":"425187","record_id":"<urn:uuid:08eeaf2f-091f-405c-8287-cd9e8e1b4b81>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00865.warc.gz"} |
GAP.jl · GAP.jl
GAP.jl is a low level interface from Julia to the computer algebra system GAP. The term "low level" means that the aim is to give Julia access to all GAP objects, to let Julia call GAP functions, and
to provide conversions of low level data (integers, Booleans, strings, arrays/lists, dictionaries/records) between the two systems.
In particular, it is not the aim of GAP.jl to provide Julia types for higher level GAP objects that represent algebraic structures, such as groups, rings, fields, etc., and mappings between such
The connection between GAP and Julia is in fact bidirectional, that is, GAP can access all Julia objects, call Julia functions, and perform conversions of low level data. This direction will become
interesting on the Julia side as soon as GAP packages provide functionality that is based on using Julia code from the GAP side.
The viewpoint of an interface from GAP to Julia is described in the manual of the GAP package JuliaInterface. | {"url":"https://oscar-system.github.io/GAP.jl/dev/","timestamp":"2024-11-08T05:22:34Z","content_type":"text/html","content_length":"10544","record_id":"<urn:uuid:2f93d520-4f98-44ee-b961-f799aac532da>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00619.warc.gz"} |
Faster circuits and shorter formulae for multiple addition, multiplication and symmetric Boolean functions
A general theory is developed for constructing the shallowest possible circuits and the shortest possible formulas for the carry-save addition of n numbers using any given basic addition unit. More
precisely, it is shown that if BA is a basic addition unit with occurrence matrix N, then the shortest multiple carry-save addition formulas that could be obtained by composing BA units are of size n
^1/p+o(1), where p is the unique real number for which the L[p] norm of the matrix N equals 1. An analogous result connects the delay matrix M of the basic addition unit BA and the minimal q such
that multiple carry-save addition circuits of depth (q + o(1)) log n could be constructed by combining BA units. On the basis of these optimal constructions of multiple carry-save adders, the
shallowest known multiplication circuits are constructed.
Dive into the research topics of 'Faster circuits and shorter formulae for multiple addition, multiplication and symmetric Boolean functions'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/faster-circuits-and-shorter-formulae-for-multiple-addition-multip","timestamp":"2024-11-12T13:19:46Z","content_type":"text/html","content_length":"47594","record_id":"<urn:uuid:58f257c0-5e4a-441d-b187-ce6d217ce000>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00727.warc.gz"} |
Bayesian Network Webserver
│Bayesian Network Webserver for Biological Network Modeling│
The Bayesian Network Web Server (BNW) is a comprehensive web server for Bayesian network modeling of biological data sets. It is designed so that users can quickly and seamlessly upload a dataset,
learn the structure of the network model that best explains the data, and use the model to understand and make predictions about relationships between the variables in the model. Many real world data
sets, including those used to create genetic network models, contain both discrete (e.g., genotypes) and continuous (e.g., gene expression traits) variables, and BNW allows for modeling of these
hybrid data sets.
BNW has recently been updated. Read more about these updates here.
An older version of BNW is available here.
How to cite BNW:
1. Ziebarth JD, Bhattacharya A, Cui Y (2013) Bayesian Network Webserver: a comprehensive tool for biological network modeling. Bioinformatics. 29(21): 2801-2803.
2. Ziebarth JD, Cui Y (2017) Precise network modeling of system genetics data using the Bayesian Network Webserver. In: Schughart K, Williams R (eds) System Genetics. Methods in Molecular Biology,
vol 1488. Humana Press, New York, NY.
Developed and maintained by: Yan Cui's Lab at University of Tennessee Health Science Center | {"url":"https://bnw.genenetwork.org/sourcecodes/home.php","timestamp":"2024-11-06T07:04:15Z","content_type":"text/html","content_length":"6829","record_id":"<urn:uuid:1bc83eec-76ba-45e2-b97d-32c7fcfd5da2>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00020.warc.gz"} |
square kilometer to barn
area surface units conversion
Amount: 1 square kilometer (km2, sq km) of area
Equals: 10,000,000,000,000,000,608,673,814,477,275,136.00 barns (b) in area
Converting square kilometer to barns value in the area surface units scale.
TOGGLE : from barns into square kilometers in the other way around.
CONVERT : between other area surface measuring units - complete list.
How many barns are in 1 square kilometer? The answer is: 1 km2, sq km equals 10,000,000,000,000,000,608,673,814,477,275,136.00 b
10,000,000,000,000,000,608,673,814,477,275,136.00 b is converted to 1 of what?
The barns unit number 10,000,000,000,000,000,608,673,814,477,275,136.00 b converts to 1 km2, sq km, one square kilometer. It is the EQUAL area value of 1 square kilometer but in the barns area unit
km2, sq km/b area surface conversion result
From Symbol Equals Result Symbol
1 km2, sq km = 10,000,000,000,000,000,608,673,814,477,275,136.00 b
Conversion chart - square kilometers to barns
1 square kilometer to barns = 10,000,000,000,000,000,608,673,814,477,275,136.00 b
2 square kilometers to barns = 20,000,000,000,000,001,217,347,628,954,550,272.00 b
3 square kilometers to barns = 30,000,000,000,000,001,826,021,443,431,825,408.00 b
4 square kilometers to barns = 40,000,000,000,000,002,434,695,257,909,100,544.00 b
5 square kilometers to barns = 49,999,999,999,999,998,431,683,053,958,987,776.00 b
6 square kilometers to barns = 60,000,000,000,000,003,652,042,886,863,650,816.00 b
7 square kilometers to barns = 70,000,000,000,000,008,872,402,719,768,313,856.00 b
8 square kilometers to barns = 80,000,000,000,000,004,869,390,515,818,201,088.00 b
9 square kilometers to barns = 90,000,000,000,000,000,866,378,311,868,088,320.00 b
10 square kilometers to barns = 99,999,999,999,999,996,863,366,107,917,975,552.00 b
11 square kilometers to barns = 110,000,000,000,000,011,307,097,977,677,414,400.00 b
12 square kilometers to barns = 120,000,000,000,000,007,304,085,773,727,301,632.00 b
13 square kilometers to barns = 130,000,000,000,000,003,301,073,569,777,188,864.00 b
14 square kilometers to barns = 140,000,000,000,000,017,744,805,439,536,627,712.00 b
15 square kilometers to barns = 150,000,000,000,000,013,741,793,235,586,514,944.00 b
Category: main menu • area surface menu • Square kilometers
Convert area surface of square kilometer (km2, sq km) and barns (b) units in reverse from barns into square kilometers.
Area units calculator
Main area or surface units converter page.
Converter type: area surface units
First unit: square kilometer (km2, sq km) is used for measuring area.
Second: barn (b) is unit of area.
15 km2, sq km = ? b
15 km2, sq km = 150,000,000,000,000,013,741,793,235,586,514,944.00 b
Abbreviation, or prefix, for square kilometer is:
km2, sq km
Abbreviation for barn is:
Other applications for this area surface calculator ...
With the above mentioned two-units calculating service it provides, this area surface converter proved to be useful also as a teaching tool:
1. in practicing square kilometers and barns ( km2, sq km vs. b ) measures exchange.
2. for conversion factors between unit pairs.
3. work with area surface's values and properties. | {"url":"https://www.traditionaloven.com/tutorials/surface-area/convert-sq-km-square-kilometer-to-barn.html","timestamp":"2024-11-03T10:35:01Z","content_type":"text/html","content_length":"46242","record_id":"<urn:uuid:6c1611b6-1de3-4da7-bb19-7c5f42b92516>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00395.warc.gz"} |
Derivative of E tetra x
Hello! First off, I have to thank everybody who contributed to the thread at
, as that was what started me on this line of reasoning. In the attached pdf (I don't know how to use TeX yet, so I did it in Mathematica), I derived what I thought was a logical derivative of e[4]x,
but upon actually running the numbers, it turned out not to work. I can't figure out why, but hopefully somebody here can. Thanks in advance for reading through this post (and hopefully the pdf as
well). :-)
11/12/2011, 09:27 AM
The attachment is broken. Somehow firefox shows it also as *.php instead of *.pdf. What's going on? (When I downloaded the file explicitely, the usual beginning of a *.pdf-file is missing, perhaps
this helps to find the source of the problem...).
Gottfried Helms, Kassel
11/12/2011, 04:24 PM
That's strange. It displayed fine for me in Chrome but not in Firefox and not when downloaded again. In that case, I'll just post the Mathematica code that was there originally.
The Derivative of E tetra x
First off, a few notes on the notation used by this paper. Also, as this is my first time trying anything like this, I apologize for any formatting errors or obvious math mistakes made here.
T[x] = n tetra x
TE[x] = E tetra x
D[f[x], x] = f'[x]
Ok, to business. To find the derivative, let's start with a basic identity.
TE[x] == TE[x]
Taking the natural log of both sides gives
Log[TE[x]] == Log[TE[x]]
One of the tetration identies is
Log[T[x]] == T[x - 1]*Log[n]
Or, using E as the base:
Log[TE[x]] == TE[x - 1]
As a result,
D[Log[TE[x]], x] == TE'[x - 1]
TE'[x]/TE[x] == TE'[x - 1]
TE'[x] == TE[x]*TE[x - 1]
Thus we have a recurrence relation for the derivative. This can be continued further.
TE'[x] == TE[x]*TE[x - 1]*TE'[x - 2]
TE'[x] == TE[x]*TE[x - 1]*TE[x - 2]*TE'[x - 3]
This is, so far, based entirely off of http://math.eretrandre.org/tetrationforu...php?tid=47. However, I took it a bit further, hoping that (for whole numbers, anyway), you could find the product of
the TE terms:
TE'[x] == TE'[0] Product[TE[x - k], {k, 0, x}]
I don't know enough about partial products to be able to know what to do in the case of non-integers here, but I figured that figuring out a general formula even only for integer values of x would be
useful, so I tried solving the product the same way (more or less) you would solve a sum of powers:
P == Product[TE[x - k], {k, 0, x}]
P == TE[x]*TE[x - 1]*TE[x - 2]*TE[x - 3] ...*TE[x - k]
E^P == TE[x + 1]*TE[x]*TE[x - 1]*TE[x - 2] ...*TE[x - k + 1]
E^P*TE[x - k] == TE[x + 1]*TE[x]*TE[x - 1]*TE[x - 2] ...*TE[x - k + 1]*TE[x - k]
E^P*TE[x - k] == TE[x + 1]*P
x - k == 0,
TE[x - k] = 1
As a result,
E^P == TE[x + 1]*P
Now, we rearrange the equation a bit.
Log[E^P] == Log[TE[x + 1]]*Log[P]
P == TE[x]*Log[P]
P == Log[P^TE[x]]
E^P == P^TE[x]
Substituting into the above equation gives
TE[x + 1]*P == P^TE[x]
TE[x + 1] == P^(TE[x] - 1)
P == TE[x + 1]^(1/(TE[x] - 1))
Now that there is a formula for the product:
TE'[x] == TE'[0]*TE[x + 1]^(1/(TE[x] - 1))
Sadly, this can be easily proven not to work. If x=2, and with the derivative recurrence equation listed above,
TE'[x] = TE[x]*TE'[x - 1]
TE'[0]*TE[x + 1]^(1/(TE[x] - 1)) == TE[x]*TE'[0]*TE[x]^(1/(TE[x - 1] - 1))
TE[x + 1]^(1/(TE[x] - 1)) == TE[x]*TE[x]^(1/(TE[x - 1] - 1))
TE[3]^(1/(TE[2] - 1)) == TE[2]^(TE[1]/(TE[1] - 1))
2.917275 != 73.71885
So after all that, it turns out not to be true. What I can't figure out is why. I'm hoping you guys could show me what's wrong with this derivation. Thanks in advance for your help.
11/12/2011, 05:07 PM (This post was last modified: 11/12/2011, 05:13 PM by Gottfried.)
(11/12/2011, 04:24 PM)Forehead Wrote: That's strange. It displayed fine for me in Chrome but not in Firefox and not when downloaded again. In that case, I'll just post the Mathematica code that
was there originally.
Hmm, just for the record, here is a bitmap of the binary beginning of the file. Surprisingly- before the "official" beginning of the pdf-file (in the picture framed by a orange frame containing the
"%pdf-1.4"-"magic stamp") there is some code which I cannot assign to something (it looks like some pdf-content-code). Usually a correct *.pdf-file begins with the "%pdf-1.4"-stamp, so the
pdf-producer-software seems to have scrambled the content of the file, or appended something unwanted...
Excerpt from the vicinity of the entry, which refers the "producing" software:
<</Producer (Mathematica PDF Export)/Creator (Mathematica)/CreationDate (D:20111112005354-05'00')/ModDate (D:20111112005354-05'00')/Title (Derivative of E tetra x.nb)>>
Gottfried Helms, Kassel
11/12/2011, 09:56 PM
(11/12/2011, 04:24 PM)Forehead Wrote: As a result,
E^P == TE[x + 1]*P
Now, we rearrange the equation a bit.
Log[E^P] == Log[TE[x + 1]]*Log[P]
I think this is the error. In going from the first to the second equation, it looks like you took, on the right hand side, \( \log(ab) = \log(a) \log(b) \). But that is wrong. Instead, \( \log(ab) =
\log(a) + \log(b) \) and so your second equation should be
Log[E^P] == Log[TE[x + 1]] + Log[P]
If we continue your steps with this corrected equation, we get
P == TE[x] + Log[P]
E^P == E^(TE[x] + Log[P])
E^P == E^TE[x] E^Log[P]
E^P == TE[x+1] P
TE[x+1] P == TE[x+1] P
a tautological equation. Though perhaps you could solve for P in the first equation via the Lambert function?
11/12/2011, 10:02 PM
(11/12/2011, 04:24 PM)Forehead Wrote: P == TE[x]*TE[x - 1]*TE[x - 2]*TE[x - 3] ...*TE[x - k]
E^P == TE[x + 1]*TE[x]*TE[x - 1]*TE[x - 2] ...*TE[x - k + 1]
I think there is a fatal problem here. In general, \( \exp(ab) \ne \exp(a) \exp(b) \).
11/13/2011, 04:21 PM (This post was last modified: 11/13/2011, 04:56 PM by Forehead.)
(11/12/2011, 10:02 PM)mike3 Wrote: I think there is a fatal problem here. In general, \( \exp(ab) \ne \exp(a) \exp(b) \).
That would do it. The question arises, then: how do you compute E raised to the power of each term in a product individually?
EDIT: I just realized that there can't be a function \( f \) such that \( f(a*b) = \exp(a)\exp(b) \) and \( f(x) = \exp(x) \). Back to the drawing board...
12/25/2015, 03:59 AM
I think the first error is on page 2, where you make the transition from Product to E^Product, the second equation is false. You can't distribute exponentiation that way. You can with the identity:
\( a = b + c \) to
\( e^a = e^b e^c \)
but that requires that you are starting with addition, but you start with multiplication. | {"url":"https://tetrationforum.org/showthread.php?pid=6206","timestamp":"2024-11-11T09:47:48Z","content_type":"application/xhtml+xml","content_length":"50633","record_id":"<urn:uuid:c185c974-0ee8-4878-878a-19df2fa4dd9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00768.warc.gz"} |
The number of F-matchings in almost every tree is a zero residue
For graphs F and G an F-matching in G is a subgraph of G consisting of pairwise vertex disjoint copies of F. The number of F-matchings in G is denoted by s(F,G). We show that for every fixed positive
integer m and every fixed tree F, the probability that s(F, T[n]) ≡ 0 (mod m), where T[n] is a random labeled tree with n vertices, tends to one exponentially fast as n grows to infinity. A similar
result is proven for induced F-matchings. As a very special special case this implies that the number of independent sets in a random labeled tree is almost surely a zero residue. A recent result of
Wagner shows that this is the case for random unlabeled trees as well.
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Geometry and Topology
• Discrete Mathematics and Combinatorics
• Computational Theory and Mathematics
• Applied Mathematics
Dive into the research topics of 'The number of F-matchings in almost every tree is a zero residue'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/the-number-of-f-matchings-in-almost-every-tree-is-a-zero-residue","timestamp":"2024-11-12T16:49:47Z","content_type":"text/html","content_length":"53871","record_id":"<urn:uuid:f7f7a5d3-9593-4ed6-9857-791ff3d6421a>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00578.warc.gz"} |
OpenStax College Physics for AP® Courses, Chapter 6, Problem 25 (Problems & Exercises)
What is the ideal banking angle for a gentle turn of 1.20 km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit?
Question by
is licensed under
CC BY 4.0
Solution video
OpenStax College Physics for AP® Courses, Chapter 6, Problem 25 (Problems & Exercises)
vote with a rating of votes with an average rating of.
Video Transcript
This is College Physics Answers with Shaun Dychko. This car is accelerating around this curve. It's going at constant speed, but it is never the less accelerating since its velocity is changing
direction. So we have centripetal acceleration in other words and that's caused by this component of the normal force which is directed towards the center of the curved path. So this angle here is
theta and we can use normal force multiplied by sine of theta to find this component in the x direction or radial component. We know this is theta because you can imagine this line here parallel to
the ramp and then you can imagine another line here which is horizontal and this would be theta because these would be interior opposite angles. With that being theta and this being 90 because this
is a normal force which is perpendicular to the surface of the ramp, this dot here plus theta has to add up to 90. But this dot here plus whatever this angle is here, let's assume we don't know it's
theta yet, also has to make 90 since this F n y is pointing straight up and that makes a 90 degree angle there. So if this dot plus theta makes 90 and this dot plus the angle in here also makes 90,
that means this must equal that and so it is theta in there. All right. So the y component of the normal force is the normal force multiplied by cosine theta because the y component is the adjacent
leg of this triangle. We know that that has to balance gravity. So that's vertical, this F n y is vertically upwards and so it has to balance the m g force of gravity downwards. That means F n cos
theta is m g. We can solve for F n by dividing both sides by cos theta . That gives F n is m g over cos theta and the reason that's useful is we can substitute that back into this formula replacing F
n in the F n x formula in order to well, eventually solve for theta. But let's take it one step at a time. So re-writing F n x and noticing that it is the centripetal force, in which case it must be
m v squared over r because that's the formula for centripetal force. Now we can say that m v squared over r is this expression we had before for F n x which is F n sine theta. But now we'll replace F
n with what we figured out in this part here in green, m g over cos theta times sine theta. Now sine theta over cos theta can be written as tan theta and we'll divide both sides by m as well. We have
v squared over r equals g tan theta. Then divide both sides by g and you get tan theta is v squared over r g. That means theta is the inverse tangent of v squared over r g. So that's the inverse
tangent of 105 kilometers per hour converted into meters per second and then square that, and divide by 1.2 kilometers converted into meters, times 9.8 meters per second squared. This all gives 4.14
degrees is the ideal banking angle for this particular speed and this radius of curvature. This assumes there is no friction by the way. | {"url":"https://collegephysicsanswers.com/openstax-solutions/what-ideal-banking-angle-gentle-turn-120-km-radius-highway-105-kmh-speed-limit-0","timestamp":"2024-11-08T16:54:46Z","content_type":"text/html","content_length":"150459","record_id":"<urn:uuid:f5e14719-aeec-40e3-b70f-fdf09bbb92c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00227.warc.gz"} |
EViews Help: uls
Unweighted least squares estimation of the factor model.
factor_name.uls(options) x1 [x2 x3...] [@partial z1 z2 z3...]
factor_name.uls(options) matrix_name [[obs] [conditioning]] [@ name1 name2 name3...]
The first method computes the observed dispersion matrix from a set of series or group objects. Simply append a period and the uls keyword to the name of your object, followed by the names of your
series and groups, You may optionally use the keyword @partial and append a list of conditioning series.
In the second method you will provide the name of the observed dispersion matrix, and optionally, the number of observations and the rank of the set of conditioning variables. If the latter is not
provided, it will be set to 1 (representing the constant in the standard centered variance calculations). You may also provide names for the columns of the correlation matrix by entering the @-sign
followed by a list of valid series names.
Estimation Options
rescale Rescale the uniqueness and loadings estimates so that they match the observed variances.
maxit=integer Maximum number of iterations.
conv=scalar Set convergence criterion. The criterion is based upon the maximum of the percentage changes in the scaled estimates. The criterion will be set to the nearest value between 1e-24
and 0.2.
showopts / [Do / do not] display the starting coefficient values and estimation options in the rotation output.
prompt Force the dialog to appear from within a program.
p Print basic estimation results.
Number of Factors Options
Number of factors: “kaiser” (Kaiser-Guttman greater than mean), “mineigen” (Minimum eigenvalue criterion; specified using “eiglimit”), “varfrac” (fraction of variance accounted
n=arg or fsmethod= for; specified using “varlimit”), “map” (Velicer’s Minimum Average Partial method), “bstick” (comparison with broken stick distribution), “parallel” (parallel analysis: number of
arg (default=“map”) replications specified using “pnreps”; “pquant” indicates the quantile method value if employed), “scree” (standard error scree method), “bn” (Bai and Ng (2002)), “ah” (Ahn and
Horenstein (2013)), integer (user-specified integer value).
eiglimit=number Limit value for retaining factors using the eigenvalue comparison (where “n=mineigen”).
varlimit=number Fraction of total variance explained limit for retaining factors using the variance limit criterion (where “n=varlimit”).
Use the unreduced matrix for parallel analysis (the default is to use the reduced matrix).
For parallel analysis only (“n=parallel”).
preps= integer Number of parallel analysis repetitions.
For parallel analysis only (“n=parallel”).
Quantile value for parallel analysis comparison (if not specified, the mean value will be employed).
For parallel analysis only (“n=parallel”).
Seed the random number generator for parallel analysis.
integer If not specified, EViews will seed the random number generator with a single integer draw from the default global random number generator.
For parallel analysis only (“n=parallel”).
arg (default= Type of random number generator for the simulation: improved Knuth generator (“kn”), improved Mersenne Twister (“mt”), Knuth’s (1997) lagged Fibonacci generator used in EViews 4
(“kn4”) L’Ecuyer’s (1999) combined multiple recursive generator (“le”), Matsumoto and Nishimura’s (1998) Mersenne Twister used in EViews 4 (“mt4”).
“kn” or method
previously set For parallel analysis only (“n=parallel”).
Maximum number of components used by selection methods: “schwert” (Schwert’s rule, default), “ah” (Ahn and Horenstein’s (2013) suggestion), “rootsize” (), “size” (, “user” (user
specified value), where
(1) For use with all components retention methods apart from user-specified (“fsmethod=user”).
(default=“user”) (2) If setting “mfmethod=user”, you may specify the maximum number of components using “rmax=”.
(3) Schwert’s rule sets the maximum number of components using the rule: let
rmax=arg (default= User-specified maximum number of factors to retain (for use when “mfmethod=user”).
Factor selection criterion (when “fsmethod=bn”): “icp1” (ICP1), “icp2” (ICP2), “icp3” (ICP3), “pcp1” (PCP1), “pcp2” (PCP1), “pcp3” (ICP3), “avg” (average of all criteria ICP1
through PCP3).
fsic=arg (default= Factor selection criterion (when “fsmethod=ah”): “er” (eigenvalue ratio), “gr” (growth ratio), “avg” (average of eigenvalue ratio and growth ratio).
Factor selection criterion (when “fsmethod=simple”): “min” (minimum of: minimum eigenvalue, cumulative eigenvalue proportion, and maximum number of factors), “max” (maximum of:
minimum eigenvalue, cumulative eigenvalue proportion, and maximum number of factors), “avg” (average the optimal number of factors as specified by the min and max rule, then round
to the nearest integer).
demeantime Demeans observations across time prior to component selection procedures, when “n=bn” or “n=ah”.
sdizetime Standardizes observations across time prior to component selection procedures, when “n=bn” or “n=ah”.
demeancross Demeans observations across cross-sections prior to component selection procedures, when “n=bn” or “n=ah”.
sdizecross Standardizes observations across cross-sections prior to component selection procedures, when “n=bn” or “n=ah”.
Initial Communalities Options
Method for obtaining initial communalities: “smc” (squared multiple correlations), “max” (maximum absolute correlation”), “pace” (noniterative partitioned covariance estimation), “frac”
priors=arg (fraction of the diagonals of the original matrix; specified using “priorfrac=”), “random” (random fractions of the original diagonals), “user” (user-specified vector; specified using
priorfrac= User-specified common fraction (between 0 and 1) to be used when “priors=frac”.
priorunique Vector of initial uniqueness estimates to be used when “priors=user”. By default, the values will be taken from the corresponding elements of the coefficient vector C.
Covariance Options
Covariance calculation method: ordinary (Pearson product moment) covariance (“cov”), ordinary correlation (“corr”), Spearman rank covariance (“rcov”), Spearman rank correlation
cov=arg (default= (“rcorr”), Kendall’s tau-b (“taub”), Kendall’s tau-a (“taua”), uncentered ordinary covariance (“ucov”), uncentered ordinary correlation (“ucorr”).
User-specified covariances are indicated by specifying a sym matrix object in place of a list of series or groups in the command.
wgt=name (optional) Name of series containing weights.
Weighting method (when weights are specified using “weight=”): frequency (“freq”), inverse of variances (“var”), inverse of standard deviation (“stdev”), scaled inverse of
wgtmethod=arg variances (“svar”), scaled inverse of standard deviations (“sstdev”).
(default = “sstdev”)
Only applicable for ordinary (Pearson) calculations. Weights specified by “wgt=” are frequency weights for rank correlation and Kendall’s tau calculations.
pairwise Compute using pairwise deletion of observations with missing cases (pairwise samples).
df Compute covariances with a degree-of-freedom correction for the mean (for centered specifications), and any partial conditioning variables.
factor f1.uls(n=map, priors=frac, priorfrac=1) x y z
declares the factor object F1 and estimates the factors for the correlation matrix of the series X, Y, and Z, by the unweighted least squares method.
f1.uls(maxit=300, conv=1e-8) group01
estimates the factors by the unweighted least squares method for the series in GROUP01 with maximum iterations 300 and convergence criterion 1e-8.
f1.uls(maxit=300, conv=1e-8) group01 @partial ser1 ser2
estimates the same specification using the partial correlation for the series in GROUP01, conditional on the series SER1 and SER2.
f1.uls(n=4) sym01 747
estimates the four factor ULS factor model using the observed matrix SYM01. The number of observations is 747.
“Factor Analysis”
for a general discussion of factor analysis. The various estimation methods are described in
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