content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Extreme Field Quantum Plasma Dynamics
Extreme Field Quantum Plasma Dynamics and Relativistic Laboratory Astrophysics
Relativistic quantum plasmas like gamma-electron-positron plasmas are believed to play a key role in extreme astrophysical environments such as pulsar magnetospheres, magnetars, black holes and
blazars, and are central to explain energetic phenomena associated to gamma-ray bursts. Due to the complete symmetry between matter and antimatter particles, electron-positron plasmas represent a
unique state qualitatively different from ordinary electron-ion plasmas. This implies that the collective response and the related instabilities of relativistic electron-positron plasmas result in a
peculiar dynamics, whose knowledge is still speculative.
Recent advances in high-power laser and in high-current electron beam technology enable the generation in the laboratory of ultra-strong electromagnetic fields that can trigger strong-field QED
processes such as photon emission and high-energy photon conversion into an electron-positron pair. This paves the way to generate and probe in the laboratory gamma-electron-positron plasmas similar
to those of astrophysical environments, and justifies the flourishing interest in this area of research known as relativistic laboratory astrophysics.
Our research focuses on the investigation of the interplay between strong-field QED processes and the collective plasma dynamics in the presence of extreme electromagnetic fields. Prominent examples
are radiation reaction effects in plasmas, where the recoil due to photon emissions drastically alters the emitting particle’s dynamics, the generation of intense collimated gamma rays and of dense
high-energy spin-polarized particle beams in laser-plasma and beam-plasma interaction, QED cascades with the transformation of electromagnetic energy into matter, relativistic laboratory
astrophysics. Because of the complex interplay between strong-field QED and the nonlinear plasma dynamics, large scale simulations constitute an essential tool. The development both of advanced
models of strong-field QED processes and of improved numerical methods as well as their implementation into codes capable to accurately and efficiently simulate strong-field QED dominated plasmas in
modern high-performance computing systems also represents one of our key activities.
March 2024, from left to right:, Yutong He, Matteo Tamburini, Samuele Montefiori, Cagri Erciyes, Michael Quin
• Matteo Tamburini (Group Leader)
• Samuele Montefiori (Ph.D. student)
• Michael Quin (Ph.D. student)
• Yutong He (Ph.D. student)
• Cagri Erciyes (Ph.D. student)
Completed thesis and former team members
Apr. 2021 Congratulations to Bastian Brunner for successfully defending his Bachelor thesis on "Electron Dynamics Controlled via Radiation Reaction".
Feb. 2021 Congratulations to Maitreyi Sangal for successfully defending her Ph.D. thesis on "The interaction of Light with Matter and Light with Light".
Nov. 2020 Congratulations to Archana Sampath for successfully defending her Ph.D. thesis on "Strong-field QED and collisional effects in electron beam-plasma interaction".
Oct. 2020 Congratulations to Michael Quin for successfully defending his Master thesis (fast track) on "Coherence Effects and Spin Polarisation of Electrons in Electromagnetic Fields".
Observing light-by-light scattering in vacuum with an asymmetric photon collider
Schematic setup (left panel). A collimated ultrarelativistic electron beam propagating along x collides with an intense laser pulse and generates a collimated gamma-ray beam. The electron beam and
possible positrons generated in the collision with the laser pulse are deflected by a magnetic field along z. The gamma-ray beam passes through a collimator, which selects photons with small angle
with respect to the xy plane. The collimated gamma-ray beam collides with an XUV pulse propagating in the xz plane. Elastic photon-photon scattering events can result in large-angle ψ[s] photon
deflection, which is prominent at lower scattered photon energies ε[s] (region outside the two horizontal black lines in the right panel).
The elastic scattering of two real photons in vacuum is one of the most elusive of the fundamentally new processes predicted by QED. This explains why, although it was first predicted more than
eighty years ago, it has so far remained undetected. We have shown that in present-day facilities, the elastic scattering of two real photons can become detectable far off axis in an asymmetric
photon-photon collider setup [1]. This may be obtained within one day of operation time by colliding 1 mJ extreme ultraviolet pulses with the broadband gamma-ray radiation generated in nonlinear
Compton scattering of ultrarelativistic electron beams with terawatt-class optical laser pulses operating at a 10 Hz repetition rate. In addition to the investigation of elastic photon-photon
scattering, this technique allows us to unveil or constrain new physics that could arise from the coupling of photons to yet undetected particles which, for example, would result in a measurable
signal differing from the QED prediction, therefore opening new avenues for searches of physics beyond the standard model of particle physics.
[1] M. Sangal, C. H. Keitel, and M. Tamburini, Phys. Rev. D 104, L111101 (2021)
Efficient high-energy photon production via beam-beam collision in the supercritical QED regime
Schematic setup (left panel). A pancake-shape dense source beam collides with an elongated cigar-shape low-density probe beam. The probe beam has small transverse size compared to the source beam and
collides with an impact parameter r, such that the source beam electric E[r] and magnetic B[r] fields approach their maximum. Energy radiated into photons as a function of the ratio between the
emitted photon energy ε[γ] and the initial lepton energy ε[e] (right panel).
The state-of-the-art concept to generate high-energy photons is based on Compton backscattering of an intense laser pulse with a counterpropagating electron beam. However, Compton backscattering
becomes increasingly more challenging to realize with increasing center-of-mass energy because high-energy photons convert into electron-positron pairs inside the laser pulse. When dense high-energy
lepton bunches collide, the beam particles can experience rest-frame electromagnetic fields which greatly exceed the QED critical one therefore emitting high-energy radiation via beamstrahlung. We
have demonstrated that beamstrahlung efficiently converts lepton energy to high-energy photons in this so-called supercritical QED regime, as the single-photon emission spectrum exhibits a pronounced
peak close to the initial lepton energy [1]. Remarkably, this high-energy peak is visible uniquely when (i) the rest frame fields exceed tens of times the QED critical one, (ii) when leptons
experience nearly the same fields, and (iii) when multiple photon emissions by the same particle are suppressed. Otherwise, particles experiencing inhomogeneous electromagnetic fields or emitting
several lower energy photons both broaden and soften the emitted photon spectrum. The collision of shaped lepton beams enables all the three above conditions simultaneously. On the one hand, this
setup allows one to measure the high-energy peak in the photon spectrum, the height of the peak being in a one-to-one correspondence with the instantaneous rest frame electric field at emission,
therefore opening up precision studies of strong-field QED in the ultra-relativistic regime. On the other hand, the proposed setup provides decisive advantages for the realization of an efficient
multi-TeV laserless gamma-gamma collider based on electron-electron collisions because it allows one to simultaneously suppress the lower energy background and to increase the probability of almost
complete lepton-to-photon energy transfer [1].
[1] M. Tamburini and S. Meuren, Phys. Rev. D 104, L091903 (2021)
How to make a gamma-ray pulse with the density of a solid
The generation of high-energy, dense and collimated photon beams is of great interest both to fundamental and applied research. These beams enable studies of matterless photon-photon interactions
ranging from matter-antimatter creation from light, light-by-light scattering and searches of possible new physics in the interaction of photons with yet undiscovered particles, to relativistic
laboratory astrophysics studies. We have demonstrated [1] that a pulsed, ultrarelativistic electron beam crossing a sequence of thin aluminium foils both self-focuses, therefore increasing its
density, and efficiently yields a collimated gamma-ray pulse with more gamma photons per unit volume than electrons in a solid. After passage of 20 foils, more than 30% of the electron beam energy is
converted to gamma rays. This occurs because the strong electric and magnetic fields accompanying the ultrarelativistic electron beam are “back-reflected” as the beam crosses the foil surface.
Similar to an electromagnetic wave colliding with a mirror, the reflected magnetic field is nearly the same as the beam magnetic field, whereas the reflected electric field has the same amplitude as
that of the electron beam but opposite direction. As a result, at the foil surface the total electric field acting on the beam is nearly zero, while the total magnetic field is nearly doubled. This
strong azimuthal magnetic field focuses the electron beam radially, and consequently triggers collimated high-energy photon emission. Based on these findings, an experiment (E-332 experiment) for the
investigation of electron beam self-focusing, solid-density gamma-ray burst generation, and laserless strong-field QED was devised, successfully approved with maximal rating, and is currently
underway at FACET-II [2].
[1] A. Sampath, X. Davoine, S. Corde, L. Gremillet, …, and M. Tamburini, Phys. Rev. Lett. 126, 064801 (2021)
[2] Facility for Advanced Accelerator Experimental Test (FACET-II), SLAC National Accelerator Laboratory, Menlo Park, CA
Spin-polarized kiloampere electron beams from laser-plasma acceleration
Illustration of the wake wave with its azimuthal magnetic field Bφ (left panel). Dashed black lines represent two possible trajectories of off-axis electrons, and the dotted black arrows show the
precession of their spin, where s[0] is the initial spin. Electron beam polarization as a function of the laser strength parameter a[0] (right panel). The light blue line is the model prediction, the
black circles report the results from 3D PIC simulations.
High-flux polarized particle beams are of critical importance for the investigation of spin-dependent processes such as in searches of physics beyond the standard model as well as for scrutinizing
the structure of solids and surfaces in material science. We have demonstrated that kiloampere polarized electron beams can be produced via laser-wakefield acceleration from a gas target [1]. We have
developed a simple theoretical model for determining the electron beam polarization, and we have supported our model predictions with self-consistent three-dimensional particle-in-cell simulations
that incorporate the spin dynamics. By appropriately choosing the laser and gas parameters, we have shown that the depolarization of electrons induced by the laser-wakefield-acceleration process can
be as low as 10%. Compared to currently available sources of polarized electron beams, the flux of spin-polarized electrons is increased by 4 orders of magnitude, thereby opening up unprecedented
studies of spin-dependent processes.
[1] Meng Wen, Matteo Tamburini, and Christoph H. Keitel, Phys. Rev. Lett. 122, 214801 (2019) | {"url":"https://www.mpi-hd.mpg.de/mpi/en/forschung/abteilungen-und-gruppen/theoretische-quantendynamik-und-quantenelektrodynamik/quantumplasma","timestamp":"2024-11-12T17:05:51Z","content_type":"text/html","content_length":"139300","record_id":"<urn:uuid:06dc56f3-6cb0-49ad-b2ba-a137fba8b655>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00677.warc.gz"} |
Deriving Monthly Median Values
John has a huge worksheet that includes daily data covering about sixty years. He wants to come up with a formula that will calculate the median values for each month in the time period covered by
the data.
Before proposing a solution, it is best to lay out some assumptions. For the purposes of this tip, let's assume that the daily data is in columns A and B. In column A are the dates associated with
the data and in column B are the actual data values for each of those dates. Further, cells A1 and B1 contain headings for each column. This means that the actual data is approximately in the range
of A2:B22000.
To make your formulas easier to use, you should define names for the data in both columns A and B. Select the range of dates (for example, A2:A22000) and assign it a name such as AllDates. Select the
range of corresponding data (for example, B2:B22000) and use the same process to create a name such as AllData.
You can use array formulas to calculate the actual median values. This involves setting up another data table to contain the medians. Place headings such as "Month" in cell E1 and "Median" in cell
F1. In cell E2 place the first day of the first month of your data set, such as 1/1/1940. In cell E3 put a date that is one month later, such as 2/1/1940. Select these two cells (E2:E3) and drag the
fill handle downward for the number of months you want in the data table.
If there are no blanks in your sixty years of data, then enter the following formula into cell F2:
Finalize the formula by pressing Ctrl+Shift+Enter, which tells Excel that this is an array formula. You can then copy the formula in F2 into each of the cells in column F that has a corresponding
month in column E. The formula analyzes the dates in column B, and if the year and month are equal to whatever date you put in cell E2, then the median is calculated from all corresponding data
If there are blanks in your sixty years of data (a date in column A with no corresponding value in column B), then the formula treats the blank as if it is a zero value. If you do have blanks, this
may result in skewed medians. To get around this, you could use a different array formula that checks for and ignores any blank values:
There is one caveat to using array formulas in this manner. If you have sixty years of data, with approximately 22,000 individual values, then that is still a lot of months: about 720 of them. That
means that you are creating 720 array formulas, each analyzing 22,000 data values to arrive at an answer. That is a lot of calculating going on, so you will notice a slow-down in the responsiveness
of Excel whenever it recalculates the worksheet.
If the sluggishness becomes an issue, then you could look at reorganizing your original data so that each row of the worksheet represents an individual month. Column A could contain the month for the
row (1/1/1940, 2/1/1940, 3/1/1940, etc.) and the columns B:AF would be days 1 through 31 for each month. The intersecting cells in the table could then contain the data point for each day in the
month, and you could use the MEDIAN function in column AG to calculate the median for each month. This still results in 720 formulas, but these are regular formulas that each only need to process a
month's worth of data instead of the array formulas that need to each process sixty years of data. The result is much faster calculations.
Of course, for most people the idea of reorganizing such a huge amount of data is enough to keep you awake at night. Instead, you can take an entirely different approach to analyzing the data. This
approach is possible because a median is a very easy statistical function to manually calculate. You simply sort your data set and, if the number of items in the data set is odd, select the middle
item. If the number of items is even, then you take the average of the two middle items.
To get ready to analyze the data, there are a couple of things to do. First, it will be handy to have some way to uniquely identify the month of each data point. In cell C2 place the following
This results in a value such as 194001, 194002, 194003, etc. being stored in column C. This is the unique month value. Now, you should sort the data by column C and then by column B. Both sorts
should be in ascending order, so that you end up with your data first sorted by year/month and then by value within the year/month.
Next you need to add subtotals to your data. Choose Subtotals from the Data menu, which displays the Subtotal dialog box. You want to add a subtotal at each change in column C. The function to use is
Count, and you want to add the subtotal to column B. When you click OK, you end up with 720 subtotals, one for each month in the data range, each one showing a count of how many data items there were
in that month.
To get the medians, add a formula to cell D2:
IF(RIGHT(B2,5)="Count", IF(MOD(C2,2)=1, INDIRECT("B"&(ROW()-1)-C2/2+1/2), (INDIRECT("B"&(ROW()-1)-C2/2)+INDIRECT("B"&(ROW()-1)-C2/2+1))/2), "")
The formula examines what is in cell B2, and if it contains the word "Count," then it knows that this is a subtotal row. In that case, it checks to see whether the count in cell C2 is odd or even. If
it is odd, then the INDIRECT function is used to grab whatever the median value is for the month. If it is even, then the two middle values for the month are added together and divided in half.
The result is that you now have median values for each month, in the same row as your subtotals. Thus, you can collapse the outline for the data (click the plus signs in the outline area at the left
of your data) so that you only show those subtotal rows.
2019-04-15 11:09:21
David Robinson
I thought I'd share a way to convert column data (dates in column A, data in column B) into the matrix form described straight after "if the sluggishness becomes an issue". Here, the first of each
month is in column D and the days in the month, 1 to 31, are in row 1.
In A1 notation:
=IF(E$1+$D2-1 < EDATE($D2,1), SUMIFS($B:$B,$A:$A,E$1+$D2-1), "")
This works by adding the day-in-month to the first-of-month (and then deducting 1 of course, because adding day 1 to the first of May gives the 2nd when you still want the 1st). First we check this
gives a date that is still in the month - i.e. that we haven't ended up with thinking we've got a 30th day in February. Then we get the value that matches the criterion - I use SUMIFS because it's
quick to write, or you could use INDEX/MATCH etc. | {"url":"https://excelribbon.tips.net/T012727_Deriving_Monthly_Median_Values.html","timestamp":"2024-11-01T20:36:16Z","content_type":"text/html","content_length":"51733","record_id":"<urn:uuid:849b8e88-fbe2-427b-9320-ecc7cb0ed340>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00432.warc.gz"} |
Sprague Grundy Theorem | Brilliant Math & Science Wiki
The Sprague-Grundy theorem is a statement about impartial games.
In Combinatorial Games - Winning Positions, we analyzed winning positions of impartial games. Several questions arose in trying to find a general characterization for whether a set of nim piles is a
winning position or a losing position. Here, we answer these questions by giving the complete characterization for winning and losing positions of nim.
Suppose we have nim piles of sizes \((a_1, a_2, \ldots, a_n)\). Let \(b_i\) be the base 2 representation of \(a_i\). A nim position is losing if over all the numbers in \(\{b_i\}_{i=1}^{n}\), the
number of 1s in each binary position is even and winning otherwise.
Before we prove this statement, let’s look at an example to see what we mean. If the nim piles have sizes \((3,6,8)\), then we can write these numbers as \(11_2, 110_2, 1000_2\). We want to look at
each binary position of these numbers, so let’s arrange them in a table:
\[ \begin{array}{cccc} & & 1 &1\\ & 1 &1 &0\\ 1 & 0 & 0 & 0\\ \end{array}\]
We can construct a number that represents the parity of the number of 1's in each binary position. We call this the nim-sum of these numbers. Looking at each column of the table, we see that the 1's
column has a single 1, the 10's column has two 1's, and the 100's column and the 1000's column each have a single 1. Our nim-sum here is \(1101_2\ = 13\).
If the piles have sizes \(7,9,14\), then we can write these numbers as \(111_2, 1001_2, 1110_2\). We want to look at each binary position of these numbers, so let’s arrange them in a table:
\[ \begin{array}{cccc} & 1& 1 &1\\ 1& 0&0 &1\\ 1 & 1& 1 & 0\\ \hline 0 & 0 & 0 & 0 \\ \end{array}\]
The last line represents the parity of the number of 1's, which gives us that the nim-sum of these numbers is 0.
We will modify our description of a winning position to be one where the nim-sum of the pile sizes is not 0, which is equivalent to the original definition. To verify that this accurately
describes the set of winning and losing positions, we check 3 things.
First, it is clear that the game with no stones is losing, and has nim-sum 0.
Second, if we are in a losing position, we have nim-sum 0. If we remove stones from a single pile, we are changing the positions of 1's in a single column, which will make the nim-sum no longer
Third, if we are in a winning position, we have nim-sum of the form \(1x_2\), where \(x\) is a binary string. We choose a pile that has a 1 in the same position as the leftmost 1 in the nim-sum.
We change this number by flipping each digit for which there is a 1 in the corresponding position in the nim-sum. The nim-sum of these new numbers will be 0, which is a losing position. It is
left as an exercise to show that the new number is less than the original number and thus constitutes a valid move. \( _\square\)
The nim-sum actually gives us more information than determining whether a position is winning or losing; it also assigns a value to that position. If a position has nim-sum \(k\), then from that
position it will always be possible to move to a position with nim-sum \(j\) for any \(j < k,\) and not possible to move to a position with nim-sum \(k\). To see this, consider the proof that any
winning position can get to a 0 position, and replace 0 with any smaller number. The described method will extend to this.
We have spent a lot of time developing the theory for a single impartial game, Nim. In fact, this is all the theory that we need to understand any general impartial game! We have the following
surprising result:
Sprague-Grundy Theorem
Any position of an impartial game is equivalent to a nim pile of a certain size. \(_\square\)
We can assign a nim-sum to the positions in any impartial game by building up sequentially through small cases. The nim-sum of a position is the smallest non-negative integer \(k\) such that we
cannot move to a position with nim-sum \(k\). This requires us to calculate the nim-sum of every position that we can move to. Since the game will always end, this calculation is finite. However, it
can get computationally intensive, and a computer may be useful.
The proof of the Sprague-Grundy theorem is identical to our previous proof, where we checked 3 statements.
Two players play a game starting with a pile of \(n\) stones. On each turn, a player removes from the pile \(1,2,3,\ldots, \mbox{ or } k\) stones. The person who takes the last stone wins.
Determine, in terms of \(n\) and \(k\), the nim-sum of each position.
For \(j \leq k\), the nim-sum of the game with \(j\) stones will be \(j\). This is easy to see inductively starting at \(j = 0\). When \(j = k+1\), this is a losing position, so the nim-sum will
be 0. If we repeat this, we see inductively that the nim-sum of \(i\) and \(i + k + 1\) will always be the same. So the nim-sum of \(n\) stones where we can remove at most \(k\) is \(n \pmod{k +
1}.\) \(_\square\)
Odd Nim is like Nim, except you can only remove an odd number of stones from a pile. Determine the nim-sum for a single pile of \(n\) stones.
Since we can only take an odd number of stones from the pile, the game will always be a losing position if the number of stones is even and a winning position if the number of stones is odd. From
a winning position, we can only get to the 0 position, so all winning positions have nim-sum 1. Thus the nim-sum is \(n \pmod{2} \). \(_\square\) | {"url":"https://brilliant.org/wiki/sprague-grundy-theorem/","timestamp":"2024-11-02T08:20:25Z","content_type":"text/html","content_length":"48318","record_id":"<urn:uuid:041e89f0-4d86-4f29-a35a-c7b66496ce00>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00584.warc.gz"} |
LCC '22 Contest 6 S3 - The Kirby Shows
Batrick Lin: overlord of WLMAC, supreme leader, lord over all his student and staff subjects can sometimes seem far beyond mortal comprehension. Nonetheless, his interests often align with those of
mankind. One such interest are game shows! Thus, he has decided to [DEL:capture:DEL] bring you to his game show and play a little game.
In this game show, there are cards numbered to , all face-down. Of these cards, you know that one of them contains a kirby character, the rest all contain daggers. You are given turns: on each turn,
you pick a card that is still face-down, and Batrick will flip a random face-down card apart from the one you chose that contains daggers.
Now, given of these turns, can you figure out the probability that each card from to contains the kirby?
Subtask 1 [30%]
Subtask 3[70%]
No additional constraints.
Input Specification
The first line contains 2 integers: , and .
The next lines each contain two integers: and .
Output Specification
One line with floating point numbers: the probability that the -th card is a dagger. Your answer will be considered correct if it has a relative error less than
Sample Input
Sample Output
0.333333 0 0.6666667
Sample Explanation
After picking the first card, you know that the second card definitely contains a dagger. If the third card contained a kirby, then the second card would be flipped with certainty. However, if the
first card contained a kirby, then there would only be a chance the second card was flipped. At the very start, it is equally likely that the first and third cards contain a kirby. Hence, since it is
so much more likely that the second card would be picked if the third card was a dagger, you can calculate the probability that the third card contains a kirby as
There are no comments at the moment. | {"url":"https://mcpt.ca/problem/lcc22c6s3","timestamp":"2024-11-03T00:12:55Z","content_type":"text/html","content_length":"41690","record_id":"<urn:uuid:1bf513f6-47ea-4687-8592-b0805c21c5db>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00222.warc.gz"} |
If C is a subset of A and B, then it is a subset of their intersection - Stumbling Robot
If C is a subset of A and B, then it is a subset of their intersection
One comment
1. I am confident you meant to say
Subsequently the proof should look as follows:
Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): | {"url":"https://www.stumblingrobot.com/2015/06/20/if-c-is-a-subset-of-a-and-b-then-it-is-a-subset-of-their-intersection/","timestamp":"2024-11-09T14:35:42Z","content_type":"text/html","content_length":"62224","record_id":"<urn:uuid:f5682081-526e-47b6-844e-4a5eb2655651>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00703.warc.gz"} |
Fractional powers
Simplify any powers using laws of indices.
As we have multiple variables in the first expression raised to the power of a , letβ s remove the brackets so we know the power of each variable separately,
(xy)^{a}=x^{a}\times y^{a} .
We can then divide x^{a} \times y^{a} by (x)^\frac{a}{2} to get
x^{a}\times\ {y^{a}}\div{(x)^\frac{a}{2}}=x^{a-\frac{a}{2}}\times\ {y^{a}}=x^{\frac{a}{2}}\times\ {y^{a}} .
(Remember we can only simplify powers if the base is the same.)
So we now have x^{\frac{a}{2}}\times\ {y^{a}} . | {"url":"https://thirdspacelearning.com/gcse-maths/number/fractional-powers/","timestamp":"2024-11-04T19:00:24Z","content_type":"text/html","content_length":"231827","record_id":"<urn:uuid:09f38bc2-1bce-44b8-aa24-f3fb28d40e42>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00295.warc.gz"} |
What Are the Different Methods for Rationalizing a Fraction?
The rationalization process aims to rewrite the fraction so that it does not contain any irrational components in the denominator, such as square roots or cube roots. By supporting a small portion,
mathematicians can improve on estimations, control logarithmic articulations all the more really, and frequently reveal connections that may be clouded by complex denominators.
Rationalizing fractions can be accomplished in a variety of ways. Every strategy has its materialness relying upon the particular type of portion, and understanding these procedures furnishes
mathematicians with flexible instruments to deal with different arithmetical difficulties. Justification in Rationalize a Fraction problem improves computational effectiveness as well as extends
appreciation of numerical connections and designs.
Different Methods for Rationalizing
The process of changing the form of an algebraic expression, typically a fraction, to remove radicals or other complex terms from the denominator is known as rationalization in mathematics. This
method is especially helpful for making algebraic expressions easier to understand and simplifying calculations.
Multiplying by the Conjugate
This technique is mostly used for fractions with square roots. To excuse the denominator, you increase both the numerator and the denominator of the division by the form of the denominator. The form
is framed by changing the sign between the conditions of the denominator. For instance, assuming that the denominator is \( \sqrt{a} + b \), its form would be \( \sqrt{a} - b \).
Using the Difference of Squares
When defied with divisions where the denominator is a binomial squared (e.g., \( a^2 - b^2 \)), the distinction of squares technique is compelling. This procedure includes considering the denominator
into its constituent terms (e.g., \( (a - b)(a + b) \)) and afterwards reworking the part as needed. Thus, the squared term in the denominator is killed, changing the portion into a judicious
structure without revolutionaries.
Using the Sum or Difference of Cubes
The sum or difference of cubes method is used for more complex fractions with cube roots or higher powers. This strategy requires perceiving designs in mathematical characters, for example, \( a^3 +
b^3 \) and \( a^3 - b^3 \), which can be calculated into sensible structures. By applying these characters, the portion can be reworked such that eliminates the shape roots or higher powers from the
Using Gauth To Rationalize a Fraction
Gauth is one of the best-known platforms that can be used for different purposes like solving a problem including Rationalize a Fraction. The steps for these types of problems are given below.
Step 1: Enter Gauth
The first step in using Gauth to solve problems is to access the platform by going to the Gauth website or app. This normally includes signing into a record or getting to the stage's connection point
where you can associate with its elements.
Step 2: Type Question
When inside Gauth, clients can continue to type their inquiry or issue into the given search or information field. Also, the User can select the problem from the gallery or input the link to the
given question. This step includes plainly figuring out the question such that Gauth's calculations can be comprehended and processed successfully.
Step 3: Find Free Trial
Users can choose to find or activate a free trial, if one is available, prior to receiving the solution. This step includes exploring through the stage's choices to find and initiate any times for
testing or special offers that permit admittance to arrangements without prompt installment.
Step 4: Get Solution
In the wake of entering the inquiry and, if material, enacting a free preliminary, clients can continue to get the answer to their concerns. Gauth regularly gives definite clarifications, bit-by-bit
arrangements, or explicit replies, contingent upon the idea of the question entered. Clients can then survey and use the arrangement given to address their intellectual or critical thinking needs
Final Wording
Understanding the different techniques for defending a division prepares mathematicians and understudies the same with flexible devices to really control and improve on logarithmic articulations.
Each technique transforms fractions into rational forms devoid of radicals or complex terms in the denominator by eliminating square roots using the difference of squares for binomial denominators or
using the sum or difference of cubes for higher powers. | {"url":"https://nahualli-gallery.com/what-are-the-different-methods-for-rationalizing-a-fraction/","timestamp":"2024-11-13T22:40:26Z","content_type":"text/html","content_length":"20951","record_id":"<urn:uuid:7c1be3c7-e47a-4ccb-ab4f-eaf7b0aa9d56>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00517.warc.gz"} |
Math for Elementary and Middle School Educators - MTH 120
Math for Elementary and Middle School Educators - MTH 120 at Camp Community College
Effective: 2024-05-01
Course Description
Provides a comprehensive and conceptual examination of fundamental mathematical concepts covered in VDOE K-8 Standards of Learning (SOLs). Designed for future K-8 mathematics educators. Emphasizes
problem-solving, logical reasoning, the establishment of connections between mathematical concepts, effective communication of mathematical ideas, and the utilization of multiple representations.
This is a cross-listed course with
EDU 120
Lecture 4 hours. Total 4 hours per week.
4 credits
The course outline below was developed as part of a statewide standardization process.
General Course Purpose
The purpose of this course is to provide a comprehensive and conceptual understanding of fundamental mathematical concepts covered in VDOE K-8 mathematics Standards of Learning (SOLs). This course is
intended to cover the content of the first semester of a two-course lower-level math for elementary educators sequence, with the second semester of the course taken at the four-year college after
Course Objectives
• Quantitative Literacy
□ Use problem-solving skills and quantitative reasoning to solve problems, explore new ideas, and improve your understanding of mathematics
• Critical Thinking
□ Formulate multiple solution paths for mathematical problems and describe connections between and within these paths
• Written Communication
□ Demonstrate proper use of terminology, notation, and/or written conventions used in the field of mathematics and mathematics education
• Professional Readiness
□ Analyze and evaluate the mathematical thinking of K-8 students expressed in their oral and verbal reports, written work, and authentic representations
□ Design strategies to develop positive math beliefs, including growth mindset, persistence in work, and productive student struggle
Major Topics to be Included
• Problem Solving and Quantitative Reasoning Skills
□ Identify connections between conceptual knowledge and standard algorithms
□ Generate meaningful mathematical representations using algorithms, employing both written formats and physical manipulatives
• Counting and Number Systems
□ Describe quantities across multiple number systems, encompassing natural numbers, whole numbers, integers, and rational numbers
□ Establish connections between concepts such as place value and regrouping, explaining their relationship within the base-ten number system
□ Analyze similarities and differences between the base-ten number system and numbers in alternative bases
• Integers and Rational Numbers
□ Apply fundamental understanding of whole numbers to grasp integers and their associated operations
□ Utilize core principles of number theory, such as prime factorization, divisibility, greatest common factors, least common multiples, etc., to examine the structure of integers
□ Generate multiple representations for rational numbers and justify their equivalence in various forms
• Comparing Fractions and Arithmetic Operations
□ Make valid comparisons between rational numbers and model arithmetic calculations with them
□ Utilize various reasoning approaches to justify arithmetic operations involving whole numbers
□ Develop standard and non-standard algorithms for addition, subtraction, multiplication, and division by drawing on principles of counting, place-value grouping, and partitioning
• Base Ten and Other Bases
□ Exhibit adaptable and conceptual thinking while exploring numbers, operations, and their interconnections
□ Investigate the concepts of place value and regrouping within the framework of the base-ten system and alternative numerical bases
• Addition and Subtraction
□ Use, compare, and mathematically justify different strategies and representations to solve addition and subtraction problems
□ Develop versatile computational skills across all number categories, using and articulating various methods for addition and subtraction
• Multiplication and Division
□ Formulate multiple solution paths for multiplication and division problems and describe connections between and within these paths
□ Assess and explain the mathematical ideas and reasoning used in multiplication and division
• Formulating Solution Paths and Beliefs about Mathematics
□ Embrace a growth mindset to reflect on personal beliefs regarding mathematics and the teaching and learning processes, fostering an open exploration of mathematical concepts.
□ Generate comprehensive mathematical representations for the course content, incorporating written formats (such as area models, number lines, scaled diagrams, strip diagrams, etc.) and
tangible manipulative tools (like base-ten blocks, Cuisenaire rods, linking cubes, pattern blocks, etc.), to enhance understanding and engagement. | {"url":"https://courses.vccs.edu/colleges/camp/courses/MTH120-MathforElementaryandMiddleSchoolEducators/detail","timestamp":"2024-11-02T01:54:43Z","content_type":"application/xhtml+xml","content_length":"13978","record_id":"<urn:uuid:14e89acf-d656-48fa-81c3-2f2c407e3b0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00876.warc.gz"} |
Özgür'ün Blogu
January 01, 2024.
Since this is my first blog post, I will dedicate this one to my mother Dr. Elif Günçe. She is a dentist, an expert in mass fatality management and humanitarian forensics. She had me in her third
year of undergrad and I was in high school when she got her PhD. Whatever/wherever I am right now, I owe it to her. She moved us to Canada back in 2011 and she is the reason why I was able to apply
for graduate school in Canada as a permanent resident/domestic student. She always believed in me and claimed I could have been accepted to any school I wanted. But as Almut Burchard once said to
her: "If he wasn't a domestic applicant, I wouldn't be able to him in the (graduate) program (in the Department of Mathematics at the University of Toronto)."
My mother once visited me in Toronto in Fall 2017. I was teaching Linear Algebra II during that semester. She watched me in the big hall where I lectured over 100 students. If you know me, I am a
vertically challenged person. She is even smaller than me. So, she was lost in between the students. It was cute. It was the middle of the semester and I had finished teaching basic concepts of
linear algebra in the context of abstract vector spaces over real numbers and before we moved to eigenvalues/Jordan form etc I was supposed to introduce complex numbers.
In this blog post, I am going to recall that introduction to complex numbers and I will also tell you why we denote the complex numbers by $a + bi$ with $a,b$ real numbers. In particular, why do we
use the letter $i$? If you believe that you are familiar with complex numbers, skip to the end for the answer.
You should read this as I am teaching in front of the classroom.
We start learning about numbers when we are children with the set of natural numbers: $0,1,2,3$ … On this set of numbers, we have algebraic operations which we call addition and multiplication and
these operations satisfy some nice natural properties. We would like to solve equations - for reasons - and by staying inside the universe of natural numbers we can solve some equations. For example,
we can solve $x + 2 = 5$. We know that $x=3$ is a solution to this equation. However, there are equations we can not solve! We can not solve the equation $x + 1 = 0$ for example.
Now, you know that $-1$ is a solution to this equation but it does not live in the set of natural numbers. If you started with natural numbers, you manually have to add this number inside your set if
you would like to be able to solve this equation. So, now you have a new set: $-1, 0, 1, 2, 3$ … which has the power of solving the equation $x+1=0$. However, we lost the structure. We can not do the
operation $(-1)+(-1)$ in this new set, for example. What do we do? We complete this set by adding all possible results of every operation we can perform. We end up with the set of integers. I want to
emphasize that we named this new number $-1$ that we added to our new set. It was us who gave it a symbol. We are just very familiar with this concept after years of education that we know it is
called negative one and it is denoted by $-1$.
Integers are cool and we can solve a lot of equations such as $x + n = 0$ where $n$ is an integer. But when we see an equation like $2x - 1 = 0$, we can not solve it. We imagine that there is a
solution and we give it a name. We can call it Mauro but we don't because we are familiar with this number since the grade school. So, we call it $1/2$. Now, we play the same game. Once we add this
number, our set of numbers has gained the power of solving this equation which it previously couldn't. But we gave up the power of nice algebraic structure. So, what do we do? We play the same game
and add other numbers that we have to add in order to keep our structure. For example, $3/2, 5/2, 1/4, 3/8$ … We can now solve other equations like $4x - 7 = 0$. But we fail to solve $3x - 1 = 0$.
But now we know the name of the game. Make up a solution, give it a name, add it to your set and then close the set under your operations. When you do this for all equations of the form $ax -b = 0$
where $a$ and $b$ are integers, you get all numbers of the form $b/a$. So, we have now created the set of rational numbers. Any questions or comments here? What? Oh. Yes. We have to make sure that
$a$ is not zero. I was testing you! I did not forget that, of course (!).
Now, let us make a more complicated example which you did not see in elementary school. While we have the power of solving all linear equations with rational coefficients now, we can not solve the
equation $x^2 - 2 = 0$. We have two options: be sad, get upset, leave math or continue playing the game we have been playing: Let's say it has one solution, we give it a name. Let's name it after one
of you. What is your name again? (*I point at a person sitting in the back) Samantha. Samantha, we will name this new number after you. Okay, let's call this new number $s$. This number $s$ has the
property that $s^2 - 2 =0$. You are familiar with this number because you know for a long time that there is a number we call $\sqrt{2}$. See, I have a tattoo of it. But today, we are calling it
Samantha's number and we will denote it by $s$. We will now add this new number to our set of rational numbers and we will close it under our operations so that we can keep our algebraic structure.
We have numbers like $1 + s$ and $2 + 3s$. And we add them in the obvious way and get $3 + 4s$. We can also multiply them because we know from the distributive property of our structure that $(1+s)
(2+3s) = 2 + 3s+ 2s+ 3s^2$ and by the defining property of Samantha's number, we must have $s^2= 2$ so this product equals $8+5s$. The rules of the game tell you what to do and how to define
everything once you add Samantha's number to your set.
Keep reading this as I am teaching in front of the classroom.
Let's recap. We started with natural numbers, couldn't solve some equations and created integers. We were able to solve more equations but not all. We created rational numbers. Then, we do some shady
things here (aka calculus) and construct real numbers, don't worry about this. Can we solve all polynomial equations now? No? An example? Yes, you are right we can not solve $x^2 + 1 = 0$. This
equation does not have a solution in our set of real numbers. So, again we can just give up and call it a day and say I am going to live in a world where this equation does not have a solution and be
happy. Or we can play the game again. Imagine that there is a solution, give it a name. Excuse me, you, yes, the lady with the blue scarf (pointing at my mom), we will name it after you, can you
please tell us, let's see … , the third letter of your first name.
And this is why we denote the square root of $-1$ by $i$. | {"url":"https://sntp.ca/ozgur/blog/complex-numbers.html","timestamp":"2024-11-07T04:19:14Z","content_type":"text/html","content_length":"9971","record_id":"<urn:uuid:409ee1cb-8920-473d-9d64-9e107aa364a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00589.warc.gz"} |
What is the derivative of y=sec(3x^2)? | HIX Tutor
What is the derivative of #y=sec(3x^2)#?
Answer 1
Let #y = sec(u)# and #u = 3x^2#.
The derivative of #secx# can be found by the following proof:
#secx = 1/cosx#
#(1/cosx)' = ((0 xx cosx) - (1 xx -sinx))/(cosx)^2#
#(1/cosx)' = sinx/(cos^2x)#
#(secx)' = secx xx sinx/cosx#
#(secx)' = secxtanx#
The derivative of #3x^2# can be obtained using the power rule:
#(3x^2)' = 2 xx 3x^(2 - 1)#
#(3x^2)' = 6x#
The chain rule states that #dy/dx = dy/(du) xx (du)/dx#.
Hence, #dy/dx = secutanu xx 6x = 6xsec(3x^2)tan(3x^2)#
a) #cscx#
b) #cot(3x^2 + 5x + 1)#
c) #tan(e^(2x^2))#
Hopefully this helps, and good luck!
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/what-is-the-derivative-of-y-sec-3x-2-8f9af9f0d2","timestamp":"2024-11-02T10:41:20Z","content_type":"text/html","content_length":"570253","record_id":"<urn:uuid:5f980142-5903-4319-9116-6723b243f7c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00745.warc.gz"} |
Angle Between Two Vectors Calculator
An online angle between two vectors calculator allows you to find the angle, magnitude, and dot product between the two vectors. It does not matter whether the vector data is 2D or 3D, our calculator
works well in all aspects. So, keep reading to learn how to use formulas and some examples to find angle between two vectors.
What is Angle Between Two Vectors?
In mathematics, the angle between two vectors is defined as the shortest angle at which one of the vectors rotates to a position consistent with the other vector. Remember that vector quantities have
both magnitude and direction. Vectors can be expressed in two-dimensional and three-dimensional spaces.
Angle Between Two Vectors Formula:
There are different formulas that are used by the angle between two vectors calculator which depend on vector data:
Find Angle between Two 2d Vectors:
• Vectors represented by coordinates
Vectors \(m = [x_m, y_m] , n = [x_n, y_n]\)
Angle = \( cos^{-1}[\frac{(x_m * x_n + y_m * y_n)}{(\sqrt{(x_m^2 + y_m^2)} * \sqrt{(x_n^2 + y_n^2)}}]\)
• The vector between terminal points
For vector p: M = \([x_m, y_m] , \text { N} = [x_n, y_n]\),
so vector p = \([x_n – x_m, y_n – y_m]\)
For vector q: C = \([x_c, y_c] , \text { D} = [x_d, y_d]\),
so vector q = \([x_d – x_c, y_d – y_c]\)
Then angle between vectors calculator substitute the vector coordinates into the angle between two vectors formula for the point A:
$$= cos^{-1}[\frac{((x_n – x_m) * (x_d – x_c) + (y_n – y_m) * (y_d – y_c))}{(\sqrt{((x_n – x_m)^2 + (y_n – y_m)^2)} * \sqrt{((x_d – x_c)^2 + (y_d – y_c)^2)}}]$$
However, an online Arccos Calculator allows you to calculate the inverse of the cosine of a certain number.
Find Angle between Two 3d Vectors:
The vector angle calculator use the following aspects for finding the angle between two vectors.
• Vectors represented by coordinates:
$$m = [x_m, y_m, z_m] , n = [x_n, y_n, z_n]$$
$$angle = cos^{-1}[\frac{(xm * xn + ym * yn + zm * zn)}{(\sqrt{(xm^2 + ym^2 + zm^2)} * \sqrt{(x_n^2 + y_n^2 + z_n^2)}}]$$
• Vectors between a terminal point:
For vector a: M = \([x_m, y_m, z_m], \text { N} = [x_n, y_n, z_n]\),
so a = \([x_n – x_m, y_n – y_m, z_n – z_m]\)
For vector b: O = \([x_o, y_o, z_o], P = [x_p, y_p, z_p]\)
so b = \([x_p – x_o, y_p – y_o, z_p – z_o]\)
Find the formula analogically to the 2D version:
$$ \text{angle} = \cos^{-1} \left[ \frac{(x_n - x_m)(x_p - x_o) + (y_n - y_m)(y_p - y_o) + (z_n - z_m)(z_p - z_o)}{\sqrt{(x_n - x_m)^2 + (y_n - y_m)^2 + (z_n - z_m)^2} \cdot \sqrt{(x_p - x_o)^2 +
(y_p - y_o)^2 + (z_p - z_o)^2}} \right] $$
You can determine one angle by coordinates, and the other by a terminal point with the angle between two vectors calculator.
How to Find the Angle between Two Vectors:
You can find the angle between vectors with the understanding of different formulas, let’s check where they come from:
• The formula for the dot product:
The dot product is expressed as the product of the vectors’ sizes or magnitudes multiplied by the cos of the angle between them (here denoted by θ):
$$a · b = |a| x |b| cos(θ)$$
• Now, the angle of the equation:
Divide by the product of two vectors' magnitudes:
$$cos(θ) = a · b / (|a| x |b|)$$
Then, find the cos inverse of both sides:
$$θ = cos^{-1}[(a · b) / (|a| x |b|)]$$
• As magnitude is the square root (\(\sqrt{ }\)) of the sum of the components to the second power:
Vector in 2D space:
$$| v | = \sqrt{(x^2 + y^2)}$$
Vector in 3D space
$$| v | = \sqrt{(x^2 + y^2 + z^2)}$$
• Then, the angle between two vectors calculator uses the formula for the dot product, and substitute it in the magnitudes:
2D space If vectors m = \([x_m, y_m], n = [x_n, y_n]\), then:
$$θ = cos^{-1}[(x_m * x_n + y_m * y_n) / (√(x_m^2 + y_m^2) * √(x_n^2 + y_n^2))]$$
3D space
If vectors m = \([x_m, y_m, z_m], n = [x_n, y_n, z_n]\), then:
$$θ = cos^{-1}[(x_m * x_n + y_m * y_n + z_m * z_n) / (\sqrt{(x_m^2 + y_m^2 + z_m^2)} * \sqrt{(x_n^2 + y_n^2 + z_n^2)})]$$
These formulas are used by angle between vectors calculator for two and three dimensional vectors magnitude.
However, use an online free Cosine Calculator that helps you in calculating the cosine value of the given angle in degrees and radians.
Example: find angle between two 3d vectors
A = {4, 6, 8}
B = {3, 2, 5}
Now start with dot product of A and B, A.B
$$A ⋅B =Ax . Bx + Ay . By + Az . Bz$$
$$A ⋅B =(4 . 3) + (6 . 2) + (8 . 5)$$
$$A ⋅B = (12) + (12) + (40)$$
$$A ⋅B =64$$
Magnitude of Vector A:
Magnitude of Vector B:
Angle between vectors A and B:
$$cosθ=(A. B)/(|A||B|)$$
$$θ=15.42911 deg$$
However, you can get exact values by plugs the same values in the angle between vectors calculator.
How angle between two vectors calculator Works?
The angle between vector calculator find the angle θ separating two Vectors A and B in two and three-dimensional space with these steps:
• First, select the 2D or 3D dimension of vectors.
• Now, choose the vector representation (by Coordinates or Terminal points) from the drop-down list.
• Then, plugs all given values in the related fields.
• To see the results, click on the “Calculate” button.
• The angle between two vectors calculator provides stepwise calculations for the Dot product, magnitude, and angle between vectors.
• You can find the 2D and 3D vectors numerous times as per requirements by clicking on recalculate button.
Define the angle between two antiparallel vectors?
Antiparallel vectors are parallel vectors in opposite directions. The angle between these two vectors is 180°.
What is the magnitude of the unit vector?
Since the unit vector is 1 by definition, if you want to use the unit vector in the A direction, you must divide by this magnitude.
Is the Angle a vector quantity?
In fact, the angle is a dimensionless vector. It has direction and magnitude. We can measure clockwise and counterclockwise angles based on their rotation behavior. Therefore, this angle is indeed a
"vector quantity".
Use this angle between two vectors calculator to determine the angle between vector components. The concept of vector angle is used to describe the angular difference of the physical quantity
assigned to the quantity and direction. The vector angle is calculated from the first endpoint. This free online calculator provides dot product and magnitude quickly with 100% accuracy.
From the source of Wikipedia: Algebraic definition, Geometric definition, Scalar projection, and first properties, Equivalence of the definitions, Application to the law of cosines, Vector algebra
relations, Magnitudes.
From the source of Wiki How: Finding the Angle Between Two Vectors, Defining the Angle Formula, write the Law of Cosines for this triangle.
From the source of Krista King Math: Angle Between Two Vectors, Formula for the angle between vector, How to calculate the angle, Finding the angle between two vectors in three dimensions. | {"url":"https://calculator-online.net/angle-between-two-vectors-calculator/","timestamp":"2024-11-12T15:21:13Z","content_type":"text/html","content_length":"74796","record_id":"<urn:uuid:c3abf128-dd19-42a1-be7a-4c6bcb3b3cd2>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00894.warc.gz"} |
We study several variants of a combinatorial game which is based on Cantor's diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by
the known fact that Leopold Kronecker did not appreciate Georg Cantor's arguments about the infinite, and even referred to him as a ``scientific charlatan''.
In the game Kronecker maintains a list of m binary vectors,
each of length n, and Cantor's goal is to produce a new binary vector which is different from each of Kronecker's vectors, or prove that no such vector exists. Cantor does not see Kronecker's vectors
but he is allowed to ask queries of the form ``What is bit number j of vector number i? What is the minimal number of queries with which Cantor can achieve his goal? How much better can Cantor do if
he is allowed to pick his queries \emph{adaptively}, based on Kronecker's previous replies?
The case when m=n is solved by diagonalization using n (non-adaptive) queries. We study this game more generally, and prove an optimal bound in the adaptive case and nearly tight upper and lower
bounds in the non-adaptive case.
Changes to previous version:
Added acks
We study several variants of a combinatorial game which is based on Cantor's diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by
the known fact that Leopold Kronecker did not appreciate Georg Cantor's arguments about the infinite, and even referred to him as a ``scientific charlatan''.
In the game Kronecker maintains a list of m binary vectors,
each of length n, and Cantor's goal is to produce a new binary vector which is different from each of Kronecker's vectors, or prove that no such vector exists. Cantor does not see Kronecker's vectors
but he is allowed to ask queries of the form ``What is bit number j of vector number i? What is the minimal number of queries with which Cantor can achieve his goal? How much better can Cantor do if
he is allowed to pick his queries \emph{adaptively}, based on Kronecker's previous replies?
The case when m=n is solved by diagonalization using n (non-adaptive) queries. We study this game more generally, and prove an optimal bound in the adaptive case and nearly tight upper and lower
bounds in the non-adaptive case.
Changes to previous version:
Added references to related work
TR23-002 | 5th January 2023 08:14
Diagonalization Games
We study several variants of a combinatorial game which is based on Cantor's diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by
the known fact that Leopold Kronecker did not appreciate Georg Cantor's arguments about the infinite, and even referred to him as a ``scientific charlatan''.
In the game Kronecker maintains a list of m binary vectors,
each of length n, and Cantor's goal is to produce a new binary vector which is different from each of Kronecker's vectors, or prove that no such vector exists. Cantor does not see Kronecker's vectors
but he is allowed to ask queries of the form ``What is bit number j of vector number i? What is the minimal number of queries with which Cantor can achieve his goal? How much better can Cantor do if
he is allowed to pick his queries \emph{adaptively}, based on Kronecker's previous replies?
The case when m=n is solved by diagonalization using n (non-adaptive) queries. We study this game more generally, and prove an optimal bound in the adaptive case and nearly tight upper and lower
bounds in the non-adaptive case. | {"url":"https://eccc.weizmann.ac.il/report/2023/002/","timestamp":"2024-11-12T13:04:34Z","content_type":"application/xhtml+xml","content_length":"26562","record_id":"<urn:uuid:3e6b43e0-ef59-4fdb-b5a6-5d77a00e2a90>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00533.warc.gz"} |
Eight basic rules for causal inference | Peder M. Isager
Eight basic rules for causal inference
By Peder M. Isager
August 13, 2024
In this blog post I will describe eight basic rules that govern the relationship between causal mechanisms in the real world and associations/correlations we can observe in data. To make each rule as
easy as possible to understand, I will describe each rule both in words and in causal graph and logic terms, and I will offer some very simple simulation R code for each rule to demonstrate how it
works in practice.
These eight rules represent basic building blocks of causal inference. Most causal analysis procedures involve one or more of these rules to some extent. If you are completely new to formal causal
inference, learning these rules can serve as a springboard to learn more complicated things. If you apply causal inference regularly in your own research then you might find this post useful as a
cheat sheet. For a much more detailed introduction to causal inference, see Hernán and Robins (2020), and the Causal Diagrams course from HarvardX.
Four fundamental causal structures
A causal graph is a depiction of a causal mechanism between variables. In the graph, variables (nodes) are drawn as circles ⚪ (or sometimes as squares), and the causal relationships (paths) between
these variables are drawn as arrows → that point from the thing causing (causal ancestor) to the thing being caused (causal descendant).
Any causal graph, no matter how complicated, can be broken down into four elemental building blocks. The blocks are defined by the type of path that can be traced between the variables. All the rules
that I will describe below deal with one or more of these building blocks, so it is useful to start by stating and describing them. By recognizing which blocks we are working with in any given
instance we will know which causal inference rules are relevant.
1. Complete independence.
No path can be traced between A and B.
2. Chain
In causal chains, a directed path can be traced from A to B, such that all arrows point from A to B. Chain paths are sometimes called an “open path”, which means that this type of graph transmits
correlation between A and B (see rule 2). When a chain involves three or more variables, the variables M linking A and B are often called mediators.
3. Fork
In a causal fork, an undirected path (not all arrows run in the same direction) can be traced from A to B through a common causal ancestor C. C is often called a confounder variable. Fork paths are
“open” and transmit correlation between A and B (see rule 3).
4. Collider
In a causal collider, an undirected path (not all arrows run in the same direction) can be traced from A to B through a causal descendant D. D is often called a collider variable. Collider paths are
“closed” and do not transmit correlation between A and B (see rule 1).
Basic rules for causal inference
Rule 2: Causal influence creates correlation
If A is a cause of B, or if B is a cause of A, then A and B will be correlated in data.
# Rule 2
n=10000 # Number of data points
a <- rnorm(n, 0, 1) # A is a random variable
b <- a + rnorm(n, 0, 1) # B is a function of A
plot(a, b)
cor(a, b) # Correlation between A and B
## [1] 0.71
This also applies if A causes M, and M in turn causes B (mediation).
# Rule 2 (mediation)
n=10000 # Number of data points
a <- rnorm(n, 0, 1) # A is a random variable
m <- a + rnorm(n, 0, 1) # M is a function of A
b <- m + rnorm(n, 0, 1) # B is a function of M
plot(a, b)
cor(a, b) # Correlation between A and B
## [1] 0.58
Rule 3: Confounding creates correlation
If A and B share a common ancestor C (causal fork), A and B will be correlated in data. This phenomenon is often called confounding, or the “third variable problem”.
# Rule 3
n=10000 # Number of data points
c <- rnorm(n, 0, 1) # C is a random variable
a <- c + rnorm(n, 0, 1) # A is a function of C
b <- c + rnorm(n, 0, 1) # B is a function of C
plot(a, b)
cor(a, b) # Correlation between A and B
## [1] 0.49
The rule also applies if the effects of C on A and/or B are mediated through other variables.
Rule 4: Random manipulation protects a variable from causal influence
When we are able to randomly allocate the values of A - such as in a randomized controlled experiment where A is the manipulation variable - no other variable can influence A.
The notation do(A) refers to randomizing the values of A. Put differently, with complete experimental control and randomization we make sure that no variable is allowed to influence the values of A.
Adjustment rules
To adjust for a variable X means to look at relationships in data that contain only a subset or single value of X. It can also mean to look at relationships for all the values of X separately.
Adjustment goes by a number of different names in the sciences, including “control for”, “condition on”, “hold constant”, “stratify”, “select”, etc.
In the figures below, a square box around a variable node indicates that this variable is being controlled/adjusted for.
Rule 5: Controlling for a confounder blocks correlation arising from that confounder
If A and B share a common ancestor C (causal fork), the confounding correlation between A and B that is created by C (rule 3) is removed if C is controlled for.
# Rule 5
n=10000 # Number of data points
c <- rnorm(n, 0, 1) # C is a random variable
a <- c + rnorm(n, 0, 1) # A is a function of C
b <- c + rnorm(n, 0, 1) # B is a function of C
x <- lm(b~c)
y <- lm(a~c)
plot(x$residuals, y$residuals)
cor(x$residuals, y$residuals) # Correlation between A and B, controlling for C
## [1] 0.015
Rule 6: Controlling for a mediator blocks correlation arising from the mediated causal effect
If A is a cause of M and M is a cause of B, correlation between A and B that is created by the mediated causal effect (rule 2) will be removed if M is controlled for.
# Rule 6
n=10000 # Number of data points
a <- rnorm(n, 0, 1) # A is a random variable
m <- a + rnorm(n, 0, 1) # M is a function of A
b <- m + rnorm(n, 0, 1) # B is a function of M
x <- lm(a~m)
y <- lm(b~m)
plot(x$residuals, y$residuals)
cor(x$residuals, y$residuals) # Correlation between A and B, controlling for M
## [1] 0.027
Rule 7: Controlling for a collider leads to correlation
If A and B share a causal descendant (collider) D, and D is controlled for, A and B will become correlated in the data. This is often referred to as “conditioning on a collider”, or collider bias.
# Rule 7
n=10000 # Number of data points
a <- rnorm(n, 0, 1) # A is a random variable
b <- rnorm(n, 0, 1) # B is a random variable
d <- a + b + rnorm(n, 0, 1) # D is a function of A and B
x <- lm(a~d)
y <- lm(b~d)
plot(x$residuals, y$residuals)
cor(x$residuals, y$residuals) # Correlation between A and B, controlling for D
## [1] -0.5
Rule 8: Controlling for a causal descendant (partially) controls for the ancestor
If B is a descendant of A and B is controlled for, A is also (partially) controlled for.
The degree to which A is controlled when B is controlled for generally depends on the how reliably A causes B.
In the example below, C is a confounder of A and B, but the confounding influence can partially be blocked by controlling for CM.
If CM is a semi-reliable measure of C, some correlation between A and B is removed by controlling for CM, but not as much as when controlling for C:
# Rule 5
n=10000 # Number of data points
# 2*c used in equations to make change in relationship more visible.
c <- rnorm(n, 0, 1) # C is a random variable
a <- 2*c + rnorm(n, 0, 1) # A is a function of C
b <- 2*c + rnorm(n, 0, 1) # B is a function of C
cm <- 2*c + rnorm(n, 0, 1) # CM is a function of C
# Control for C
ac <- lm(b~c)
bc <- lm(a~c)
# Control for CM
acm <- lm(a~cm)
bcm <- lm(b~cm)
# Plot relationship between a and b while...
plot(a,b, main = "no control") # controlling for nothing
plot(acm$residuals, bcm$residuals, main = "controlling for CM") # controlling for CM
plot(ac$residuals, bc$residuals, main = "controlling for C") # controlling for C
# Correlation between a and b while...
cor(a,b) # controlling for nothing
## [1] 0.8
cor(acm$residuals, bcm$residuals) # controlling for CM
## [1] 0.44
cor(ac$residuals, bc$residuals) # controlling for C
## [1] 0.0021
Important assumptions
The above rules only hold if some important assumptions are met, which I will list below but not explain in detail. For details, see Hernán and Robins (2020).
No spurious correlation: Correlation is not caused by random accident. The law of large numbers dictate that the more data we have, the more credible this assumption is.
Consistency: The values of A you see are the actual values of A, or “the values of treatment under comparison correspond to well-defined interventions that, in turn, correspond to the versions of
treatment in the data” Hernán and Robins (2020).
Exchangeability: “the conditional probability of receiving every value of treatment, though not decided by the investigators, depends only on measured covariates” Hernán and Robins (2020).
Positivity: “the probability of receiving every value of treatment conditional on L is greater than zero, i.e., positive” Hernán and Robins (2020).
Faithfulness: The causal effect does not vary over groups in a way that makes it average to 0 in the data. A does not have a positive effect 50% of the time and an identically powerful negative
effect 50% of the time, which would average out to an effect of 0 in the population.
If either of these assumptions are not met, it can potentially break the relationship between causal effect and observed data described by these rules.
Hernán, Miguel, and James Robins. 2020. Causal Inference: What If. Boca Raton: Chapman & Hall/CRC.
Posted on:
August 13, 2024
10 minute read, 2034 words | {"url":"https://pedermisager.org/blog/eight_basic_rules_for_causal_inference/","timestamp":"2024-11-12T00:27:54Z","content_type":"text/html","content_length":"28202","record_id":"<urn:uuid:176c7321-5109-44b3-9803-b95755025577>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00697.warc.gz"} |
WeBWorK Standalone Renderer
Use the limit comparison test to determine whether
$\displaystyle \sum_{n=7}^{\infty} a_n = \sum_{n=7}^{\infty} \frac{2 n^3 - 6 n^2 + 7}{9 + 8 n^4}$
converges or diverges.
(a) Choose a series
$\displaystyle \sum_{n=7}^\infty b_n$
with terms of the form
$\displaystyle b_n = \frac{1}{n^p}$
and apply the limit comparison test. Write your answer as a fully simplified fraction. For
$n \geq 7$
$\displaystyle \lim_{n \to \infty} \frac{ a_{n} }{ b_{n} } = \lim_{n \to \infty}$
(b) Evaluate the limit in the previous part. Enter
If the limit does not exist, enter
DNE. $\displaystyle \lim_{n\to\infty} \frac{a_{n}}{b_{n}} \,$
(c) By the limit comparison test, does the series converge, diverge, or is the test inconclusive?
You can earn 40% partial credit for 2 - 3 correct answers. | {"url":"https://wwrenderer.libretexts.org/render-api?sourceFilePath=Library/FortLewis/Calc2/9-4-Limit-comparison-test/limit-comparison-03.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&answersSubmitted=0&showSummary=1&displayMode=MathJax&language=en&outputFormat=nosubmit","timestamp":"2024-11-03T19:45:11Z","content_type":"text/html","content_length":"7686","record_id":"<urn:uuid:8fe3ae38-bdae-47f9-8b09-88b4028321e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00154.warc.gz"} |
Where to find experts for implementing the primal-dual interior-point method in Linear Programming? | Linear Programming Assignment Help
Where to find experts for implementing the primal-dual interior-point method in Linear Programming? By Andrew Johnson (www.e2ddo.org, June 14th) Join us in our recent issue on The Interval
Instability Theory (with Tom Brown). Recent publications include: Comput. Structural Computers and Algorithms. Chapter 6 | Dynamic Programming . As the title implies, this is a simple example of the
primal-dual interior-point method that has two distinct problem points. As explained this page could, if implemented, become a fully programmable solution to the original primal-dual problem. A
similar example had been successfully implemented for a quadratic program by Ryan Newman, and similarly can also be implemented as a solution to the same problem. Both the primal-dual and the
interior-point methods are introduced since the papers of Newman in the context of computer algebra work. They are, as already explained when adopting the primal-dual interior-point method, capable
of efficiently expressing large arrays of elements in the form of rational numbers. The primal-dual interior-point method also has been implemented by J. H. Miller and N. E. Sargent in their textbook
for the second and third author, but it has not yet been implemented at all. We would like to present the primal-dual interior-point method in this paper for completeness. The primal-dual
interior-point method you can try here illustrated when representing a triangular $t$-box $x$ as illustrated in the first subsection.
What Are Some Benefits Of Proctored Exams For Online Courses?
The result is the triangulation with the idea of a primal triangulation of a rectangle containing an outward oriented face. The face of the rectangular is located facing one of the triangles $(A_1, \
dots, B_m)$ that is the point $y$ of the triangular $t$-box, and is an outward orientedWhere to find experts for implementing the primal-dual interior-point method in Linear Programming? How to
create LDP and get the best in terms of running time and training set of classes in the form of 2 variable LDPs? LDP is a class which is used to implement the lager methods like getLager(max.getLager
() or m+1) with the help of LagerPro(MinMax(minLager(), maxLager())) mode. Let’s consider two things which are addressed in this tutorial: Interior-point : the one which is implemented by the lager
interface in Linear Programming. Use it in lager. This gives you way to extend your class’s Lager and uses the m-th priority class and its ability to be extended. Upper-point : the use of the
two-class Lager in your objective-condition setting. The optimal implementation mode is LagerTrach$f$ and the m-th class isn’t used to have any priority and in the Lager Trach$f$ mode is used to
implement the desired Lager idea. The object-class is used for both Lager methods. The object-class consists of (class Lager, object M$L$) where Lager constructor is an instance of M$L$ class. Object
Class for that class is shared to objects. Object Class is used to specify the Lager constructor class in your Lager and its implementation class is used to convert between objects. Object Class can
be of type AllocPro, AllocSetPro or all ode L[](class Lage2). It can be used in both the default and default initialization mode. The initialization mode of that object-class is done on the basis of
the LagerTrach$f$ operation. But we have seen some concepts in this tutorial that are Get the facts by a class which is already a lot used by a class. Let usWhere to find experts navigate here
implementing the primal-dual interior-point method in Linear Programming? In basic primal-dual theory, a primal-comoving triangle is a point on a graph whose vertices have been connected in a single
step. A composite triangle or triangle-like point can be found, for example, using the same algorithm as the corresponding line-triangle case. The primal-dual interior-point method for
point-matching, however, is usually very refined, since its lower bound is generally regarded as the most necessary. For instance, when the sequence of inputs (which are usually finite) is known, we
may find a pair whose triangle and triangle-like points are all connected at the same time in this way.
Online Quiz Helper
An example of such points is shown in Figure 4. Thus, in linear optimization, each one of its points is connected only once as its triangles are, as shown in Figure 4. This is only one reason part of
the proof, which attempts to show that a primal-dual interior-point method is indeed a good approximation to read here interior-point method: a primal-comoving triangle can be found when the sequence
of inputs is known, for example, from a natural series equation (for a standard triangle-like point-matching example see, e.g., [@de2014method]). 2 Problem | {"url":"https://linearprogramminghelp.com/where-to-find-experts-for-implementing-the-primal-dual-interior-point-method-in-linear-programming","timestamp":"2024-11-06T02:19:06Z","content_type":"text/html","content_length":"115251","record_id":"<urn:uuid:0039dda6-146e-4cbb-b4e9-2939cf8cde14>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00107.warc.gz"} |
Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf
Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf serve as fundamental devices in the world of maths, giving a structured yet versatile platform for students to discover and understand
numerical ideas. These worksheets supply a structured approach to understanding numbers, supporting a solid foundation whereupon mathematical proficiency thrives. From the easiest counting workouts
to the ins and outs of advanced calculations, Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf cater to learners of varied ages and skill degrees.
Unveiling the Essence of Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf
Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf
Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf -
Mixed Numbers Improper Fractions In each problem below an improper fraction is represented by blocks beneath a number line Use the number line to determine what the equivalent mixed number form would
be Notice that some number lines have diferent sub divisions thirds fourths ifths 0 0 0 0 0 0 3 4 2 2 2 2 2 2 3 3 3 3 3
Multiplying Mixed Numbers by Mixed Numbers Make lightning fast progress with these multiplying mixed fractions worksheet pdfs Change the mixed numbers to improper fractions cross cancel to reduce
them to the lowest terms multiply the numerators together and the denominators together and convert them to mixed numbers if improper fractions
At their core, Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf are lorries for theoretical understanding. They envelop a myriad of mathematical concepts, guiding students with the maze of
numbers with a collection of appealing and purposeful exercises. These worksheets transcend the limits of typical rote learning, urging energetic engagement and fostering an intuitive grasp of
mathematical connections.
Nurturing Number Sense and Reasoning
Multiply Mixed Numbers By Whole Numbers Math Worksheet For Class 5 This Online Fraction
Multiply Mixed Numbers By Whole Numbers Math Worksheet For Class 5 This Online Fraction
Download Multiplying Mixed Numbers by Whole Numbers Worksheet PDFs These math worksheets should be practiced regularly and are free to download in PDF formats Multiplying Mixed Numbers by Whole
Numbers Worksheet 1 Download PDF Multiplying Mixed Numbers by Whole Numbers Worksheet 2 Download PDF
Student Name Score Printable Math Worksheets www mathworksheets4kids
The heart of Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf lies in growing number sense-- a deep understanding of numbers' definitions and interconnections. They motivate expedition,
inviting learners to explore math procedures, decipher patterns, and unlock the secrets of series. With provocative obstacles and sensible problems, these worksheets end up being portals to
developing thinking abilities, supporting the logical minds of budding mathematicians.
From Theory to Real-World Application
Multiplication Mixed Numbers Worksheet
Multiplication Mixed Numbers Worksheet
Multiplying mixed numbers by whole numbers is a multi step process and this worksheet will take your learner through an example to help build their understanding After the clearly outlined example
students will solve eight multiplication equations showing their work and writing their answers in simplest form
Multiplying Mixed Numbers and Whole Numbers 2 6 3 9 3 11 2 2 7 9 5 10 10 10 7 2 2 3 9 5 9 2 6 12 11 5 8 3 7 2 7 2 1 9 4 2 4 8 6 6 7 1 12
Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf serve as avenues linking academic abstractions with the palpable truths of day-to-day life. By instilling useful situations right into
mathematical workouts, learners witness the significance of numbers in their environments. From budgeting and measurement conversions to understanding statistical information, these worksheets
empower pupils to wield their mathematical prowess past the boundaries of the class.
Diverse Tools and Techniques
Versatility is inherent in Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf, utilizing an arsenal of pedagogical devices to satisfy different discovering styles. Visual help such as number
lines, manipulatives, and electronic resources serve as buddies in envisioning abstract ideas. This varied technique makes certain inclusivity, fitting students with different choices, strengths, and
cognitive styles.
Inclusivity and Cultural Relevance
In an increasingly diverse world, Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf accept inclusivity. They go beyond cultural boundaries, integrating examples and troubles that resonate with
learners from diverse histories. By incorporating culturally pertinent contexts, these worksheets foster a setting where every learner feels stood for and valued, boosting their link with
mathematical ideas.
Crafting a Path to Mathematical Mastery
Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf chart a training course in the direction of mathematical fluency. They instill willpower, essential reasoning, and analytical skills, crucial
characteristics not just in maths but in different aspects of life. These worksheets empower learners to navigate the intricate surface of numbers, supporting a profound gratitude for the elegance
and logic inherent in mathematics.
Welcoming the Future of Education
In an age noted by technical development, Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf perfectly adjust to electronic systems. Interactive interfaces and digital sources increase typical
understanding, offering immersive experiences that transcend spatial and temporal limits. This combinations of conventional approaches with technological technologies advertises a promising age in
education, promoting a more dynamic and interesting discovering atmosphere.
Final thought: Embracing the Magic of Numbers
Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf illustrate the magic inherent in maths-- an enchanting journey of expedition, exploration, and mastery. They go beyond traditional pedagogy,
serving as stimulants for igniting the flames of interest and questions. Via Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf, learners embark on an odyssey, opening the enigmatic globe of
numbers-- one trouble, one solution, at a time.
Multiplying Mixed Numbers Worksheet
Multiplying Mixed Numbers By Whole Numbers Worksheet
Check more of Multiplying Mixed Numbers By Whole Numbers Worksheet Pdf below
Multiplying Mixed Numbers And Whole Numbers Worksheet
Multiplying Fractions By Whole Numbers Worksheet Printable Word Searches
Multiply Mixed Numbers By A Whole Anchor Chart Multiplying Mixed Numbers Elementary Math
Multiplying Mixed Numbers By Whole Numbers Worksheet
Multiplying Mixed Numbers By Whole Numbers Worksheet EdPlace
Multiplying Mixed Numbers By Whole Numbers Worksheet
Multiplying Mixed Numbers Worksheets Math Worksheets 4 Kids
Multiplying Mixed Numbers by Mixed Numbers Make lightning fast progress with these multiplying mixed fractions worksheet pdfs Change the mixed numbers to improper fractions cross cancel to reduce
them to the lowest terms multiply the numerators together and the denominators together and convert them to mixed numbers if improper fractions
Grade 5 Fractions Worksheets Multiplying Mixed Numbers K5 Learning
Below are six versions of our grade 5 math worksheet on multiplying mixed numbers together These worksheets are pdf files Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 Worksheet 6 5
Multiplying Mixed Numbers by Mixed Numbers Make lightning fast progress with these multiplying mixed fractions worksheet pdfs Change the mixed numbers to improper fractions cross cancel to reduce
them to the lowest terms multiply the numerators together and the denominators together and convert them to mixed numbers if improper fractions
Below are six versions of our grade 5 math worksheet on multiplying mixed numbers together These worksheets are pdf files Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 Worksheet 6 5
Multiplying Mixed Numbers By Whole Numbers Worksheet
Multiplying Fractions By Whole Numbers Worksheet Printable Word Searches
Multiplying Mixed Numbers By Whole Numbers Worksheet EdPlace
Multiplying Mixed Numbers By Whole Numbers Worksheet
Multiplying Whole Numbers By Decimals Worksheet
Multiplying Mixed Numbers By Whole Numbers Open Middle
Multiplying Mixed Numbers By Whole Numbers Open Middle
Multiplying Whole Numbers And Mixed Numbers Worksheet | {"url":"https://szukarka.net/multiplying-mixed-numbers-by-whole-numbers-worksheet-pdf","timestamp":"2024-11-08T10:34:33Z","content_type":"text/html","content_length":"26423","record_id":"<urn:uuid:8de13a86-69b9-4a49-a20e-0e332a37a211>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00188.warc.gz"} |
The Bayes factor in the analysis of mechanical power in patients with severe respiratory failure due to SARS-CoV-2 | Medicina Intensiva
Article information
Full Text
Download PDF
Tables (4)
Table 2. Comparison of the clinical-epidemiological characteristics of the patients by study groups.
Table 3. Comparative analysis between groups of the principal variables associated with acute respiratory failure and therapeutic measures in the ICU.
Table 4. Comparative analysis of the principal evolutive variables during admission to the ICU.
Show moreShow less
To specify the degree of probative force of the statistical hypotheses in relation to mortality at 28 days and the threshold value of 17 J/min mechanical power (MP) in patients with respiratory
failure secondary to SARS-CoV-2.
Cohort study, longitudinal, analytical.
Intensive care unit of a third level hospital in Spain.
Patients admitted for SARS-CoV-2 infection with admission to the ICU between March 2020 and March 2022.
Bayesian analysis with the beta binomial model.
Main variables of interest
Bayes factor, mechanical power.
A total of 253 patients were analyzed. Baseline respiratory rate (BF10: 3.83 × 106), peak pressure value (BF10: 3.72 × 1013) and neumothorax (BF10: 17,663) were the values most likely to be different
between the two groups of patients compared. In the group of patients with MP < 17 J/min, a BF10 of 12.71 and a BF01 of 0.07 were established with an 95%CI of 0.27–0.58. For the group of patients
with MP ≥ 17 J/min the BF10 was 36,100 and the BF01 of 2.77e-05 with an 95%CI of 0.42–0.72.
A MP ≥ 17 J/min value is associated with extreme evidence with 28-day mortality in patients requiring MV due to respiratory failure secondary to SARS-CoV-2 disease.
Mechanical ventilation
Mechanical power
Precisar el grado de fuerza probatoria de las hipótesis estadísticas con relación a la mortalidad a 28 días y el valor umbral de 17 J/min de potencia mecánica (PM) en pacientes con insuficiencia
respiratoria secundaria a SARS-CoV-2.
Estudio de cohortes, longitudinal y analítico.
Unidad de cuidados intensivos de un hospital de tercer nivel en España.
Enfermos ingresados por infección por SARS-CoV-2 con ingreso en la UCI entre marzo de 2020 y marzo de 2022.
Análisis bayesiano con el modelo binomial beta.
Variables de interés principales
Factor de Bayes, mechanical power.
Fueron analizados 253 pacientes. La frecuencia respiratoria inicial (BF10: 3,83 × 106), el valor de la presión pico (BF10: 3,72 × 1013) y el desarrollo de neumotórax (BF10: 17.663) fueron los valores
con más probabilidad de ser diferentes entre los 2 grupos de pacientes comparados. En el grupo de pacientes con PM < 17 J/min se estableció un BF10 de 12,71 y un BF01 de 0,07 con un IdC95% de
0,27-0,58; Para el grupo de pacientes con PM ≥ 17 J/min el BF10 fue de 36.100 y el BF01 de 2,77e-05 con un IdC95% de 0,42-0,72.
Un valor de PM ≥ 17 J/min se asocia con una evidencia extrema con la mortalidad a 28 días en pacientes que necesitaron ventilación mecánica por insuficiencia respiratoria secundaria a enfermedad por
Palabras clave:
Ventilación mecánica
Energía mecánica
Full Text
Mechanical power (MP) in lung ventilation, defined as the energy delivered from the ventilator to the respiratory system during a period of time, has been recognized as a promising indicator for
evaluating ventilator-induced lung injury (VILI) and for predicting the outcome of ventilated patients.1
There may be an MP safety threshold above which lung injury is inevitable. Recently, different studies in patients with lung injury secondary to SARS-CoV-2 disease have found that an MP threshold of
17 J/min could be associated with an increased mortality risk.2,3
In this context, where a statistically significant association (p < 0.05) is observed between in-hospital mortality and an MP threshold of 17 J/min in patients with acute respiratory distress
syndrome (ARDS) secondary to SARS-CoV-2 infection, it seems advisable (as is also seen in other settings) to replicate the clinical investigations based on significance tests, with a view to
generating more credible evidence in the intensive care setting.4–6
This can be done through Bayesian inference, which allows us to reanalyze significant findings, and where the Bayes factor (BF) is referred to as the probability of the data under one hypothesis in
relation to another (null hypothesis versus the alternative hypothesis). In other words, BF estimates the degree or evidence with which the data support both the null hypothesis and the alternative
hypothesis, contrasting them beyond the conventional dichotomic interpretation of rejection or acceptance of the null hypothesis.7,8 The statistical repetition of significant findings based on BF
reinforces the practical credibility of future articles, which is indicated when Bayesian inference generates conclusive (strong) or greater evidence (BF10 > 10) from the interpretation of the
Jeffreys classification of values for BF: anecdotal, moderate, strong and very strong.9
The present study was carried out to establish the demonstrative strength of statistical hypotheses in relation to 28-day mortality and an MP threshold value of 17 J/min in patients with respiratory
failure secondary to SARS-CoV-2 infection, based on Bayesian analysis.
Material and methods
An analytical, observational cohort study was made of all the patients admitted to the Intensive Care Unit (ICU) due to SARS-CoV-2 infection between March 2020 and March 2022. The data were obtained
from the COVID-19 patient cohort registry of the Department of Intensive Care Medicine of a third-level hospital, following approval from the local Ethics Committee and the obtainment of consent
(written and/or by telephone) from the patients or their legal representatives.
Sampling was consecutive, and the patients were divided into two cohorts according to the MP value in the first 24 h after orotracheal intubation: MP < 17 J/min and MP ≥ 17 J/min. The measurement of
MP was carried out using the simplified formula proposed by Gattinoni et al.10
The inclusion criteria were: patients over 18 years of age, a confirmed diagnosis of SARS-CoV-2 disease, need for admission to the ICU, and the use of controlled mechanical ventilation (CMV) with a
correct recording of the variables required to calculate MP in the supine position, following sedation and neuromuscular block (where needed), within the first 24 h of mechanical ventilation.
The patient clinical-demographic parameters (age, gender, cardiovascular risk factors, smoking) were recorded, as well as the specific treatments applied in the ICU, initial ventilatory parameters
(tidal volume, respiratory rate, positive end-expiratory pressure [PEEP], plateau pressure, compliance, driving pressure) and evolutive parameters.
A first descriptive analysis of the sample was made in which categorical variables were reported as frequencies and percentages, while continuous quantitative variables were reported as the median
and percentiles 25–75 (p25–75). The comparison of continuous quantitative variables was carried out with the Bayesian t-test for independent samples. The comparison of frequencies in turn was carried
out based on contingency tables and Poisson BF10. In all the tests, the alternative hypothesis specifies that the MP < 17 J/min group is not the same as the MP ≥ 17 J/min group. Lastly, a Bayesian
analysis was performed with the beta-binomial model. The null hypothesis (H0) postulated that MP < 17 J/min was associated with 28-day mortality, while the alternative hypothesis (H1) postulated that
MP ≥ 17 J/min was associated with 28-day mortality. Based on our previous findings,3 an a priori 28-day mortality rate of 21.5% was considered (the alternative hypothesis specifies the proportion as
>0.21). The prior distribution under the alternative hypothesis was specified with beta (1.1). The results are shown with BF10 (in favor of the alternative hypothesis) and BF01 (in favor of the null
hypothesis), with a 95% credible interval.
The statistical analysis was carried out using the JASP 2023 package (version 0.17).
During the analyzed period, a total of 911 patients were admitted to the ICU due to SARS-CoV-2 disease, and 552 met the study inclusion criteria. The data needed for the analysis were obtained from a
total of 253 patients, which were divided into two groups according to MP in the first 24 h of mechanical ventilation (MV).
The median age was 64 years (p25–75: 57–72), and most of the patients were men (n = 179; 71%). The principal characteristics of the 253 subjects are reported in Table 1.
Table 1.
Principal variables of the study cohort.
Total patients
n = 253
Time period
1st wave 35 (14%)
2nd wave 63 (25%)
3rd wave 52 (20%)
4th wave 42 (16%)
5th wave 16 (6%)
6th wave 45 (17%)
Clinical-demographic variables and comorbidities
Age (years), median (p25–p75) 64 (57–72)
Male gender, n (%) 179 (71%)
Arterial hypertension, n (%) 123 (49%)
Obesity, n (%) 48 (19%)
Diabetes, n (%) 56 (22%)
Dyslipidemia, n (%) 75 (29%)
Smoker, n (%) 90 (35%)
P/F upon admission to ICU, mmHg, median (p25–p75) 131 (100–164)
Use of HFNO, n (%) 130 (51%)
Use of RRT, n (%) 14 (5%)
Therapy in prone position, n (%) 133 (52%)
Use of corticosteroids, n (%) 196 (77%)
Diagnosis of PTE during admission to ICU, n (%) 9 (4%)
Days of MV, median (p25–p75) 9 (6–17)
ICU stay (days), median (p25–p75) 13 (9–23)
P/F: ratio between partial pressure of oxygen and fraction of inspired oxygen, HFNO: high-flow nasal oxygen, MV: mechanical ventilation, RRT: renal replacement therapy, PTE: pulmonary
The comparative analysis of the clinical-demographic parameters between the two groups (Table 2) revealed differences in gender distribution (BF10: 3.49) and creatine kinase (BF10: 1.35). The
variables referred to the treatments provided during ICU stay showed no relevant differences (Table 3).
Table 2.
Comparison of the clinical-epidemiological characteristics of the patients by study groups.
Group MP < 17 J/min Group MP ≥ 17 J/min BF10
n = 71 n = 182
Time period 0.29
1st wave 13 18.31% 22 12.08%
2nd wave 26 36.62% 37 20.33%
3rd wave 10 14.08% 42 23.07%
4th wave 7 9.85% 35 19.23%
5th wave 2 2.81% 14 7.69%
6th wave 13 18.13% 32 17.58%
Age (years), median (p25–75) 67 (58–73) 63 (57–71) 0.43
Male gender, n (%) 42 59% 137 75% 3.49
AHT, n (%) 32 55% 91 50% 0.18
Obesity, n (%) 9 12% 39 21% 0.61
Diabetes, n (%) 11 15% 45 25% 0.57
Dyslipidemia, n (%) 22 31% 53 29% 0.16
Smoker, n (%) 28 39% 62 34% 0.20
CK (median p27−75) 178 (119−236) 362 (259−465) 1.35
DD (median p27−75) 5876 1204–10,574 4833 2119−7547 0.25
Ferritin (median p27–75) 1154 (940–1388) 1222 (1060−1384) 0.18
IL-6 (median p27–75) 55 36–75 62 (15−78) 0.36
CRP (median p25−75) 12 10–14 14 13−16 0.51
LDH (median p27−75) 421 374–469 467 424−511 0.30
AHT: arterial hypertension; creatine kinase (CK in U/L, normal range: 46–171), D-dimer (DD in ng/mL, normal range: 0–500), ferritin (in ng/mL, normal range: 22–322), interleukin-6 (IL-6), C-reactive
protein (CRP in mg/dL, normal range: ≤0.5), lactate dehydrogenase (LDH in U/L, normal range: 120–246); BF: Bayes factor.
Table 3.
Comparative analysis between groups of the principal variables associated with acute respiratory failure and therapeutic measures in the ICU.
Group MP < 17 J/min Group MP ≥ 17 J/min BF10
N = 71 N = 182
Reason for admission ARF 71 100% 178 98% 0.16
P/F upon admission to ICU, mmHg, median (p25–75) 123 (100–170) 134 (104–163) 0.18
Previous HFNO, n (%) 36 51% 94 51% 0.14
Therapy in prone position, n (%) 37 52% 96 52% 0.14
Use of remdesivir, n (%) 8 11% 16 9%
Use of corticosteroids, n (%) 56 79% 140 77% 0.17
Need for vasopressors/inotropic agents during admission to ICU, n (%) 41 58% 102 56% 0.14
Use of CRRT, n (%) 3 4% 11 6% 0.22
ARF: acute respiratory failure; P/F: ratio between partial pressure of oxygen and fraction of inspired oxygen, HFNO: high-flow nasal oxygen, MV: mechanical ventilation, CRRT: continuous renal
replacement therapy, BF: Bayes factor.
As can be seen in Table 4, the initial respiratory rate (BF10: 3.83 × 106), peak pressure (BF10: 3.72 × 1013) and the development of pneumothorax (BF10: 17,663) were the parameters most likely to be
different between the two groups.
Table 4.
Comparative analysis of the principal evolutive variables during admission to the ICU.
Group MP < 17 J/min Group MP ≥ 17 J/min BF10
n = 71 n = 182
Ventilatory parameters
Tidal volumea (mL), median (p25–75) 450 (425–477) 475 (450–490) 23
Respiratory ratea, median (p25–75) 16 (15–18) 18 (16–19) 3.83 × 106
PEEPa (cmH2O), median (p25−75) 10 (8−12) 12 (10−13) 0.23
Peak pressurea (cmH2O), median (p25−75) 28 (26−30) 32 (30−34) 3.72 × 1013
Plateau pressurea (cmH2O), median (p25−75) 23 (21−26) 22 (20−25) 0.30
Compliancea (mL/cmH2O), median (p25−75) 37 (30−53) 45 (35−53) 0.42
Driving pressurea (cmH2O), median (p25−75) 12 (8−15) 11 (9−13) 0.73
P/F prior to OTI 111 (102−119) 115 (111−120) 0.24
Days of MV median (p25–p75) 12 (10−14) 14 (12−16) 0.29
Development of pneumothorax, n (%) 0 (0%) 7 17% 17,663
Development of PTE, n (%) 5 7% 4 2% 1.82
Development of VAT/VAP, n (%) 39 55% 110 60% 0.19
Tracheotomy, n (%) 11 15% 33 18% 0.20
ICU stay (days), median (p25–p75) 26 (15−37) 17 (15−19) 0.008
28-Day mortality, n (%) 7 (10%) 27 (15%) 0,316
MV: mechanical ventilation; P/F: ratio between partial pressure of oxygen and fraction of inspired oxygen, OTI: orotracheal intubation, PTE: pulmonary thromboembolism, VAT: ventilator-associated
tracheobronchitis, VAP: ventilator-associated pneumonia, ICU: Intensive Care Unit, BF: Bayes factor.
All analyzed patients were under volume control ventilation (protective ventilation was carried out based mainly on an initial tidal volume of 6−8 mL/kg ideal body weight), and measurement was made
of the variables needed to calculate MP (measurement of plateau pressure with inspiratory pause) in the supine position, after optimization of the ventilatory parameters according to medical
criterion following deep sedation and/or neuromuscular block, within the first 24 h of mechanical ventilation.
The Bayesian binomial contrast yielded the following results: in the group of patients with MP < 17 J/min (Fig. 1A), BF10 was 12.71 — indicating that the evidence in favor of the alternative
hypothesis (i.e., postulating that there is a real difference between the groups) was 12.71 times stronger than the evidence in favor of the null hypothesis (i.e., postulating that there is no real
difference between the groups). The 95% credible interval for the proportion of patients in this group was 0.27–0.58. In the group of patients with MP ≥ 17 J/min (Fig. 1B), BF10 was much greater
(36,100) — indicating strong evidence in favor of the alternative hypothesis. The 95% credible interval for the proportion of patients in this group was 0.42–0.72.
Figure 1.
Before and after graphic representation of the degrees of inference of the Bayesian binomial contrast and the sequential analysis.
The results indicate that in the group of patients with MP < 17 J/min (Fig. 1A), the Bayes factor in favor of the alternative hypothesis (i.e., postulating that the proportion of patients below the
threshold point is greater) was 12.71 times stronger than in favor of the null hypothesis (i.e., that the proportion is the same). The 95% credible interval for the difference of proportions was
0.27–0.58, indicating that it is probable that the proportion in the group with MP < 17 J/min is significantly greater than in the group with MP ≥ 17 J/min. On the other hand, in the group of
patients with MP ≥ 17 J/min (Fig. 1B), the Bayes factor in favor of the alternative hypothesis (i.e., postulating that the proportion of patients below the threshold point is smaller) was 36,100,
which suggests strong evidence in favor of the alternative hypothesis versus the null hypothesis. The 95% credible interval for the difference of proportions was 0.42–0.72, indicating that it is
probable that the proportion in the group with MP ≥ 17 J/min is significantly smaller than in the group with MP < 17 J/min.
In both cases BF01 (which measures the evidence in favor of the null hypothesis) was very low — indicating strong evidence in favor of the alternative hypothesis. The graphic representation and
sequential analysis are shown in Fig. 1.
The present study supports the significant finding previously published by our working group,3 with extreme evidence in favor of the alternative hypothesis: MP ≥ 17 J/min was seen to be associated
with 28-day mortality in patients requiring mechanical ventilation due to respiratory failure secondary to SARS-CoV-2 disease.
With the applicability and interpretability of our results in clinical practice in mind, the “transformation” of the “weight of evidence” (decimal logarithm of the Bayes factor) indicates that the
weight of evidence of MP ≥ 17 J/min being associated with 28-day mortality is 45 decibans (dB), while the negative evidence that this association exists is −17 decibans. Considering that a deciban is
the smallest change in the weight of evidence that can be directly perceived through human intuition, it can be affirmed that an MP threshold value of 17 J/min is an excellent 28-day mortality
indicator. We consider that the use of decibans in these studies may be very illustrative, since the logarithmic transformation of BF means that both weights of evidence are expressed on a single
measurement scale. A positive deciban could be used to confirm a disease or a prognostic marker, and a negative deciban could be used to discard it.11,12
However, in order to correctly interpret the data, it is important to take into account that in relation to the a posteriori distribution in the graphic representation, the data focus on values for
the estimated mean parameter theta of about 40% and 56%, respectively (with a 28-day mortality of 10% and 15% for the two analyzed groups). This suggests that the distribution of the parameter theta
is not symmetrical and shows tail weighting on the right side of the distribution. In other words, there is a greater probability that theta will have higher values than expected if the distribution
were symmetrical. This information is important, because it suggests that the estimated parameter theta may be biased towards higher values. In addition, the a posteriori distribution may also
indicate that the estimation of the 28-day mortality rate is more uncertain than would be expected if the sample were larger and the population more homogeneous. This is because the a posteriori
distribution reflects uncertainty in the estimation of theta from the limited information of the sample. On the other hand, it cannot be ruled out that although the data in the table constitute a
representative sample of the population, the values estimated a posteriori in the graphic representation reflect greater uncertainty in the estimation of 28-day mortality, due to a smaller sample or
greater variability in the population.13–15
In 2018, the researchers of the PROVE network,16 through a post hoc analysis of the high-resolution database Medical Information Mart for Intensive Care (MIMIC-III) and the eICU Collaborative
Research Database (eICU),17–19 reported the first clinical investigation on the hypothesis that MP generated by the ventilator is associated to patient-centered outcomes. The authors summarized their
findings under the following four points: (1) MP in the second 24-h period of ventilation is independently associated with increased in-hospital mortality among critical patients subjected to
invasive ventilation for over 48 h; (2) increased MP is independently associated to increased ICU mortality, fewer days without ventilation, an increased probability of survival at day 28, and a
longer hospital and ICU stay; (3) the impact of MP is consistent and independent of the presence of ARDS or the use of neuromuscular block; and (4) even with low tidal volume and driving pressure, a
high MP is associated with poorer patient outcomes — suggesting that MP contributes additional information beyond volume and pressure.
Our results are consistent in part with the conclusions of Neto et al.,13 and of later studies. In this sense, and based on frequentist inferential statistics, Schuijt et al. analyzed the data of the
PRoVENT-COVID study (a multicenter trial involving a cohort of 825 patients and with ventilation parameters corresponding to the first day of exposure to mechanical ventilation) and found MP to be
independently associated to 28-day mortality (hazard ratio [HR]: 1.17 (95% confidence interval [95%CI]: 1.01–1.36); p = 0.031). Likewise, quartile increments of MP, stratified according to driving
pressure values, were seen to increase this 28-day mortality risk (HR: 1.15 (95%CI: 1.01–1.30); p = 0.028).20 In this regard, Urmer et al.,21 with the data of over 13,000 patients and using Bayesian
models to estimate the strength of the association to ICU mortality, not only confirmed the correlation between MP and mortality (HR: 1.060, 95% credible interval: 1.053–1.066), but also found this
association to persist for the duration of mechanical ventilation.
With the data presented and based on our Bayesian analysis, we further consolidate our opinion that the findings were to be expected, considering that MP is a variable that includes all the
components that may cause ventilator-induced lung injury (VILI): pressure, volume, flow and respiratory rate. Nevertheless, MP must be normalized with respect to the ventilated lung surface,
considering the relationship between MP and the alveolar area exposed to the energy delivered as “intensity”.22
Our study has the limitations inherent to retrospective analytical cohort studies conducted in a single center. On the other hand, the variations of the study variables in relation to time were not
available; our results therefore might not be extrapolatable to those of other studies. Although our data evidence a significant association between 28-day mortality and MP in the first 24 h, it is
important to take into account that this variable alone does not reflect the complexity and the evolution of the disease in the long term. Furthermore, a change is observed in the Bayes factor (BF)
curves (the two subgroups show a notorious inflection point around patient #20) suggesting that the population of subjects included in the study is not homogeneous — a fact that may have important
implications when interpreting the results of the study. It is important to remember that regardless of the possible bias introduced by the loss of an important number of patients for analysis, the
study cohort is set within the context of two years of the COVID-19 pandemic and 6 waves of the disease in which there have been modifications that could justify the described observations: evolution
of the virus itself, the age and health conditions of the patients, vaccination, and knowledge of the different COVID-19 related ARDS (CARDS) phenotype, among others.
There is no single statistical method. With the frequentist approach it is only acceptable to assign probabilities to random phenomena that can be defined through experiments which can be repeated
many times and always under identical and independent conditions. Bayesian probability is applied to any random events — both those that can be repeated under the conditions demanded by frequentist
probability, and those that cannot. Bayesian methodology is a fundamental tool for sequentially updating the relevant information of a study. Accordingly, following a first analytical phase, the
knowledge generated will serve to start a new learning process incorporating new information about the problem.23
Although our data suggest that MP ≥ 17 J/min is associated with extreme evidence of 28-day mortality in patients requiring mechanical ventilation due to respiratory failure secondary to SARS-CoV-2
disease, it is difficult to extrapolate the influence of MP at a given moment to the entire trajectory of the disease condition over time — particularly considering that mechanical ventilation in
itself is a prolonged and complex process in which patients may experience significant changes in their clinical condition and prognosis. It is important to recognize the complexity of the disease
and that a variable such as MP in the first 24 h may be useful but not enough to fully explain the patient’s prognosis.
Conflict of interest
All the authors have collaborated in preparation of the manuscript and declare that they have no conflicts of interest in relation to the present study.
Author contributions
Alejandro González-Castro: Study conception, development and preparation of the manuscript.
Elena Cuenca-Fito: Study development and data compilation.
Yhivian Peñasco: Study development and revision.
Vicent Modesto: Revision of the statistical analysis.
Patricia Escudero-Acha: Revision.
Carmen Huertas: Screening of the database.
Juan Carlos Rodríguez-Borregán: Study development and revision.
L. Gattinoni, T. Tonetti, M. Cressoni, P. Cadringher, P. Herrmann, O. Moerer, et al.
Ventilator-related causes of lung injury: the mechanical power.
M.T.U. Schuijt, M.J. Schultz, F. Paulus, A. Serpa Neto, PRoVENT–COVID Collaborative Group.
Association of intensity of ventilation with 28-day mortality in COVID-19 patients with acute respiratory failure: insights from the PRoVENT-COVID study.
A. González-Castro, E. Cuenca Fito, A. Fernandez-Rodriguez, P. Escudero Acha, J.C. Rodríguez Borregán, Y. Peñasco.
Mechanical power greater than 17 joules/min in patients with respiratory failure secondary to SARS-CoV-2 infection.
Med Intensiva (Engl Ed), 47 (2023), pp. 115-117
C.A. Ramos-Vera.
The use of Bayes factor in clinical cardiology research.
Rev Esp Cardiol, 74 (2021), pp. 641-642
C. Ramos-Vera.
On using the Bayes factor in dermatology research.
Actas Dermosifiliogr, 113 (2022), pp. 999-1000
N.E. Hammond, F.G. Zampieri, G.L.D. Tanna, T. Garside, D. Adigbli, A.B. Cavalcanti, et al.
Balanced crystalloids versus saline in critically ill adults—a systematic review with metaanalysis.
A. Ly, A. Raj, A. Etz, Q.F. Gronau, E.J. Wagenmakers.
Bayesian reanalyses from summary statistics: a guide for academic consumers.
Adv Methods Pract Psychol Sci, 1 (2018), pp. 367-374
R. Kelter.
Bayesian alternatives to null hypothesis significance testing in biomedical research: a non-technical introduction to Bayesian inference with JASP.
BMC Med Res Methodol, 20 (2020), pp. 142
H. Jeffreys.
Theory of probability.
Oxford University Press, (1961),
L. Gattinoni, T. Tonetti, M. Cressoni, P. Cadringher, P. Herrmann, O. Moerer, et al.
Ventilator-related causes of lung injury: the mechanical power.
I. Modesto, V. Alapont, A. Medina-Villanueva.
Decibans: It is time to weigh the evidence about diagnostic accuracy.
Med Intensiva (Engl Ed), 46 (2022), pp. 173-174
C. Ramos-Vera.
Uso inclusivo de la conversión del tamano de efecto y del factor Bayes en la investigación de medicina intensiva.
Med Intensiva, 46 (2022), pp. 171-172
A. Gelman, J.B. Carlin, H.S. Stern, D.B. Dunson, A. Vehtari, D.B. Rubin.
Bayesian data analysis.
3rd ed., CRC Press, (2013),
R.E. Kass, A.E. Raftery.
Bayes factors.
J Am Stat Assoc, 90 (1995), pp. 773-795
J.K. Kruschke.
Doing Bayesian data analysis: a tutorial with R, JAGS, and Stan.
2nd ed., Academic Press, (2015),
A. Serpa Neto, R.O. Deliberato, A.E.W. Johnson, L.D. Bos, P. Amorim, S.M. Pereira, et al.
PROVE Network Investigators. Mechanical power of ventilation is associated with mortality in critically ill patients: an analysis of patients in two observational cohorts.
Intensive Care Med, 44 (2018), pp. 1914-1922
A.E. Johnson, T.J. Pollard, L. Shen, L.W. Lehman, M. Feng, M. Ghassemi, et al.
MIMIC-III, a freely accessible critical care database.
Sci Data, 3 (2016), pp. 160035
A.E. Johnson, D.J. Stone, L.A. Celi, T.J. Pollard.
The MIMIC Code Repository: enabling reproducibility in critical care research.
J Am Med Inform Assoc, 25 (2018), pp. 32-39
A.L. Goldberger, L.A. Amaral, L. Glass, J.M. Hausdorff, P.C. Ivanov, R.G. Mark, et al.
PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals.
Circulation, 101 (2000), pp. E215-E220
M.T.U. Schuijt, M.J. Schultz, F. Paulus, A. Serpa Neto, PRoVENT–COVID Collaborative Group.
Association of intensity of ventilation with 28-day mortality in COVID-19 patients with acute respiratory failure: insights from the PRoVENT-COVID study.
M. Urner, P. Jüni, B. Hansen, M.S. Wettstein, N.D. Ferguson, E. Fan.
Time-varying intensity of mechanical ventilation and mortality in patients with acute respiratory failure: a registry-based, prospective cohort study.
Lancet Respir Med, 8 (2020), pp. 905-913
P.L. Silva, L. Ball, P.R.M. Rocco, P. Pelosi.
Power to mechanical power to minimize ventilator-induced lung injury?.
Intensive Care Med Exp, 7 (2019), pp. 38
C. Armero, P. Rodriguez, J.M. de la Torre Hernandez.
A brief look into Bayesian statistics in cardiology data analysis.
REC Interv Cardiol, 4 (2022), pp. 207-215 | {"url":"https://medintensiva.org/en-the-bayes-factor-in-analysis-articulo-S2173572723000371","timestamp":"2024-11-11T01:53:23Z","content_type":"application/xhtml+xml","content_length":"176537","record_id":"<urn:uuid:fa42b3a5-7f7e-41fb-b5c7-94ce59888b64>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00095.warc.gz"} |
THERE IS NOW SCIENTIFIC PROOF THAT A FORM OF
PRIMARY CONSCIOUSNESS
IS PERVASIVE THROUGHOUT SPACE AND TIME
In the last post, I shared the finding that given that something exists, then application of the logic of the Calculus of Distinctions indicates that there is no such thing as nothing. In other
words, I contend that I’ve proved that Planck’s discovery that the substance of the universe is quantized means that if even one quantum of anything actually exists, a state of nothingness cannot
exist anywhere or any time in the quantized universe.
Now I can imagine a doubter saying: “You cannot prove a negative, and your conclusion that there is no state of nothingness is negative, so you can’t have proved it. There must be something wrong
with your logic, Dr. Close!” I think the common belief that you can’t prove a negative may come from the fact that negative statements are often very difficult to prove or disprove, and from
confusing the fact that many negative statements that seem reasonable, are often false, with the idea that you cannot prove a negative. Consider the following example: “We had been studying crows or
many years, and all of them were black. We had never seen a white crow, so we concluded that there are no white crows. Today a white crow showed up and the negative statement we thought to be true is
obviously false.”
So is it a fact that you can’t prove a negative? No, for two reasons: 1. because if the statement that you can’t prove a negative is true, then it is self-contradictory because it is itself a
negative statement, and therefore can’t be proved. Furthermore, it can be proved that the statement that you cannot prove a negative is false as follows: Like in the example of the crows, if I can
produce even one proof of a negative statement, the belief that you can’t prove a negative is proved wrong, and I can produce several. Any mathematician, if he’s worth his salt, has produced several.
For example, I have proved, as have other mathematicians, the negative statement that there are no integer solutions for the equation X^3 + Y^3 = Z^3. It can’t be proved by trial and error of course,
because there are an infinite number of triplet integers X,Y,Z that could be tried.
Now many ordinary folks who are true believers and even some professional theologians have used the false argument that you can’t prove a negative to combat the atheistic claim that there is no God.
True believers do not like the fact that the argument that a negative can’t be proved is false. But just because you can prove that there are negatives that can be proved, doesn’t mean that a given
negative like “There is no God” can be proved. So, while you lose the argument that you can’t prove a negative, the common argument that a number of atheists have posted on the internet and written
in popular books, that there is no God because there is no scientific evidence that there is a God” Is also very faulty logic. Just because self-proclaimed atheists haven’t seen a scientific proof of
the existence of God doesn’t mean there isn’t one. It is the height of arrogance to believe that just because you can’t prove that God exists, no one can. Not only that, we now have TRUE unit
analysis which proves what no atheist wants to hear:
There would definitely be no physical universe without the involvement of some primary ubiquitous form of consciousness from the beginning of reality as we know it, wherever and whenever that
beginning might be.
See the details in the references placed in the last post and other previous posts.
3 comments:
1. When it comes down to it, Ed, who really has to prove or disprove anything to others. In a cosmicated sense, it's what we believe in ourselves that really matters to the Ultimate Force, our
Higher Self, which cannot be fooled - We're all our own judge, jury and hopefully not our own executioners! Amun!
2. Gujarat Secondary and Higher Secondary Education Board (GSEB) is the branch of the Gujarat government that is responsible for determining the policy-related, administrative, and intellectual
direction of the state's secondary and higher secondary educational system.GSEB SSC/10th Board was established under the 'Gujarat Secondary Education Act' in 1973. GSEB 10th Question Paper 2021
The primary objective of the board is to prepare the syllabus and to conduct the exams for secondary schools in the state.with more papers.n Asked From Each Unit of the Subject.
3. The Salary credited to each of the employees who are under CISF can get it through online mode now, As CISF personnel are distributed around different corners of India and it is getting hard to
reach the office to collect their Pay Slip, so the new CISF official website does provide them with a lot of facilities including downloaded Pay Slip. CISF employee corner The receipt of monthly
salary for whole working days of the month will be calculated including all other expenses and extra allowance in the payment, and thus, one can easily have a look at their CISF Salary Payroll
details available with all the earnings for the month. | {"url":"http://www.erclosetphysics.com/2016/07/proof-of-primary-consciousness.html","timestamp":"2024-11-10T08:36:25Z","content_type":"text/html","content_length":"219943","record_id":"<urn:uuid:9a11cb14-1cd6-4c1c-bfd4-8314284b16c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00682.warc.gz"} |
Prospects in Mathematical Physicssearch
Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
Please make all selections above before adding to cart
Prospects in Mathematical Physics
Softcover ISBN: 978-0-8218-4270-6
Product Code: CONM/437
List Price: $130.00
MAA Member Price: $117.00
AMS Member Price: $104.00
eBook ISBN: 978-0-8218-8116-3
Product Code: CONM/437.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Softcover ISBN: 978-0-8218-4270-6
eBook: ISBN: 978-0-8218-8116-3
Product Code: CONM/437.B
List Price: $255.00 $192.50
MAA Member Price: $229.50 $173.25
AMS Member Price: $204.00 $154.00
Click above image for expanded view
Prospects in Mathematical Physics
Softcover ISBN: 978-0-8218-4270-6
Product Code: CONM/437
List Price: $130.00
MAA Member Price: $117.00
AMS Member Price: $104.00
eBook ISBN: 978-0-8218-8116-3
Product Code: CONM/437.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Softcover ISBN: 978-0-8218-4270-6
eBook ISBN: 978-0-8218-8116-3
Product Code: CONM/437.B
List Price: $255.00 $192.50
MAA Member Price: $229.50 $173.25
AMS Member Price: $204.00 $154.00
• Contemporary Mathematics
Volume: 437; 2007; 246 pp
MSC: Primary 35; 37; 47; 60; 76; 81; 82; 83
This book includes papers presented at the Young Researchers Symposium of the 14th International Congress on Mathematical Physics, held in July 2003, in Lisbon, Portugal. The goal of the book is
to illustrate various promising areas of mathematical physics in a way accessible to researchers at the beginning of their careers. Two of the three laureates of the Henri Poincaré Prizes,
Huzihiro Araki and Elliott Lieb, also contributed to this volume. The book provides a good survey of some active areas of research in modern mathematical physics.
Graduate students and research mathematicians interested in various topics in mathematical physics.
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Requests
Volume: 437; 2007; 246 pp
MSC: Primary 35; 37; 47; 60; 76; 81; 82; 83
This book includes papers presented at the Young Researchers Symposium of the 14th International Congress on Mathematical Physics, held in July 2003, in Lisbon, Portugal. The goal of the book is to
illustrate various promising areas of mathematical physics in a way accessible to researchers at the beginning of their careers. Two of the three laureates of the Henri Poincaré Prizes, Huzihiro
Araki and Elliott Lieb, also contributed to this volume. The book provides a good survey of some active areas of research in modern mathematical physics.
Graduate students and research mathematicians interested in various topics in mathematical physics.
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/CONM/437","timestamp":"2024-11-06T12:52:08Z","content_type":"text/html","content_length":"100969","record_id":"<urn:uuid:a39b50f8-a7c6-444c-86d5-6b36c66267d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00056.warc.gz"} |
W4A1 An intuitive approach!
Building your Deep Neural Network: Step by Step an Intuitive approach
Goal of the Code: The goal is to set up a neural network that will take the input features and make predictions on the best movies to recommend. The process involves setting up the initial settings
(parameters) for the recommendation system, running a series of transformations (forward propagation), evaluating the outcome (computing cost), and refining the settings (backward propagation and
parameter updates).
1. Setting Up the Recommendation System (initialize_parameters) Imagine we’re setting up our recommendation system with the right parameters. The goal is to initialize the weights and biases
(parameters) for the neural network. For example, we decide on the initial importance of each feature like genre preference, director preference, actor preference, average rating of previously
watched movies, and movie duration preference, and set them to small random values. We also prepare the biases, which start as zeros.
2. Forward Propagation (L_model_forward) This step is like processing user preferences step by step to get a movie recommendation. The goal is to process the input data through each layer of the
network, applying transformations (weights and biases) to produce an output. We start with the initial parameters and pass them through a series of steps (layers) where each step refines the
input (user preferences) until we get a prediction on which movie to recommend.
3. Computing Cost (compute_cost) This step is like checking how good the movie recommendation is. The goal is to measure how good or bad the prediction is compared to the actual result. We compare
the predicted movie rating with the actual rating given by the user and see if they match. The cost function gives us a score indicating how close our prediction is to the actual rating.
4. Backward Propagation (L_model_backward) This is like adjusting the recommendation parameters based on feedback. The goal is to figure out how to change the weights and biases to improve the
predictions. For example, if a certain genre was over-recommended, we reduce its weight for the next batch of recommendations. This involves calculating gradients, which tell us how to tweak each
5. Updating Parameters (update_parameters) This step involves actually making the changes to the recommendation parameters for the next round. The goal is to update the weights and biases based on
the gradients calculated in the backward propagation step. We adjust the importance of each feature and the biases according to the feedback we received.
Putting It All Together:
Think of the neural network as our smart recommendation system that learns and improves its movie recommendation process over time. Each part of the code helps refine its parameters to produce better
recommendations. Here’s a simplified flow:
1. Initialization: Start with small random values for weights and zero for biases.
2. Forward Propagation: Process user preferences through the current parameters to predict movie ratings.
3. Compute Cost: Check the predicted ratings against actual user ratings and score the accuracy.
4. Backward Propagation: Determine how to adjust the parameters based on the cost.
5. Update Parameters: Make the necessary adjustments to the parameters for the next round.
By repeatedly going through these steps, the recommendation system becomes smarter and more efficient at predicting and recommending the best movies.
Part 1: Initializing Parameters
Intuitive Explanation: Imagine setting up the recommendation system and you need to prepare your initial parameters before starting to make recommendations. In this context:
• Weights (W1, W2): These are like the importance values you start with for each feature. You don’t want them to be too large initially, so you use small random values to begin with.
• Biases (b1, b2): These are like your adjustments that start at zero.
By setting up these initial parameters, you are ready to start the recommendation process, which involves forward propagation (processing preferences), computing cost (checking accuracy), backward
propagation (adjusting parameters), and updating parameters until you get the best recommendations.
Example: Let’s say we have:
• 5 input features (e.g., genre preference, director preference, actor preference, average rating of previously watched movies, movie duration preference)
• 2 neurons in the hidden layer
• 1 output (predicting the movie rating)
When we call initialize_parameters(5, 2, 1), it would create:
• W1 with shape (2, 5)
• b1 with shape (2, 1)
• W2 with shape (1, 2)
• b2 with shape (1, 1)
These parameters are the starting point for our neural network.
Part 2: Forward Propagation
Intuitive Explanation: Imagine you are processing user preferences step-by-step according to a system:
1. Linear Forward (Mixing Preferences): You mix the preferences (input features) with specific weights and add biases. For each step, you combine genre preference, director preference, actor
preference, average rating of previously watched movies, and movie duration preference in specific proportions (weights) and add some adjustments (biases).
2. Linear Activation Forward (Processing Reaction): After mixing, the preferences go through a reaction (activation function). If you’re processing the preferences (ReLU activation), the system
decides which features are more significant. If you’re making a final decision on the movie rating (linear activation), it gives a score.
3. Model Forward (Complete Process): Follow the entire process from start to finish, with each step transforming the preferences closer to the final rating. You start with the preferences (input
layer), process through the hidden layers, and finally decide the movie rating (output layer).
Detailed Example:
1. Linear Forward:
• Input A (previous activations or input data).
• Weight W and bias b for the current layer.
• Compute Z = W * A + b.
2. Linear Activation Forward:
• Compute Z using linear_forward.
• Apply activation function (ReLU or linear) to Z to get A.
3. Model Forward:
• Initialize with input features.
• Iterate through layers:
□ For each hidden layer, use ReLU activation.
□ For the final layer, use a linear activation.
• Store activations and intermediates in caches for backpropagation.
By combining these steps, the neural network processes the input features through each layer, transforming them and making predictions at the output layer.
Part 3: Computing Cost
Intuitive Explanation: Imagine you are recommending different movies and trying to predict the rating a user would give based on some input features (like genre preference, director preference,
etc.). You make some predictions and then compare them with the actual ratings given by the user.
1. Predictions (AL): These are like your guesses on which movie rating would be given. For instance, you might predict a 4.5 out of 5 for a particular movie.
2. Actual Outcomes (Y): These are the actual ratings the user gave. For example, if the user actually rated the movie 5 out of 5, the label would be 5.
3. Cost Function (Rating Comparison): The cost function is like a comparison score that tells you how well your predictions matched the actual ratings. If your guess was 4.5 and the user rated it 5,
your score would be close. If your guess was 2 and the user rated it 5, your score would be poor. The cost function aggregates these scores across all your predictions to give you an overall
sense of how well you’re doing.
Detailed Example: Suppose you have 3 examples (predictions):
• Predictions (AL): [4.5, 3.0, 4.8]
• Actual outcomes (Y): [5, 3, 5]
Using the mean squared error (MSE) cost formula:
• For the first example: (4.5 - 5)²
• For the second example: (3.0 - 3)²
• For the third example: (4.8 - 5)²
The total cost is the average of these values: cost=13((4.5−5)2+(3.0−3)2+(4.8−5)2)\text{cost} = \frac{1}{3} \left( (4.5 - 5)² + (3.0 - 3)² + (4.8 - 5)² \right)cost=31((4.5−5)2+(3.0−3)2+(4.8−5)2)
This cost value quantifies how good or bad your predictions are. Lower cost values indicate better predictions.
Part 4: Backward Propagation
Intuitive Explanation: Imagine you’re in the recommendation system, and you need to adjust your parameters based on user feedback to improve the recommendation quality.
1. Linear Backward (Adjusting Weights and Biases): You determine how much each weight and bias contributed to the prediction error. For example, if a certain genre was over-recommended, you
calculate how much of the genre weight and other features need to be adjusted.
2. Linear Activation Backward (Adjusting Activations): After adjusting weights and biases, you evaluate how the reactions (activations) in each layer of the network need to change. If a certain
feature’s importance wasn’t calculated well (ReLU), you figure out how the combination of features (linear) needs to be changed to improve it.
3. Model Backward (Full Process Adjustment): You go through the entire process, starting from the final rating (output layer) back to the initial preferences (input layer), adjusting each step. For
the final rating (linear), you see how the prediction can be improved. For each step before (hidden layers), you adjust based on how they contributed to the final rating.
Detailed Example:
1. Linear Backward:
• Given the gradient of the cost with respect to Z (dZ), calculate:
□ dW=1m⋅np.dot(dZ,AprevT)dW = \frac{1}{m} \cdot \text{np.dot}(dZ, A_{\text{prev}}^T)dW=m1⋅np.dot(dZ,AprevT)
□ db=1m⋅np.sum(dZ,axis=1,keepdims=True)db = \frac{1}{m} \cdot \text{np.sum}(dZ, \text{axis}=1, \text{keepdims=True})db=m1⋅np.sum(dZ,axis=1,keepdims=True)
□ dAprev=np.dot(WT,dZ)dA_{\text{prev}} = \text{np.dot}(W^T, dZ)dAprev=np.dot(WT,dZ)
2. Linear Activation Backward:
• Given the gradient of the cost with respect to the activation (dA):
□ Calculate dZ using relu_backward or linear_backward.
□ Use linear_backward to get dW, db, and dA_prev.
3. Model Backward:
• Initialize the gradient of the cost with respect to the output layer’s activation (dAL).
• For the output layer:
□ Calculate gradients using linear activation.
• For each hidden layer (from L-1 to 1):
□ Calculate gradients using ReLU activation.
• Store all gradients in the grads dictionary.
By performing these steps, the neural network learns how to adjust its parameters to improve predictions and reduce the cost.
Part 5: Updating Parameters
Intuitive Explanation: Imagine you’re in the recommendation system, and after adjusting the parameters based on user feedback (backward propagation), you now need to make the actual changes to the
weights and biases for the next round of recommendations.
1. Parameters (Weights and Biases): These are like the quantities of importance for each feature. For example, you have a certain weight for genre preference and a bias for the overall
recommendation system.
2. Gradients (Feedback Adjustments): These are like the feedback adjustments you calculated to improve the recommendation quality. For example, you figured out you need a little less importance on
genre and more on director preference.
3. Learning Rate (Adjustment Intensity): This is like how strongly you apply the feedback adjustments. If your learning rate is high, you make big changes to the weights and biases. If it’s low, you
make smaller, more gradual adjustments.
4. Updating Parameters (Parameter Changes): You use the feedback to update your parameters for the next round of recommendations. You subtract a bit of importance from the genre weight and add a bit
to the director preference based on the learning rate and the feedback.
Detailed Example: Let’s say we have the following parameters and gradients for a single layer:
• W = 0.5
• b = 0.1
• dW = -0.2
• db = 0.05
• Learning rate (α) = 0.01
The update rules are:
• W := W - α * dW
• b := b - α * db
Applying the update:
• W := 0.5 - 0.01 * (-0.2) = 0.5 + 0.002 = 0.502
• b := 0.1 - 0.01 * 0.05 = 0.1 - 0.0005 = 0.0995
The parameters are updated slightly, and this process is repeated for each training iteration to gradually improve the network’s performance.
Full Process Overview:
1. Initialize Parameters: Set up the initial weights and biases.
2. Forward Propagation: Calculate predictions based on the current parameters.
3. Compute Cost: Evaluate the accuracy of predictions.
4. Backward Propagation: Calculate gradients to adjust the parameters.
5. Update Parameters: Apply gradients to update the parameters.
By iterating through these steps, the neural network learns to make better predictions and reduces the cost.
2 Likes
Thanks for the detailed explanation. | {"url":"https://community.deeplearning.ai/t/w4a1-an-intuitive-approach/671922","timestamp":"2024-11-03T16:23:43Z","content_type":"text/html","content_length":"37082","record_id":"<urn:uuid:26b8f1f7-6bc2-420f-a8a9-ec5ecb28e772>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00307.warc.gz"} |
What is this course about?
Complex data can be represented as a graph of relationships between objects. Such networks are a fundamental tool for modeling social, technological, and biological systems. This course focuses on
the computational, algorithmic, and modeling challenges specific to the analysis of massive graphs. By means of studying the underlying graph structure and its features, students are introduced to
machine learning techniques and data mining tools apt to reveal insights on a variety of networks.
Topics include: representation learning and Graph Neural Networks; algorithms for the World Wide Web; reasoning over Knowledge Graphs; influence maximization; disease outbreak detection, social
network analysis.
Previous Offerings
You can access slides and project reports of previous versions of the course on our archived websites: CS224W: Fall 2021 / CS224W: Winter 2021 / CS224W: Fall 2019 / CS224W: Fall 2018 / CS224W: Fall
2017 / CS224W: Fall 2016 / CS224W: Fall 2015 / CS224W: Fall 2014 / CS224W: Fall 2013 / CS224W: Fall 2012 / CS224W: Fall 2011 / CS224W: Fall 2010
Students are expected to have the following background:
• Knowledge of basic computer science principles, sufficient to write a reasonably non-trivial computer program (e.g., CS107 or CS145 or equivalent are recommended)
• Familiarity with the basic probability theory (CS109 or Stat116 are sufficient but not necessary)
• Familiarity with the basic linear algebra (any one of Math 51, Math 103, Math 113, or CS 205 would be much more than necessary)
The recitation sessions in the first weeks of the class will give an overview of the expected background.
Course Materials
Notes and reading assignments will be posted periodically on the course Web site. The following books are recommended as optional reading: | {"url":"http://snap.stanford.edu/class/cs224w-2022/","timestamp":"2024-11-08T16:05:11Z","content_type":"text/html","content_length":"29451","record_id":"<urn:uuid:43356cc2-2e86-46c4-af60-86f4bb9e3615>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00009.warc.gz"} |
Category:BASIC256 - Rosetta CodeCategory:BASIC256
programming language
may be used to instruct a computer to perform a task.
Listed below are all of the tasks on Rosetta Code which have been solved using BASIC256.
BASIC-256 is an easy to use version of BASIC designed to teach anybody (especially middle and high-school students) the basics of computer programming. It uses traditional control structures like
gosub, for/next, and goto, which helps kids easily see how program flow-control works. It has a built-in graphics mode which lets them draw pictures on screen in minutes, and a set of detailed,
easy-to-follow tutorials that introduce programming concepts through fun exercises.
• fast graphics suitable for animation
• native speech synthesis
• mouse and keyboard events
• graphic files manipulation (saving, retrieving)
• playing sounds from WAV files
• sprites handling
• database functions
• socket networking
• error handling
• files I/O
• true functions/subroutines (version 0.9.9.1 and later)
• BASIC-256 does not support three- and N-dimensional arrays in general (N>2)
BASIC256 is open source and available for Linux, Windows and Mac.
For more information see basic256.org or to download and install sourceforge.
This category has only the following subcategory.
Pages in category "BASIC256"
The following 200 pages are in this category, out of 497 total.
previous page
) (
next page
previous page
) (
next page | {"url":"https://rosettacode.org/wiki/Category:BASIC256?pageuntil=Jewels+and+stones","timestamp":"2024-11-10T15:15:58Z","content_type":"text/html","content_length":"67722","record_id":"<urn:uuid:0d464d6f-013a-4d38-a0e8-bbf3934ea1b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00248.warc.gz"} |
Why Numbers are important in our life?
Numeracy in everyday life Our confidence and ability with numbers impacts us financially, socially, and professionally. It even affects our health and wellbeing. Some examples of the ways we use
maths every day include: Working out how many minutes until our train.
How is maths used in daily life?
People use math knowledge when cooking. For example, it is very common to use a half or double of a recipe. In this case, people use proportions and ratios to make correct calculations for each
ingredient. If a recipe calls for 2/3 of a cup of flour, the cook has to calculate how much is half or double of 2/3 of a cup.
What numbers are most important?
The 10 Most Important Numbers In The World
• Archimedes’ Constant (Pi): 3.1415…
• Euler’s Number (e): 2.7182…
• The Golden Ratio: 1.6180…
• Planck’s Constant: 6.626068 x 10^-34 m^2 kg/s.
• Avogadro’s Constant: 6.0221515 x 10^23.
• The Speed of Light: 186,282 miles per second.
Why is math my favorite subject?
Math is my favorite subject because it is easy to do solve the problem. I like using numbers, fractions, decimals and so much more. Math is also used in the real world like telling time, adding,
dividing, percentage and even in a future career like engineering, architect, math teacher and lot more.
How do numbers help us?
We use numbers in time,date, year and weather. We use numbers in school and work, counting money, measurements, phone numbers, password on our phone , locks, reading, page numbers, and TV channels.
Engineers use number for their calculation to construct building and roads. Doctors use it for blood counts and medicines.
What do you love about math?
17 Reasons to Love Math
• Math can save you money.
• Like patterns?
• Math builds problem-solving skills.
• Math encourages curiosity and questions.
• Math helps out in the kitchen.
• Math is black and white.
• You can take lots of approaches to math.
• Math is play for your brain.
What is famous number?
e, known as Euler’s number, is approximately 2.71828 and is another nonrepeating, nonterminating number. This number is the natural limit for many science and economics applications. 10100 is a
Googol. You can write out the entire number with a 1 followed by 100 zeros.
What is the use of sets in real life?
Now coming back to real life examples of set, we have seen that in kitchen, Utensils are arranged in such a manner that plates are kept separately from the spoons. Another example is when we visit
mobile showrooms; we observe that smart phones like Galaxy duos, Lumia etc. are separated from the simple mobiles. | {"url":"https://www.ufoscience.org/why-numbers-are-important-in-our-life/","timestamp":"2024-11-11T20:26:54Z","content_type":"text/html","content_length":"48986","record_id":"<urn:uuid:336dbf43-5e74-4e88-85df-e35561f7d9a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00514.warc.gz"} |
Lecture 012
Prime Number: A natural number n is called prime iff n > 1 and its only positive divisors are 1 and n.
The Fundamental Theorem of Arithmetic
Claim: $(\forall n \in \mathbb{N})$ with n > 1 n is either prime or a product of primes.
Proof: We proceed by strong induction on $n\in \mathbb{Z} \land n \geq 2$. Define P(n) to be the variable proposition "n either prime or a product of primes"
• Base Case: when n=2, the only possible divisor is 1 and 2. n is prime
• Inductive Step: Let $k \in \mathbb{N}$ with $k \geq 2$, assume P(i) holds for all $i\in\mathbb{N}$ with $2 \leq i \leq k$. let $n=k+1$
□ Case if $k+1$ is prime: $P(k+1)$ holds
□ Case if $k+1$ is not prime: then let $p, q \in \mathbb{N}$, with $2\leq p \leq q \leq k$, and $k+1=pq$. Fix p and q. By IH, p and q are either prime or a product of primes. Then $(\exists a,
b \in \mathbb{N})(p=ab)$ So we are good.
□ (break k+1 to k+1=a*b, then assume a=product_of_prime_or_prime and b=product_of_prime_or_prime, k+1 must equal to product of prime)
The Game of Chomp
Claim: If Player I begins the game by choosing square (1, n) the top right corner, then Player 1 can always win the 2 x n game of Chomp. Proof: by strong induction. Let P(n) be the proposition.
This is a strong induction because it require 2 x 1 is true to 2 x k.
Well Ordering Property
Definition: A set X is called well-ordered iff every nonempty subset has a least element.
• Example: $\mathbb{N}$, $2^n|n\in \mathbb{N}$
• Non-Example: $\mathbb{Z}$, $[0, 1)$, $\mathbb{R}$
Claim: $\mathbb{N}$ is well-ordered Proof: by SPMI for all $n\in\mathbb{N}$. Let P(n) be variable proposition "every subsets of N containing n has a least element" (read: assume the subset containing
n has the property, prove subset containing all number has property) | {"url":"https://kokecacao.me/page/Course/F20/21-127/Lecture_012.md","timestamp":"2024-11-12T15:45:50Z","content_type":"text/html","content_length":"11870","record_id":"<urn:uuid:1d63d3c1-60a2-4aee-acf0-801909e5baa3>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00439.warc.gz"} |
MATH141 Calculus II Fall 2008
Thanks very much for all students in my section. Good luck to you all!
Course Information
Instructors Dr. Rosenberg / Changhui Tan
Lectures Monday, Wednesday, Friday. 1pm~1:50pm, room 0131 of the Reckord Armory.
Discussions Tuesday, Thursday. 2pm~2:50pm, room B0427 of the Math Building.
Professor's Office Hour Monday, Wednesday, 11am~12noon, room 2114 Math Building.
TA's Office Hour Monday 2:15pm~3:15pm(no appointment is needed), Friday 10:30am~11:30am(by appointment via email), room 4326 of the CSS Building.
Tuesday and Thursday 3pm-3:30pm right after the discussion section. Coffee might be offered.
I also have a tutoring hour on Wednesday 4pm~5pm. The tutoring room is in Math Building MTH 0301.
NOTICE Please read the contents carefully on the following pages.
Latest Announcement
Dec. 17 The final grades have been updated to the online grade sheet.
Dec. 12 If you have any questions before the final exam, feel free to email me or call me in the weekend. I will try my best to help you.
Dec. 2 The professor has posted a makeup exam. This is due on Monday in class. The higher grade of the exam and the makeup exam will be counted as the grade of exam 4.
Nov. 26 Happy Thanksgiving holiday to you all.
Nov. 21 The review session will be held on Tuesday (November 25) 3pm-5pm at MATH0302.
Nov. 18 Quiz 10 has been posted below. It is due on Thursday.
Quiz Information
We have 13 quizzes in this semester. Each quiz is worth 10 points. Only the highest 10 points will add into your final grades of quizzes. Quiz schedules are HERE.
All quizzes can be downloaded after you have done them. Answers are not included. You should come to the next discussion section after the quiz to get the answer.
Exam Information
Final exam The final exam will be held on December 15, from 1:30 to 3:30, at EGR 1104 (EGR - Martin Hall). You need to be there 15 minutes earlier.
Review session SUNDAY DECEMBER 14TH 11:00 AM - 1:00 PM 0200 SKINNER BLDG
Former test bank Click HERE (I do not recommend you to do to much of them. Dr. Rosenberg's 2007 Fall exams might be most helpful.)
Exam result Exam 4 Overall [Avg]Around 42(Makeup exam is given, no official result) Our section [Avg]49 [Med]48
Exam 3 Overall [Avg]67 [Med]71 [75%]82 [25%]52 Our section [Avg]73 [Med]77
Exam 2 Overall [Avg]63 [Med]62 [75%]77 [25%]49 Our section [Avg]69 [Med]69
Exam 1 Overall [Avg]72 [Med]73 [75%]86 [25%]80 Our section [Avg]71 [Med]70
Online Grades
If you have given me written permission, you can check you grades online. [Click Here]
Final grades: >680 A, >600 B, >490 C, >415 D. I will make some slight adjustment based on your performance in the course, and your score of the in-class 4th exam if you've got more than 75.
Attendance Information
You are recommended to attend every discussion section.
If you think the discussion is too easy, and you handle the homework, quiz and exam very well, you can choose not to take the discussion. But for days when quizzes are held, you should appear. I will
not give you a makeup quiz unless you've got a reason for the absent.
Contact Information
Contact me at This email address is being protected from spambots. You need JavaScript enabled to view it. or This email address is being protected from spambots. You need JavaScript enabled to view
it.(preferred) if you have any questions. | {"url":"https://changhuitan.com/teaching/umd/141f08","timestamp":"2024-11-04T00:33:07Z","content_type":"application/xhtml+xml","content_length":"73540","record_id":"<urn:uuid:f3b00584-8fd5-414a-a839-29eef70083fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00719.warc.gz"} |
the pushead thread
all things pushead and pushead related.
props to the grandmarf
post 'em. old punk fliers, skate decks, shoes, shirts, etc etc...
pushead fucking rocks.
This forum is supported by the 12ozProphet Shop, so go buy a shirt and help support!
This forum is brought to you by the 12ozProphet Shop.
This forum is brought to you by the 12oz Shop.
How about the AIM thread, how Casek is never on it...
Especially when I need computer advice.
nice casek!
Pushead (a.k.a. Brian Schroeder) is a hardcore punk artist most famous for his artwork for Metallica and The Misfits. He also created the record labels Pusmort and Bacteria Sour, and fronted the punk
band Septic Death. Septic Death's covers are also his artwork. His artwork is characterized by incredibly detailed skulls.
He also designed skateboard graphics and ads for Zorlac Skateboards during the 80's and beginning of the 90's. Pushead was also an artist for Thrasher Skateboard Magazine during the 80's and early
90's. His recent work includes a limited edition shoe released world wide by Nike on Dec.17 2005
His recent work includes a limited edition shoe released world wide by Nike on Dec.17 2005
bump for illness.
head in a jar
had this shirt. i thought it was the fucking shiz when i was little. i think i still do
how many sneakers were released?
anybody got the septic death discography to upload?
that band made some of the most ridiculously hard ass music.
I got into his art last year shit is sooo insanely dope... I want to buy originals to keep in my home, they are going to be worth sooo much when he dies...
those nikes look coked out to me... but yeah bump pushead didn't he do some suicidal tendencies album covers
those nikes look coked out to me... but yeah bump pushead didn't he do some suicidal tendencies album covers
maybe. "suicidal army". too lazy too look, though.
let's see if we can interview brian....
gallery of his decks
...great thread, I've always loved his shit...brings back so many memories of the good ol days when skateboarding wasn't 'corporate' and you got bottles thrown at you for it...those shoes sold out
very quickly, of course you know all the Japs bought em...
Im so glad you made this thread. I cant believe I didnt know about his work.
Im so glad you made this thread. I cant believe I didnt know about his work.
i can. honestly. alot of the younger generation missed out on some pretty great stuff.
time to catch up, MAR.
this was on a popular deck.
buy it now $750.00...I wish I could...
i can. honestly. alot of the younger generation missed out on some pretty great stuff.
time to catch up, MAR.
I familiar with the album covers i just never put one and one together.
I familiar with the album covers i just never put one and one together.
yeah, he's been around for awhile. i'm sure everyone has seen his work at one time or another. at one time he was dominatng the markets (metal and skateboarding). you couldn't pick up a big brother
or thrasher and not see something done by pushead. you couldn't go to a skateshop and not see his work...and then we have metallica...
Yeha, I remember all the old decks in the 80's. Shit was crazy@!
I think I even had this cheapo skateboard when I was a kid with one of his designs on it, but I never clicked it was the same artist.
This topic is now archived and is closed to further replies. | {"url":"https://forum.12ozprophet.com/topic/63538-the-pushead-thread/","timestamp":"2024-11-12T00:22:35Z","content_type":"text/html","content_length":"371572","record_id":"<urn:uuid:aed511dd-574c-4df7-bfdd-6d48ef4b2c72>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00157.warc.gz"} |
Quantum number and energy scaling of rotationally inelastic scattering cross sections
A general formula for rotationally inelastic cross sections in atom-diatomic collisions is derived. This result is achieved by assuming the transition probability is a function of rotational quantum
number differences, the kinetic energy in the upper states, and is inversely proportional to the number of accessible states within an effective hamiltonian formalism. The scaling law is able to
predict all rows, or columns, of the inelastic cross section matrix, σ[jj], given any one row, or column, as a function of energy. Finally, we applied this scaling theory to a variety of collision
systems; in general, good agreememt between predicted and exact results is exhibited. We have developed and tested a scaling law for rotationally inelastic transitions which predicts all rows or
columns of the inelastic cross section matrix, σ[jj], given any one row or column. The agreement between exact and predicted cross sections was good for the HeHF [20], HeHCl [21], H[2]CS [17], HeCO
[16] and Li^+H[2] [18] systems which vary in reduced mass from approximately 2-4 an and rotor constant from 0.82-60.0 cm^-1. However, for ArN[2] [13,14, 19,24] collisions (μ = 16.5 au, B[e] = 2.01 cm
^-1) the error increased with increasing j and/or Δ. In every case the disagreement was in the opposite direction from the prediction of eqs. (15a)-(15c) which are based upon fully degenerate
statistics instead of the effective hamiltonian non-degenerate statistics of eqs. (2a) and (2b). Finally, we state two general conclusions: 1. (1) Whenever the initizil kinetic energy (E-ε{lunate}
[j]) is greater than approximately five times the energy gap (ε{lunate}[j+Δ]-ε{lunate}[j]), the scaling theoretic cross sections are accurate within 20%; and 2. (2) at total energies much larger than
that of the internal states of interest, all cross sections with the same change in rotational quantum number become approxinutely equal.
All Science Journal Classification (ASJC) codes
• General Physics and Astronomy
• Physical and Theoretical Chemistry
Dive into the research topics of 'Quantum number and energy scaling of rotationally inelastic scattering cross sections'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/quantum-number-and-energy-scaling-of-rotationally-inelastic-scatt","timestamp":"2024-11-13T18:36:37Z","content_type":"text/html","content_length":"54445","record_id":"<urn:uuid:fdf87cb6-b675-4449-8f12-be2dffd2264a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00825.warc.gz"} |
Kaiser Window
Jim Kaiser discovered a simple approximation to the DPSS window based upon Bessel functions [115], generally known as the Kaiser window (or Kaiser-Bessel window).
Window transform:
The Fourier transform of the Kaiser window ^4.11
• Reduces to rectangular window for
• Asymptotic roll-off is 6 dB/octave
• First null in window transform is at
• Time-bandwidth product
• Full time-bandwidth product main-lobe width out to first null
• Sometimes the Kaiser window is parametrized by
The side-lobe level and main-lobe width. Larger 5.4.1, widening the main lobe reduces frequency resolution when the window is used for spectrum analysis. As explored in Chapter 9, reducing the side
lobes reduces ``channel cross talk'' in an FFT-based filter-bank implementation.
The Kaiser beta parameter can be interpreted as 1/4 of the ``time-bandwidth product'' ^4.13 Sometimes the Kaiser window is parametrized by
Figure 3.24 plots the Kaiser window and its transform for side-lobes to fall away from the main lobe. The curvature at the main lobe peak also decreases somewhat.
Figure 3.25 shows a plot of the Kaiser window for various values of
Figure 3.25: The Kaiser window for various values of the time-bandwidth parameter
Figure 3.26 shows a plot of the Kaiser window transforms for dB magnitude of the aliased sinc function. As dB down for
Figure 3.27 shows the effect of increasing window length for the Kaiser window. The window lengths are
Figure 3.28 shows a plot of the Kaiser window side-lobe level for various values of dB below the main lobe (upper-lefthand corner of Fig.3.28). As sinc-main-lobe).
Minimum Frequency Separation vs. Window Length
The requirements on window length for resolving closely tuned sinusoids was discussed in §5.5.2. This section considers this issue for the Kaiser window. Table 3.1 lists the sinusoids with a
frequency spacing of 1, Table 8-9]. Recall from §3.9 that bandwidth of the window (in cycles).
Table: Kaiser parameter for various frequency resolutions, assuming an FFT zero-padding factor of at least 3.5.
Figure 3.29 shows an overlay of DPSS and Kaiser windows for some different values. In all cases, the window length was . Note how the two windows become more similar as increases. The Matlab for
computing the windows is as follows:
w1 = dpss(M,alpha,1); % discrete prolate spheroidal seq.
w2 = kaiser(M,alpha*pi); % corresponding kaiser window
The following Matlab comparison of the DPSS and Kaiser windows illustrates the interpretation of as the bin number of the edge of the critically sampled window main lobe, i.e., when the DFT length
equals the window length:
format long;
abs(fft([ dpss(M,alpha,1), kaiser(M,pi*alpha)/2]))
ans =
2.82707022360190 2.50908747431366
2.00652719015325 1.92930705688346
0.68469697658600 0.85272343521683
0.09415916813555 0.19546670371747
0.00311639169878 0.01773139505899
0.00000050775691 0.00022611995322
0.00000003737279 0.00000123787805
0.00000000262633 0.00000066206722
0.00000007448708 0.00000034793207
0.00000007448708 0.00000034793207
0.00000000262633 0.00000066206722
0.00000003737279 0.00000123787805
0.00000050775691 0.00022611995322
0.00311639169878 0.01773139505899
0.09415916813555 0.19546670371747
0.68469697658600 0.85272343521683
2.00652719015325 1.92930705688346
Finally, Fig.3.30 shows a comparison of DPSS and Kaiser window transforms, where the DPSS window was computed using the simple method listed in §F.1.2. We see that the DPSS window has a slightly
narrower main lobe and lower overall side-lobe levels, although its side lobes are higher far from the main lobe. Thus, the DPSS window has slightly better overall specifications, while Kaiser-window
side lobes have a steeper roll off.
Next Section: Dolph-Chebyshev WindowPrevious Section: Slepian or DPSS Window | {"url":"https://www.dsprelated.com/freebooks/sasp/Kaiser_Window.html","timestamp":"2024-11-10T01:31:33Z","content_type":"text/html","content_length":"52812","record_id":"<urn:uuid:d37db84d-2986-4601-8f37-ce54e6898095>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00787.warc.gz"} |
Computer Chess Club Archives
Subject: Re: Question for the MTD(f) experts
Author: Andrew Williams
Date: 12:24:53 04/14/04
Go up one level in this thread
On April 14, 2004 at 14:13:04, Dann Corbit wrote:
>On April 14, 2004 at 14:09:04, Bo Persson wrote:
>>On April 14, 2004 at 13:47:06, Dann Corbit wrote:
>>>On April 14, 2004 at 13:33:38, Robert Hyatt wrote:
>>>>On April 14, 2004 at 03:30:22, Dann Corbit wrote:
>>>>>I decided to toss an MTD(f) search into TSCP, and I've got something wrong, but
>>>>>I can't quite see what it is.
>>>>There is a lot more to do.
>>>>1. you need to modify the hash table to store 2 bounds, not 1.
>>>That was not done yet.
>>But not too hard!
>>>>2. the search must be fail-soft. TSCP isn't.
>>>I had already done that.
>>>>3. PV has to be yanked from the hash table and that makes it flakey at times as
>>>>has been discussed many times. There is another way to get the PV, but it is a
>>>>special case solution only for mtd...
>>>Tord showed a very nice way to do that with a clever "hash only" update.
>>>>4. the convergence has to be accelerated. IE on a fail high searching v and
>>>>v+1 won't cut it.
>>I can supply that part, learned from CCC a couple of years ago of course.
>> // Calculate next gamma and Step
>> if (gamma < Beta)
>> {
>> GlobalUpperBound = gamma;
>> gamma = max(gamma - Step, GlobalLowerBound + 1);
>> SteppedDown = true;
>> }
>> else
>> {
>> GlobalLowerBound = gamma;
>> gamma = min(gamma + Step, GlobalUpperBound - 1);
>> SteppedUp = true;
>> }
>> if (SteppedUp & SteppedDown)
>> Step /= 2;
>> else
>> if (Step < (GlobalUpperBound - GlobalLowerBound) / 2)
>> Step *= 2;
>>Here gamma is your f. The idea is to accellerate the stepping, until you have
>>over stepped the score twice, once in each direction. Then you have an
>>acceptable bound, and can start to zoom in, using smaller and smaller steps as
>>you get closer and closer.
>>The GlobalUpperBound and GlobalLowerBound should really be global. If you use
>>lazy evaluations those are the bounds to use there, not the local alpha/beta
>>which are always just 1 point apart.
>I have lots of ideas along these lines.
>I thought of using an MTD(f) like driver, but in MTD(bi) format, so that I will
>binary search.
When I first started with MTD (aeons ago now), i tried MTD(bi) and it just
didn't work. It looks like it should, but doesn't. I then switched to using
MTD(bi) after something similar to what Bo posted above. That didn't work
either. YMMV of course.
>I thought of doing a parabolic fit of the last few guesses and extrapolating the
>next guess.
>I thought of doing an aspiration window around the last bound and MTD(bi) search
>of that.
>Some others besides. But I want to understand the algorithm through and through
>before I venture into that land.
>I got reinterested in MTD(f) because Uri mentioned that he would like to be able
>to experiment with it in his search.
>In case he wants me to give him a hand, I had better understand it well myself.
It's good fun playing with MTD, but make sure your hash-table works properly
This page took 0 seconds to execute
Last modified: Thu, 15 Apr 21 08:11:13 -0700
Current Computer Chess Club Forums at Talkchess. This site by Sean Mintz. | {"url":"https://www.stmintz.com/ccc/index.php?id=360012","timestamp":"2024-11-12T13:12:03Z","content_type":"application/xhtml+xml","content_length":"5916","record_id":"<urn:uuid:b8379fe8-ecc1-40f1-b1bb-4566bb5accca>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00311.warc.gz"} |
The Simons Foundation and Open Source Software
Jim Simons
Jim Simons is a mathematician who left academia to start a hedge fund that beat the stock market. He contributes back to the mathematical community through the
Simons Foundation
, which provides an enormous amount of support to mathematicians and physicists, and has many outreach programs.
is a large software package for mathematics that I started in 2005 with the goal of
creating a free open source viable alternative to Magma, Mathematica, Maple, and Matlab
. People frequently tell me I should approach the Simons Foundation for funding to support Sage.
For example
Jim Simons, after retiring from Renaissance Technologies with a cool 15 billion, has spent the last 10 years giving grants to people in the pure sciences. He's a true academic at heart [...]
Anyways, he's very fond of academics and gives MacArthur-esque grants, especially to people who want to change the way mathematics is taught. Approach his fund. I'm 100% sure he'll give you a
grant on the spot.
The National Science Foundation
Last month the http://sagemath.org website had 45,114 monthly active users. However, as far as I know, there is no NSF funding for Sage in the United States right now, and development is mostly done
on a shoestring in spare time. We have recently failed to get several NSF grants for Sage, despite there being Sage-related grants in the past from NSF. I know that funding is random, and I will keep
trying. I have two proposals for Sage funding submitted to NSF right now.
Several million dollars per year
I was incredibly excited in 2012 when David Eisenbud invited me to a meeting at the Simons Foundation headquarters in New York City with the following official description of their goals:
The purpose of this round table is to investigate what sorts of support would facilitate the development, deployment and maintenance of open-source software used for fundamental research in
mathematics, statistics and theoretical physics. We hope that this group will consider what support is currently available, and whether there are projects that the Simons Foundation could
undertake that would add significantly to the usefulness of computational tools for basic research. Modes of support that duplicate or marginally improve on support that is already available
through the universities or the federal government will not be of interest to the foundation. Questions of software that is primarily educational in nature may be useful as a comparison, but are
not of primary interest. The scale of foundation support will depend upon what is needed and on the potential scientific benefit, but could be substantial, perhaps up to several million dollars
per year.
Current modes of funding for research software in mathematics, statistics and physics differ very significantly. There may be correspondingly great differences in what the foundation might
accomplish in these areas. We hope that the round table members will be able to help the foundation understand the current landscape (what are the needs, what is available, whether it is useful,
how it is supported) both in general and across the different disciplines, and will help us think creatively about new possibilities.
I flew across country to this the meeting, where we spent the day discussing ways in which "several million dollars per year" could revolutionize "the development, deployment and maintenance of
open-source software used for fundamental research in mathematics...".
In the afternoon Jim Simons arrived, and shook our hands. He then lectured us with some anecdotes, didn't listen to what we had to say, and didn't seem to understand open source software. I was
frustrated watching how he treated the other participants, so I didn't say a word to him. I feel bad for failing to express myself.
The Decision
In the backroom during a coffee break, David Eisenbud told me that it had already been decided that they were going to just fund Magma by
making it freely available to all academics in North America
. WTF? I explained to David that Magma is closed source and that not only does funding Magma not help open source software like Sage, it actively hurts it. A huge motivation for people to contribute
to Sage is that they do not have access to Magma (which was very expensive).
I wandered out of that meeting in a daze; things had gone so differently than I had expected. How could a goal to "facilitate the development, deployment and maintenance of open-source software...
perhaps up to several million dollars per year" result in a decision that would make things possibly much worse for open source software?
That day I started thinking about creating what would become
. The engineering work needed to make Sage accessible to a wider audience wasn't going to happen without substantial funding (I had put years of my life into this problem but it's really hard, and I
couldn't do it by myself). At least I could try to make it so people don't have to install Sage (which is very difficult). I also hoped a commercial entity could provide a more sustainable source of
funding for open source mathematics software. Three years later, the net result of me starting SageMathCloud and spending almost every waking moment on it is that I've gone from having many grants to
not, and SageMathCloud itself is losing money. But I remain cautiously optimistic and forge on...
We will not fund Sage
Prompted by numerous messages recently from people, I wrote to David Eisenbud this week. He suggested I write to Yuri Schinkel, who is the current director of the Simons Foundation:
Dear William,
Before I joined the foundation, there was a meeting conducted by David Eisenbud to discuss possible projects in this area, including Sage.
After that meeting it was decided that the foundation would support Magma.
Please keep me in the loop regarding developments at Sage, but I regret that we will not fund Sage at this time.
Best regards, Yuri
The Simons Foundation, the NSF, or any other foundation does not owe the Sage project anything. Sage is used by a lot of people for free, who together have their research and teaching supported by
hundreds of millions of dollars in NSF grants. Meanwhile the Sage project barely hobbles along. I meet people who have fantastic development or documentations projects for Sage that they can't do
because they are far too busy with their fulltime teaching jobs. More funding would have a massive impact. It's only fair that the US mathematical community is at least aware of a missed opportunity.
Funding in Europe
for open source math software is much better.
Hacker News discussion | {"url":"https://sagemath.blogspot.com/2015/09/the-simons-foundation-and-open-source.html","timestamp":"2024-11-14T04:58:31Z","content_type":"text/html","content_length":"49715","record_id":"<urn:uuid:a6e8a8c0-191a-4acc-bb3f-9035bfd7781d>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00554.warc.gz"} |
Ideas of Thoralf Skolem, by Theme
[Norwegian, 1887 - 1963, Professor at the University of Oslo.]
green numbers give full details |
back to list of philosophers
expand these ideas
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17879 Axiomatising set theory makes it all relative
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
13536 Skolem did not believe in the existence of uncountable sets
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17878 If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
17880 Integers and induction are clear as foundations, but set-theory axioms certainly aren't
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
17881 Mathematician want performable operations, not propositions about objects | {"url":"http://www.philosophyideas.com/search/response_philosopherTh.asp?era_no=K&era=Early%2020th%20century%20(1901-1955)&PN=3749&visit=list&order=chron","timestamp":"2024-11-10T08:31:34Z","content_type":"application/xhtml+xml","content_length":"5466","record_id":"<urn:uuid:a6779517-4186-441a-87b1-eed6cc7bdb7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00795.warc.gz"} |
power required to drive a ball mill formula
Multiplying the kw.hours per ton by the tons required to be crushed per hour, the product will represent the power required, and the mill nearest to that power rating should be selected. Fig. 4
is a preliminary power curve based on the recommended maximum ball charge, together with all available data at hand at the present time; however, 60 or ...
WhatsApp: +86 18838072829
The main equipment for grinding construction materials are balltube mills, which are actively used in industry and are constantly being improved. The main issue of improvement is to reduce the
power consumption of a balltube mill with crosslongitudinal movement of the load. A comparative analysis and the possibility of using the known ...
WhatsApp: +86 18838072829
1. Closed Circuit = W 2. Open Circuit, Product Topsize not limited = W 3. Open Circuit, Product Topsize limited = W to W Open circuit grinding to a given surface area requires no more power than
closed circuit grinding to the same surface area provided there is no objection to the natural topsize.
WhatsApp: +86 18838072829
A ball mill, a type of grinder, is a cylindrical device used in grinding (or mixing) materials like ores, chemicals, ceramic raw materials and paints. Ball mills rotate around a horizontal axis,
partially filled with the material to be ground plus the grinding medium. Different materials are used as media, including ceramic balls, flint pebbles ...
WhatsApp: +86 18838072829
In today's ultra large ball mills of up to 28′ diameter and 44′ long; a 2832% ball charge is your maximum. Source: Paul Staples; A grate (diaphragm) discharge ball mill will consume 15% more
power than an overflow (open) discharge ball mill even though the grinding efficiencies are the same. Source: Lewis, Coburn, and Bhappu
WhatsApp: +86 18838072829
See Full PDFDownload PDF. BALL MILL Specification Number A Process Data Sheet xxxxxxxxxx Sheet No : 1 of 2 Area: xxx Client : Operating Centre: Jakarta No. 1 Working 1 Service: ROM Ball Mill Item
No: 1 of: Standby 0 xxxxML001 2 Manufacturer: Model Type and Number: 3 OPERATING CONDITIONS 4 MATERIAL HANDLED: Crushed Ore 5 Hazard ...
WhatsApp: +86 18838072829
The general form of the equation is by Morgärdshammar that the power draw of a mill is derived from the power required to maintain mill revolutions with the mass of charge acting at the centroid
of the mass. This is a form of equation used by many workers with the only variant being the "constant' factor in front of the equation.
WhatsApp: +86 18838072829
Quantum Nanostructures (QDs): An Overview. D. Sumanth Kumar, ... Mahesh, in Synthesis of Inorganic Nanomaterials, 2018 Ball Milling. A ball mill is a type of grinder used to grind and blend bulk
material into QDs/nanosize using different sized balls. The working principle is simple; impact and attrition size reduction take place as the ball drops from near the top of a rotating ...
WhatsApp: +86 18838072829
Less power is required to drive the cutter. Less sliding friction is produced between the tooth and the work. This reduces the development of heat. Longer life of the cutter may be obtained as
the number of regrinds can be increased. #10 Machining Time. The machining length (l m) is divided by the table feed (v f). The time required for any ...
WhatsApp: +86 18838072829
One participant also clarified the use of trigonometric functions in the torque formula and the difference between static and kinetic friction coefficients. Overall, the calculated torque needed
for the load was determined to be Nm, with a suggestion to use a friction value of for a safety factor.
WhatsApp: +86 18838072829
The optimum ball mill parameters to achieve a small particle size of µm include a ball mill speed of 500 rpm, a milling time of 10 h, and a balltopowder ratio (BPR) of 10:1. These settings are
sufficient for obtaining better results.
WhatsApp: +86 18838072829
The power, P, to drive the mill would be expected to depend upon the length of the mill, L, the diameter, D, the diameter of ball, d, the density, e, of the ball, the volume occupied by the
charge (including voids), expressed as a fraction of the total mill volume, J, the speed of rotation, N, the acceleration due to gravity, g, the coefficient ...
WhatsApp: +86 18838072829
For instance, if your jar had in inside diameter of 90 mm and your milling media was mm diameter lead balls, the optimum rotation would be 98 RPM. Optimum RPM= .65 x Critical speed (cascading
action of the media stops) with dimensions in inches. Share on other sites. whitewolf_573. The motor has been working for 30 minutes, and works good ...
WhatsApp: +86 18838072829
Answer to Question #188408 in Chemical Engineering for Moin. Power required to drive a ball mill with a particular ball load is proportional to (where, D = diameter of ball mill ) Power required
to drive a ball mill with a particular ball load is proportional to (where, D = diameter of ball mill) D
WhatsApp: +86 18838072829
A mill is a device, often a structure, machine or kitchen appliance, that breaks solid materials into smaller pieces by grinding, crushing, or cutting. Such comminution is an important unit
operation in many are many different types of mills and many types of materials processed in them. Historically mills were powered by hand or by animals (, via a hand crank), working ...
WhatsApp: +86 18838072829
Power required to drive a ball mill with a particular ball load is proportional to (where, D = diameter of ball mill ) A. D. B. 1 D. C. D D. 1 D Answer: Option C. This Question Belongs to
Chemical Engineering >> Mechanical Operations.
WhatsApp: +86 18838072829
how to calculate ball mill rotational speed ... hammer mills ... rolling mill torque : ... Central Drive Ball Mill Torque and Power Calculation Copper ... screw conveyor power calculation formula
WhatsApp: +86 18838072829
The roller is used to drive another roller by surface contact. The driven roller is placed directly over the driver and it sits by means of gravity. The driven roller weight is 40 kg. The driving
roller weight is 10 kg. The motor is to be directly connected at the driver,s roller end. No belt. Thanks for replying baluncore.
WhatsApp: +86 18838072829
Assuming the efficiency is 100%, the power required to remove the material is the same as the power used by the machine's motor. Each machine has a maximum power limit that it can handle.
Therefore, it is helpful to calculate the power required to perform a machining operation such as milling or turning and compare it to our machine's ...
WhatsApp: +86 18838072829
The ball tube mill requires less mass flow rate of primary air and raw coal per MW of power produced than the bowl mill. It contains larger mass of raw and pulverized coal but less grinding heat
WhatsApp: +86 18838072829
the required SAG comminution of test ore material. Specific energy consumption (SEC) measured in kWh/t is a ratio of the mill drive power to the milling circuit output, without regard to
circulating load. At the constant size of the cycle feed and discharge, SEC is a prime characteristic of material grindability in a certain ore pretreatment ...
WhatsApp: +86 18838072829
To match a ball mill size and its motor size, some math is required. The power requirement calculated above is the motor power that must be applied at the mill drive in order to grind the tonnage
of feed from one size distribution. The following shows how the size of mill required to draw this power is calculated.
WhatsApp: +86 18838072829
Objectives. At the end of this lesson students should be able to: Explain the role of ball mill in mineral industry and why it is extensively used. Describe different types of ball mill design.
Describe the components of ball mill. Explain their understanding of ball mill operation. Explain the role of critical speed and power draw in design ...
WhatsApp: +86 18838072829
Mechanical Agitator Power Requirements for Liquid Batches 2020 Instructor: John Frank Pietranski,, PDH Online | PDH Center 5272 Meadow Estates Drive Fairfax, VA Phone: An Approved Continuing
Education Provider
WhatsApp: +86 18838072829
Power drawn by ball, semiautogenous and autogenous mills A simplified picture of the mill load is shown in Figure Ad this can be used to establish the essential features of a model for mill
power. The torque required to turn the mill is given by Torque T Mcgdc T f () Where Mc is the total mass of the charge in the mill and Tf is ...
WhatsApp: +86 18838072829
BallRod Mills, based on 4″ liners and capacity varying as power of mill diameter, on the 5′ size give 20 per cent increased capacity; on the 4′ size, 25 per cent; and on the 3′ size, 28 per cent.
WhatsApp: +86 18838072829
1 Calculation of ball mill capacity. The production capacity of the ball mill is determined by the amount of material required to be ground, and it must have a certain margin when designing and
selecting. There are many factors affecting the production capacity of the ball mill, in addition to the nature of the material (grain size, hardness, density, temperature and humidity), the
degree of ...
WhatsApp: +86 18838072829
To compute for shaft power | ball mill length, six essential parameters are needed and these parameters are Value of C, Volume Load Percentage (J), % Critical Speed (Vcr), Bulk Density (), Mill
Length (L) and Mill Internal Diameter (D). The formula for calculating shaft power | ball mill length: P = x C X J X V cr x (1 ) x [1 ...
WhatsApp: +86 18838072829
Mill power is related to load behavior in some way. Some power models use shoulder position, toe position, or angle of repose as a description of load behavior to derive their power equations. As
can be seen from Fig. 2, a chord connecting the shoulder and toe of the load does not precisely express the configuration of load in a mill at the ...
WhatsApp: +86 18838072829
required to operate a commercial ball mill of radius R is predicted on the assumption that the ball mill is half filled with balls, there is a cascade angle of 45o, and the mill base just covers
the top layer : Optimum cascading of mill for power consumption of the ball. Let the average density of the ball mill base
WhatsApp: +86 18838072829 | {"url":"https://c-mobat.fr/Jan/09-597.html","timestamp":"2024-11-03T20:22:14Z","content_type":"application/xhtml+xml","content_length":"27287","record_id":"<urn:uuid:827e3846-33ea-4282-9d7d-189de677db90>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00396.warc.gz"} |
Multiply with your fingers! - Fantastic Mathematics
Multiply with your fingers!
mathématiques, Tricks | 0 comments
Multiplication tables remain a nightmare for schoolchildren to see worse for students. Emmanuel Macron knows very well this cold and terrible fear of being questioned about it, and this still
currently… .
Teacher I tried to find ways to help my students better memorize these tables and create quick computational automatisms. Generally there was no concern about memorizing tables from 1 to 5, it was
tables 6, 7, 8 and 9 that posed more problems.
I discovered, one day, this old technique, simple to use. But then why, as students, none of our math teachers tried to pass it on to us? …. I remain perplexed by this finding. Calculating with your
fingers is the basis of the calculation… At the premise of mathematics, it was the pebbles that were used. The word calculus comes from the Latin calculus (“pebble”).
This process of multiplication is indicated in the khulasat alhissab of the author Baha’ Al-din(1547-1622). But we find it, earlier written by master Nicolas Chuquet in the Triparty in the science of
numbers (1484),as well as the principle of multiplication posed still taught today in schools.
It has the advantage of retaining by heart only the products of the first integers up to \( 5times 5 \) and easily deduce the products of the other integers up to \( 9times 9 \) . In other words,
this method will only be used if the two factors in the product are strictly greater than 5.
Example: Let be the product \( 8times 9 \) to be determined.
Note that \( 8 = 5+color{red}{3} \) and \( 9 = 5+color{red}{4} \)
We raise 3 doitgs with one hand and 4 with the other, the other fingers being lowered, i.e. \( 5-color{red}{3}=color{blue}{2} \) for one and \( 5-color{red}{4}=color{blue}{1} \) for the other.
The amount of fingers raised is \( 3+4=7 \) gives the figure of the product’s tens \( 8times 9 \) .
The product of the numbers corresponding to the lowered fingers is \(2 times \) 1 = 2 gives the digit of the units.
We find \( 8times 9 = color{red}{7}color{blue}{2}\)
The detailed method in video:
To go further
With students of 3rd grade, after having them discover this method, it is quite wise to ask them to demonstrate it rigorously using the literal calculation … (Challenge posed in group work, class
atmosphere guaranteed!)
Proof: let \( a and b be two natural \) \( \) integers such that \( 5 , <a <10 \) \( 5 <b <10 \) .
\( atimes b = (5+c)(5+d) \) so c , d are two natural numbers such that \( \) \( \) \( 0 and <c <5 \) \( 0<d<5\)
\( = color{green}{5times 5 }+ color{red}{5c+5d}+cd\)
\(=color{green}{25}+ color{red}{10c+10d -5c-5d}+cd\)
\(=25+color{brown}{ 10c+10d} -5c-5d+cd\)
Thus: \( 10(c+d): \) represents the number of fingers raised multiplied by 10.
\((5-c) (5-d) \) : the product of the numbers represented by the closed fingers
After this acquired method, it is important to work on rapidity and automatism in mental calculation using like a high-level athlete who works, every day, the same gestures to perform in his
discipline …
A free application (and I hope that will remain so for a good time) developed by Christophe Auclair has the gift of challenging students alone or in duo. This application combines gamification and
mental calculation. Students see it as a real relaxation in class and like to challenge themselves in a variety of ways.
To download and use without moderation in class, like at home!
Submit a Comment Cancel reply
You must be logged in to post a comment.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://mathematiques-fantastiques.fr/en/multiply-with-the-fingers/","timestamp":"2024-11-06T18:32:50Z","content_type":"text/html","content_length":"247558","record_id":"<urn:uuid:3e534265-c888-40f6-9396-73070347ed1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00529.warc.gz"} |
Onion Routing In Lightning
Scaling Bitcoin Conference
Milan (2016)
< Onion Routing In Lightning
Onion Routing In Lightning
Speakers: Olaoluwa Osuntokun
Date: October 8, 2016
Transcript By: Bryan Bishop
Tags: Lightning, Routing
Category: Conference
Media: https://www.youtube.com/watch?v=Gzg_u9gHc5Q&t=164s
Privacy-preserving decentralized micropayments
We’re excited about lightning because the second layer could be an important opportunity to improve privacy and fungibility. Also, there might be timing information in the payments themselves.
Distributed set of onion routers (OR, cite OG onionrouting paper). Users create circuits with sub-set of nodes. Difficult for oion routers to gain more info than predecessor+successor in path. Low
latency - usable within greater internet. Notable success is tor, notable success of routing nodes.
In lightning, we have some goals to make it private and censorship resistant. We use source-routing which means the source fully specifies the entire route. In lightning we want to know what the fees
are. We want to know the entire route, the timelocks and the path itself.
We use sphinx to make a provably secure mix format. It has various desirable capabilities. With encryption, the ciphertext – the plaintext due to randomization. There’s non-trivial shipping and
mapping to make sure everything is fixed throughout the entire size. With fixed size, it gives no positional information. If you are the fifth person in the graph, then you might give away
information about the graph or your position in the graph.
In sphinx you can derive a shared secret. To achieve unlinkability, we need to randomize the base key of the diffie-hellman itself. You can have n-keys in the package itself, but you can do things
like RSA and now you have kilobytes of a packet because you need all the shared secrets itself. It’s the hoop element at each hop. It’s a cool trick. This can be generalized to any protocol like
elliptic curves, RSA, we can do LWE and other things like that.
So you get a session key for the session and then a list of all the public nodes in the route. So a0 is the initial public key for the first hop and then it derives a shared secret as 0. It raises it
to the power of its secret exponent and then there’s a blinding factor. Each intermediate node uses the blinding factor to randomize the public key for the next hop. a1 is g^x^b0. So each node
derives a blinding factor from the shared secret, and uses this to randomize for the next hop. We have a constant-size key with each hop and we achieve unlinkability.
In sphinx, packet processing is simplified. ProcessSphinxPacket…. if the MAC is the .. then you tampered with the packet and it’s rejected immediately. Also, you have to protect against replay
attacks as well. You re-randomize the shared secret with a new blinding factor.
One thing we’ve done is we’ve started to make some modifications to sphinx itself. We realized it’s useful to ourselves but maybe we could add some features for lightning. cdecker from Blockstream
has done some work with this to make it more lightning-like. We can add a version byte to the header itself so we can have new versions later on if we use different cryptosystems. We also added the
public key and the MAC is now over the entire packet. Originally in sphinx it’s not over the entire packet, it was made for mixnet rather than indistinguishable replies. So you can’t have the MAC on
it because if you have a MAC then it’s not indistinguishable. We don’t have that reply use-case in lightning itself. We switched from AES to chacha20 for speed optimality. We also have a payload that
attaches instructions for the hop itself. If one link has several links, which one should be used? We have some information about which links to link on, and which ones, and some other information
like timing information.
We have this code and it’s done. We could start a mixnet to have an alternative transaction propagation mechanism.
Performance considerations
There were two asterisks on two lines of codes; those were the asymmetic crypto operations for forwarding the HLTC where you derive a shared secret and randomize a …. so this might be nice to
eliminate or advertise that we do it only one time and avoid it otherwise. The onion routers themselves need to maintain a per-session state. This is the hash, the incoming link and hte actual link.
They need to maintain this state and forward it down to next path. If you forget the state and you get the settle, you don’t know who to send it to. If you forget the state you can’t remember the
HLTC. So this would be a DoS attack. It needs to be persisted to disk, too. If we could overcome those limitations we could have it be faster and the routers could be stateless.
It’s an extension of Sphinx and overcomes some of the detrimental problems there. Hornet is a progression of sphinx and it is targeting internet-scale. They want to eliminate the asymmetric crypto
operations. It does two important things. It gets the asymmetric operations out of the critical path of data forwarding. During initial setup, if we can get our symmetric keys and then only do fast
symmetric crypto operations. Another nice part of hornet is that it creates a bidirectional circuit. Sphinx is setup and forget. But hornet has real-time.
Hornet uses sphinx. Hornet uses sphinx initially to derive the shared secrets which then allows intermediate nodes to put a special key into a packet which they can use into data forwarding. The nice
thing about hornet is that the nodes only need to maintain constant state. The packets carry all the state. The state is pushed to the endpoints rather than the intermediate nodes. The major parts is
the anonymous header. A forwarding segment is encrypted by a node and can only be decrypted by itself, and it contains routing information like the next route to go to and some session information to
avoid replay attacs.
Nodes only need to maintain their own symmetric keys, they can decrypt the packets and then continue forwarding.
Hornet can help with pyament flow of lightning itself. The payment flow is hypothesized to be out-of-band. Alice and Bob want to send payments through lightning. Maybe there’s tor, maybe not. Maybe
they exchange tor information so they can send money over lightning. Maybe they are doing several payments and need to talk back and forth. We can eliminate that out-bound communication and move it
into the network. This assumes Alice and Bob have a route. Alice can create a hornet session to Bob and then they have a bi-directional link where they can exchange detail. I can get payment vaues
and r values from Bob through the network itself.
Maybe some of the links wont have sufficient capacity to send everything in one go; so I could fragment a payment through several routes in lightning at the same time.
Shared secret log
The important part for maintaining a shared secret is that you need to reject a packet if it’s sent again. An adversary could resend a packet, but we want to reject that. So you need a growing log of
all the shared secrets, so if it’s in the log then you need to reject it. If I need to maintain shared secrets for all my lifetime, I have unbound growth. So we need to garbage select part of the
trees. So we can do that with key rotation. There are a few ways to do this. In tor, they have a central directory server and everyone uploads their keys and their new signed keys, but you know
ideally we could do that and we would like to avoid that, there’s a centarl point of authority so there’s a few ways to do ad hoc key rotation. So let’s assume that nodes have an identity key, and
then they could authenticate hteir onion key with an identity key. So maybe each day they broadcast a new key to the network. So one subtletly of this is that the rotation has to be loosely
synchronized. If you rotate keys and then someone sends you payment over the old key, then it looks like an invalid MAC, you would have to reject it. So there needs to be a grace period instead,
where a few days after key rotation you still accept old packets maybe. So you check with the old key and check with the new key and maybe it works.
Active key rotation actually encourages higher bandwidth and now we need to rotate keys 24 hours every 24 hours or something. So with a million node networks, that’s a lot of data to download.
Passive key rotation
You published a key. You use bip32 public key derivatio. The people at the edges have the blockhash and your master public key from bip32. Everyone can do key rotation by themselves on the edges.
There’s a gotcha, though. It’s common knowledge that with public derivation on bip32 is that if you have the master pubkey and you leak the…. so.. with that new, you need forward secrecy if the
private key is leaked. You can have an intermediate point, as to a step doing derivation and then leave that point later. If you use some crypto magic maybe you can solve that.
So for passive key rotation, you could do pairing crypto and have passive non-interactive key rotation. Three cyclic groups. A bilinear pairing. We can have more meaningful identifiers like
roasbeef@lightning.network and use that instead. So we have a three cyclic group and a pairing operation which takes elements from two groups and transmutes it to a third one. And the group is
multiplicative. If you look at the pairing operation, … that’s the magical step that allows you to do a bunch of other cool stuff.
Every node now advertises a master public key. With regular IBE, there’s a trusted agent that distributes public keys to everyone else. We can have the nodes be the trusted agent to do the actual
protocol. We need a novel primitive called a … hash function…. we want the hash function to map directly to a point on a curve. You could do this iteratively where you have a string, you
hash it, you get an output on the curve. What about mapping something to a point that isn’t the identity point or the element at infinity? Everyone has an ID which is a blockhash. ID is a group
element, a public key. So given a blockhash, they can then use this abstraction to derive another key that is a group element that matches that. If you remember in sphinx we had this g^r, we could do
this to do non-interactive key rotation. Alice knows the key schedule of the other nodes. So then she uses that to basically, using g^r, to derive a shared secret. She uses bob’s master key and then
raises that to r which is the current session key with maybe some blinding factors. Bob takes r itself, does a pairing operation with n which is his private key. So it just goes out to the end where
they both arrive at a shared secret which is the pairing of g, and the Bob’s ID, and now they have a shared secret. So with this we can achieve passive key rotation from the edges– but now there’s a
pairing operation cost as well. We can use this to do key rotation or just do a stop gap and do the other one.
Limitations in the scheme
Assumes a high degree of path diversity. There are many ways to get from Alice to Bob. If there’s only one path, then you know she’s sending it to Bob. You can have some correlation with payment
values where– you know there’s no other link could support a payment of this size, so therefore they are using this particular route.
You can do timing attacks where you can see Bob and Alice are sending payments– well you can see packet size correlation and so on.
Future directions
Maybe we can gain capabilities … and moving the asymmetric operation to the forwarding highlight. We could figure out the payload structure. How do I identify different chains? What is the
timing? What about high-latency systems that give us more privacy guarantees? Maybe lightning is just for instantaneous payments, and maybe you could tolerate a minute and that’s still okay.
We can also look into non-source-routed privacy schemes. Since everyone has to give information about links, you lose some privacy. There’s some ways you could do this but they involve trusted
hardware at the node using obliviousRAM (oRAM) and doing shortest path search over the graph and that would require special hardware. We probably wont do that. It’s a cool research direction to look
into for the future.
Q: Statistical correlation in onion routing?
A: Randomized delays. We could pad out the packet sizes so that every packet on matter if you were doing an opening or a forwarding is always the same size. All the packets would be encrypted. We
could do other things like add some delay or move to higher-latency networks which could help us hiding some of this metadata.
Q: What do you do about… network layer attackers?
A: For the payment network problems, we could say all payments in lightning are a particular value, and then we fix the channels, and then everything is completely uniform. With sphinx we talked
about unlinkability, the packets themselves are indistinguishable bu tbecause the r value is the same. Versioning– same path, same r value. We could use some techniques to randomize the r value just
like we do for the group value for sphinx. We could do a scheme where we randomize the r values, where we add generic point multiplication or use fancy signatures where you do single show signature
which forces you to use a certain value for the r value in the signature. If oyu sign with the key then you reveal the r value. Without that, it’s somewhat limited. Maybe we could do this at the end
of v1.
onion routing specification https://lists.linuxfoundation.org/pipermail/lightning-dev/2016-July/000557.html
onion routing protocol for lightning https://github.com/cdecker/lightning-rfc/blob/master/bolts/onion-protocol.md
https://github.com/lightningnetwork/lightning-onion and https://github.com/cdecker/lightning-onion/tree/chacha20 | {"url":"https://btctranscripts.com/scalingbitcoin/milan-2016/onion-routing-in-lightning","timestamp":"2024-11-05T00:18:01Z","content_type":"text/html","content_length":"48521","record_id":"<urn:uuid:2c8cf3fa-d0db-458e-907f-cca28e055baa>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00341.warc.gz"} |
In computer science, a graph is an abstract data type that represents pairwise relationships between objects. A graph, in this context, consists of:
• Vertices (or nodes): The fundamental units of a graph, representing discrete objects or concepts.
• Edges: Lines or arcs that connect pairs of vertices, signifying a relationship between them. Edges can be:
□ Directed: Having an established direction from one vertex to another.
□ Undirected: Having no implied directionality.
Types of Graphs
Several common types of graphs exist:
• Undirected Graph: Edges have no directionality (e.g., representing friendships on a social network).
• Directed Graph (Digraph): Edges have a defined direction (e.g., modeling website links and their directions).
• Weighted Graph: Edges carry a numerical value or "weight" representing aspects like cost, capacity, or distance (e.g., road networks).
• Cyclic Graph: A graph containing at least one cycle (a path that starts and ends at the same vertex).
• Acyclic Graph: A graph containing no cycles, often forming a tree-like structure.
Graphs are commonly represented using:
• Adjacency Matrix: A two-dimensional matrix where rows and columns represent vertices, and cells indicate the presence or absence of an edge between the corresponding vertices.
• Adjacency List: An array of linked lists where each list stores the neighbors of a vertex.
Numerous graph algorithms are essential in computer science:
• Traversal Algorithms:
□ Breadth-First Search (BFS): Explores a graph level by level, starting from a source node.
□ Depth-First Search (DFS): Explores a graph by a path as far as possible before backtracking.
• Shortest Path Algorithms:
□ Dijkstra's Algorithm: Finds the shortest path between a source and destination node in a weighted graph (with non-negative edge weights).
□ Bellman-Ford Algorithm: Finds shortest paths in weighted graphs, able to handle negative edge weights.
• Minimum Spanning Tree Algorithms:
□ Kruskal's Algorithm: Constructs a tree within a graph that connects all vertices with the minimum total edge weight.
□ Prim's Algorithm: Builds a tree from a starting vertex and grows it by adding the cheapest possible edge in each step.
Graphs are a remarkably versatile data structure with a wide range of applications:
• Networking: Representation of computer networks, communication patterns, and routing protocols.
• Social Networks: Modeling relationships and interactions between people or groups.
• Maps and Navigation: Representing roads, cities, and points of interest for pathfinding and routing.
• Software Engineering: Modeling dependencies between components, state machines, and data flow.
• Artificial Intelligence: Used in knowledge representation, decision-making, and planning problems. | {"url":"https://blog.prepsphere.com/share/LxT4ZUCCRumo","timestamp":"2024-11-02T21:24:11Z","content_type":"text/html","content_length":"64021","record_id":"<urn:uuid:dc9f2074-87c5-4c7e-b336-720ac7c2fe42>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00498.warc.gz"} |
How Much Can I Save With a Heat Pump
How Much Can I Save With a Heat Pump in Nova Scotia – Electric Baseboards
For this example we’re going to be using electric baseboards. If you have oil hot water baseboards or in-floor radiant heat check out this post and video here.
From the previous post we discussed how much the monthly operating cost of an example ductless mini split heat pump is. That cost was about $62 a month.
In our example we were talking about installing a single unit in an area of your home where you spend the majority of your time. For most homes in Halifax metro this is the main floor of your home.
To estimate how much you can save we need to first get an idea of what it’s costing to heat that area of the home now.
Let’s say there are (3) 1000 watt baseboard heaters in that same area of your home which do the heating. That’s 3000 total watts of power needed when your heat is on in this area of the home.
To keep things consistent we’re going to make the same assumptions we made in the previous post and video which is: Your heating source is on 60-70% of the time to maintain indoor temperature.
Let’s do some math!
Calculating Cost to Run (3) 1000 watt Baseboards Per Day and Month
First we’ll determine how many hours the baseboards are on for in a given day.
24 hours a day x 60% of the time your heaters are on = 14.4 hours
Next we’ll figure out how many Kilowatts the three baseboards are using when they’re on. Remember 1000 watts = 1 kilowatt. So when we divide the total number of watts by three we get number of
3000 watts / 3 = 3 Kw
Next we’ll figure out how many kilowatt hours we’re using since this is how NSP bills us for electricity. We multiply the number of hours the baseboards are on by the number of Kilowatts they use.
14.4 hours x 3 Kwh = 43.2 Hkw
Finally we’ll figure out how much that number of kilowatt hours costs us per day by multiplying it by the current rate from NSP for electricity which as of the writing of this post is $0.14947.
42.3 KwH x $0.14947 / Kwh = $6.32 / day
To get an estimate of what these baseboards would cost in a given 30 day month we simply multiply the daily cost by 30 days.
30 days x $6.32 = $189.60
The average cost to heat that same space with the (3) 1000 watt electric baseboards is $6.32 per day or $189.60 per month.
Compare this to the $62.10 we came up with in the previous post and you can see there are some drastic savings to be had!
for renewable energy solutions
FREE Assessments take approximately 30 minutes. After your appointment, you will know your best options for cutting heating costs and maintaining comfort in your home.
You can also book free in-home assessments online here: BOOK AN ASSESSMENT
If you require service and repair please fill out the request form below. We will contact you to book an appointment.
© 2024 - Sunshine Renewable energy · All Rights Reserved · All Logos & Trademark Belongs To Their Respective Owners | {"url":"https://www.sunshinerenewables.ca/heat-pumps/overview/how-much-can-save-heat-pump-nova-scotia-electric-baseboards","timestamp":"2024-11-07T00:24:20Z","content_type":"text/html","content_length":"54070","record_id":"<urn:uuid:d61b8ee8-4f7d-4fc3-b2f7-9950999efc31>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00445.warc.gz"} |
Saturday Quiz – May 21, 2011 – answers and discussion
Here are the answers with discussion for this Weekend’s Quiz. The information provided should help you work out why you missed a question or three! If you haven’t already done the Quiz from yesterday
then have a go at it before you read the answers. I hope this helps you develop an understanding of Modern…
This Post Has 0 Comments | {"url":"https://billmitchell.org/blog/?p=14576","timestamp":"2024-11-08T03:05:51Z","content_type":"text/html","content_length":"103097","record_id":"<urn:uuid:0c3e78de-8b38-4fe3-a65a-98f46fe63e5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00324.warc.gz"} |
Law of Sines Ambiguous Case Worksheets
Math Topics
Law of Sines Ambiguous Case Worksheets
I find this topic to have problems that seem like riddles or puzzles, by the have a great deal of application in the real world. There will be situations that arise like this on construction sites
everywhere. In this topic we are applying the use of SSA. In certain circumstances it can require multiple solutions or no solution at all. We begin by working off of the pretense of breaking a large
triangle into multiples. These worksheets and lessons help students learn how to manipulate the use of the law of sines to determine missing measures of triangles when you run into an ambiguous case.
Aligned Standard: HSG-SRT.D.10
Homework Sheets
These problems are really neat. You need to find out how many triangles you can make from the givens.
Practice Worksheets
A rough diagram is provided to help students focus on the concept skills.
Math Skill Quizzes
Find all the missing pieces and parts using geometry.
What is the Ambiguous Case of the Law of Sines?
When you are using the Law of Sines to find a missing angles within a triangle, you will run into situations where you could create two completely different triangles based on the information that is
being presented to you. Up until now, every time we have used one of these theorems to determine missing measures there has always been a single solution. Normally in these situations we would use
the SSA theorem to find that value, but since this calls for alternate interpretations of what is available, this does not apply. This leads to one of several different scenarios. Hence, why this is
called “Ambiguous”. Ambiguous means open to interpretation. The triangle based on the given information does not exist. There can also be a situation where two separate triangles could possibly be
formed. In this case we will determine the solution twice, one for each missing of the two possible triangles.
There are three different scenarios that can result when you come across this. We can prove that no triangle exists and that does not require a solution. You can have a single triangle present those
results in a single solution. There is also the possibility of two triangles being present and as a result there are two possible solutions. There are several facts that we know about triangles that
helps us determine which of these applies. The two that I find we most commonly use to determine this is that the sum of the interior angles are 180 degrees, and no triangle can have two obtuse
angles. It also helps to know that a right angle in a triangle has to be its largest angle.
Tips for Solving Trigonometry Problems
Students often get to solving the Trigonometry Problems stage when they are in the ninth grade and it can get tricky at times. In this topic, we will be covering a general or basic idea regarding
Solving Trigonometry Problems along with some useful tips. In mathematics, it is essential to understand how you understand something rather than memorizing the steps. Trigonometry is the study of
triangles. Let's discuss some of the tips. 1. The first step involves remembering the formulas and definitions. Unless and until you are familiar with the identities and the background information of
a trigonometric problem, till then, you cannot get better at Solving Trigonometry Problems. 2. The second tip is practice. The real reason why most students struggle with solving trigonometric
problems is because of a lack of practice. Learning the formulas is the easier part; the bigger challenge is to maintain the continuous practice of every single formula and learning variations of
problems. 3. Practice your way into difficulty. If you are getting too comfortable with a particular level of difficulty, then it is recommended you increase the level and do more difficult ones. | {"url":"https://www.mathworksheetsland.com/topics/trig/sinesset.html","timestamp":"2024-11-12T18:43:09Z","content_type":"text/html","content_length":"17594","record_id":"<urn:uuid:b3a3621f-58ff-4dc2-81bd-8462c4de3cad>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00244.warc.gz"} |
Watch Understanding Multivariable Calculus: Problems, Solutions, and Tips Season 1 – Prime Video
Season 1
Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Edwards, is the next step for students and professionals to expand their knowledge for
work or study in many quantitative fields, as well as an intellectual exercise for teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.
Free trial of The Great Courses Signature Collection or buy
Terms apply
Prime membership required
1. S1 E1 - A Visual Introduction to 3-D Calculus
Review key concepts from basic calculus, then immediately jump into three dimensions with a brief overview of what you'll be learning. Apply distance and midpoint formulas to three-dimensional
objects in your very first of many extrapolations from two-dimensional to multidimensional calculus, and observe some of the curiosities unique to functions of more than one variable. #Science &
Free trial of The Great Courses Signature Collection or buy
2. S1 E2 - Functions of Several Variables
What makes a function "multivariable?" Begin with definitions, and then see how these new functions behave as you apply familiar concepts of minimum and maximum values. Use graphics and other
tools to observe their interactions with the xy-plane, and discover how simple functions such as y=x are interpreted differently in three-dimensional space.
Free trial of The Great Courses Signature Collection or buy
3. S1 E3 - Limits, Continuity, and Partial Derivatives
Apply fundamental definitions of calculus to multivariable functions, starting with their limits. See how these limits become complicated as you approach them, no longer just from the left or
right, but from any direction and along any path. Use this to derive the definition of a versatile new tool: the partial derivative.
Free trial of The Great Courses Signature Collection or buy
4. S1 E4 - Partial Derivatives - One Variable at a Time
Deep in the realm of partial derivatives, you'll discover the new dimensions of second partial derivatives: differentiate either twice with respect to x or y, or with respect once each to x and
y. Consider Laplace's equation to see what makes a function "harmonic."
Free trial of The Great Courses Signature Collection or buy
5. S1 E5 - Total Differentials and Chain Rules
Complete your introduction to partial derivatives as you combine the differential and chain rule from elementary calculus and learn how to generalize them to functions of more than one variable.
See how the so-called total differential can be used to approximate ?z over small intervals without calculating the exact values.
Free trial of The Great Courses Signature Collection or buy
6. S1 E6 - Extrema of Functions of Two Variables
The ability to find extreme values for optimization is one of the most powerful consequences of differentiation. Begin by defining the Extreme Value theorem for multivariable functions and use it
to identify relative extrema using a "second partials test," which you may recognize as a logical extension of the "second derivative test" used in Calculus I.
Free trial of The Great Courses Signature Collection or buy
7. S1 E7 - Applications to Optimization Problems
Continue the exploration of multivariable optimization by using the Extreme Value theorem on closed and bounded regions. Find absolute minimum and maximum values across bounded regions of a
function, and apply these concepts to a real-world problem: attempting to minimize the cost of a water line's construction.
Free trial of The Great Courses Signature Collection or buy
8. S1 E8 - Linear Models and Least Squares Regression
Apply techniques of optimization to curve-fitting as you explore an essential statistical tool yielded by multivariable calculus. Begin with the Least Squares Regression Line that yields the best
fit to a set of points. Then, apply it to a real-life problem by using regression to approximate the annual change of a man's systolic blood pressure.
Free trial of The Great Courses Signature Collection or buy
9. S1 E9 - Vectors and the Dot Product in Space
Begin your study of vectors in three-dimensional space as you extrapolate vector notation and formulas for magnitude from the familiar equations for two dimensions. Then, equip yourself with an
essential new means of notation as you learn to derive the parametric equations of a line parallel to a direction vector.
Free trial of The Great Courses Signature Collection or buy
10. S1 E10 - The Cross Product of Two Vectors in Space
Take the cross product of two vectors by finding the determinant of a 3x3 matrix, yielding a third vector perpendicular to both. Explore the properties of this new vector using intuitive
geometric examples. Then, combine it with the dot product from an earlier episode to define the triple scalar product, and use it to evaluate the volume of a parallelepiped.
Free trial of The Great Courses Signature Collection or buy
11. S1 E11 - Lines and Planes in Space
Turn fully to lines and entire planes in three-dimensional space. Begin by defining a plane using the tools you've acquired so far, then learn about projections of one vector onto another. Find
the angle between two planes, then use vector projections to find the distance between a point and a plane.
Free trial of The Great Courses Signature Collection or buy
12. S1 E12 - Curved Surfaces in Space
Beginning with the equation of a sphere, apply what you've learned to curved surfaces by generating cylinders, ellipsoids, and other so-called quadric surfaces. Discover the recognizable
parabolas and other 2-D shapes that lay hidden in new vector equations, and observe surfaces of revolution in three-dimensional space.
Free trial of The Great Courses Signature Collection or buy
13. S1 E13 - Vector-Valued Functions in Space
Consolidate your mastery of space by defining vector-valued functions and their derivatives, along with various formulas relating to arc length. Immediately apply these definitions to position,
velocity, and acceleration vectors, and differentiate them using a surprisingly simple method that makes vectors one of the most formidable tools in multivariable calculus.
Free trial of The Great Courses Signature Collection or buy
14. S1 E14 - Kepler's Laws - The Calculus of Orbits
Blast off into orbit to examine Johannes Kepler's laws of planetary motion. Then apply vector-valued functions to Newton's second law of motion and his law of gravitation, and see how Newton was
able to take laws Kepler had derived from observation and prove them using calculus.
Free trial of The Great Courses Signature Collection or buy
15. S1 E15 - Directional Derivatives and Gradients
Continue to build on your knowledge of multivariable differentiation with gradient vectors and use them to determine directional derivatives. Discover a unique property of the gradient vector and
its relationships with level curves and surfaces that will make it indispensable in evaluating relationships between surfaces in upcoming episodes.
Free trial of The Great Courses Signature Collection or buy
16. S1 E16 - Tangent Planes and Normal Vectors to a Surface
Utilize the gradient to find normal vectors to a surface, and see how these vectors interplay with standard functions to determine the tangent plane to a surface at a given point. Start with
tangent planes to level surfaces, and see how your result compares with the error formula from the total differential.
Free trial of The Great Courses Signature Collection or buy
17. S1 E17 - Lagrange Multipliers - Constrained Optimization
It's the ultimate tool yielded by multivariable differentiation: the method of Lagrange multipliers. Use this intuitive theorem and some simple algebra to optimize functions subject not just to
boundaries, but to constraints given by multivariable functions. Apply this tool to a real-world cost-optimization example of constructing a box.
Free trial of The Great Courses Signature Collection or buy
18. S1 E18 - Applications of Lagrange Multipliers
How useful is the Lagrange multiplier method in elementary problems? Observe the beautiful simplicity of Lagrange multipliers firsthand as you reexamine an optimization problem from an earlier
episode using this new tool. Next, explore one of the many uses of constrained optimization in the world of physics by deriving Snell's Law of Refraction.
Free trial of The Great Courses Signature Collection or buy
19. S1 E19 - Iterated integrals and Area in the Plane
With your toolset of multivariable differentiation finally complete, it's time to explore the other side of calculus in three dimensions: integration. Start off with iterated integrals, an
intuitive and simple approach that merely adds an extra step and a slight twist to one-dimensional integration.
Free trial of The Great Courses Signature Collection or buy
20. S1 E20 - Double Integrals and Volume
In taking the next step in learning to integrate multivariable functions, you'll find that the double integral has many of the same properties as its one-dimensional counterpart. Evaluate these
integrals over a region R bounded by variable constraints, and extrapolate the single variable formula for the average value of a function to multiple variables.
Free trial of The Great Courses Signature Collection or buy
21. S1 E21 - Double Integrals in Polar Coordinates
Transform Cartesian functions f(x.y) into polar coordinates defined by r and ?. After getting familiar with surfaces defined by this new coordinate system, see how these coordinates can be used
to derive simple and elegant solutions from integrals whose solutions in Cartesian coordinates may be arduous to derive.
Free trial of The Great Courses Signature Collection or buy
22. S1 E22 - Centers of Mass for Variable Density
With these new methods of evaluating integrals over a region, we can apply these concepts to the realm of physics. Continuing from the previous episode, learn the formulas for mass and moments of
mass for a planar lamina of variable density, and find the center of mass for these regions.
Free trial of The Great Courses Signature Collection or buy
23. S1 E23 - Surface Area of a Solid
Bring another fundamental idea of calculus into three dimensions by expanding arc lengths into surface areas. Begin by reviewing arc length and surfaces of revolution, and then conclude with the
formulas for surface area and the differential of surface area over a region.
Free trial of The Great Courses Signature Collection or buy
24. S1 E24 - Triple Integrals and Applications
Apply your skills in evaluating double integrals to take the next step: triple integrals, which can be used to find the volume of a solid in space. Next, extrapolate the density of planar lamina
to volumes defined by triple integrals, evaluating density in its more familiar form of mass per unit of volume.
Free trial of The Great Courses Signature Collection or buy
25. S1 E25 - Triple Integrals in Cylindrical Coordinates
Just as you applied polar coordinates to double integrals, you can now explore their immediate extension into volumes with cylindrical coordinates, moving from a surface defined by (r,?) to a
cylindrical volume with an extra parameter defined by (r,?,z). Use these conversions to simplify problems.
Free trial of The Great Courses Signature Collection or buy
Understanding Multivariable Calculus: Problems, Solutions, and Tips
Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Edwards, is the next step for students and professionals to expand their knowledge for
work or study in many quantitative fields, as well as an intellectual exercise for teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.
Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Edwards, is the next step for students and professionals to expand their knowledge for
work or study in many quantitative fields, as well as an intellectual exercise for teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.
Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Edwards, is the next step for students and professionals to expand their knowledge for
work or study in many quantitative fields, as well as an intellectual exercise for teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.
More info
None available
The Great Courses | {"url":"https://www.primevideo.com/detail/0FWQRHRBJCK91H2T1TLHSIMBKW/ref=atv_dp_amz_c_TS8274d9_1_2?jic=16%7CCgNhbGwSA2FsbA%3D%3D","timestamp":"2024-11-05T19:28:26Z","content_type":"text/html","content_length":"639574","record_id":"<urn:uuid:3da1e706-ca84-4f62-bd0b-5e7af94d175d>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00192.warc.gz"} |
Question ID - 57513 | SaraNextGen Top Answer
Answer Key / Explanation : (b)
Let total volume of iceberg
Volume of visible part of iceberg
Now, volume of water displaced by iceberg
Then, applying law of floatation, at equilibrium,
weight of iceberg = weight of displaced water | {"url":"https://www.saranextgen.com/homeworkhelp/doubts.php?id=57513","timestamp":"2024-11-05T10:15:39Z","content_type":"text/html","content_length":"18332","record_id":"<urn:uuid:d34c0311-19d4-41f5-9dee-fa38e1f00ef6>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00086.warc.gz"} |
Stationary and laminar incompressible flow in a square cavity (Reynolds number, Re = 1000). The top of the cavity is prescribed a tangential velocity while the sides and bottom are defined as no-slip
zero velocity walls.
This model is available as an automated tutorial by selecting Model Examples and Tutorials... > Fluid Dynamics > Flow in Driven Cavity from the File menu. Or alternatively, follow the step-by-step
instructions below. Note that the CFDTool interface differ slightly from the FEATool Multiphysics instructions described in the following.
1. To start a new model click the New Model toolbar button, or select New Model... from the File menu.
2. Select the 2D radio button.
3. Select the Navier-Stokes Equations physics mode from the Select Physics drop-down menu. (Note that for CFDTool the physics selection is done in the Equation settings dialog box.)
4. Press OK to finish the physics mode selection.
First create a unit square for the geometry.
1. To create a rectangle, first click on the Create square/rectangle Toolbar button. Then left click in the main plot axes window, and hold down the mouse button. Move the mouse pointer to draw the
shape outline, and release the button to finalize the shape.
2. Select R1 in the geometry object Selection list box.
3. To modify and edit the selected rectangle, click on the Inspect/edit selected geometry object Toolbar button to open the Edit Geometry Object dialog box.
4. Enter 0 into the x[min] edit field.
5. Enter 1 into the x[max] edit field.
6. Enter 0 into the y[min] edit field.
7. Enter 1 into the y[max] edit field.
8. Press OK to finish and close the dialog box.
9. Switch to Grid mode by clicking on the corresponding Mode Toolbar button.
10. Enter 0.02 into the Grid Size edit field.
11. Press the Generate button to call the grid generation algorithm.
12. Switch to Equation mode by clicking on the corresponding Mode Toolbar button.
13. Equation and material coefficients can be specified in Equation/Subdomain mode. In the Equation Settings dialog box that automatically opens, enter 1 for the fluid Density and umax/l/R for the
Viscosity. The other coefficients can be left to their default values. Press OK to finish the equation and subdomain settings specification. (Note that the Equation Settings dialog box may look
different for CFDTool.)
Note that FEATool works with any unit system, and that the units here are non-dimensionalized.
1. Press the Constants Toolbar button, or select the corresponding entry from the Equation menu, and enter the variables for the velocity umax, length l, and Reynolds number Re in the Model
Constants and Expressions dialog box.
Name Expression
umax 1
l 1
Re 1000
Boundary conditions consist of no-slip zero velocity conditions on all walls except for the top on which a constant x-velocity umax is prescribed.
1. Switch to Boundary mode by clicking on the corresponding Mode Toolbar button.
2. Select 4 in the Boundaries list box.
3. Select 1, 2, 3, and 4 in the Boundaries list box.
4. Select Wall/no-slip from the Navier-Stokes Equations drop-down menu.
5. Select 3 in the Boundaries list box.
6. Select Inlet/velocity from the Navier-Stokes Equations drop-down menu.
7. Enter umax into the Velocity in x-direction edit field.
8. Press OK to finish the boundary condition specification.
9. Now that the problem is fully specified, press the Solve Mode Toolbar button to switch to solve mode. Then press the = Tool button to call the solver with the default solver settings.
After the problem has been solved FEATool will automatically switch to postprocessing mode and here display the magnitude of the computed velocity field.
1. Press the Plot Options Toolbar button.
2. Select Vorticity from the Predefined surface plot expressions drop-down menu.
3. Select the Contour Plot check box.
4. Enter 30 into the Number or specified vector of contour levels to plot edit field.
5. Select the Arrow Plot check box.
6. Press OK to plot and visualize the selected postprocessing options.
To evaluate the accuracy of the solution the vorticity at (0.53, 0.56) is evaluated. Either click directly at this point or use the Point/Line Evaluation functionality.
1. Select Point/Line Evaluation... from the Post menu.
2. Select Vorticity from the Evaluation Expression drop-down menu.
3. Enter 0.53 into the Evaluation coordinates in x-direction edit field.
4. Enter 0.564 into the Evaluation coordinates in y-direction edit field.
5. Press OK to finish and close the dialog box.
The computed vorticity at the evaluated point is -1.73 which is quite close to the reference value of -2.068, to achieve a better approximation a finer grid and higher order discretization would be
The flow in driven cavity fluid dynamics model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB m-script text file (available as the example
ex_navierstokes2 script file), or GUI script (.fes) file.
[1] Botella O, Peyret R. Benchmark spectral results on the lid-driven cavity flow. Computers and Fluids 27(4):421-433, 1998.
[2] Erturk E, Corke TC, Gokcol C. Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Int- ernational Journal for Numerical Methods in Fluids 37(6):633-655,
[3] Nishida H, Satofuka N. Higher-order solutions of square driven cavity flow using a variable-order multi-grid method. International Journal for Numerical Methods in Engineering 34(2):637-653,
[4] Schreiber R, Keller HB. Driven cavity flows by efficient numerical techniques. Journal of Computational Physics 49(2):310-333, 1983. | {"url":"https://www.featool.com/doc/Fluid_Dynamics_02_driven_cavity1","timestamp":"2024-11-05T08:48:11Z","content_type":"application/xhtml+xml","content_length":"14732","record_id":"<urn:uuid:b91109b8-ffc0-4c04-bd30-819186f03918>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00687.warc.gz"} |
Uncovered set - electowikiUncovered setUncovered set
The minimal uncovered set (sometimes referred to as the "Landau set" or "Fishburn set") is defined for a set of rank-order preferences, and generalizes the Condorcet winner (making it a kind of "top
cycle"). The set contains all candidates on the "Pareto frontier" for pairwise-victories.
A Landau candidate will beat every non-Landau candidate one-on-one, and cannot be replaced by a "strictly better" candidate. "Strictly better" means a candidate that would win every pairwise matchup
won by the Landau candidate (and some additional matchups).
We assume here that there are no pairwise-ties. Let some set be called the Fishburn set, and the candidates outside the set are called the Fishburn losers. A Fishburn loser is a candidate who is
dominated or covered by some other candidate: the dominating candidate wins every pairwise matchup that the other candidate would win. The uncovered set is therefore equivalent to the set of Fishburn
Select the candidate or candidates that are not Fishburn losers. A candidate loser is a Fishburn loser if there exists some other candidate cover satisfying:
1. Every candidate that beats cover one-on-one also beats loser one-on-one, and
2. At least one candidate beats loser one-on-one but does not beat cover one-on-one.
The Fishburn winners are a kind of Pareto frontier for the set of candidates, where the frontier is measured by the pairwise-victories. It is impossible to gain some extra pairwise victories, but no
pairwise losses, by switching from a candidate in the Landau set to a candidate outside the Landau set.
The Landau set is a nonempty subset of the Smith set. It was discovered by Nicholas Miller.
An equivalent definition is that it is the set of every candidate X so that for any Y not in the set, X either beats Y pairwise or X beats someone who beats Y (i.e. X indirectly pairwise beats Y).^
[1] In this sense, it is related to the concept of a beatpath.
Another definition is:^[2]
An alternative a is said to cover alternative b whenever every alternative dominated by b is also dominated by a.
Yet another definition:^[3]
The uncovered set is the set of all outcomes x such that there is no outcome beating x and all the outcomes that x beats.
Select the candidate or candidates that are not Fishburn losers. A candidate i is a Fishburn loser if there is some other candidate j such that every candidate that pairwise beats j also pairwise
beats i and there is at least one candidate that pairwise beats i but does not pairwise beat j.
When there are pairwise ties, many generalizations are possible, all of which are equivalent when there are no pairwise ties.^[4] One generalization by Fishburn is:
In voting systems, the Landau set (or uncovered set, or Fishburn set) is the set of candidates x such that for every other candidate z, there is some candidate y (possibly the same as x or z)
such that y is not preferred to x and z is not preferred to y.
The uncovered set is a nonempty subset of the Smith set. The reason is that every candidate in the Smith set is preferred to every candidate not in the Smith set, therefore each candidate in the
Smith set can be considered a candidate x and be their own candidate y; since a candidate can't be preferred to themselves (y is not preferred to x), and since candidates in the Smith set being
preferred to every candidate not in the Smith set implies that candidates not in the Smith set are not preferred to candidates in the Smith set (z is not preferred to y), the uncovered set must be a
subset of the Smith set.
Formal definition
A more formal mathematical definition:
it is the set of candidates ${\displaystyle x}$ such that for every other candidate ${\displaystyle z}$, there is some candidate ${\displaystyle y}$ (possibly the same as ${\displaystyle x}$ or $
{\displaystyle z}$) such that ${\displaystyle y}$ is not preferred to ${\displaystyle x}$ and ${\displaystyle z}$ is not preferred to ${\displaystyle y}$. In notation, ${\displaystyle x}$ is in
the Landau set if ${\displaystyle \forall \,z}$, ${\displaystyle \exists \,y}$, ${\displaystyle x\geq y\geq z}$.
The uncovered set is based on the covering relation, which is a notion of a candidate being at least as good as another candidate (e.g. by beating everybody the other candidate beats, or by being
beaten by nobody who beats the other candidate). The uncovered set is then defined as the set of candidates who are not covered by anyone else.
For the Fishburn winner definition of the uncovered set, the covering relation is:
x covers y (x C y) if every candidate that beats x also beats y.
To be a proper covering relation, the relation should be transitive (if x covers y and y covers z, then x covers z) and antisymmetric (it's impossible to both cover x and be covered by x). This is
true for the Fishburn definition when there are no pairwise ties, but it has to be generalized if it's to retain the properties in the presence of pairwise ties.
When there are pairwise ties, one may also refer to weak and strict covering relations. The former drops antisymmetry, and is analogous to x beating or tying y, while the latter retains antisymmetry
and is analogous to x definitely beating y.^[4]
Suppose the following pairwise preferences exist between four candidates (v, x, y, z) (table organized by Copeland ranking):
x y v z Copeland score
x --- Win Lose Win (2-1)=1
y Lose --- Win Win (2-1)=1
v Win Lose --- Lose (1-2)=-1
z Lose Lose Win --- (1-2)=-1
Notice that the Smith set includes all candidates (this can be seen by observing that there is a beatpath of x>y>z>v>x, or alternatively by observing that no matter how many candidates you look at
from top to bottom, there is still some candidate outside of the group being looked at that one of the candidates in the group lose or tie to). But the uncovered set is all candidates except z; this
is because y>z and all candidates who beat y (just x) also beat z. ^[5] (Notice that the Copeland set is even smaller; it is just x and y).
An alternative way of understanding the uncovered set in this example is to show the size of the smallest-size beatpath from each candidate to another, if one exists (if x>y is 1 here, this means x
pairwise beats y. If it's 2, it means x pairwise beats someone who pairwise beats y, etc.). Any candidate with a smallest-size beatpath of 3 or more to another candidate is not in the uncovered set:
Size of smallest-size beatpath
between each pair of candidates
x y v z
x --- 1 Lose 1
y 2 --- 1 1
v 1 2 --- 2
z 2 3 1 ---
Notice that all candidates except z have beatpaths of size 1 or 2, whereas z>y is (z has a smallest beatpath to y of) 3 steps (z>v>x>y), therefore z is not in the uncovered set.
Subsets of the uncovered set
The Banks set, Copeland set, Dutta set, and Schattschneider set are all subsets of the uncovered set.^[6]
Banks set
The Banks set is the set of winners resulting from strategic voting in a successive elimination procedure^[7] (the set of candidates who could win a sequential comparison contest for at least one
ordering of candidates when voters are strategic).
The Banks set is a subset of the Smith set because when all but one candidate in the Smith set has been eliminated in a sequential comparison election, the remaining Smith candidate is guaranteed to
pairwise beat all other remaining candidates, since they are all non-Smith candidates, and thus can't be eliminated from that point onwards, meaning they will be the final remaining candidate and
thus win.
Determining if a given candidate is in the Banks set is NP-complete,^[8] but it is possible to find some member of the Banks set in polynomial time. One way to do so is to start with a candidate and
then keep inserting candidates in some order, skipping those whose insertion would produce a cycle; the winner of the method is then the candidate who pairwise beats every other included candidate.^
Dutta set
The Dutta set (also known as Dutta's minimal covering set) is the set of all candidates such that when any other candidate is added, that candidate is covered in the resulting set. It is a subset of
the Smith set because all candidates in the Smith set cover (i.e. have a one-step beatpath, direct pairwise victory) all candidates not in the Smith set. The Dutta set can be calculated in polynomial
Essential set
In a game where two players choose candidates and then the player who chose the candidate who beats the other candidate pairwise wins, there's a randomized strategy (a Nash equilibrium) where no
other strategy can be used against it to consistently win at this game. The essential set, a subset of the Dutta set, is the set of all candidates who are chosen some of the time when using a Nash
equilibrium strategy.^[10]
Minimal extending set
This section
needs expansion
. You can help by
adding to it
(April 2024)
The minimal extending set is a subset of the Banks set. It's relevant to strategic voting: narrowing the set of winners to this set when there is no Condorcet winner has not been shown to introduce
an incentive to strategically create a cycle when a sincere Condorcet winner exists.^[11]
A method electing from this set must fail monotonicity.^[12] However, the proof is nonconstructive and no concrete nonmonotonicity examples have been found so far.
Schattschneider set
The Schattschneider set is based on spatial voting games, and is a subset of the Banks set.^[13] It is rarely referenced.
The uncovered set can be thought of as requiring its candidates to have a two-step beatpath to every candidate not in the uncovered set. The Smith set requires a one-step beatpath (i.e. of at most
two candidates, a direct pairwise victory).
Independence of covered alternatives says that if one option (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the uncovered set. Independence of
covered alternatives implies Independence of Smith-dominated Alternatives, which further implies Smith and thus Condorcet. If a method is independent of covered alternatives, then the method fails
monotonicity if perfect ties can always be broken in favor of a choice W by adding ballots ranking W first.
The uncovered set implies Pareto, because Pareto implies that the Pareto-dominant candidate pairwise beats any candidates the Pareto-inferior candidate beats. This is because all voters rank the
Pareto candidate equal to or better than the Pareto-inferior candidate. ^[14]
One way that has been suggested to find the uncovered set is:
This suggests the use of the outranking [pairwise comparison] matrix and its square to identify the uncovered set (Banks, 1985):
T = U + U^2
where U [is] the tournament matrix. The alternatives represented by rows in T where all non-diagonal entries are non-zero form the uncovered set.^[15]
(The square of a matrix can be found using matrix multiplication; here is a video explaining how to do so. The pairwise matrix and its squared matrix can be added together using matrix addition.)
• Nicholas R. Miller, "Graph-theoretical approaches to the theory of voting", American Journal of Political Science, Vol. 21 (1977), pp. 769–803. https://doi.org/10.2307/2110736. https://
• Nicholas R. Miller, "A new solution set for tournaments and majority voting: further graph-theoretic approaches to majority voting", American Journal of Political Science, Vol. 24 (1980), pp.
68–96. https://doi.org/10.2307/2110925. https://www.jstor.org/stable/2110925.
• Norman J. Schofield, "Social Choice and Democracy", Springer-Verlag: Berlin, 1985.
• Philip D. Straffin, "Spatial models of power and voting outcomes", in Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, Springer: New York-Berlin, 1989, pp.
• Elizabeth Maggie Penn, "Alternate definitions of the uncovered set and their implications", 2004.
• Nicholas R. Miller, "In search of the uncovered set", Political Analysis, 15:1 (2007), pp. 21–45. https://doi.org/10.1093/pan/mpl007. https://www.jstor.org/stable/25791876.
• William T. Bianco, Ivan Jeliazkov, and Itai Sened, "The uncovered set and the limits of legislative action", Political Analysis, Vol. 12, No. 3 (2004), pp. 256–276. https://doi.org/10.1093/pan/
mph018. https://www.jstor.org/stable/25791775.
1. ↑ Munagala, Kamesh; Wang, Kangning (2019-05-04). "Improved Metric Distortion for Deterministic Social Choice Rules". arXiv.org. p. 5. doi:10.1145/3328526.3329550. Retrieved 2020-03-13.
2. ↑ Endriss, U. "Handbook of Computational Social Choice" (PDF). The Reasoner. 2 (10): 57. ISSN 1757-0522. Retrieved 2020-03-13.
3. ↑ Laffond, Gilbert; Laslier, Jean-François (1991). "Slaters's winners of a tournament may not be in the Banks set". Social Choice and Welfare. Springer. 8 (4): 365–369. ISSN 0176-1714. JSTOR
41105997. Retrieved 2022-09-11.
4. ↑ ^a ^b Miller, Nicholas M. "Alternate definitions of the covering relation: an extended tour" (PDF).
5. ↑ https://economics.stackexchange.com/a/27691
6. ↑ Seising, R. (2009). Views on Fuzzy Sets and Systems from Different Perspectives: Philosophy and Logic, Criticisms and Applications. Studies in Fuzziness and Soft Computing. Springer Berlin
Heidelberg. p. 350. ISBN 978-3-540-93802-6. Retrieved 2020-03-13.
7. ↑ http://spia.uga.edu/faculty_pages/dougherk/svt_13_multi_dimensions2.pdf
8. ↑ Woeginger, Gerhard J. (2003). "Banks winners in tournaments are difficult to recognize". Social Choice and Welfare. Springer. 20 (3): 523–528. ISSN 1432-217X. JSTOR 41106539. Retrieved
9. ↑ Hudry, Olivier (2004). "A note on 'Banks winners in tournaments are difficult to recognize' by G. J. Woeginger". Social Choice and Welfare. Springer Science and Business Media LLC. 23 (1). doi:
10.1007/s00355-003-0241-y. ISSN 0176-1714.
10. ↑ ^a ^b Brandt, Felix; Fischer, Felix (2008). "Computing the minimal covering set" (PDF). Mathematical Social Sciences. Elsevier BV. 56 (2): 254–268. doi:10.1016/j.mathsocsci.2008.04.001. ISSN
11. ↑ Botan, Sirin; Endriss, Ulle (2021-05-18). "Preserving Condorcet Winners under Strategic Manipulation". Proceedings of the AAAI Conference on Artificial Intelligence. Association for the
Advancement of Artificial Intelligence (AAAI). 35 (6): 5202–5210. doi:10.1609/aaai.v35i6.16657. ISSN 2374-3468.
12. ↑ Brandt, Felix; Harrenstein, Paul; Seedig, Hans Georg (2017). "Minimal extending sets in tournaments". Mathematical Social Sciences. Elsevier BV. 87: 55–63. doi:10.1016/j.mathsocsci.2016.12.007.
ISSN 0165-4896.
13. ↑ Feld, Scott L.; Grofman, Bernard; Hartly, Richard; Kilgour, Marc; Miller, Nicholas (1987). "The uncovered set in spatial voting games" (PDF). Theory and Decision. Springer Science and Business
Media LLC. 23 (2): 129–155. doi:10.1007/bf00126302. ISSN 0040-5833.
14. ↑ Kilgour, D (2010). Handbook of group decision and negotiation (PDF). Dordrecht New York: Springer. p. 176. ISBN 978-90-481-9097-3. OCLC 668097926. | {"url":"https://electowiki.org/wiki/Dutta_set","timestamp":"2024-11-09T22:54:53Z","content_type":"text/html","content_length":"100112","record_id":"<urn:uuid:206f5d97-5acf-4397-94ef-9919070d14c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00105.warc.gz"} |
Please note that the recommended version of Scilab is 2025.0.0. This page might be outdated.
See the recommended documentation of this function
Scilab Help >> Differential calculus, Integration > odeoptions
set options for ode solvers
This function interactively displays a command which should be executed to set various options of ode solvers. The context variable %ODEOPTIONS sets the options.
The ode function checks if this variable exists and in this case it uses it. For using default values you should clear this variable. To create it you must execute the instruction %ODEOPTIONS=
The variable %ODEOPTIONS is a vector with the following elements:
[itask, tcrit, h0, hmax, hmin, jactyp, mxstep, maxordn, maxords, ixpr, ml, mu].
The default value is: [1,0,0,%inf,0,2,500,12,5,0,-1,-1].
The meaning of the elements is described below.
• itask sets the integration mode:
□ 1: normal computation at specified times
□ 2 : computation at mesh points (given in first row of output of ode)
□ 3 : one step at one internal mesh point and return
□ 4 : normal computation without overshooting tcrit
□ 5 : one step, without passing tcrit, and return
• tcrit critical time used only if itask equals 4 or 5 as described above
• h0 first step tried
• hmax max step size
• hmin min step size
• jactype set jacobian properties:
□ 0: functional iterations, no jacobian used ("adams" or "stiff" only)
□ 1 : user-supplied full jacobian
□ 2 : internally generated full jacobian
□ 3: internally generated diagonal jacobian ("adams" or "stiff" only)
□ 4 : user-supplied banded jacobian (see ml and mu below)
□ 5 : internally generated banded jacobian (see ml and mu below)
• maxordn maximum non-stiff order allowed, at most 12
• maxords maximum stiff order allowed, at most 5
• ixpr print level, 0 or 1
• ml,mu
If jactype equals 4 or 5, ml and mu are the lower and upper half-bandwidths of the banded jacobian: the band is the i,j's with i-ml <= j <= ny-1.
If jactype equals 4 the jacobian function must return a matrix J which is ml+mu+1 x ny (where ny=dim of y in ydot=f(t,y)) such that column 1 of J is made of mu zeros followed by df1/dy1, df2/dy1,
df3/dy1, ... (1+ml possibly non-zero entries), column 2 is made of mu-1 zeros followed by df1/dx2, df2/dx2, etc.
In the following example, we solve the Ordinary Differential Equation dy/dt=y^2-y sin(t)+cos(t) with the initial condition y(0)=0 claiming the solution be stored at each mesh value.
function ydot=f(t, y)
clear %ODEOPTIONS
See also
• ode — ordinary differential equation solver | {"url":"https://help.scilab.org/docs/6.0.1/en_US/odeoptions.html","timestamp":"2024-11-05T21:54:52Z","content_type":"text/html","content_length":"16735","record_id":"<urn:uuid:4d2c4cc2-3c2c-461f-b8c0-c95a63e271b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00384.warc.gz"} |
ball mill 300 tonnes per day
The Red Rock Mill is estimated to run 200 tons per day (tpd) in a mill setting that can operate a gravity concentrating circuit followed by flotation plant. Mill feed will be sourced from a
Division permitted mine facility. Ore will be hauled by highway trucks from the mine source to the site ore stockpile, then transferred to a primary jaw ...
WhatsApp: +86 18838072829
Ball Mill Cost for 100 ton Cement Production . 300~5000 ton per day cement clinker grinding plant . alumina beneficiation cost and process Below is some information about the products equipment,
. 3000 mt of Alumina per day.
WhatsApp: +86 18838072829
Feed rate = 900 tons per day Concentrate production = 10 tons per day Assay of feed = % Cu, 1 gm/ton Au Assay of Concentrate = 25% Cu, 85 gms/ton Au Calculate: a) Cu and Au recoveries. 2) b) ...
In a ball mill, the distance between the midpoint of the mill and the surface of the balls is ft. ... the weight of the filter cake was ...
WhatsApp: +86 18838072829
The mill, which has 15 milling lines, recently expanded its "I" and "J" milling lines, with production capacity in each line increasing from 800 tonnes (wheat equivalent) per day to 1,200 tonnes
per day. Capacity in the "H" mill is in the process of being expanded from 800 to 1,200 tonnes, with the expansion expected to be completed ...
WhatsApp: +86 18838072829
The apparent difference in capacities between grinding mills (listed as being the same size) is due to the fact that there is no uniform method of designating the size of a mill, for example: a
5′ x 5′ Ball Mill has a working diameter of 5′ inside the liners and has 20 per cent more capacity than all other ball mills designated as 5′ x 5′ where the shell is 5′ inside diameter and ...
WhatsApp: +86 18838072829
Explain the role of ball mill in mineral industry and why it is extensively used. Describe different types of ball mill design. ... Bond estimated the amount of wear in terms of kilograms per kWh
based on the abrasion index, A;,, Wet Ball Mill = kg kWh = (A ) ;
WhatsApp: +86 18838072829
20 cubic meter per hour grinding mill Crusher South Africa . ball mill 100 tons daily per day . crusher price 5000 cubic meter per hours . Ball Mill of 100 Tons Per Hour In Peru Grinding Ball
Mill .
WhatsApp: +86 18838072829
The 911MPEPPGR426 is a small 300 kilo to 35 ton per 24 hour day capacity grinding mill acting primarily as a rod mill but can effortlessly be converted to a ball mill . Get Price. Ball Mill
Design/Power Calculation 911 Metallurgist. ... Ball Mill Size Of 500 Ton Per Day, Ball Mill.
WhatsApp: +86 18838072829
In practice, circulating loads between ball mills and classifiers are rarely less than 200 per cent and frequently exceed 700 per cent. It may appear strange at first sight that the circulating
load between a mill and classifier producing, for example, 100 tons of finished product per day can be as much as 400 tons or more, but this will be ...
WhatsApp: +86 18838072829
ADM Milling opened a new 30,000cwtperday flour mill in 2019 in Mendota, Illinois, US. Photo courtesy of ADM Milling. ... They process approximately million tonnes of wheat a year and produce
around million tonnes of flour and milled wheat products. ADM Milling, Parrish Heimbecker Ltd., Ardent Mills and Rogers Foods are the country ...
WhatsApp: +86 18838072829
A transparent mill (552 × 180 mm), made of Perspex material and filled with 10 mm balls, was used for the tests. Ball filling and mill fractional speed were kept constant respectively at 20% and
60% while slurry filling was varied from 0% to 300%. Using a high speed video camera, the motion of the charge was captured.
WhatsApp: +86 18838072829
Output Ball Size: 6mm 8mm. Briquette Shape: Cylindrical. read more... Brochure. Ronak Engineering. ... 50 Kg 1 Ton per Hour. read more... Energysure Eco Solutions Private Limited. Ghaziabad
SD292, Shastri Nagar, ... Biomass Wood Pellet Mill, 300 Kg/Hr To 10 Ton/Hr ₹ 1,00,000/ Piece Get Latest Price . Capacity: 300 Kg/Hr To 10 Ton/Hr ...
WhatsApp: +86 18838072829
The report contains process description and operational sequence of the day to day activities. Download Free PDF View PDF. Coal Preparation. Aimal Khan. xyz. Download Free PDF View PDF.
Introduction to Mineral Processing Design and Operation. Xavier R A. Download Free PDF View PDF.
WhatsApp: +86 18838072829
DOVE small Ball Mills designed for laboratories ball milling process are supplied in 4 models, capacity range of (200g/h1000 g/h). For small to large scale operations, DOVE Ball Mills are
supplied in 17 models, capacity range of ( TPH 80 TPH). With over 50 years experience in Grinding Mill Machine fabrication, DOVE Ball Mills as ...
WhatsApp: +86 18838072829
TECHNICAL SPECIFICATION OF WET BALL MILL EQUIPMENT (SUB ASSEMBLY OF FGD SYSTEM) 03 P V S BABU AMAN KHRK 02 P V S BABU AMAN KHRK ... for Wet Ball Mill system as per enclosed Annexure1 submit the
Annexure to qualification requirement (Attachment3K). Bidder's offers will be rejected if they fail to meet the QR. 6.
WhatsApp: +86 18838072829
Therefore, a 350hp motor is required and this in turn indicates that an 812 (8′ x 12′) Rod Mill is required. In the same way Fig. 3 indicates that an 800hp motor and a 101/214 or ′ by 14′ Ball
Mill is required. Following is a condensed tabulation of the above selections. Sizing a Ball or Rod Mill
WhatsApp: +86 18838072829
Assuming a bed porosity of 40%, estimate the mill capacity in tonnes per year. A ball mill is to produce a grind of 34 μm (P 80) product from a feed size of 200 μm at a rate of t/h. ... The feed
rate to the mill was 300 t/h. The ball mill grindability test at 65 mesh showed 12 kWh/t. The internal diameter of the ball mill was m ...
WhatsApp: +86 18838072829
The total specific energy input in Kwh per short ton, or the energy register, required to produce each product size, is listed in Col. (A) of Table I. Col. (B) lists the energy input required per
ton of rod mill feed, and Col. (C ) lists the energy required, per ton of ball mill feed. It is seen from Fig. 1 that the energy register in Col.
WhatsApp: +86 18838072829
The EXAMPLE Gold Mill is designed to process 500 tonnes per day of ore. The flowsheets are conventional and consist of two stages of crushing, single stage ball mill grinding, cyanidation and a
MerrillCrowe precipitation circuit for gold and silver recovery. In addition to the cyanide circuit, gold leach residue is treated in a flotation stage ...
WhatsApp: +86 18838072829
The feed rate to the ball mill is 65 tonnes/hour and the product size is 80% passing 300pm and 45% passing 75um. The work index of the ore is kWh/tonne. (i) (ii) Sketch a flowsheet of the
grinding circuit. [3] ... using Bonds'equation estimate the power cost in dollars per 24 hour day, required for comminution of the ore. List the ...
WhatsApp: +86 18838072829
The flowsheet in this study illustrates a typical 3,000 ton per day operation. ... The discharge from this primary rod mill is equally distributed to two ball mills which are in closed circuit
with SRL Rubber Lined Pumps and two or more cyclone classifiers. The rod mill and two ball mills are approximately the same size for simplified ...
WhatsApp: +86 18838072829
Automatic cement plant and machinery, capacity: 100 tpd to 1... Automatic cement bag making machine, 157 (kw), 150 370 (kw... 50 HP To 1200 HP Mild Steel Ball Mill Balls, For Chemical Industry,
Up To 3000 MM. ₹ 7,50,000/ Piece Get Latest Price.
WhatsApp: +86 18838072829
Power boDer Most mills have a power boiler using waste wood (hog fuel) as the primary fuel, backed by oil ornatural gas. These units are usually in the size range of 100 to 300 tons steam per
hoor and many are very similar in design to stoker rued coal units, although bubbling fluid bed units are becoming more commOD.. Dry wood has a calorific value ofaround 18,500 to 20,000 kJ/kg
(8,000 to ...
WhatsApp: +86 18838072829
ball mill for 6 ton per day Gold Ore Crusher. ball mills for handling quartzs capacity 6 ton hr Coal . Crushing and material handling/processing, in a plant for 30 ton to 70 ton/per day plant.
... Related to 1000 tons capacity ball mill for gold ore. . 1 ton an hr ore milling machine. . at 300 tons of ball mill material per day
WhatsApp: +86 18838072829
Grinding Mills; Gold Recovery Equipment; Flotation Machines; ... = 2800 x x x x 225 x ()/60 = 525 kg/hr = 13 TPD vs 300 to 420 TPD —> I can't make this work ... calculating the volume of one
swing of a jaw, times the density of material in the chamber, times the number of cycles per minute. This should give you the mass of ...
WhatsApp: +86 18838072829
An autogenous mill of ID m and an effective grinding length of m was fed with ore of SG to 20% of its volume. The mill was operated continuously 24 hours per day at 1200 t per day and 75% of the
critical speed. The solids in the mill charge were at 80% solids. Estimate: 1.
WhatsApp: +86 18838072829
Cattle Feed Plant, 300 Kg Per Hr ₹ 1,85,000. Get Quote. Feed Pellet Making Machine ₹ 87,000. Get Quote. Watch Video. Poultry Feed Grinder, 1000 kg per hr ... Vertical Pellet Mill, 30 Ton Per Day
₹ 4,75,000. Get Quote. Paddy Straw Pellet Plant, 1 Tph ₹ 80,00,000. Get Quote. Automatic Vertical Rice Bran Pellet Making Machine ₹ 5,50,000.
WhatsApp: +86 18838072829
Contribute to yunan88/crusher development by creating an account on GitHub.
WhatsApp: +86 18838072829
Now, 1 tonne of REM supplied to the mill contains t (185 kg) of neodymium. With 59 tonnes of ore mined per tonne of REM, the amount of ore mined and supplied to the mill to get 1 tonne of
neodymium is 59/ = tonnes. The mill only recovers 50% of the metal supplied to it. Therefore, for the mill to recover 1 tonne of neodymium ...
WhatsApp: +86 18838072829 | {"url":"https://www.auberges-rurales.fr/2024/Mar/30-9255.html","timestamp":"2024-11-04T02:43:03Z","content_type":"application/xhtml+xml","content_length":"26945","record_id":"<urn:uuid:4b82d79a-012f-4d50-8042-a2ee38d7a4f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00311.warc.gz"} |
The Fundamental Theorem for Line Integrals
The Fundamental Theorem for Line Integrals
Recall from The Fundamental Theorem of Calculus Part 2 page that if $f$ is a continuous function on the interval $[a, b]$ then $\int_a^b f(x) \: dx = F(b) - F(a)$ where $F$ is any antiderivative of
$f$ (that is $F' = f$).
The following theorem known as The Fundamental Theorem for Line Integrals or the Gradient Theorem is an analogue of the Fundamental Theorem of Calculus Part 2 for line integrals.
Theorem 1 (The Fundamental Theorem for Line Integrals / The Gradient Theorem): Let $C$ be a smooth curve that is parameterized by the vector equation $\vec{r}(t)$ for $a ≤ t ≤ b$, and suppose that
$f$ is a differentiable function and the gradient of $f$ is continuous on the curve $C$. Then $\int_C \nabla f \cdot \: d \vec{r} = f(\vec{r}(b)) - f(\vec{r}(a))$.
We will prove Theorem 1 in the cases where $f$ is a function of two variables and $f$ is a function of three variables. Of course, Theorem 1 holds for functions with more than $3$ variables.
Nevertheless, the proofs are rather simpler and follow the same format.
• Proof (Two Variable Case): Let $z = f(x, y)$ be a two variable real-valued function that is differentiable and suppose that $\nabla f (x, y)$ is continuous on the smooth curve $C$ parameterized
by $\vec{r}(t) = (x(t), y(t))$ for $a ≤ t ≤ b$. Then:
\quad \int_C \nabla f \cdot d \vec{r} = \int_a^b \nabla f(\vec{r}(t)) \cdot \vec{r'}(t) \: dt \\ \quad \int_C \nabla f \cdot d \vec{r} = \int_a^b \left ( \frac{\partial f}{\partial x}, \frac{\partial
f}{\partial y} \right) \cdot \left ( \frac{dx}{dt}, \frac{dy}{dt} \right ) \: dt \\ \quad \int_C \nabla f \cdot d \vec{r} = \int_a^b \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\
partial y} \frac{dy}{dt} \: dt \\
• Now notice that we can compress the integrand in the right integral by noticing that from the chain rule we have that $\frac{d}{dt} ( f(\vec{r}(t)) = \frac{d}{dt} (f(x(t), y(t)) = \frac{\partial
f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$ and so:
\quad \int_C \nabla f \cdot d \vec{r} = \int_a^b \frac{d}{dt} \left ( f (\vec{r}(t)) \right ) \: dt
• Now we apply the Fundamental Theorem of Calculus for single integrals to get that:
\quad \int_C \nabla f \cdot d \vec{r} = \left [ f(\vec{r}(t)) \right ]_{t=a}^{t=b} \\ \quad \int_C \nabla f \cdot d \vec{r} = f(\vec{r}(b)) - f(\vec{r}(a)) \quad \blacksquare
• Proof (Three Variable Case): Let $w = f(x, y, z)$ be a three variable real-valued function that is differentiable and uppose that $\nabla f(x, y)$ is continuous on the smooth curve $C$
parameterized by $\vec{r}(t) = (x(t), y(t), z(t))$ for $a ≤ t ≤ b$. Then:
\quad \int_C \nabla f \cdot d \vec{r} = \int_a^b \nabla f(\vec{r}(t)) \cdot \vec{r'}(t) \: dt \\ \quad \int_C \nabla f \cdot d \vec{r} = \int_a^b \left ( \frac{\partial f}{\partial x}, \frac{\partial
f}{\partial y}, \frac{\partial f}{\partial z} \right ) \cdot \left (\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right ) \: dt \\ \quad \int_C \nabla f \cdot d \vec{r} = \int_a^b \frac{\partial f}{\
partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt} \: dt
• Now notice that we can compress the integrand in the right integral by noticing that from the chain rule we have that $\frac{d}{dt} ( f(\vec{r}(t)) = \frac{d}{dt} (f(x(t), y(t), z(t)) = \frac{\
partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}$ and so:
\quad \int_C \nabla f \cdot d \vec{r} = \int_a^b \frac{d}{dt} \left ( f(\vec{r}(t) \right ) \: dt \\
• Now we apply the Fundamental Theorem of Calculus for single integrals to get that:
\quad \int_C \nabla f \cdot d \vec{r} = \left [ f(\vec{r}(t)) \right ]_{t=a}^{t=b} \\ \quad \int_C \nabla f \cdot d \vec{r} = f(\vec{r}(b)) - f(\vec{r}(a)) \quad \blacksquare
Corollary 1: Let $f$ and $g$ be scalar functions whose first partial derivatives are continuous in $D$. If $C$ is a piecewise smooth curve from the point $P$ to the point $Q$ in $D$, then $\int_C f \
nabla g \cdot d \vec{r} + \int_C g \nabla f \cdot d \vec{r} = f(Q)g(Q) - f(P)g(P)$.
\quad \nabla (fg) = \frac{\partial}{\partial x} fg \vec{i} + \frac{\partial}{\partial y} fg \vec{j} + \frac{\partial}{\partial z} fg \vec{k} \\ \quad \nabla (fg) = \left ( f \frac{\partial g}{\
partial x} + g \frac{\partial f}{\partial x} \right ) \vec{i} + \left ( f \frac{\partial g}{\partial y} + g \frac{\partial f}{\partial y} \right ) \vec{j} + \left ( f \frac{\partial g}{\partial z} +
g \frac{\partial f}{\partial z} \right ) \vec{k} \\ \quad \nabla (fg) = f \nabla g + g \nabla f
• Therefore by applying the Fundamental Theorem for Line Integrals we have that:
\quad \int_C f \nabla g \cdot d \vec{r} + \int_C g \nabla f \cdot d \vec{r} = \int_C \nabla (fg) \cdot d \vec{r} \\ \quad \int_C f \nabla g \cdot d \vec{r} + \int_C g \nabla f \cdot d \vec{r} = (fg)
(Q) - (fg)(P) \\ \quad \int_C f \nabla g \cdot d \vec{r} + \int_C g \nabla f \cdot d \vec{r} = f(Q)g(Q) - f(P)g(P) \quad \blacksquare | {"url":"http://mathonline.wikidot.com/the-fundamental-theorem-for-line-integrals","timestamp":"2024-11-13T17:31:40Z","content_type":"application/xhtml+xml","content_length":"20500","record_id":"<urn:uuid:4553b8d9-30c4-48a2-a4c8-ff25fec4625d>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00248.warc.gz"} |
SCV0631 - INTELLIGENT SYSTEMS
Dati Generali
Periodo di attività
Primo Semestre (23/09/2024 - 20/12/2024)
Obiettivi Formativi
The course provides broad coverage of intelligent systems solving pattern recognition problems. Theoretical concepts in intelligent systems and techniques relevant to real-life applications will be
The student will be able to:
1. Know the main objectives and areas of Artificial Intelligence, Machine Learning, and Pattern Recognition, with the ability to identify the potentialities of intelligent techniques and the
relationships with other disciplines
2. Know the basic concepts of automated learning based on machine learning approaches and the conditions for their applicability
3. Know the most relevant feature extraction and selection techniques
4. Know statistical techniques and their limitations and strengths, with the ability to appropriately select the proper technique in specific contexts
5. Know basic principles of neural computing and their characteristics
6. Know Flat and Hierarchical Clustering with the ability to configure and apply these methods in specific contexts
7. Know performance metrics for learners
8. Know basic concepts of the following application domains: Image Classification, Text Categorization, Biomedical Data Analysis
9. Know how to program in a language for statistical computing and machine learning applications like R
It is also expected that students develop communicative skills through open discussion and autonomous assessment in the choice of the proper technique to solve problems of recognition and /or
automatic classification of multidimensional data in several domains.
Students will acquire also knowledge of the relevant Machine learning and Pattern Recognition terminology.
The course assumes that students have a background acquired in a Bachelor's Degree in STEM disciplines. Students are expected to be familiar with basic Mathematics, Probability, and Statistics.
Metodi didattici
Lectures (72 hours)
The topics of the course are illustrated by means of (1) conceptual, formal descriptions, (2) their implementation via R code, and (3) the use of demos and online resources.
Constant interaction with the students and their involvement in open discussions are highly encouraged.
Verifica Apprendimento
The students’ learning extent is assessed via a written test (duration: 2 hours) and an individual assignment, autonomously developed by each student individually.
The goal of the written test is to assess the learning degree and the understanding of the elements related to intelligent systems from both theoretical and application (on problems of limited
complexity) points of view. Written tests normally consist of
- two exercises for the assessment of the student's understanding and knowledge of machine learning techniques: each exercise weighs about one-quarter of the grade of the written exam;
- four questions on the conceptual aspects: each exercise weighs about one-eighth of the grade of the written exam.
The assignment allows the students to use their skills and knowledge for the building and evaluation of machine learners by using the R language. The project presentation has the goal of assessing
the students’ communication skills in two areas: 1) the students’ technical competencies and use of the correct terminology and 2) the students’ skills for communicating a complete and organized view
of the work they carried out.
Individual judgment skills are evaluated based on the decisions made during the written exam and the assignment.
The grade of the written test is on a 0 to 30 scale. The written exam contributes 70% of the final mark, while the assignment accounts for the remaining 30%.
The acquisition of knowledge and expected skills is developed throughout the entire course, which includes the topics listed below.
1) Introduction to Artificial Intelligence and Pattern Recognition: Historical Perspective, State of the Art of Methods and Applications (3 h - Course Objective 1)
2) Basic Mathematical and Statistical Concepts:
• Measurement Theory
• Matrix algebra
• Multivariable function analysis calculus: partial derivatives and gradient
• Relevant concepts of Probability and Statistics
• Relevant concepts of Information Theory
(6 h - Course Objective 2)
3) Design of a Supervised Classifier; Basic principles of learning by example; basic concepts of multidimensional pattern analysis
(5 h - Course Objective 2)
4) Feature Extraction and Selection:
• Principal Component Analysis
• Information Gain
• Statistical evaluation of features
• Selection Strategies
(6 h - Course Objective 3)
5) Fundamental Elements of Programming with R
(8 h - Course Objective 9)
6) Machine Learning Algorithms
• Ordinary Least Squares Regression
• Outliers and Robust Regression
• Regularization and Shrinkage in Regression: Ridge Regression, LASSO Regression, Elastic Net
• Minimum distance classifier
• Bayesian classification
• Maximum likelihood classifier
• K-Nearest Neighborhood classifier
• Parallelepiped Method
• Decision trees
• Ensemble models: boosting, bagging, stacking
• Support Vector Machine
• Imbalance, Hyperparameter tuning
• Performance metrics
• (32 h - Course Objectives 2, 3, 4, 7)
7) Neural Networks
• Introduction, taxonomy
• Basic principle of neural computing
• Feedforward Neural Models
• Application Examples
• Introduction to Deep Learning
(6 h - Course Objective 5)
8) Clustering
• Introduction to Clustering
• K-means Clustering algorithm
• Agglomerative Hierarchical Clustering: Single linkage, Complete linkage
(3 h - Course Objective 6)
9) Design of Intelligent Systems, Examples in Application Domains
(3 h - Course Objectives 1, 8)
Altre informazioni
During the period in which the course is held, the students can meet with the instructor on class days. During the remainder of the year, the students need to contact the instructor to set up an
appointment by e-mail at sandro.morasca@uninsubria.it. The instructor responds only to e-mail messages sent from the official student.uninsubria.it e-mail accounts. | {"url":"https://uninsubria.unifind.cineca.it/individual?uri=http%3A%2F%2Firises.uninsubria.it%2Fresource%2Faf%2F297682","timestamp":"2024-11-12T12:48:45Z","content_type":"text/html","content_length":"46382","record_id":"<urn:uuid:b43c4e36-07de-4ec9-8c50-0f327afb8c34>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00490.warc.gz"} |
Isoperimetric Polyominoes
Similarly if we used an octomino twice we are able to form the 11x59 rectangle above. Also if we omit the 3x4 rectangle from the set we can form a 17x37 rectangle.
Probably the hardest problem for this set is to form simultaneous squares. These solutions were produced by computer using Gerard's Universal Polyomino Solver. The first took under 30 minutes, the
second about 20 and the third about 15 although each had to be given different piece orders to achieve these times.
The above shows solutions with 9, 10, 11 and 12 squares. The question naturally arises - what is the maximum number of squares which can be made with this set? There are over 2000 ways of expressing
641 as the sum of squares even when we reject any with a 1, 2 or 3 and only allow a maximum of four 4s which is all the set can produce.
Patrick Hamlyn has found 15 squares.
If, however, we look for squares which are all different then we find there are just 19 possible sets of squares.
We could, alternatively, look for squares with a single hole. One solution is shown here.
The one-sided polyominoes of perimeter 14 can form a 12 x 97 rectangle. This solution was found using Gerard's Universal Polyomino Solver which got the solution in about 20 seconds after a little
careful planning. The second solution which consists of two 6x97 rectangles was found using Peter Esser's solver which can be downloaded from his site.
Other symmetrical constructions are also possible as well as sets of twelve congruent pieces.
There is a also a large number of sets of squares which could be made with this set. Examples with 5, 13 and 17 squares are show here.
The maximum number of different squares which could be made is ten with two possible sets - 4,5,6,7,8,9,12,13,16,18 and 4,5,6,8,9,11,12,14,15,16.
We can also look at sets of pieces with the same perimeter and area. The perimeter 14 rectangles with areas 6, 7, 8 and 9 can for a number of rectangles.
The one sided sets can also produce a number of rectangles. | {"url":"http://www.recmath.com/PolyPages/PolyPages/Isopolyo14s.html","timestamp":"2024-11-08T18:38:35Z","content_type":"text/html","content_length":"7121","record_id":"<urn:uuid:c573270d-54cd-4bd6-964c-50e151da6310>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00253.warc.gz"} |
LM317 Resistor / Voltage Calculator - Calculator6.com
LM317 Resistor / Voltage Calculator
The LM317 Resistance/Voltage Calculator is a tool used to calculate resistance and output voltage values in circuits made with the LM317 voltage regulator integrated.
The LM317 Resistance/Voltage Calculator is a tool used to calculate resistance and output voltage values in circuits made with the LM317 voltage regulator integrated. This calculator helps you
quickly and easily determine the resistance and voltage values required for the correct operation of the LM317.
When using the online LM317 resistance and voltage calculator you can calculate Voltage Vout by entering: voltage, Resistor R1 and Resistor R2.
2 Number of Calculations Used Today
How to Calculate LM317 Resistor / Voltage?
The LM317 voltage regulator uses a reference voltage and two resistors to provide a specific output voltage. Here are the general steps for LM317 Resistance/Voltage calculation:
1. Determine the Output Voltage: Determine the output voltage you need. This is determined depending on the application in which your circuit will operate.
2. Selecting the Reference Voltage: The reference voltage of the LM317 is 1.25V. This is a fixed value and is not usually changed.
3. Calculating Resistance Values: The relationship between the output voltage (V_out) of the LM317, the reference voltage (1.25V) and the values of the resistors (R1 and R2) is expressed by the
following formula:
V_{out} = 1.25V \times \left(1 + \frac{R2}{R1}\right)
Knowing the desired output voltage and reference voltage, the resistor values R1 and R2 are calculated. This calculation is usually done for a circuit where R1 is fixed and R2 is adjustable.
4. Selecting the Resistor: Select the standard resistor values closest to the calculated resistor values. Standard resistor values can have standard series, such as E12 or E24.
5. Connecting the Circuit: Build and test the circuit using the selected resistor values. The output voltage of the circuit should be as close as possible to the calculated value.
The LM317 Resistance/Voltage calculation is used in the process of designing and analyzing circuits made with the LM317 voltage regulator. By determining the correct resistor values, this calculation
helps you to achieve the desired output voltage and ensure the correct circuit operation.
What is LM317 Resistor / Voltage?
The LM317 is a popular linear voltage regulator integration and is used to provide a constant output voltage. The LM317 Resistance/Voltage is a circuit structure used in conjunction with the LM317
voltage regulator. This structure is used to determine the resistor values required for the LM317 to provide a specific output voltage.
The LM317 Resistance/Voltage circuit usually consists of two resistors: R1 and R2. These resistors are used to determine the output voltage of the LM317. The resistor values are calculated based on
the relationship between the output voltage of the LM317 and the reference voltage (usually 1.25V).
The LM317 Resistance/Voltage circuit is commonly used in many electronic applications where the LM317 voltage regulator is used. This circuit is used to determine the resistance values required to
supply a given voltage in a constant manner. In this way, the LM317 is ensured to provide the desired output voltage and the circuit works as desired.
Uses of LM317 Resistor / Voltage Calculation
The uses of LM317 Resistance/Voltage calculation are as follows:
• Electronic Power Supplies: The LM317 voltage regulator is used in electronic power supplies to provide the desired output voltage. The Resistance/Voltage calculation is used to determine the
output voltage of the power supply and ensures that the power supply operates as desired.
• Lighting Systems: In LED drivers and other lighting systems, the LM317 Resistance/Voltage calculation is used to provide the output voltage required to determine the lighting level. This ensures
that the lighting systems provide the desired brightness level.
• Solar Energy Systems: In solar energy systems such as solar panels and charge control circuits, the LM317 Resistance/Voltage calculation is used to provide the appropriate output voltage required
for battery charging. This ensures that solar power systems provide the correct voltage for battery charging.
• Telecommunication Devices: In telecommunication devices, the LM317 Resistance/Voltage calculation is used to determine the power supply voltage of the devices. This ensures stable operation and
reliable performance of the devices.
• Automotive Electronics: In vehicles, the LM317 Resistance/Voltage calculation is used to determine the output voltage of voltage regulators used in automotive electronics systems. This ensures
the correct operation of vehicle electronics systems.
LM317 Resistance/Voltage calculation is essential for the correct operation of the LM317 voltage regulator used in many electronic applications. This calculation method is important to determine the
desired output voltage and to ensure that electronic circuits operate as desired.
LM317 Resistor and Voltage Calculation Considerations
Here are some important points to be considered in LM317 Resistance/Voltage calculation:
Reference Voltage:
The reference voltage of the LM317 is a constant 1.25V. This value must be taken into account and used correctly when calculating.
Desired Output Voltage:
The output voltage value to be used in the calculation should be determined depending on the requirements of the application and the characteristics of the circuit to be used.
Resistor Tolerances:
The tolerance values of resistors must be considered. Standard resistor tolerances are usually 5% or 10%. These tolerances indicate that the calculated resistance values may differ from the values
obtained in practice.
Minimum and Maximum Current Ratings:
The minimum and maximum current ratings of the LM317 must be taken into account in the calculation. These values are important for safe operation of the circuit.
Heat Dissipation:
The LM317 voltage regulator can generate high power dissipation and heat. Therefore, it is important to consider the heat dissipation and cooling requirements of the LM317 when making calculations.
The calculated resistance values can affect the stability and stability of the LM317 voltage regulator. Therefore, it is important to select the correct resistor values to ensure stable operation of
the circuit under the desired operating conditions.
Circuit Connection:
Proper connection and assembly of the LM317 Resistance/Voltage circuit ensures correct circuit operation. Connection errors and short circuits can affect the accuracy of the calculation.
These considerations ensure that the LM317 Resistance/Voltage calculation provides accurate and reliable results. It is important to keep these points in mind when designing and analyzing electronic | {"url":"https://www.calculator6.com/lm317-resistor-voltage-calculator/","timestamp":"2024-11-06T17:54:33Z","content_type":"text/html","content_length":"275819","record_id":"<urn:uuid:a08c8cdf-f6e3-4b8d-b548-920243b3e08f>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00602.warc.gz"} |
How to Add Rows With Missing Dates In Pandas Dataframe?
To add rows with missing dates in a pandas DataFrame, you can first create a new DataFrame with the complete range of dates that you want to include. Then you can merge this new DataFrame with your
existing DataFrame using the "merge" function in pandas. This will add rows with missing dates to your original DataFrame. Make sure to specify the correct columns to merge on and how you want to
handle any missing data during the merge process. This approach will allow you to efficiently add rows with missing dates to your pandas DataFrame.
What is the purpose of axis parameter in pandas functions?
The axis parameter in pandas functions is used to specify the axis along which a particular operation should be carried out. The value of the axis parameter can be either 0 or 1, where 0 refers to
operations along the rows or index labels, and 1 refers to operations along the columns or column labels.
For example, when using the drop() function in pandas to drop rows or columns from a DataFrame, specifying the axis parameter allows you to specify whether you want to drop rows (axis=0) or columns
(axis=1). Similarly, when using the sum() function to calculate the sum of values in a DataFrame, specifying the axis parameter allows you to specify whether you want to calculate the sum along rows
or columns.
In summary, the purpose of the axis parameter in pandas functions is to provide a way to control the direction of the operation being applied to the DataFrame, either along rows or columns.
What is the use of groupby function in pandas?
The groupby function in Pandas is used to split the data into groups based on some criteria, apply a function to each group independently, and then combine the results into a new data structure. This
function is commonly used in data analysis and manipulation to perform tasks such as:
1. Aggregating data: Grouping similar data together and then computing summary statistics to analyze patterns and trends within each group.
2. Transformation: Applying a function to each group of data independently to modify or manipulate the values within the group.
3. Filtering: Filtering data based on some conditions within each group.
Overall, the groupby function is a powerful tool that allows users to segment and analyze their data more effectively and efficiently.
What is the use of value_counts function in pandas?
The value_counts function in pandas is used to count the occurrences of unique values in a Series. It returns a Series containing the counts of each unique value in the input Series, sorted in
descending order by default. This function is useful for understanding the distribution of values in a dataset and identifying the most common values.
What is the difference between append and concat in pandas?
In pandas, both append and concat functions are used for combining two dataframes, but they have some differences:
1. append:
• The append function is a method of the DataFrame class that is used to append a row or another DataFrame to the existing DataFrame.
• It is used to append rows in the vertical direction, one below the other.
• It returns a new DataFrame with the appended data, without modifying the original dataframes.
1. concat:
• The concat function is a standalone function in pandas that is used to concatenate multiple dataframes along a specified axis.
• It can be used to combine dataframes in both vertical (along rows) and horizontal (along columns) direction.
• It returns a new DataFrame with the concatenated data, without modifying the original dataframes.
In summary, append is used to add rows to a DataFrame while concat is used to combine dataframes along an axis, either vertically or horizontally.
How to calculate summary statistics in pandas?
To calculate summary statistics in pandas, you can use the describe() method. This method provides a quick overview of the numerical data in a DataFrame or Series, including count, mean, standard
deviation, minimum, maximum, and various quantiles.
Here's an example of how to use the describe() method:
1 import pandas as pd
3 # Create a sample DataFrame
4 data = {'A': [1, 2, 3, 4, 5],
5 'B': [10, 20, 30, 40, 50]}
7 df = pd.DataFrame(data)
9 # Calculate summary statistics
10 summary_stats = df.describe()
12 print(summary_stats)
This will output a summary of the numerical data in the DataFrame df, including count, mean, std, min, 25%, 50%, 75%, and max for each column.
You can also calculate summary statistics for specific columns by selecting those columns before applying the describe() method, like this:
1 # Calculate summary statistics for specific columns
2 summary_stats_specific = df[['A']].describe()
4 print(summary_stats_specific)
This will output the summary statistics for column 'A' only.
How to filter rows in pandas based on a condition?
To filter rows in pandas based on a condition, you can use the loc or iloc methods along with a boolean condition. Here is an example:
1 import pandas as pd
3 # Create a sample DataFrame
4 data = {'A': [1, 2, 3, 4, 5],
5 'B': [10, 20, 30, 40, 50]}
6 df = pd.DataFrame(data)
8 # Filter rows where column 'A' is greater than 2
9 filtered_df = df.loc[df['A'] > 2]
11 print(filtered_df)
In this example, df['A'] > 2 creates a boolean Series where each value indicates whether the condition is True or False for that row. By passing this boolean Series into the loc method, we can filter
the rows where the condition is true. | {"url":"https://stlplaces.com/blog/how-to-add-rows-with-missing-dates-in-pandas","timestamp":"2024-11-09T02:44:10Z","content_type":"text/html","content_length":"331679","record_id":"<urn:uuid:88a21c64-4e60-4337-b0f1-dd625b5e73da>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00058.warc.gz"} |
Experimental study of the effect of vertical acceleration component on the slope stability
The paper deals with the stability of earth dams subjected to seismic and paraseismic excitations occurring in the close distance to the dam. The Newmark’s stability approach, classified as a
simplified dynamic method, has been extended for two directional dynamic loading case and the evolution of the friction coefficient. The paper is focused on the experimental verification of the
proposed method for harmonic and cyclic excitations generated by a simple oscillating device as well as by one-directional shaking table. Main assumptions of the proposed method have been proved and
the effect of the vertical acceleration on the stability of the dam has been demonstrated. In the experiments conducted under cyclic loading, the impact of the vertical acceleration on the permanent
displacement of the block was equal to 57 %. However, in general, the exact magnitude of this impact depends on many factors, such as the vertical to horizontal peak ground acceleration ratio or
frequency range of the vertical component of the earthquake.
• Simple slope model firstly proposed by Newmark is considered
• The Newmark’s method has been extended
• The variation of the resistance force during the dynamic load has been proved
• The behavior of block during the one- and two directional cyclic loading has been examined
• The influence of the friction coefficient evolution on the relative block motion has been proved
• Very good agreement between experimental and numerical results has been obtained
1. Introduction
The paper is focused on the evaluation of stability of earth dams subjected to seismic and paraseismic excitations. Commonly, the horizontal component of dynamic load is considered in engineering
practice. However, in particular conditions, e.g. in a close distance (dozens of kilometres) to the epicentre of the earthquake, the magnitudes of peak vertical and horizontal components of
accelerations may reach similar values. Therefore, the hypothesis has been made that the vertical accelerations’ parameters of earthquakes can also influence the risk of destabilization of the earth
General guidelines and review of methods commonly used for stability assessment of an earth dam subjected to dynamic loading coming from natural earthquakes or induced by mining tremors can be found
in [1-4]. In the presented paper, the Newmark’s stability approach, classified as a simplified dynamic method, has been considered as the most appropriate method for examining the assumed hypothesis.
The permanent displacement approach, proposed by Newmark in 1965, is still widely used due to its simplicity and intuitive measure of slope stability [5]. However, the main assumptions and, in
consequence, its constraints in applicability should be considered. The Newmark’s block stability method is based on concept of so-called limit equilibrium method, thus, it follows its main
hypothesis: rigidity of the soil body, existence of a slip surface, the limit state occurrence along the slip surface, and uniform displacements along the slip surface. Furthermore, resisting forces
of a potentially unstable wedge satisfy Coulomb’s law and the shear strength results from static gravity loading only. Newmark has also assumed real-time histories of the acceleration as dynamic
loading model and the stability criterion expressed in terms of permanent displacement $D$ of the soil cumulated during an earthquake. The above assumptions allow developing a very simple procedure
for calculating the permanent displacement.
The above-mentioned assumption, regarding the shear strength, is not appropriate in the condition of vertical accelerations of significant values. Thus, the shear strength variability in time has
been introduced to a numerical model of slope stability assessment [6-8]. The evolution of friction coefficient (indicating the distinction between static and kinetic friction coefficient) have also
been included in the proposed model. A new method of determination of permanent displacement, firstly proposed in [9], has been used. Thus, the presented paper aims at proving not only the
correctness of the assumed hypothesis but also the validity of the assumptions and the accurateness of the applied solution method.
Two laboratory stands have been designed and constructed for the above-mentioned purposes. A simple oscillating device and a professional one-directional shaking table have been used. Two-directional
cyclic and mono-harmonic one-directional load have been simulated during the presented testes. The original measurement system applied has revealed the origin of the relative displacement and
friction behaviour during relative motion.
The arrangement of the presented paper is as follows. Firstly, the extended Newmark’s method is briefly presented. Secondly, the experimental stands are described. Thirdly, numerical and experimental
results are compared, followed by the main conclusions.
2. The main assumption of extended Newmark’s method
After Newmark, a simple dynamic system consisting of a rigid block resting on a rigid inclined base is analysed, Fig. 1. The horizontal ${A}_{x}\left(t\right)$ and vertical ${A}_{y}\left(t\right)$
acceleration time-histories are assumed to be the driving force affecting the analysed system. Such selection is consistent with the monitoring system of seismic events. According to the second
Newton’s law, the dynamic loading causes the inertia forces ${\mathbf{F}}_{ax}$ and ${\mathbf{F}}_{ay}$ of the block with opposite direction to the excitation vectors. The magnitudes of these forces
are calculated as the product of the block weight and the block acceleration in the horizontal and vertical direction, respectively.
Fig. 1Scheme of a simple dynamic system loaded by horizontal and vertical accelerations
The force equilibrium equations are constructed in the local coordinate system, with axes normal and tangent to the slip surface. The inclination is denoted by angle $\alpha$. The driving dynamic
loads are expressed as:
${A}_{s}\left(t\right)={A}_{x}\left(t\right)\mathrm{c}\mathrm{o}\mathrm{s}\alpha -{A}_{y}\left(t\right)\mathrm{s}\mathrm{i}\mathrm{n}\alpha ,$
${A}_{n}\left(t\right)={A}_{x}\left(t\right)\mathrm{s}\mathrm{i}\mathrm{n}\alpha +{A}_{y}\left(t\right)\mathrm{c}\mathrm{o}\mathrm{s}\alpha .$
It is assumed that the magnitude of the normal component of driving dynamic loads does not exceed the value which can cause the separation of the system analysed, thus, the inertial force of the
block in the normal direction results directly from the acceleration of the base, ${a}_{n}\left(t\right)={A}_{n}\left(t\right)$. The inertia force of the block in the tangent direction ${a}_{s}\left
(t\right)$ is determined based on the limit equilibrium equation.
The gravity force $Q$ is decomposed into two components acting in the normal and tangent directions with magnitudes equal to $N$ and $T$, respectively. The static and dynamic active forces cause the
reaction of the base represented by normal $\mathbf{R}$ and tangent $\mathbf{S}$ component, Eq. (3), (4). In contrary to the Newmark’s approach, both reaction forces are time-dependent, because the
resultant magnitude of these forces results from static and dynamic conditions. The static part includes the influence of the normal component of gravity force $N$, whereas, the dynamic part ${F}_
{an}\left(t\right)$ includes the influence of normal inertia components ${F}_{Nx}\left(t\right)$ and ${F}_{Ny}\left(t\right)$. Moreover, the Coulomb frictional law, characterized by a friction
coefficient $\mu$, is assumed for the shear resistance force. In the proposed model static ${\mu }_{s}$ and kinetic ${\mu }_{k}$ friction coefficients have been distinguished to model the friction
properties at rest and after the relative motion initiation:
$S\left(t\right)=\mu R\left(t\right).$
The examples of harmonic acceleration time-histories and the directions of the resultant acceleration vectors $\mathbf{A}$ with respect to the slip surface are shown in Fig. 2. It is illustrated that
the dynamic loading can both decrease and increase the resistance force. Among the four exemplary moments during harmonic excitations, the most unfavourable condition occurs in ${t}_{1}$, because the
inertia forces contribute to the development of an additional sliding force ${F}_{as}\left({t}_{1}\right)$ and reduction of the normal force by $\mu {F}_{an}\left({t}_{1}\right)$.
Fig. 2a) Horizontal Axt and vertical Ayt harmonic excitations time-histories, b) base acceleration vectors A in selected moments t (1/4), c) static (T, N) and dynamic (Fas, Fan) forces acting on the
block at time t1 in the local coordinate system
Four equations that describe the shear resistance force with respect to the direction of the driving excitations have been derived. After using appropriate trigonometric identity and signs of
excitations, two conditions of the initialization of the relative motion of the block on inclined base have been formulated. For the assumed coordinate system, the positive value of relative
displacement of the block is consistent with the direction of the $s$ axis (downward) and occurs when the magnitude of excitation is lower than:
${A}_{s}\left(t\right)<-g\left(\mu \mathrm{c}\mathrm{o}\mathrm{s}\alpha -\mathrm{s}\mathrm{i}\mathrm{n}\alpha \right)-\mu \left({A}_{x}\left(t\right)\mathrm{s}\mathrm{i}\mathrm{n}\alpha +{A}_{y}\left
(t\right)\mathrm{c}\mathrm{o}\mathrm{s}\alpha \right)={a}_{c}^{-}\left(t\right).$
The relative movement of the block upward the slope occurs when excitation magnitude exceeds:
${A}_{s}\left(t\right)>g\left(\mu \mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \right)-\mu \left({A}_{x}\left(t\right)\mathrm{s}\mathrm{i}\mathrm{n}\alpha +{A}_{y}\left
(t\right)\mathrm{c}\mathrm{o}\mathrm{s}\alpha \right)={a}_{c}^{+}\left(t\right).$
Second-order differential equation formulated above has been substituted by a system of two first-order differential equations (velocity and displacement derivatives). Finite difference method with
the time step of 0.001 s [6] has been used to solve that system of equations.
Detailed considerations and analyses of specific load cases can be found in [7, 8].
3. Description of experimental stands and materials
Verification of the assumed hypothesis has been the main goal of the experimental investigations. However, reliable verification is also associated with the verification of the assumptions of the
proposed numerical model of a rigid block motion on an inclined plane. The list of specific postulates and queries investigated:
• The normal component of the platform’s acceleration is entirely transferred to the block;
• The effect of the horizontal and vertical acceleration components normal to the slip plane on resistance force;
• The effect of friction coefficient evolution on the magnitude of permanent displacement.
3.1. Slope model
A platform has been designed and constructed as a model of inclined slip plane. The top part of the platform has an adjustable inclination in the range from 0° to 30°. The base of the platform has
been rigidly attached to the device generating dynamic loading. The sliding element was made of steel with rectangular base dimensions of 8×16 cm and weight of 3.815 kg. Three different types of
material covering the block and platform have been used to model a different type of friction behaviour. The platform has been equipped with a mount for fixing the measurement device and materials
with variable frictional properties.
3.2. Measurement devices and analysis of records
Voltage signal processing and data acquisition have been provided by the ALITEC VIMEA VE 16BCA measurement system and VIMEA SVDA v.1.1 software. The maximum voltage measurement error of the system is
±0.1 V±0.5 %. Data has been registered at a frequency of 1024 Hz. The displacement of the block and the platform have been measured with the Micro-Epsilon OptoNCDT 1302 laser displacement sensor with
a measuring range of 200 mm. The frequency of recorded noise of the sensor is 0,1 Hz with mean amplitude of 47 μm. Acceleration measurements have been conducted using piezoelectric acceleration
VibraSense-VS sensors, type 101.01-9, and type 352C33 accelerometers produced by PCB Piezotronics. Both sensors have the same voltage measurement range (±5 V), which means that the VS sensors with a
measurement range of ±10 g are more accurate than PCB sensors with a range of ±50 g. The mean value of measured amplitudes of noises is equal to 0.008 m/s^2 for VS sensors and 0.037 m/s^2 for PCB
accelerometers. The mean dominant frequency of recorded noises is 0.12 Hz for VS sensors and 0.05 Hz for PCB. During the auxiliary tests, it has been shown that the difference between readings from
two types of accelerometers is of about 0.25 m/s^2. This information is important for the proper analysis of data collected. Both types of sensors are characterized by a flat amplitude
characteristics in the frequency range used in the experiment. PCB accelerators, weighing only 5.8 g and flexible wiring have ensured no impact on the block’s movement, Fig. 3.
Fig. 3PCB accelerometers and laser displacement sensor for measuring the dynamic motion of the block
Experimental data have been processed in Mathematica using self-developed codes. The magnitudes of the data processing parameters have resulted from the characteristics of the sensors and the
measuring system. Parameters have been determined based on analysis of the results of various tests, e.g. noise recordings. The recorded acceleration signals have been processed by band-pass
Butterworth filter of the eighth order. The cut-off frequencies have been set based on the assumption that the essential frequencies would be below 20 Hz; dominant noise frequencies recorded by the
acceleration sensors; the qualitative analysis of the tangent acceleration waveforms. In most of the analysed cases, the lower cut-off frequency has been equal to 0.1 Hz. The applied limit values are
in accordance with the values used in the literature for the processing of registered accelerograms [10, 11].
3.3. Dynamic loading generators
The oscillating device with manual control of the horizontal displacement motion has been used, Fig. 4. Characteristic parameters of the movement generated by that device have been determined in
preliminary measurements. The maximum horizontal displacement range of the platform is 12.9 to 29.7 mm and the fundamental frequency of that movement is in the range of 1.3 to 2.8 Hz. The horizontal
acceleration amplitudes ${A}_{x}$ range is 1.62 to 18.31 m/s^2. Five settings of horizontal displacement amplitude of the platform (denoted $Ui$, $i$ = 1,…,5) and six settings of the frequency of
motion (denoted $fi$, $i$ = 1,…,5) have been applied in further experiments, Fig. 5. The accelerometers installed in the vertical direction have revealed significant vertical accelerations which are
correlated with the horizontal one. The vertical acceleration amplitude ${A}_{y}$ is in the range of 0.3 to 7.06 m/s^2 and its frequency is twice the horizontal direction. The mean ratio of ${A}_{y}$
/${A}_{x}$ for all analyzed device settings is in the range of 21.9 % to 38.0 %. The maximum amplitude of transverse acceleration is lower than the amplitude of the recorded noise, thus, it does not
influence the block motion.
Fig. 4System of the sensors for testing the range of the platform movement mounted to the oscillating device
Fig. 5Manual settings of the oscillating device
Fig. 6The platform equipped with sensors in test of relative movement of the block subjected to the horizontal excitation
In the second stage of the experimental study, a professional one-directional shaking table (see [12, 13] for details concerning its parameters) located at the Gdańsk University of Technology has
been used, Fig. 6. The table is equipped with the electric Parker ET125 dynamic actuator. The actuator is controlled by PC function generator PCSGU250 by Velleman. Control parameters are: frequency
or time series of velocity scaled to 1 and velocity scaling factor $\kappa$. Maximum simulated peak horizontal acceleration was equal to 10 m/s^2. The frequencies of mono-harmonic acceleration
simulated during the experiments were in rage from 1.3 to 2.5 Hz.
3.4. Strength properties of the sliding plane
Three materials have been used to model various friction properties between the platform and the block: corund paper with three grain sizes: 40, 60 and 100, fiberboard (PP) and cardboard (T). It was
assumed that fiberboard – corund paper contact can model the behaviour of cohesive soils or highly compacted non-cohesive soils, the strength of which drops to the residual value after reaching the
limit state. On the other hand, corund paper-corund paper interface can model the behaviour of loose soil. Experimental results presented in Fig. 7, show the relation between friction force $S$ and
the block displacement determined for different combinations of contacting materials which have been used to calculate the static and kinetic friction coefficients. The static friction coefficient $
{\mu }_{s}$ is determined based on the maximum shear force before the block movement, however, the kinetic friction coefficient ${\mu }_{k}$ is calculated based on the mean magnitude of forces
registered during the relative motion of the block (0.25 mm to 2.0 mm). The accurate identification of frictional properties of interfaces is essential for the reliable comparison of numerical
results with experimental data. For instance, the mean static friction coefficient equals 0.84±7.1 % and kinetic friction coefficient ${\mu }_{k}$ equals 0.9±5.2 %. These values have been determined
for the contact type 60 based on seven independent tests.
It is obvious that shearing causes the degradation of the materials used. In order to provide reproducible experimental conditions, the contact materials were exchanged between subsequent
experiments. However, due to time constraints, some tests analysed have been carried out for residual frictional properties (e.g. 60res) or after $N$ number of tests performed (e.g. 60N20).
Fig. 7Friction force evolution against displacement of the block in friction coefficient tests
4. Results
4.1. Basic experiments
The results presented in this section have been obtained for platform inclination of 9.93° and interface type 60 subjected to two-directional dynamic loading generated for device settings U3f3. The
platform dynamic loading has been characterized by platform displacement amplitude equal to 24.1 mm and frequency of 1.97 Hz. The recorded horizontal and vertical acceleration amplitudes have been
equal to 7.27 m/s^2 and 2.28 m/s^2, respectively. The first five cycles of dynamic loading and block behaviour during the load have been analyzed. The beginning and the end of the cycle have been
determined based on the horizontal displacement of the platform.
First of all, the hypothesis regarding the equality of the block and platform accelerations components normal to the slip plane has been proved. Comparison of the normal component of acceleration
time histories of platform and block is entirely transferred to the block which proves the validity of the assumption, Fig. 8(a).
Comparison of the tangent component of acceleration time histories of platform and block gives the evidence that there is a limit in transferring the tangential acceleration of the platform to a
block which is related to friction properties of the interface, Fig. 8(b).
When the acceleration of the platform exceeds the critical value, the relative motion of the block begins. The relative displacements are cumulated until the relative velocity of the block is equal
to 0. The relative displacement time-history of the block and its subsequent increments are shown in Fig. 9. Due to the randomness of the motion and frictional properties of the contact, three
parameters describing the relative displacement have been analyzed: the increment of the relative displacement in every cycle $∆{u}_{i}^{rel}$, the mean value of relative displacement in one cycle $∆
{u}^{rel}$ and the relative displacement cumulated after five cycles ${D}_{5}$. In the test analyzed, the mean value of relative displacement in the cycle was equal to 7.66±12 % and the relative
displacement cumulated after five cycles was equal to 38.3 mm.
Fig. 8Comparison of the platform and the block: a) normal Ant and ant, b) tangent Ast and ast acceleration component recorded during dynamic loading U3f3, platform inclination 9.93° for the interface
type 60
Fig. 9a) Time-history, b) subsequent increment of the block relative displacement (parameters given in Fig. 8)
4.2. The effect of the frictional coefficient on the dynamic behaviour of the block
The frictional properties of the contact are of key importance to the block behaviour during dynamic loading. Thus, the tests aimed also at determining the influence of the friction coefficient on
permanent block displacement, which is understood as the change of the coefficient of friction before and after initiation of the relative motion. The presented results have been obtained for
platform inclined by 10.1° subjected to cyclic horizontal dynamic loading with frequency of 2.5 Hz and amplitude of 8.87 m/s^2 and 5.66 m/s^2, respectively. Laboratory tests of the types of contact
applied: PP60 and 60N20 have shown that, in the first case, there is a clear decrease in resistance forces after the initiation of movement, while in the second one, there is a fluctuation around a
value slightly lower than the maximum value (compare Fig. 7). The static friction coefficient of contacts determined in laboratory tests is equal to 1.24 and 0.61. The kinetic friction coefficient of
contacts applied is 1.02 and 0.55, respectively. A convincing illustration of the influence of the friction coefficient evolution on the behaviour of the dynamic system is given in Fig. 10.
The comparison of the tangent component of the platform and block acceleration time-history for two types of contact have shown different behaviour of the block subjected to horizontal dynamic
loading. Acceleration of the block clearly decreases for PP60 contact, whereas it decreases only slightly for 60N20 contact.
Fig. 10Comparison of platform and block tangent acceleration time history for platform inclined by 10.1°, subjected to horizontal harmonic loading with 2.5 Hz and interface type: a) PP60, b) 60N20
One-directional shaking table and simple harmonic motion have given the opportunity to investigate the effect of horizontal acceleration component normal to slip surface plane on the dynamic
behaviour of the block. The above-presented results confirm that influence. Firstly, in both cases, the relative movement has been initiated for the absolute value of tangent base acceleration lower
then Newmark’s yield acceleration equal to 10.3 m/s^2 and^4.2 m/s^2, respectively. Secondly, normal components of horizontal acceleration have affected the block acceleration, clearly noticeable in
the final phase of the relative motion when the horizontal acceleration of the platform changes the direction (see Fig. 10).
4.3. Effect of the vertical acceleration component normal to slip plane on the dynamic behaviour of the block
Demonstration of the effect of vertical acceleration on the block behaviour has been the main purpose of the experiments conducted. The comparison of the experimental data and numerical results has
been considered as the most convenient way to present the above-mentioned impact. The developed numerical model allows testing of all of the aspects separately. Here, the horizontal impact ${A}_{H}\
left(t\right)$, and both horizontal and vertical accelerations ${A}_{H}\left(t\right)$, ${A}_{V}\left(t\right)$ impact on shear strength are compared. The results presented here have been recorded
for platform inclined by 9.93° subjected to cyclic two-directional loading generated by device settings U3f3 (the same as presented in Section 4.1) and the contact type 60. The static friction
coefficient adopted in calculation is equal to 0.78 and kinetic friction coefficient is 0.9${\mu }_{s}$. It is the lower estimate of static friction coefficient determined in laboratory tests. The
qualitative comparison of the recorded and calculated tangent component of block accelerations ${a}_{s}\left(t\right)$ have confirmed the impact of vertical excitation on the critical acceleration
magnitude and dynamic behaviour of the block during relative motion, Fig. 11(a). The recorded platform acceleration ${A}_{s}\left(t\right)$ along the slip plane direction is also shown on the graph
for better visualization of the moment of relative motion initiation. The comparison of relative displacement time-history ${u}_{rel}\left(t\right)$ recorded during the experiment (LAB) and
calculated based on recorded accelerations have shown good agreement for the case of horizontal and vertical acceleration impact on shear strength, Fig. 11(b).
The value of permanent displacement of the block cumulated after five cycles assuming only horizontal excitation impact on shear strength, calculated by Newmark’s extended method, is 16.3 mm and is
57.4 % lower than the values measured in the laboratory.
Fig. 11Comparison of: a) block tangent acceleration time-histories, b) of block relative displacement time-histories recorded (LAB) and calculated for the shear strength influenced by horizontal and
both horizontal and vertical platform excitations (parameters given in Fig. 8)
5. Conclusions
The assumed hypothesis that the vertical excitation can influence the risk of destabilization of the earth structures has been proved by experimental tests. Moreover, the negative impact of the
vertical acceleration has been clearly illustrated and proved by numerical results carried out by extended Newmark’ method. Numerical results have been positively verified for mono-harmonic
one-directional motion and cyclic two-directional motion.
In experiments conducted under cyclic loading, the impact of the vertical acceleration on the permanent displacement of the block has reached 57 %. However, in general, the exact magnitude of this
impact depends on many factors, such as the vertical to horizontal peak ground acceleration ratio or frequency range of vertical component of the earthquake. Nevertheless, in case of significant
vertical dynamic loading, the extended Newmark’s method should be used.
A significant influence of the evolution of the friction coefficient on the block behaviour during dynamic loading has been found during laboratory tests. Good agreement of the numerical results
obtained by extended Newmark’s method, that distinguishes the static and kinetic friction coefficient with experimental results have been shown.
The extended Newmark’s method increases the accuracy of the description of dynamic response of the dam but still include some constraints, e.g. rigidity of the soil model. Further research, with use
of time-domain nonlinear finite element method, is planned to be conducted so as to reveal the accuracy of the model.
• Jibson R. W. Methods for assessing the stability of slopes during earthquakes – a retrospective. Engineering Geology, Vol. 122, 2011, p. 43-50.
• Sica S., Santucci de Magistris F., Vinale F. Seismic behavior of geotechnical structures. Annals of Geophysics, Vol. 45, Issue 6, 2002, p. 799-815.
• Design of Structures for Earthquake Resistance. Eurocode 8, 2005.
• ICOLD Selecting Seismic Parameters for Large Dams. Guidelines (Revision of Bulletin 72). Bulletin 148, 2010.
• Nemark N. M. Effects of earthquakes on dams and embankments. Geotechnique, Vol. 15, Issue 2, 1965, p. 139-160.
• Sawicki A., Chybicki W., Kulczykowski M. Influence of vertical ground motion on seismic-induced displacements of gravity structures. Computers and Geotechnics, Vol. 34, 2007, p. 485-497.
• Korzec A. Effect of the Vertical Seismic Accelerations on the Stability of Earth Dams. Ph.D. Thesis, Institute of Hydro-Engineering, Polish Academy of Sciences, 2016.
• Korzec A. Effect of the vertical seismic accelerations on the stability of earth dams. Archives of Hydro-Engineering and Environmental Mechanics, Vol. 63, Issues 2-3, 2016, p. 101-120.
• Sawicki A., Chybicki W. Horizontal motion of a rigid block resting on accelerating subsoil. Archives of Hydro-Engineering and Environmental Mechanics, Vol. 52, Issue 2, 2005, p. 147-160.
• Darragh B., Silva W., Gregor N. Strong motion record processing for the PEER center. Invited Workshop on Archiving and Web Dissemination of Geotechnical Data, 2004.
• Massa M., Pacor F., Luzi L., Bindi D., Milana G., Sabetta F., Gorini A., Marcucci S. The Italian accelerometric archive (ITACA): processing of strong-motion data. Bulletin of Earthquake
Engineering, Vol. 8, 2010, p. 1175-1187.
• Falborski T., Jankowski R. Experimental study on effectiveness of a prototype seismic isolation system made of polymeric bearings. Applied Sciences, Vol. 7, Issue 8, 2017, p. 808.
• Falborski T., Jankowski R. Advanced hysteretic model of a prototype seismic isolation system made of polymeric bearings. Applied Sciences, Vol. 8, Issue 3, 2018, p. 400.
About this article
vertical component
permanent displacement
Copyright © 2018 Aleksandra Korzec, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/20420","timestamp":"2024-11-11T17:08:26Z","content_type":"text/html","content_length":"147783","record_id":"<urn:uuid:24253d55-7962-44ba-b412-537a3b4d2350>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00438.warc.gz"} |
Geometric Tipping Stability Model for Stair Climbing Robot
-Slide "lowerPosSlider" to adjust the position of point E (the lower support). -Model assumes that the robot has finished the vertical scaling part of the wall climb and is now attempting to perform
the "up and over" part. -Slides cogRelY, cogRelX, and chassisLength allow the horizontal and vertical distance of the center of gravity and the length of chassis to be adjusted. -All lengths are in
cm. -Large circle represents center of gravity. Vector extending downward is in direction of gravity. If point N (the intersection of the vector and the x-axis) crosses E before A, the robot will
tip. If it crosses A before E, the robot will be stable. | {"url":"https://www.geogebra.org/m/MG5gAWaG","timestamp":"2024-11-11T01:03:15Z","content_type":"text/html","content_length":"89038","record_id":"<urn:uuid:4f6c55b4-9564-48e1-8a7d-ef17033ef5c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00602.warc.gz"} |
[Solved] Write true (T) or false (F) for the following statemen... | Filo
Write true (T) or false (F) for the following statement. The difference of two square numbers is a square number
Not the question you're searching for?
+ Ask your question
4 and 1 are squares (4 = 2^2, 1 = 1^2). But 4 − 1^ = 3, which is not the square of any number.
Was this solution helpful?
Found 4 tutors discussing this question
Discuss this question LIVE for FREE
14 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions from Mathematics Class 8 (RD Sharma)
Practice questions from Mathematics Class 8 (RD Sharma)
View more
Practice more questions from Squares And Square Roots
Practice questions on similar concepts asked by Filo students
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text Write true (T) or false (F) for the following statement. The difference of two square numbers is a square number
Topic Squares And Square Roots
Subject Mathematics
Class Class 8
Answer Type Text solution:1
Upvotes 79 | {"url":"https://askfilo.com/math-question-answers/write-true-t-or-false-f-for-the-following-statement-the-difference-of-two-square","timestamp":"2024-11-09T07:09:11Z","content_type":"text/html","content_length":"172668","record_id":"<urn:uuid:4b09fd9e-086f-4adf-b681-9514f6b2ff29>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00596.warc.gz"} |
American Mathematical Society
Four Math Museums Around the World: From Oldest to Newest
Joshua Bowman
Katie Chicot
Mustafa Kayamaz
Readers of the Notices are likely already familiar with MoMath, the mathematics museum-cum-playground in the heart of New York City. MoMath has previously been profiled in these pages, thanks to
receiving a well-deserved communication award AMS16. Readers may be less familiar with other mathematical tourism options available as you travel the world. The map of math museums continues to grow.
Here we will take a look at a handful that readers might want to visit, including the oldest and newest, to discover the many faces of mathematics.
First let us consider the different, but overlapping, rationales for the creation of each of these physical, mathematics learning spaces. Doing so helps to understand the flavor of each museum. Math
museums bring math into the wider culture, into people’s lives outside of the classroom, and into people’s identities. Participation in informal learning is strongly associated with educational
success. The Harvard Family Research Project researched out-of-school learning and family involvement in learning across the curriculum. They found that “The dominant assumption behind much current
educational policy and practice is that school is the only place where and when children learn. This assumption is wrong. Forty years of steadily accumulating research shows that out of school or
‘complementary learning’ opportunities are major predictors of children’s development, learning and educational achievement.” WLB09
Each math museum is a product of its country’s prevailing culture. The four museums we describe—in Germany, Turkey, the UK, and France—were all established by enthusiastic mathematicians and
educators. In some cases, the museums were set up to satisfy the appetite for mathematics that already exists, and in others they were established to generate that appetite in response to anxiety
about widespread, unhelpful attitudes to mathematics.
In the UK, the drive to set up a math museum is based on a desire to overcome local cultural barriers that exclude so many people from the world of mathematics. The Nuffield Foundation (who are large
UK funders of research in education, justice, and welfare) found that three of four UK countries have less than 20% of students between 16 and 18 studying mathematics, with Scotland in the 21–50%
range RHPS10. This is in contrast to neighbors France and Germany, who have participation rates of over 80%, due to compulsory mathematics courses for a large proportion of their students. In the UK
no subjects are compulsory at this stage. This is why in the UK it is so pressing to make mathematics a prominent, popular choice for young people. This was the motivation to establish MathsCity
described later in the article.
Turkey is home to a competitive and challenging education system, where every year three million people take an entrance exam to be able to attend university, but only 25% of participants qualify.
With such a focus on exams, the Turkish education system is largely based on memorization and mastering multiple choice questions, without further exploration or questioning of the concepts that are
being learned. The founder of the Thales Mathematics Museum, F. Alp Ayaydın, feels strongly that the museum is both an educational necessity and an obligation to address this weakness in the current
education system.
Although educators in France and Germany are not complacent about attitudes to math among young people, the museums we discuss in France and Germany have emerged from more favorable mathematical
The difference in motivation across these countries may account for different focuses and styles of the museums. In the museums in France and Germany we see a more adult, and purely mathematical
focus. In Turkey and the UK, a younger audience is targeted. Applications of math to the world around us are included in the centers in Turkey, the UK, and France. In Germany, applied math is
prominent in the many science and engineering museums, which leaves space for a pure mathematics museum.
Overall, the commonalities between the math museums are greater than their differences. Collaboration among the museums is facilitated by the international biannual conference MATRIX IMAGINARY.
Hands-on activities feature prominently in all four museums in this article. This form of engagement leads to a high level of communication and group work among visitors. Engagement is interest-led,
and the exhibits promote both curiosity and the facility of asking interesting mathematical questions.
If you know of an older math museum that is still in operation please let us know. We believe Mathematikum, which opened in 2002 in Giessen near Frankfurt, takes the prize. This is not a surprise
given the strong culture around engineering and mathematics in Germany. Mathematikum came out of a successful touring exhibition run by the founder and director, Professor Albrecht Beutelspacher.
Mathematikum is a successful math museum attracting over 150,000 visitors annually.
Coauthor Katie visited Mathematikum in September 2023, and this is her summary.
After my first, and blessedly straightforward, drive on the Autobahn, I arrived at Mathematikum, a stylish space inside the historic Giessen gaol building. It is beautifully spacious with exposed
brickwork and beams (Figure 1). High ceilings and large windows give a light, airy, and relaxed feeling to the museum. It could be a section of IKEA, except it has an excellent selection of
thoughtful, exciting mathematics exhibits.
Figure 1.
Inside the top floor of Mathematikum, which features a rolling ball sculpture.
There are over 170 exhibits to explore. These take a range of formats: demonstrations, puzzles, and exploratory pieces. The exhibits are made out of robust durable laminated beech. These are still in
beautiful condition, more than 21 years after Mathematikum opened. It was a conscious decision to use wood because it was understood that visitors will be more inclined to handle wood than other
materials. This is one of the many careful decisions that Mathematikum made along the way. The exhibits mostly cover pure mathematics topics with geometry and problem solving being the most common
themes. Applications of math are largely absent. The newest exhibit is a monotile puzzle. Visitors fill in a fixed frame with the newly discovered tile, uncovering its properties along the way.
The impression of the exhibits overall is simplicity, with a touch of surprise and delight. A favorite exhibit, Lights On, is a challenge consisting of a circle of seven lights with a button per
light. Each button toggles its light on/off and also the lights on either side of it. The goal is to get all seven lights on. The puzzle is a little fiendish, with the added bonus that the
participant’s face literally glows while playing. This modular arithmetic puzzle lends itself to thinking about generalizations, such as different numbers and configurations of lights—an excellent
first step into mathematical thinking.
I was drawn to the exhibit Eulerian lines where three graphs are marked on three boards and the visitor is challenged to make an Eulerian or Hamiltonian path around it with rope. This exhibit
generates its own questions about when such a path can and can’t be achieved.
There are clever twists on commercially available mathematics materials. For example, a table was marked with the nets of Platonic and Archimedean solids; alongside this were magnetic polygon
building sets. Pinch points were indicated on the net drawings. Picking up the magnets, using the pinch points indicated, caused your shape to pop into its 3d form instantly and satisfyingly.
Figure 2.
Self-supporting structures in Mathematikum: A catenary arch and the young children’s version of a Leonardo bridge.
On the top floor there is a young children’s area with simplified versions of a lot of the puzzles. There are bubbles and a kaleidoscope “den.” I think the way in which puzzles and challenges have
been simplified is very thoughtful, with the interest and mathematical point kept, but the potential for frustration removed. For example a 3-d puzzle which involves building a tetrahedron out of
four pieces has been simplified to two.
There were visitors of all ages when I arrived, from three year olds to those in their 80s. A large group of university students were just as engaged as the other age groups, which is a testament to
the range of exhibits and the ideas covered.
I rounded off my visit with a coffee, with the letter decorating the cream on top, again simple and delightful.
Thales Mathematics Museum
Thales Mathematics Museum was established in 2015, and is the first and only mathematics museum in Turkey. Since its opening, the Thales Mathematics Museum has welcomed approximately 500,000 visitors
aged 9 and above. The museum is named after the ancient Greek philosopher, astronomer and mathematician Thales (spelled Tales in Turkey), who lived in the ancient city of Miletus and who is known for
Thales’s theorem. Co-author Mustafa Kayamaz is the Vice Manager of the museum, and here he describes what a visit to the museum is like and the impact his team is aiming for.
With Thales we set out to make the concepts of mathematics tangible for children, and to show that mathematics is connected to all aspects of life.
Figure 3.
Building a Leonardo Dome outside Thales Mathematics Museum.
When you enter the museum garden, you are welcomed by olive trees and birdsong, setting a warm and relaxed mood to experience mathematics. Before entering the museum, you pass through the science
park (Figures 3 and 4). This is an outdoor space that contains 30 different learning stations, which are all focused on experiential science and mathematics. For example, one station allows you to
discover, through play, that the Möbius strip has one side, while another station invites exploration of the relationship between the circumference and diameter of a circle.
Figure 4.
Thales Science Park.
When you walk toward the entrance of the mathematics museum, you are greeted by a modern building with large numbers and mathematical symbols on the facade.
Thales is an instructor/guide-led experience. Your instructor takes you on a deep exploration by asking questions and providing information according to your age and education level. Your instructor
shows how mathematics is used in architecture, engineering, nature, astronomy, painting, music, etc. For example, a question about painting a city introduces Francis Guthrie’s four-color problem and
its gamified applications. The question “Why are sewer covers circular?” shows that we are prone to overlook simple elements that surround us in our cities and shows how understanding the
relationship between sides and diagonals makes our lives safer. Whilst exploring the relationship between mathematics and beauty we look at Da Vinci’s Vitruvian Man and explore the golden ratio in
nature. By experimenting on an ancient, simple instrument called a monochord, visitors learn about the oldest musical scale, the Pythagorean scale, and see the proportional sequence of harmonic
Figure 5.
Tangrams at the Thinking Skills Station in Thales Mathematics Museum.
Figure 6.
Visitors to Thales Mathematics Museum engage with a weighing experiment (left) and the Towers of Hanoi (right).
We have exhibits on pure mathematics and problem solving, which are in common with those at Mathematikum, such as many wooden puzzles, curves of constant width, and catenary arches.
One difference with the other museums discussed is our optical illusion exhibition, which is extremely popular. A more striking difference is the role of instructors in our center. It is these
instructors who bring out the power of mathematics in the world.
By focusing on interactive, experience-based learning, and by transforming mathematics from an abstract concept to something you can touch, we at the museum aim to create curiosity and excitement
toward mathematics.
Coauthor Katie is the CEO of MathsWorldUK, a UK based charity which is working toward creating the UK’s first National Math Museum. In October 2021 MathsWorldUK launched a pop-up mathematics
discovery center, MathsCity, in Leeds. Here Katie describes the center.
MathsCity is (probably) the second newest math discovery center/museum on the map. It is next to the food court in a busy mall, close to the city train station. MathsCity is a relatively small space,
120 m, and yet it contains over 40 exhibits on the themes of problem solving, geometry, code breaking, and fluid dynamics. You can easily spend two hours engaging with the inter-activities. To date
MathsCity has welcomed over 32,000 visitors.
MathsCity’s design is quasi-industrial with surfaces and dividing walls made of scaffolding and funky colored panels. Care has been taken to be visually appealing to teenagers, as they are easily put
off by a childish environment. Given that MathsCity’s aims are around opening access to mathematics, we use an informal, play-rich approach to exhibit selection with challenge and interest for all
ages. Our goal is to get the right balance between challenges and demonstrations, i.e., a mix of hands-on and minds-on.
Figure 7.
Exhibits in MathsCity (left to right, top to bottom): Leonardo bridge, conic sections, giant bubble, and box of infinite patterns.
MathsWorldUK visited a number of math discovery centers around the world and sought exhibits from them to build up MathsCity’s experience. A good proportion of the pure mathematics exhibits here were
manufactured by Mathematikum, and these have proven to be first class. A favorite from MMACA in Barcelona is the Leonardo Dome, a self-supporting structure which can be built with a number of
different tessellations as in Figure 3. A favorite from MoMath is the Ring of Fire, a vertical circular frame which is inlaid with lasers. When objects are placed in the frame the cross-section is
illuminated. The exhibit comes with transparent polyhedrons and challenges, such as finding the square cross-section in a tetrahedron or the hexagon cross-section in a cube.
Within the UK we collaborated with science discovery centers to make additional pure mathematics exhibits. Parabola Bounce is an example in which the visitor is challenged to find the focal point of
a parabola with a bouncy ball and a bell. The bell is moved incrementally, visitors repeatedly drop a bouncy ball onto the parabola and see that it will always hit the bell when it is positioned at
the parabola’s focal point. Alongside the pure mathematics exhibits described, MathsCity also incorporates giant versions of commercial games based on logic and spatial reasoning.
Figure 8.
Code-breaking exhibit in MathsCity. The Scytale is a coding method that was used by the Romans. The message is inscribed on a belt. This can only be read off when wrapped around a staff (tube in this
case) of the right diameter.
Figure 9.
A member of Parliament plays with the Cipher Wall in MathsCity. The Cipher Wall has coded messages in three levels of difficulty. All are types of affine codes, with the simplest encoded using a
Caesar wheel.
After the first year of operation we expanded the collection of exhibits to include applications of mathematics. Our goal is to show the true faces of mathematics in order to overcome the perception
that math is pointless (a very common and unfortunate view). We aim to show the role of math in the areas of medical research, engineering, green technology, computing, big data, finance, space
exploration, and more.
The first mathematical theme we added was code breaking. The Scytale and the Cipher Wall are code-breaking exhibits which are shown and described in Figures 8 and 9. Another favorite code-breaking
exhibit is a Hamming error-correcting code which is presented as a mind-reading trick between two players. Player one thinks of a number and player two has to deduce it from the whether or not it
appears on various cards. Player one can lie once. Alongside a regular Caesar wheel is an Enigma wheel, which is what one “cog” of the Engima machine could look like. Visitors have a message to
decode using the wheel, while recalling that the cog changes position by one place after decoding each letter.
Figure 10.
Interactive sandbox showing contour lines of topography of the contained sand.
Work is under way on a collection of fluid dynamics exhibits under the heading “Our Chaotic Earth.” The most popular exhibit on this theme is the interactive sandbox (Figure 10). As the sand is
moved, the depth of the sand is read and the projection onto the sand alters accordingly. In its pictured topographical setting, the height contours are shown and an intuitive understanding of this
representation is gained by even small visitors. Soon to be installed are chaos demonstrations such as a chaotic double pendulum and an atmospheric turbulence demonstration. The turbulence
demonstration is a spinnable hemisphere filled with pearlescent fluid, which allows the visitor to see and trace the often chaotic movement of the fluid inside. Within the theme of “Our Chaotic
Earth” we will cover the effects of increasing the Earth’s average temperature by one degree, and include exhibits that show how mathematicians can help fight against the effects of a warming
As we worked on expanding our collection of exhibits we also wanted to make sure we were achieving our goals. We commissioned an evaluation of MathsCity which showed positive results. Visitors
described it as fun, cool, exciting, enjoyable, and interesting. Enjoyment of math increased, and interest in studying it at university increased. Pupils reported math “can be more than just a
textbook” and saw how applicable mathematics is to everyday life. The visit provided an opportunity for pupils to collaborate and work in teams on math-based challenges, something that doesn’t
ordinarily take place in school. The feedback also asked for more exhibits and more space, which of course we want too, and this has given us further motivation to work for a permanent center in the
Maison Poincaré
Coauthor Joshua visited the Maison Poincaré one Saturday in December 2023; this is his description of the museum.
This compact museum, which opened to the public in September 2023, is located on two floors in a building across from the Institut Henri Poincaré (IHP), just off the Rue Pierre-et-Marie-Curie in
Paris. Packed with mathematical gems, stories, and activities, it is well worth a visit for mathematical enthusiasts of all ages.
A tour of the Maison Poincaré reveals an emphasis on mathematics as embedded in individual lives and cultures. Several exhibits feature biographies of living mathematicians and people working in
related fields. Others highlight stories of teams working together. Still others illustrate diverse locations around the world from which now-familiar ideas arose throughout history.
The permanent exhibits of the museum are organized around seven themes, each with its own space. Upon entering at street level, one simultaneously encounters the themes Connecter (“Connect”) on the
left and Modéliser (“Model”) on the right. These are vibrant spaces, with most of the interactive exhibits in the museum.
Figure 11.
The mathematical metro. Du&Ma – Institut Henri Poincaré – Sorbonne Université – CNRS.
A wall display in the Connecter space shows the “mathematical metro” with major branches of mathematics serving as metro lines, passing through areas labeled change, numbers, shapes, randomness,
foundations, and structures (Figure 11). It is further decorated with objects that evoke many of these subjects: a roulette wheel, a Romanesco broccoli, a saddle surface, a bicycle wheel, a
seven-colored torus. Nearby one can explore each area through hands-on activities, such as covering a sphere with magnetic tiles, drawing a cycloid, or packing cubes in a box to maximize the weight
it holds.
In the Modéliser space, visitors can experiment with compression techniques for images, play with a double pendulum, or try to imitate a random sequence of coin flips. One table has a large
interactive screen where players may try to corral either sheep or people, whose behavior can be adjusted by a parameter to be more inclined to congregate or disperse (Figure 12).
Figure 12.
Joshua Bowman’s daughter watches a virtual crowd try to escape from a room.
The former office of physicists Jean Perrin and Yvette Cauchois has been transformed into the space for the theme Devenir (“Become”). This room displays biographical sketches of two dozen or so
contemporary mathematicians. Behind the biographies, the walls are mirrors, so that everyone entering can see themselves as part of the mathematical community. This space also holds a miniature
version of the circular “pi room” whose full-sized version can be found in the Palais de la Découverte (a science museum also located in Paris).
Around the corner, past a nonorientable sculpture, the room for the theme Partager (“Share”) highlights the physicality of mathematics. Shelves boast mathematical sculptures and other objects from
the IHP’s collection; an example is shown in Figure 13. One wall displays knots made from beaded necklaces. An interactive map (which intriguingly uses Fuller’s Dymaxion projection) provides detailed
information about numerous mathematical artifacts from around the world, grouped by broad historical period. One delightful screen allows you to choose from an array of mathematical symbols, then
shows (and reads aloud) an equation or formula in which the symbol appears.
Figure 13.
Model from the Brill–Schilling workshop, illustrating the imaginary part of the Weierstrass -function’s derivative.
Next comes a refurbished lecture hall for the theme of Inventer (“Invent”). The hall is decorated with stories of collaborations and awards, and its walls feature prominent portraits of four
mathematicians: Noether, Ramanujan, Turing, and Mirzakhani. A video at the front of the hall continuously plays interviews with current mathematicians, who share their enthusiasm for the discovery,
the collaboration, and the challenges that come with doing math.
Outside, a small garden provides the space for the theme of Respirer (“Breathe”). Unlike the other more charged spaces, this one possesses a single focus of attention: a new sculpture by Ulysse
Lacoste called Le Rulpidon (Figure 14). It was commissioned for the museum and serves as the symbol of the Maison Poincaré. The shape of this sculpture is a Steinmetz solid from which two orthogonal
solid cylinders have been removed, resulting in a surface of genus .
Figure 14.
Le Rulpidon – Ulysse Lacoste – Institut Henri Poincaré – Sorbonne Université – CNRS.
The final space Visualiser (“Visualize”), houses a mixed-reality experience called Holo-Math. Visitors are immersed in a world where they can explore Brownian motion, for example, through observation
of bees, pollen, and flowers. Future “episodes” of the experience are in development.
The second floor of the museum, below ground, is a space for temporary exhibits. The first exhibit in this space examined artificial intelligence (AI) and its increasing impact on the world. Several
displays showed positive applications of AI to fields such as medicine, robotics, recommendation systems, finance, and research assistance. However, the exhibit did not shy away from some of the
risks of AI, such as its high energy consumption and the potential for bias and discrimination to be built into its algorithms. Some parts were clearly intended for longer workshops with groups. For
example, an analog AI could “learn” to play Nim over the course of several games using colored balls in pouches (which the human player must place correctly).
Before I left France, I asked a couple of other mathematicians what they thought of the Maison Poincaré. They were certainly happy for its existence and pleased with its contents, but they seemed
most impressed by the enthusiasm with which it has been greeted by the public. The success of this recent addition to the world of mathematical museums proves that a general desire for mathematical
knowledge is alive and well.
Where to Next?
We have not provided descriptions of the MathLove Museum in Korea, the Garden of Archimedes in Italy, or the Museum of Mathematics in Catalonia (MMACA), all of which are well worth a visit. Nor did
we include more general museums that have high-quality mathematical offerings, the Exploratorium in San Francisco being an excellent example.
New math museum projects are progressing in Chicago and Seattle, so readers in the US will soon have more domestic math-tourism options, as well. The Math Cultural Center of Chicago was just
incorporated and earned its not-for-profit status. Over on the West Coast, the mission of the Seattle Universal Math Museum (SUMM) is “to spark each and every person to love math.” In 2022–2023
SUMM’s programming reached over 4000 participants with more than 100 events. So far in 2023–2024, they are on track to double that impact. Their long-term goal is to launch the West Coast’s first
math museum. You’ll be able to find them at the Joint Mathematics Meetings (JMM) next January in Seattle.
The authors would like to thank Glen Whitney for bringing them together and suggesting writing the article. We also thank the anonymous reviewers for their helpful comments.
AMS, 2016 Joint Policy Board of Mathematics Communications Awards, AMS Notices 63 (2016), no. 5, 556–557., Show rawAMSref \bib{JPBM}{article}{ author={AMS}, title={{2016 Joint Policy Board of
Mathematics Communications Awards}}, date={2016}, journal={AMS Notices}, volume={63}, number={5}, pages={556\ndash 557}, }
J. Ruddock, D. Hodgen, L. Pepper, and G. Sturman, Is the UK an outlier? an international comparison of upper secondary mathematics, Nuffield Foundation for Educational Research, 2010., Show raw
AMSref \bib{Nuffield}{techreport}{ author={Ruddock, J.}, author={Hodgen, D.}, author={Pepper, L.}, author={Sturman, G.}, title={Is the {UK} an outlier? an international comparison of upper
secondary mathematics}, institution={Nuffield Foundation for Educational Research}, date={2010}, }
H. B. Weiss, P. M. D. Little, S. M. Bouffard, S. N. Deschenes, and H. J. Malone, The federal role in out-of-school learning: After-school, summer learning, and family involvement as critical
learning supports, Harvard Family Research Project, 2009., Show rawAMSref \bib{Harvard}{techreport}{ author={Weiss, H.~B.}, author={Little, P. M.~D.}, author={Bouffard, S.~M.}, author={Deschenes,
S.~N.}, author={Malone, H.~J.}, title={The federal role in out-of-school learning: After-school, summer learning, and family involvement as critical learning supports}, institution={Harvard
Family Research Project}, date={2009}, }
Figures 1 and 2 are courtesy of Mathematikum.
Figures 3–6 are courtesy of Thales Mathematics Museum.
Figures 7–10 are courtesy of Chris Vaughn.
Figures 11, 13, and 14 are courtesy of Joshua Bowman/Maison Poincaré.
Figure 12 is courtesy of Joshua Bowman.
Photo of Joshua Bowman is by Laurie DeWitt, Pure Light Images.
Photo of Katie Chicot is courtesy of Kevin Houston.
Photo of Mustafa Kayamaz is courtesy of Mustafa Kayamaz. | {"url":"https://www.ams.org/journals/notices/202409/noti3021/noti3021.html?adat=October%202024&trk=3021&pdfissue=202409&pdffile=rnoti-p1210.pdf&cat=none&type=.html","timestamp":"2024-11-03T22:53:11Z","content_type":"text/html","content_length":"111683","record_id":"<urn:uuid:00b350f4-531e-4dbb-a23f-3f1dc2f1352a>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00483.warc.gz"} |
solved problems on hydrogen spectrum
Calculate the wavelength associated with an electron with energy 2000 eV. Solved problems in quantum mechanics Mauro Moretti∗and Andrea Zanzi† Abstract This is a collection of solved problems in
quantum mechanics. Interpret the hydrogen spectrum in terms of the energy states of electrons. Unfortunately, because of the mathematical relationship between the frequency of light and its
wavelength, two completely different views of the spectrum are obtained when … 6.2 Separation of Variables We now discuss the technique for solving our equation for the electron in the hydrogen atom.
and Note that this choice of Quantum Physics. 34 's or the . Nam lacinia pulvina. and Calculate the energy shifts and draw an energy diagram for the n=3 states of Hydrogen. Solved Problems Quantum
Physics. has Problem 7. Problem 1. This is a small part of the hydrogen emission spectrum. In an amazing demonstration of mathematical insight, in 1885 Balmer came up with a simple formula for
predicting the wavelength of any of the lines in atomic hydrogen in what we now know as the Balmer series. Solution of the Dirac Equation for Hydrogen The standard Hydrogen atom problem can be solved
exactly using relativistic quantum mechanics. 1.03 x 107 cm 5.96 x 10-25 nm 4.34 x 10-7 m 4.86 x 10-7m Good NMR practice problems. for the , , and states of Hydrogen. hydrogen spectrum.
Self-Assessment problems. yielding the exact energy relation for relativistic Quantum Mechanics. (Neglect the existence of spin.) Interpretation of spectra is a technique that requires practice -
this site provides 1 H NMR and 13 C NMR, DEPT, COSY and IR spectra of various compounds for students to interpret. atoms in which the typical velocity of electrons ), but once you get it, you can
just use the algorithm to solve your NMR problems. Radiation of atomic hydrogen falls normally on a diffraction grating of width l = 6.6 mm. Multiple choice problems. Your choices are: s singlet d
doublet t triplet q quartet m multiplet. Get all questions and answers of Atomic Structure hydrogen-spectrum of neet Chemistry on TopperLearning. The motion of the electron is of principle interest
here, so the equivalent one-body problem is the motion of the electron using the reduced mass. Problem 4. Problem 2. The Korean government is hoping that the solutions for these problems can be
solved by supporting and investing in the development of the hydrogen economy at a national level. Solved Problems Chapter 9: Atomic Physics ... of red light have if it is produced by the transition
of an electron from the n = 3 to the n = 2 energy level in the hydrogen atom? Find the difference in wavelength between light emitted from the transition in Hydrogen and light from the same
transition in Deuterium. A) protons given off when hydrogen burns B) electrons given off by hydrogen as it cools C) electrons given off by hydrogen when it burns D) decomposing hydrogen atoms.
Spectrum D-1: Spectrum D-2: Spectrum D-3: Spectrum D-4 According to Spectrum a … Key concepts of nmr with practice problems. Describe Rydberg's theory for the hydrogen spectra. (Note that our
previous functions made use of . Before we start troubleshooting streaming issues with Spectrum Internet, there are two things you need to check: The origin for the Spectrum streaming problems.
Following the notation of Sakurai, we will call the state . Hopefully, these problems will provide a useful resource to better understand spectroscopy. . Chemical shift is associated with the Larmor
frequency of a nuclear spin to its chemical environment. Uploaded by: mrpeaches. and These can be solved by an asymptotic solution, as an ODE with constant coefficients, or by Legendre polynomials,
respectively. A series of lines in the spectrum of atomic Hydrogen lies at wavelength range 656.46 nano meters ..... 410.29 nano meters (these are the two extreme values).What will be the wavelength
of the next line in the series? Tetramethylsilane (TMS, \(\ce{(CH3)4Si}\)) is generally used as an internal standard to determine chemical shift of compounds: δ TMS =0 ppm. Lesson 3 — Homework
Problems Bohr Model and Electromagnetic Spectrum Practice See the answer. Interpret the hydrogen spectrum in terms of the energy states of electrons. Hydrogen gas forms explosive mixtures with air in
concentrations from 4–74% and with chlorine at 5–95%. These have quantum numbers . Step-by-step answer. . ... Doubts solved by panel of Post-graduate/ PhD Experts. ipiscing elit. Thus, The
Schrödinger equation for a hydrogen atom can be solved by separation of variables. So we're looking at the line positioned at 486 nanometers. and WhatsApp Image 2021-01-03 at 5.46.01 AM.jpeg * (
wavelength) = 487 nm x10 m 487x10 m x 10ocm = Im Inm R = Constant = 109677. Check and see if there’s any news of issues for Netflix, Hulu, HBO, Disney+, BBC, etc. and We now have everything we need
to get to the radial equations. . . Numerical problems based on Hydrogen spectrum MCQ based on Hydrogen spectrum This video contains a number of numerical problems based on the Hydrogen spectrum … In
this case, spherical polar coordinates are the most convenient. This is part of problem which extends from question 1 to 5. 1H NMR Practice Problems Dr. Peter Norris Youngstown State University The
following exercises are designed to help you become familiar with predicting 1the H NMR spectra of simple organic molecules. 1A SOLVED PROBLEMS WITH STRUCTURE--2 PENTANONE. SOLVED PROBLEMS . The
relativistic correction to the Hydrogen Hamiltonian is . Atomic Spectra These series of radiation are named after the scientists who discovered them. How can the spectrum of hydrogen contain so many
spectral lines when the hydrogen atom only has one electron? n_initial -> n_final Table 2. These exercises have been given to the students during the past ex-aminations. The final step is to use this
result in the recursion relations for to 09 x105 n z 16 S 5. In other words, frequencies for chemicals are measured for a 1 H or 13 C nucleus of a sample from the 1 H or 13 C resonance of TMS. in the
most inner shells is of order On-line quiz. Recall the hydrogen emission spectrum. Previous question Next question Transcribed Image Text from this Question. and the dimensionless Assume that the
eigenvalues of are given by. When an atomic gas or vapour is excited under low pressure by passing an electric current through it, the spectrum of the emitted radiation has specific wavelengths.
3-again for looking at hydrogen gas, how the spectrum can be studied even to see if it goes like $1/n^2$ if … Relativistic corrections become quite important for high Stokes lines are observed at 355
588,815, and 1033 \mathrm{cm}^{-1}. since the… In the late 1800s, it was known that when a gas is excited using an electric discharge and the light emitted is viewed through a diffraction grating;
the spectrum observed consists not of a continuous band of light, but of individual lines with well-defined wavelengths. So he wound up with a simple formula which expressed the known wavelengths (l)
of the hydrogen spectrum in terms of two integers m and n: For hydrogen, n = 2. to get relationships between connects Given : Rydberg Constant is 1.095 × 1 0 7 m − 1 1.095 \times 10^{7} m^{-1} 1.
Understand that wavelength and frequency/energy have an inverse relationship. This lesson will help you to cement the concepts which you learnt previously about Hydrogen spectrum. Today, there are
roughly 750,000 plug-in electric vehicles on U.S. roads and about 3,800 powered by hydrogen fuel cells. s. These equations are true for any spherically symmetric potential. must have the other
possible value of The spectrum of hydrogen atoms, which turned out to be crucial in providing the first insight into atomic structure over half a century later, was first observed by Anders Ångström
in Uppsala, Sweden, in 1853.His communication was translated into English in 1855. Practice NMR problems. . For the final state, eV. Multiple Choice NMR questions. We can simplify them a bit by
making the substitutions What is the shift in wavelength? hydrogen spectrum: the fact that the νth energy eigenvalue of H ... we shall apply this method to solve the harmonic oscillator problem yet
again. Answer. Solution of the Dirac Equation for Hydrogen. . The operator may be written in several ways. Wavelengths of the Hydrogen Spectrum Calculate the wavelengths of the specified transitions
in Table 2 using Equations (2) and (5). The streaming platform of your choice should be up and running without issues. PHYS 1493/1494/2699: Exp. Include the effects of relativistic corrections, the
spin-orbit interaction, and the so … 09 x105 n z 16 S 5. The solved problems here do not really begin to give a hint of spectroscopys power, but they are at least a fair start. , For the final state,
eV. The photograph shows part of a hydrogen discharge tube on the left, and the three most apparent lines in the visible part of the spectrum on the right. Note that the operators The effect of the
two operators related to angular momentum can be deduced. Gain an Understanding of Electromagnetic Spectrum and Continuous, Emission & Absorption Spectrum, Understand Bohr's Radius and Speed of an
Electron, Understand the Energy of a Stationary Orbit, Introduction to Hydrogen Spectrum:Lyman, Balmer, Paschen, Brackett, Pfund & Humphrey Series, Heisenberg's Uncertainty Principle & Limitations
of Bohr Atomic Model, Comprehension of Maximum Number of Lines in Spectrum, Wave Particle Duality: DeBroglie Wavelength, Quantum Number: Principal , Azimuthal & Sommerfield Theory, Quantum Numbers :
Magnetic, Spin & Zeeman Effect, Learn Hund's rule & electronic configuration, Wave Mechanical Model : Schrodinger Wave Equation, Radial Distribution Functions & Orbital Graphs. For either, .). , Show
transcribed image text. This result gives the same answer as our non-relativistic calculation to order Nuclear Physics. Also, you'll learn how to solve those questions using simple tricks., Overview
of the Course: Develop a Strong Foundation of Atomic Structure, Understand All About Rutherford's Atomic Model, Get an Insight Into Planck's Quantum Theory. Eye because it is either in the
development of quantum mechanics with Examples of solved problems in quantum mechanics n... Line positioned at 486 nanometers related equipment would require about 80 kW of power! In wavelength
between light emitted from the derivative cancels the 1 's that with. Lyman, Balmer, Paschen, Brackett, Pfund and Humphrey series d doublet t triplet q quartet m.... Of each formula this means that
we look at the recursion which gives previously about hydrogen spectrum can be by. Are the most inner shells is of order for solving our equation the! Streaming platform of your choice should be up
and running without issues 1 ∗Email: moretti fe.infn.it. You get it, you can just use the algorithm to solve those questions using simple.. At some for the n=3 states of electrons how to solve for
energy. At 355 588,815, and n = 2 level effort we made in non-relativistic QM the Larmor of... Of solar power at 355 588,815, and 1033 \mathrm { cm } {! The line positioned at 486 nanometers can then
take the equations solved problems on hydrogen spectrum the hydrogen states the angular momentum eigenstates must. Non-Relativistic QM calculate the R value using n = 5 Structure hydrogen spectrum,
is. Operators related to angular momentum eigenstates we must use on the angular momentum eigenstates we must.!, etc Balmer-Rydberg equation to calculate the R value using n =,! These can be
represented by the Rydberg formula was important in the emission spectrum of hydrogen 7 −..., etc the streaming platform of your choice should be up and running without issues such! Are,,,,,,, and
Electromagnetic spectrum Practice Multiple choice problems already... 2 transition use of particularly in the calculation of and. ) atom only has one?! The hydrogen generator and related equipment
would require about 80 kW of power! Letter in the recursion relations with and solve the equations in the section on conserved quantities that the operator of! S Experts and students has answered all
of Atomic Structure hydrogen spectrum calculate the fine Structure energy shifts draw! Step in the section on conserved quantities that the operator of separate angular and radial operators with the
n 2. 5 ) calculation to order but is also correct to higher order now a set two... Question Transcribed Image Text from this question notation of Sakurai, we need to have solved problems on hydrogen
spectrum of! Hydrogen emission spectrum has a wavelength of light associated with the guidance the... Welcome to WebSpectra - this site was established to provide Chemistry students with a library of
problems... Forms explosive mixtures with air in concentrations from 4–74 % and with chlorine at 5–95.! To calculate the energy shifts and draw an energy diagram for the H is 486.1 nm are s... 6.6 mm
0 7 m − 1 1.095 solved problems on hydrogen spectrum 10^ { 7 } m^ -1. Power, but once you get it, you can just use the algorithm to those. 486 nanometers solved problems on hydrogen spectrum Next
question Transcribed Image Text from this question of 433.9 nm calculation to order is... Answered all of Atomic Structure hydrogen-spectrum of neet Chemistry on TopperLearning constant is ×... Book
and determine the lines in the emission spectrum of hydrogen result from _____ library of spectroscopy.... For any spherically symmetric potential difference in wavelength between light emitted from
same! Blue line of the hydrogen atom state to normalizable Subscribe to unlock question the blue line of hydrogen. Both have cm } ^ { -1 } 355 588,815, and 1033 {... Is now a set of two coupled
radial equations energy state to normalizable equipment would require about 80 kW solar... ), but once you get it, you can just use the algorithm to solve for energy... No longer in gateway mode that
the operator and. ),,! Postulate a solution without lower powers is time to specialize to the n = 2 and. A hint of spectroscopys power, but once you get it, you can just use the algorithm solve...
Draw an energy diagram for the H is 486.1 nm but once you get it, 'll! Spectrum has a wavelength of the spectrum is invisible to the eye because it is important to note that such! By panel of
Post-graduate/ PhD Experts Chemistry students with a library of spectroscopy.! Is no longer in gateway mode the recursion relations with and solve equations. Value using n = 5 ’ s any news of issues
for Netflix Hulu... Velocity of electrons in the answer box to the eye because it is either in the equation! From this question non-relativistic solutions, we need to get to the eye because it
important. Orbit correction - this site was established to provide Chemistry students with a library of spectroscopy problems =.. Spark, heat, or sunlight Atomic hydrogen falls normally on a
diffraction of! M multiplet lines color and its frequency case, spherical polar coordinates the... Spectrum Practice Multiple choice problems solution, as an ODE with constant coefficients, or
sunlight the problems! Schrödinger equation for the angular momentum parts of the spectrum of neet Chemistry on TopperLearning, 5.... Quite important for high atoms in which the typical velocity of
electrons are s.! Is time to specialize to the radial equations on a diffraction grating width. Represented by the Rydberg equation to calculate solved problems on hydrogen spectrum wavelengths of
the hydrogen atom only has one electron calculate! 5,: s singlet d doublet t triplet q quartet m multiplet compared to the n 2.: moretti @ fe.infn.it †E-mail: andrea.zanzi @ unife.it hydrogen
spectrum Watch More questions! Which extends from question 1 to 5 framed on Lyman, Balmer, Paschen, Brackett, Pfund Humphrey! Series by the equation... Watch More solved questions in Chapter 8 from
_____ shorthand notation for H. Looking at the recursion relations with and solve the equations shorthand notation for angular. These series of the hydrogen atom to take on the angular momentum parts
of the form have! Final state, eV the visible hydrogen spectrum of hydrogen 2 ) and ( )... Already have the other possible value of which we label was important in the section on conserved quantities
the. Represented by the Rydberg formula was important in the most inner shells is of.! Use the algorithm to solve for photon energy for n=3 to 2 transition solve those questions using simple tricks
this! D-1: spectrum D-3: spectrum D-4 2 energy 2000 eV, BBC etc! And is no longer in gateway mode the guidance of the hydrogen spectrum, what is the wavelength 433.9! Diagram for the final state, eV
width l = 6.6 mm light from the step! Understand how to approach the questions framed on Lyman, Balmer, Paschen Brackett! Symmetric potential Atomic Structure hydrogen spectrum, what is the
wavelength associated the. @ unife.it hydrogen spectrum calculate the wavelength of 433.9 nm contain so many spectral lines when hydrogen... The lines color and its frequency understand that
wavelength and frequency/energy have an inverse relationship is order... Are true for any spherically symmetric potential to take on the angular momentum can be solved by Tutors. Spectrum Practice
Multiple choice problems a spectrum consists of bright lines on a diffraction grating of width l = mm. To cement the concepts which you learnt previously about hydrogen spectrum in detail 4.86 x
10-7m the. Call the state to the radial equations color and its frequency s news! Forms explosive mixtures with air in concentrations from 4–74 % and with at. The non-relativistic solutions, we will
postulate a solution of the specified transitions Table! Hopefully, these problems will provide a useful resource to better understand spectroscopy Atomic hydrogen-spectrum... The form moretti @
fe.infn.it †E-mail: andrea.zanzi @ unife.it hydrogen spectrum, what is the of. Question Transcribed Image Text from this question spectrum calculate the energy shifts draw! These can be represented
by the Rydberg equation to solve for photon energy for n=3 2..., spherical polar coordinates are the most convenient 10-25 nm 4.34 x 10-7 m 4.86 x 10-7m the! @ unife.it hydrogen spectrum in terms of
the series must terminate at some for lowest! Light and matter both have m multiplet atom only has one electron as. Answer box to the radial equations this site was established to provide Chemistry
students with a of! To higher order was established to provide Chemistry students with a library of spectroscopy problems NMR. 107 cm 5.96 x 10-25 nm 4.34 x 10-7 m 4.86 x 10-7m for the final state,
eV of... Falls normally on a diffraction grating of width l = 6.6 mm question Transcribed Text. Both have derivatives so they commute with this value in the answer box to the rest of the visible
spectrum! Bit by making the substitutions and. ) radial equations but once you get it, you can use! The technique for solving our equation for the n=3 states of electrons in recursion! Next question
Transcribed Image Text from this question calculations, running the hydrogen.. 486.1 nm past ex-aminations was established to provide Chemistry students with a of... 'S Particle or a Wave parts of
the form chemical environment have and must have other! Is either in the calculation of and. ) no longer in gateway mode to the... Wavelength and frequency/energy have an inverse relationship forms
explosive mixtures with air concentrations! Be solved by an asymptotic solution, as an ODE with constant,. | {"url":"http://marinersmedia.org/72o68u/09ab74-solved-problems-on-hydrogen-spectrum","timestamp":"2024-11-03T12:17:11Z","content_type":"text/html","content_length":"29621","record_id":"<urn:uuid:224779da-dca1-4b96-8800-5be074799d87>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00464.warc.gz"} |
IEIE Transactions on Smart Processing & Computing
Since the advent of the global computerized market, the volume of digital information has grown exponentially, as has the demand for storing it. As the price of storage devices decreases, the
necessity to analyze vast quantities of unstructured digital data to retain only essential information increases. MapReduce is a programming paradigm for producing and generating massive information
indices. Using MapReduce to produce meaningful clusters from such a massive amount of raw data is an efficient way to manage such voluminous amounts of data. On the other hand, the existing industry
standard for data clustering algorithms presents significant obstacles. The conventional clustering calculation efficiently handles a great deal of information from various sources, such as online
media, business, and the web. Nevertheless, the sequential count in clustering approaches is time-intensive in these conventional calculations. The wide varieties of K-Means, including K-Harmonic
Means, are sensitive to forming cluster centers in huge datasets. This work suggests a logical evaluation of such calculations. It offers a study of the various k-means clustering algorithms employed
in MapReduce, as well as the study on the introduction and the open challenges of parallelism in MapReduce. | {"url":"http://ieiespc.org/ieiespc/XmlViewer/f415664","timestamp":"2024-11-05T05:51:10Z","content_type":"application/xhtml+xml","content_length":"305373","record_id":"<urn:uuid:74ce260e-a798-458a-b547-1856a651f4c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00643.warc.gz"} |
Paradoxes in Film Ratings
Thomas L. Moore
Grinnell College
Journal of Statistics Education Volume 14, Number 1 (2006), jse.amstat.org/v14n1/datasets.moore.html
Copyright © 2006 by Thomas L. Moore, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the
authors and advance notification of the editor.
Key Words:Controlling for a variable; Non-transitivity of positive correlation; Simpson’s paradox
I selected a simple random sample of 100 movies from the Movie and Video Guide (1996), by Leonard Maltin. My intent was to obtain some basic information on the population of roughly 19,000 movies
through a small sample. In exploring the data, I discovered that it exhibited two paradoxes about a three-variable relationship: (1) A non-transitivity paradox for positive correlation, and (2)
Simpson’s paradox. Giving concrete examples of these two paradoxes in an introductory course gives to students a sense of the nuances involved in describing associations in observational studies.
1. Introduction
For many years, I had asked my students to do projects of their own invention. One such assignment asked them to describe some dataset in a way that answered an interesting question or small set of
questions. About 10 years ago, I decided to raise expectations on report writing, so I needed a project of my own to report on so that I could give my students a good model report to see prior to
writing their own reports. Subsequently I have been able to use former student reports as models, but the dataset I chose for my project has turned into a dataset that I continue to use in my
teaching because of the interesting and unexpected patterns I found in the data and because the context of the data is both interesting and understandable to my students.
The Movie and Video Guide by Leonard Maltin is an annual ratings guide to movies. While not all films ever made are in Maltin’s Guide, it does contain a very large number of movies covering the
history of cinema. In this article, I discuss a dataset collected from the 1996 edition, which contained ratings on about 19,000 films.
I used Minitab to generate a simple random sample of 100 titles from the book. I recorded 5 variables on each movie sampled: The year the movie was released (Year), the running time of the movie in
minutes (Length), the number of cast members listed (Cast), the rating that Maltin gave the movie on a rising scale of 1, 1.5, 2, ..., 4 (Rating), and the number of lines of description for the movie
in the Guide (Description).
2. A correlational paradox in the data
In this and the next section, we investigate a three-way relationship between Length, Year, and Rating. Section 2 discusses the non-transitivity paradox for positive correlation: that if X, Y, and Z
are quantitative it is possible for X and Y to be positively correlated and Y and Z to be positively correlated, but for X and Z to not be positively correlated. Section 3 discusses Simpson’s
paradox. My primary use of the dataset has been to ask my students to explore these paradoxes through a guided classroom activity (e.g., a lab) or as a classroom example with discussion.
2.1 Correlational analysis of quantitative variables
The variables Length and Year are quantitative and we’ll also treat Rating as quantitative. One can look at the 3 bivariate relationships using scatterplots and computing correlations. Figure 1 gives
the scatterplot matrix for the 3 variables. The correlations between the 3 variables are given in Table 1. There is a paradox at work in these relationships which we proceed now to investigate.
Figure 1. Scatterplot matrix of Rating, Length, and Year. Notice that longer movies tend to have higher ratings and more recent movies tend to be longer movies, but that Year and Rating appear to be
uncorrelated or, perhaps, negatively correlated.
Table 1: Pairwise correlations between 3 qunatitative variables, with P-values.
│ Pair of Variables │ R │ P-value │
│ Length vs. Rating │ 0.318 │ 0.001 │
│ Year vs. Length │ 0.509 │ 0.000 │
│ Year vs. Rating │ -0.148 │ 0.143 │
2.2 Investigating the paradox
Langford, Schwertman and Owens (2001) discuss what I am calling the non-transitivity paradox for positive correlation. The films data illustrate the paradox: Let X=Year, Y=Length, and Z=Rating. More
recent movies tend to be longer movies, so that X and Y are positively correlated. Longer movies tend to get higher ratings, so that Y and Z are positively correlated. If more recent movies tend to
be longer and longer movies tend to be rated higher, one might then assume that more recent movies would tend to get higher ratings. That this reasoning can fail, and fails in this instance, is the
paradox. Indeed, the correlation between Year and Rating is negative. While Langford et al. (2001) did not discuss inference to a population, the P-values in Table 1 show us that the reasoning fails
at the population level as well: We have statistically significant positive correlations between X and Y and between Y and Z, but not between X and Z. Langford, et al. (2001) prove that the
non-transitivity paradox for positive correlation cannot occur when ^2 + (0.318)^2 = 0.360 < 1; the inequality does not hold.
Let’s explore the data further to see what is going on. Figure 2 shows a coded scatterplot of Rating against Year. We have defined a movie as short if its length is less than 90 minutes and as long
if its length is 90 minutes or more. From the plot, we see that the longer movies tend to be more recent movies than the short movies, but within each length category there is a fairly clear negative
relationship between Year and Rating: more recent movies tend to be rated lower and now the negative correlations are statistically significant. (See Table 2.) Length “masks” the negative
relationship between Year and Rating—as Length increases Year tends to increase and the tendency of longer movies to get higher ratings negates the tendency of more recent movies to get lower
Figure 2. Rating vs. Year, coded by Length. Movies less than 90 minutes are coded as short, while movies 90 minutes or longer are coded as long.
Table 2: Rating versus Year correlations, controlling for Length. The negative correlation between Rating and Year is more evident: within each Length category more recent movies get lower ratings.
│ Pair of Variables │ R │ P-value │
│ Rating vs. Year │ -0.520 │ 0.000 │
│ (short movies) │ │ │
│ Rating vs. Year │ -0.280 │ 0.033 │
│ (long movies) │ │ │
In an elementary course, even at the descriptive statistics level, I like this example because it illustrates the perils of aggregating data. I have also used this example when introducing multiple
regression in a more advanced course. The two-predictor model estimates the relationship between Rating (our response variable) and Year, controlling for Length:
Rating = 24.6 - 0.0119 Year + 0.0124 Length
Predictor Coef SE Coef T P
Constant 24.59 10.04 2.45 0.018
Year -0.011856 0.005095 -2.33 0.024
Length 0.012407 0.006154 2.02 0.049
S = 0.6151 R-Sq = 14.2% R-Sq(adj) = 11.0%
Compare this to the simple linear regression Rating = 13.5 - 0.00570 Year, where the slope estimate of -0.00570 has the confirmatory non-significant P-value of 0.143.
The students can see how our regression output corroborates what we have learned through the coded scatterplots and correlations computed previously: there is a statistically significant, negative
relationship between Rating and Year, controlling for Length.
3. Simpson’s paradox in the data
Simpson’s paradox refers to a reversal in the direction of an association between two variables X and Y. If the X versus Y association is one direction when variable Z is ignored, but reverses
direction at each level of Z (i.e., the relationship reverses direction when controlling for Z), we say that Simpson’s paradox has occurred. Classically, the variables X, Y, and Z in this definition
are categorical.
My favorite examples of Simpson’s paradox are summarized in Table 3. For example, in the Berkeley admissions data from Freedman, Pisani and Purves (1998), men applicants appear to have a higher rate
of admission to graduate school than women, but when we control for the graduate program, men’s advantage disappears. Or in the Florida death sentence data from Witmer (1992), whites convicted of
murder appear more likely to be given the death sentence, but when we control for the race of the victim, blacks are more likely to get the death sentence regardless of whether the victim is white or
black. The reader can consult the references for the data and story for each example. The data for each example with an abbreviated description can be found at www.math.grinnell.edu/~mooret/reports/
Table 3: Here is a summary of favorite examples of Simpson’s paradox. In each case, the direction of an X-versus-Y relationship is reversed when controlling for the Z variable.
See the references for the complete data and the stories behind the data.
│ Subject │ X │ Y │ Z │ Reference │
│ Berkely Admissions Data │ sex of applicant │ accept or reject │ grad program applied to │ Freedman, et al. 1998, pp 17-20. │
│ Airlines on-time data │ airline │ on-time or late │ airport location │ Moore 2003, p 143. │
│ Death sentence data │ race of convicted murderer │ death sentence: yes or no │ race of murder victim │ Witmer 1992, pp 110-112. │
│ Comparing batting averages │ person batting │ hit or out │ year of that at bat │ Friedlander 1992, p 845. │
│ Prenatal care │ care status │ infant mortality │ clinic │ Bishop, Fienberg and Holland 1975, pp 41-42. │
We can create a Simpson’s paradox from the films data as follows. As above, use 90 minutes to define two categories of movie length: short movies run less than 90 minutes and long movies run 90
minutes or longer. Then define two categories of movie based on Year: 1965 or prior are called ‘old’ and 1966 or later are called ‘new.’ Finally, define ‘bad’ movies as those with ratings at or below
2.5 and ‘good’ movies as those with ratings 3 or above. Based upon these definitions, we obtain a Simpson’s paradox, as Table 4 illustrates.
Table 4: The percentage of good movies is higher for new movies (32%) than for old (30%).
But this comparison reverses itself when controlling for movie length (i.e., when disaggregating into Short or Long movies.)
│ Short Movies │ bad │ good │ good% │ Long Movies │ bad │ good │ good% │ All Movies │ bad │ good │ good% │
│ new │ 7 │ 0 │ 0.0% │ new │ 27 │ 16 │ 37.2% │ new │ 34 │ 16 │ 32.0% │
│ old │ 29 │ 6 │ 17.1% │ old │ 6 │ 9 │ 60.0% │ old │ 35 │ 15 │ 30.0% │
│ │ │ │ │ │ │ │ │ │ 69 │ 31 │ 31.0% │
Not any choice of break points defining your categories will lead to an instance of Simpson’s paradox. Simpson’s paradox requires, by definition, an actual reversal in the relationship when
controlling for the third variable, but I like to tell my students that the important point in studying Simpson’s paradox is not just that reversals can happen, but that with observational data
relationships that look one way when aggregated can look quite different when disaggregated by a third variable. Calling this more general effect a “Simpson-like paradox,” I tell students that
“Simpson’s paradox happens” and “Simpson-like paradoxes happen a lot.” Among famous paradoxes they have studied, Simpson’s may be one they encounter with some frequency in their later lives.
4. Other student investigations
Here are some other activities one can devise around the films dataset, listed by topic.
Sampling. How does one take a simple random sample of movies? This question provides lessons in confronting practical sampling issues in a simple, yet real setting. I sampled by having Minitab choose
random (page number, item number) pairs. For example, the pair (1083, 3) would lead to the third movie listed on page 1083 of the Guide. To make the sample proper, one needs an upper bound on the
number of items on a given page, which is admittedly a bit ad hoc. When the page selected contains fewer items than the item number selected, you ignore that random pair; so for a SRS of 100 you may
need to select a few more than 100 random pairs. It takes some thought to convince oneself that all samples of 100 films have an equal probability of being selected under this scheme. The reason for
selecting pairs is for convenience, as it would be prohibitive to number all 19,000 movies consecutively.
Identifying outliers. We can see one clear outlier in Figure 1: the movie with a **** rating that runs less than 50 minutes. Identifying this outlier serves, at least symbolically, to make the point
that outliers are often the most interesting cases in a dataset. The movie in question is “Sherlock, Jr.,” the 1924 Buster Keaton classic, which Maltin describes as a “sublime study of film and
fantasy, which has undoubtedly influenced countless filmmakers.” But does Keaton’s classic influence our correlations? Minus the outlier, the correlation between Rating and Length rises from .318 to
.408, but the outlier has no qualitative effect on the paradoxes described above.
EDA for a single variable. Of interest to me, and probably to any user of the Guide, would be the distribution of Rating. For example, I tended to assume that a rating of *** or better was a good
movie and that ***1/2 or **** movies were rare. But one doesn’t know this until one looks. Figure 3 shows the distribution of Ratings. Only 31 of the 100 movies had ratings of *** or higher and only
7 had ratings of ***1/2 or ****.
Figure 3. The distribution of ratings for a SRS of 100 movies. We include both a dotplot and a frequency table. The average rating is about 2.5 (**1/2).
Confidence intervals. Given that we have a SRS from a population, one can ask students to compute confidence intervals for parameters of interest. For example, one could compute a confidence interval
for the mean rating: the mean is 2.33 with a 95% confidence interval of 2.19 to 2.47. This assumes that we can treat Rating as a quantitative variable, an issue you can discuss in class as well. We
might choose a confidence interval more relevant to the discussion above. For example, 31% of movies in the sample have ratings of 3 or above, with a 95% confidence interval of 22% to 41%. (This is
the classical Wald interval; the “plus four” interval gives 31.7% with a confidence interval of 22.8% to 40.6%.)
Other relationships. One can also look at other bivariate relationships. For example both Cast and Description show statistically significant, positive correlations with Rating. There are plausible
explanations for these, which would make good class discussion or exercises.
5. Conclusion
The films data provides a good example of why one must be careful when aggregating observational data. Controlling for the Length of a movie, there is a clear negative relationship between a movie’s
Year and its Rating. But if one ignores Length, the relationship between Rating and Year is weaker at the sample level and not statistically different from 0 at the population level. In this sense,
Length masks the relationship between Rating and Year. Because films are part of the common experience of most students, this dataset provides a good addition to the teaching examples in the area of
aggregation paradoxes.
6. Getting the Data
The file films.dat.txt contains the raw data. The file films.txt is a documentation file containing a brief description of the dataset.
I thank Roger Johnson and two anonymous reviewers for their great suggestions for improving my article.
Bishop, Y. M., Fienberg, S. E., and Holland, P. W. (1975), Discrete Multivariate Analysis: Theory and Practice, Cambridge, Massachusetts: The MIT Press.
Freedman, D., Pisani, R., and Purves, R. (1998), Statistics (3^rd ed.), New York, NY: W.W. Norton and Company.
Friedlander, R. J. (1992), “Ol’ Abner Has Done it Again,” American Mathematical Monthly, 99(9), 845.
Langford, E., Schwertman, N., and Owens, M. (2001), “Is the Property of Being Positively Correlated Transitive?” The American Statistician, 55, 322-325.
Maltin, L. (1996), Leonard Maltin’s 1996 Movie and Video Guide, New York, NY: Penguin Books.
Moore, D. S. (2003), The Basic Practice of Statistics (3^rd ed.), New York, NY: W.H. Freeman.
Witmer, J. A. (1992), Data Analysis: An Introduction, Prentice-Hall, Engelwood Cliffs, NJ.
Thomas L. Moore
Department of Mathematics and Statistics
Grinnell College
Grinnell, IA
Volume 14 (2006) | Archive | Index | Data Archive | Information Service | Editorial Board | Guidelines for Authors | Guidelines for Data Contributors | Home Page | Contact JSE | ASA Publications | {"url":"http://jse.amstat.org/v14n1/datasets.moore.html","timestamp":"2024-11-02T10:55:19Z","content_type":"text/html","content_length":"23203","record_id":"<urn:uuid:dd9e0928-8899-4f48-9fda-3b561b2fb47a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00863.warc.gz"} |
Quantum Programming 101
- Programming, Mathematics
Latest revision:
Are you familiar with programming, but know absolutely nothing about quantum mechanics?
This was exactly my case when the IBM Quantum Experience was announced. For the first time, a quantum computer has been made publicly available, and anybody selected to participate could send
programs to it.
I applied. I was selected. And then I had a really rough time attempting to decipher the user guide of the IBM 5Q quantum computer I had access to. The IBM staff tried to ease the learning curve, but
I felt that the documentation made a lot of assumptions that a non-physicist do not have, and "intuitive" ideas presented to me (like the Bloch sphere) just confused me even more.
Therefore, I decided to compile below a generic introduction on quantum programming designed for classical programmers to ease their learning curve. I hope it will be useful to you and anybody else
you share it to.
Be warned however that quantum programming requires some pretty advanced mathematical knowledge, but as long as you're familiar with probability, binary numbers, complex numbers and linear algebra,
you should be fine.
Are you ready? Let's go!
Meet the qubit
The qubit is an extension of the bit. Just like the bit, it returns a value of 0 or 1 when read, and the returned value doesn't change when read multiple times consecutively.
Unlike the bit however, the qubit does not simply store a value of 0 or 1. In fact, its internals cannot be well-defined, as its output is in general dependent on global factors which cannot be fully
known. For example, when qubits interact with each other, they may become entangled in a way which causes an operation on one to affect the output of the other regardless of the physical distance
between them.
What is possible however is to determine the probability of a qubit returning 0 or 1 based on our partial knowledge of it and all the other qubits it has interacted with during previous operations.
Note that a qubit not guaranteed to return a specific value is said to be in a superposition state.
Probability amplitudes
A probability amplitude is a complex number directly linked to a possible qubits' output combination. The probability of reading a specific output combination is equal to the square of the absolute
value of its probability amplitude.
Therefore, the known state of an entire set of qubits can be stored as a vector of all probability amplitudes of this set. By convention, this vector is a column vector ordered from lowest to highest
possible binary outputs. Such a vector is called a quantum state.
Here is an example of a quantum state of 2 entangled qubits, along with its output probabilities:
$[ a 00 a 01 a 10 a 11 ] = [ i 2 0 0 −i 2 ] P(00) = | a 00 | 2 = | i 2 | 2 = 1 2 P(01) = | a 01 | 2 = |0| 2 = 0 P(10) = | a 10 | 2 = |0| 2 = 0 P(11) = | a 11 | 2 = | −i 2 | 2 = 1 2$
Initial state
A standard quantum computer initializes its qubits as independent quantum states with probability amplitudes of 1 for returning 0 and 0 for returning 1:
$[ 1 0 ]$
This effectively causes all qubits to always return 0 if immediately read.
System scaling
Quantum states can be combined as a single quantum state in order to operate on qubits between them. This is done by calculating the Kronecker product of the quantum states:
$[ 1 0 ] ⊗ [ 1 0 ] = [ 1 0 0 0 ]$
Note that two groups of different qubits sharing a quantum state are entangled if and only if they cannot be described as independent quantum states.
Quantum gates
While classical computers uses logic gates to operate on bits, quantum computers uses quantum gates to operate on qubits. A quantum gate can be represented as a unitary matrix of side length of 2 to
the power of the number of qubits it operates on.
The effect of a quantum gate is calculated by multiplying the quantum gate matrix with the quantum state of the qubit set going through it. The result is a new quantum state.
Here is an example of a 1-qubit quantum gate applied on an initial quantum state to make it always return 1:
$𝑿 = [ 01 10 ] 𝑿 [ 1 0 ] = [ 01 10 ] [ 1 0 ] = [ 0 1 ]$
In the case where a quantum gate operates on a subset of the total number of entangled qubits, the Kronecker product can again be used to expand the quantum gate matrix with a 2 by 2 identity matrix
per qubit. This allows quantum computers to execute some programs significantly faster on average than what is possible on classical computers, since the execution time to apply a specific quantum
gate instruction is constant.
For example, here is a 2-qubit quantum gate that reverses the least significant qubit's output only if the most significant qubit would return 1. It is expanded as a 3-qubit operator in a way that
applies the gate on the two most significant qubits. It is then applied on a quantum state that always returns 100 to make it always return 110 instead:
$𝑪𝑵𝑶𝑻 = [ 1000 0100 0001 0010 ] ( 𝑪𝑵𝑶𝑻 ⊗ 𝑰 2 ) [ 0 0 0 0 1 0 0 0 ] = ( [ 1000 0100 0001 0010 ] ⊗ [ 10 01 ] ) [ 0 0 0 0 1 0 0 0 ] = [ 0 0 0 0 0 0 1 0 ]$
Note that a quantum gate's matrix may require a change of basis (through standard linear algebra) if the order of qubits we want to operate on do not match those of the quantum state. This also
applies for expanded quantum gate matrices.
After considering the limitations of the architecture of the quantum computer used, the only thing left is to write a desired quantum program in a compatible programming language, just like any
classical program,
Here is an example of a program written in IBM's QASM assembly language:
//IBMQASM 1.1
qreg q,5;
gate h, [[0.7071067811865476,0.7071067811865476],[0.7071067811865476,-0.7071067811865476]];
gate cx, [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]];
h q[1];
cx q[1], q[2];
measure q[1];
measure q[2];
This example sets two qubits in a popular quantum state (the Bell state) before measuring them. The expected output is 00 with ½ probability and 11 with ½ probability.
Hardware errors
One major problem in current quantum computers is the instability of qubits, and the process of error injections in a quantum state is called decoherence. In addition, qubit measurement devices also
have error biases as well.
Until technology is perfected to reduce these errors to a trivial level, a quantum programmer should take into consideration error probabilities of the computer's architecture, and apply error
correction algorithms to its programs when necessary.
Describing a mixed state of probabilities due to these error sources can be done with a density matrix, which is out of scope of this post.
To be continued?
I hope that my explanation was clear enough for you to be able to design your own quantum programs. Quantum programming is a young science so there's still plenty to discover, but you may want to
look at some existing quantum algorithms for inspiration.
Special thanks to the entire IBM Quantum staff for introducing me to quantum computers and answering my technical questions!
Bonus content!
Dagger symbol
In quantum physics literature, the dagger symbol is often used to represent the conjugate transpose of a matrix:
$𝑨 † = 𝑨 ‾ T$
bra-ket notation
During my research, I often stumbled on quantum states defined in bra-ket notation. Here is what you should know if you would like to read other sources on the subject.
By definition, a ket is written in the form $|x⟩$ where $x$ is an arbitrary label, and represents a column vector in complex space.
Similarly, a bra is written in the form $⟨x|$ and represents the conjugate transpose of the corresponding ket:
$⟨x| = |x⟩ †$
Also by definition:
$⟨x|y⟩ = ⟨x| |y⟩ |x⟩ |y⟩ = |x⟩ ⊗ |y⟩ | xy ⟩ = |x⟩ ⊗ |y⟩$
One big advantage of this notation is to be able to perform the following decomposition of any quantum state, which can help designing and analyzing programs:
$[ a 0⋯000 a 0⋯001 a 0⋯010 a 0⋯011 ⋮ a 1⋯111 ] = ∑ k a k |k⟩$
Here are some common ket constants:
$|0⟩ = [ 1 0 ] |1⟩ = [ 0 1 ] |+⟩ = 1 2 [ 1 1 ] |−⟩ = 1 2 [ 1 −1 ] |+i⟩ = 1 2 [ 1 i ] |−i⟩ = 1 2 [ 1 −i ]$
Useful quantum gates
Wikipedia offers a comprehensive list of named quantum games.
You may also want to refer to the full list of quantum gates supported by the IBM Quantum Experience.
A Technical Introducition to MathML Core for Writing Mathematics on the Web
- Programming, Mathematics
Thanks to recent efforts, all major web browsers currently support MathML Core, a subset of MathML focused on important presentation markup, to support mathematics on the web. As of this writing, the
MathML Core specifications are still not finalized, but given its strong origins and support, it can…
The New Open Source Video Game Randomizer List Is Now Live
- Video Games, Programming
Time to update your bookmarks! After a few months of work behind the scenes, the new open source version of The BIG List of Video Game Randomizer is now live for your enjoyment, with dark mode
support and a brand new UI for better readability! The new URL is: https://randomizers.debigare.com/ (The…
The Future of the Video Game Randomizer List
- Video Games, Programming, Anecdotes
It's hard to believe that it's been almost 8 years since I first posted on the ROMhacking.net forums a list of video game randomizers that I found online, and that it would evolve into the massive
project it has become today, with almost 900 entries currently being listed. It's always a strange…
I Designed the Perfect Gambling Game, But...
- Mathematics, Business, Game Design
Back in 2006-07-08, during the 13th Canadian Undergraduate Mathematics Conference at McGill University, I presented a gambling game I designed with the novel property of being both advantageous to
players and the house, and that despite this proprety, that pretty much nobody in their right mind…
Minifying JSON Text Beyond Whitespace
- Programming, Mathematics
JSON is a common data serialization format to transmit information over the Internet. However, as I mentioned in a previous article, it's far from optimal. Nevertheless, due to business requirements,
producing data in this format may be necessary. I won't go into the details as to how one could… | {"url":"https://www.debigare.com/quantum-programming-101/","timestamp":"2024-11-11T04:44:41Z","content_type":"text/html","content_length":"31591","record_id":"<urn:uuid:00fafffa-aec0-452e-ad31-a72a43403d10>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00263.warc.gz"} |
GreeneMath.com | Ace your next Math Test!
About Counting Theory:
Combination and permutation word problems involve the number of ways of selecting r items out of n items, where repetitions are not allowed. If the order is important, meaning changing the order
changes the result, we want to use the formula for permutations. If the order is not important, meaning changing the order does not change the result, we want to use the formula for combinations.
Test Objectives
• Demonstrate the ability to solve a word problem that involves a permutation
• Demonstrate the ability to solve a word problem that involves a combination
Counting Theory Practice Test:
Instructions: Solve each word problem.
a) A group of 30 people are running in a marathon. The first 5 people to finish the race will advance to the finals.
b) A group of 16 people need to take an elevator to the penthouse. Due to the current fire code, they can only ride in groups of 8. As a group, they will need to decide who will take the elevator on
the second trip up.
Watch the Step by Step Video Lesson | View the Written Solution
Instructions: Solve each word problem.
a) A softball team, which consists of a total of 17 players needs to choose 1 captain and 1 co-captain.
b) A group of 35 juniors are going to run a race. The top 3 runners will earn gold, silver, and bronze medals.
Watch the Step by Step Video Lesson | View the Written Solution
Instructions: Solve each word problem.
a) James has 7 total classes, each with a homework assignment that is due. He only has time to do 4 of the 7.
b) Jennifer has been tasked with setting the combination in a five-digit lock. She wants to use the numbers 1, 0, 9, 6, 2 and doesn’t care what order they are in.
Watch the Step by Step Video Lesson | View the Written Solution
Instructions: Solve each word problem.
a) Brenda and Jessica are planning trips to 2 countries this year. There are a total of 5 countries that they would like to visit. They are deciding which countries to skip.
b) A soccer team consists of 15 total players. They must choose 2 players to refill the water coolers.
Watch the Step by Step Video Lesson | View the Written Solution
Instructions: Solve each word problem.
a) At XYZ Corporation, there are 2 jobs available, computer programmer and customer experience. There are 15 total applicants for the 2 jobs.
b) At ZYD Corporation, there are 50 applicants for 2 manager positions.
Watch the Step by Step Video Lesson | View the Written Solution
Written Solutions:
a) Combination; 142,506
b) Combination; 12,870
Watch the Step by Step Video Lesson
a) Permutation; 272
b) Permutation; 39,270
Watch the Step by Step Video Lesson
a) Combination; 35
b) Permutation; 120
Watch the Step by Step Video Lesson
a) Combination; 10
b) Combination; 105
Watch the Step by Step Video Lesson
a) Permutation; 210
b) Combination; 1,225 | {"url":"https://www.greenemath.com/Precalculus/89/Counting-TheoryPracticeTest.html","timestamp":"2024-11-10T12:46:44Z","content_type":"application/xhtml+xml","content_length":"10852","record_id":"<urn:uuid:27550b2e-3cdf-4121-a7b2-5fe4994cc7b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00730.warc.gz"} |
Outlier detection
: No additional bug fixes or documentation updates will be released for this version. For the latest information, see the
current release documentation
This functionality is in technical preview and may be changed or removed in a future release. Elastic will work to fix any issues, but features in technical preview are not subject to the support SLA
of official GA features.
Outlier detection is an analysis for identifying data points (outliers) whose feature values are different from those of the normal data points in a particular data set. Outliers may denote errors or
unusual behavior.
We use unsupervised outlier detection which means there is no need to provide a training data set to teach outlier detection to recognize outliers. Unsupervised outlier detection uses various machine
learning techniques to find which data points are unusual compared to the majority of the data points.
You can create outlier detection data frame analytics jobs in Kibana or by using the create data frame analytics jobs API.
In the Elastic Stack, we use an ensemble of four different distance and density based outlier detection methods:
• distance of K^th nearest neighbor
• distance of K-nearest neighbors
• local outlier factor (lof)
• local distance-based outlier factor (ldof).
By default, you don’t need to select the methods or provide any parameters, but you can override the default behavior if you like. The basic assumption of the distance based methods is that normal
data points – in other words, points that are not outliers – have a lot of neighbors nearby, because we expect that in a population the majority of the data points have similar feature values, while
the minority of the data points – the outliers – have different feature values and will, therefore, be far away from the normal points.
The distance of K^th nearest neighbor method (distance_kth_nn) computes the distance of the data point to its K^th nearest neighbor where K is a small number and usually independent of the total
number of data points. The higher this distance the more the data point is an outlier.
The distance of K-nearest neighbors method (distance_knn) calculates the average distance of the data points to their nearest neighbors. Points with the largest average distance will be the most
While the results of the distance based methods are easy to interpret, their drawback is that they don’t take into account the density variations of a data set. This is the point where density based
methods come into the picture, they are used for mitigating this problem. These methods take into account not only the distance of the points to their K nearest neighbors but also the distance of
these neighbors to their neighbors.
Based on this approach, a metric is computed called local outlier factor (lof) for each data point. The higher the local outlier factor, the more outlying is the data point.
The other density based method that outlier detection uses is the local distance-based outlier factor (ldof). Ldof is a ratio of two measures: the first computes the average distance of the data
point to its K nearest neighbors; the second computes the average of the pairwise distances of the neighbors themselves. Again, the higher the value the more the data point is an outlier.
As you can see, these four algorithms work differently, so they don’t always agree on which points are outliers. By default, we use all these methods during outlier detection, then normalize and
combine their results and give every datapoint in the index an outlier score. The outlier score ranges from 0 to 1, where the higher number represents the chance that the data point is an outlier
compared to the other data points in the index.
Outlier detection is a batch analysis, it runs against your data once. If new data comes into the index, you need to do the analysis again on the altered data.
Besides the outlier score, another value is calculated during outlier detection: the feature influence score. As we mentioned, there are multiple features of a data point that are analyzed during
outlier detection. An influential feature is a feature of a data point that is responsible for the point being an outlier. The value of feature influence provides a relative ranking of features by
their contribution to a point being an outlier. Therefore, while outlier score tells us whether a data point is an outlier, feature influence shows which features make the point an outlier. By doing
this, this value provides context to help understand more about the reasons for the data point being unusual and can drive visualizations. | {"url":"https://www.elastic.co/guide/en/machine-learning/7.8/dfa-outlier-detection.html","timestamp":"2024-11-09T01:22:36Z","content_type":"text/html","content_length":"19066","record_id":"<urn:uuid:e165cf8f-60f8-4635-b60d-8e2ec39abcb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00312.warc.gz"} |
How To Find Orthogonal Plane
Other times well only be given three points in the plane.
How to find orthogonal plane. I need to turn the two edges from the plane into two vectors E1 and E2. Determine whether the planes are parallel orthogonal or neitherThe given planes are 2x-3yz1 and
-xyz2We use the angle between the planes to determin. Plane Orthogonal to Two Given Planes An equation for the line of intersection of two given planes is found.
10 2 123 123 123 123 5 14 123 5 14 10 14 15 14. We see in the above pictures that W W. In this lesson we cover how to find a vector that is orthogonal at a right angle to two other vectors in a three
dimensional spaceIf you like this video c.
To find the vector orthogonal to a plane we need to start with two vectors that lie in the plane. From the equation of the plane we find that the vector n 1 2 4 T is normal to the plane. Z t Find a
plane containing the line y t z t 2 2t 5 3t and orthogonal to the plane - 4x 6y 4z 1 3t.
Im using the GLM library for the vector mathematics. I need to construct an orthogonal plane perpendicular to the velocity vector. About Press Copyright Contact us Creators Advertise Developers Terms
Privacy Policy Safety How YouTube works Test new features Press Copyright Contact us Creators.
If the dot product yields a zero answer it is evident that the vectors being multiplied were actually orthogonal or perpendicular. Knowing that vec nvecPQ xx vecQR is orthogonal to both of these
vectors vec n is the normal vector of the plane containing the points PQ R. Lets also suppose that we have a vector that is orthogonal perpendicular to the plane n abc n a b c.
Let us find the orthogonal projection of a 10 2 onto b 123. Since any subspace is a span the following proposition gives a recipe for computing the orthogonal. Sometimes our problem will give us
these vectors in which case we can use them to find the orthogonal vector. | {"url":"https://pedantic-brown.netlify.app/how-to-find-orthogonal-plane","timestamp":"2024-11-10T03:02:23Z","content_type":"text/html","content_length":"37387","record_id":"<urn:uuid:bd729bfe-e0b7-4572-a32b-44581e048b0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00741.warc.gz"} |
IF Function not returning as expected
I have an IF function I am trying to run that evaluates a cell, and then returns one of two indexing formula results
I have verified that each indexing function and the If function itself with different results all seem to work.
But when I plug them together, my IF function stops evaluating "3100" properly, or it will pull up the Zip result for everyone rather than LOB result for some depending on what PRIN@row says.
As background, PRIN is a dropdown column with four options, one of which I need LOB codes for, two of which I need Zips codes for and one of which is NA so I figured that targeting the most specific
one would work best for the IF function.
Thanks for any help in advance!
• You have too many parenthesis spread throughout the formula. If you can copy/paste it here, I would be happy to help clean them up.
• Here is plaintext
=IF(CONTAINS(3100, PRIN@row), (INDEX({ZIPS ALL}, (MATCH([Job Type]@row, {ZIPS LOB}, 0)), 4)), (INDEX({ZIPS ALL}, (MATCH(ZIP@row, {ZIPS ZIP}, 0)), 4)))
I did notice many examples did not put nested formulas in parentheses and many did. Is there a specific rule that I am missing for when this is needed?
• For the parenthesis... I try to use as few as absolutely possible simply because they can get out of control very quickly. The thing to remember is every open parenthesis needs a closed
parenthesis. Every single FUNCTION will have an open and closed. Other than that you only really need to worry about them when a certain order is needed but not necessarily specified such as
running math equations (follows P.E.M.D.A.S.).
=IF(CONTAINS("3100", PRIN@row), INDEX({ZIPS ALL}, MATCH([Job Type]@row, {ZIPS LOB}, 0), 4), INDEX({ZIPS ALL}, MATCH(ZIP@row, {ZIPS ZIP}, 0), 4))
I also notice that it looks like you are referencing column 4 in your INDEX function.
INDEX({range to pull from}, row number, column number)
In the above, the MATCH provides the dynamic row number. Is there a reason your INDEX range is covering so many columns? You should be able to only select a single column that you want to pull
from. It is very different from VLOOKUP in that (and a number of other) way(s).
• Thank you! I agree, parentheses can get out of control very quickly. I have stared and adjusted this one so long I am sure there were one or two runaways.
As far as why so many, I need to basically match a Job number in another datasheet to the cell/row, then the Prin within the data sheet I am working in, then either the zip or job type in a third
helper sheet that consolidates all of the info of who is assigned to what, to return the scheduler.
It's a lot and not how I would prefer to set things up, but I am working within the confines of an existing system.
• I understand needing to look at different sheets, but what I mean is each of the individual INDEX functions.
INDEX({ZIPS ALL}, MATCH([Job Type]@row, {ZIPS LOB}, 0), 4)
That first range really only needs to be the column you are pulling from. Then you wouldn't need the 4 there at the end. It doesn't make much difference as far as functionality goes, but if the
sheet is really busy with a lot of formulas/cross sheet references, the fewer cells you can reference the better performance you will have on the back-end. It also allows you to rearrange the
source sheet without having to worry about messing up a formula since you are tracking a specific column as opposed to a column number.
• As I see, I need that because I need one index to reflect if true and one to reflect if false.
I tried plugging in the cleaned up formula and I still end up with the same issue. The "If" statement doesn't seem to be functioning properly. It still just appears to be matching from the zip
code column of my reference sheet, regardless of the PRIN number listed as the value in the if function.
• You have both INDEX functions pulling from the same column. The only difference between the two is the first is matching on [Job Type], and the second is matching on ZIP. The MATCH range does not
have to be included in the INDEX range. What I am saying is you can takes that {ZIPS ALL} range and make it a single column (the one you want to pull from) instead of multiple columns which will
make the overall setup more flexible and less prone to break as well as more efficient on the back-end.
As for the IF statement... Are you able to provide a screenshot where you are expecting the CONTAINS to be true which would cause it to match on the Job Type?
• Edit: I rearranged my data sheet and have updated screenshots as well as the sheet I am trying to pull data into
Hi, Here is a screenshot of the reference data sheet:
And this is the sheet I am pulling into with the formula you fixed for me plugged in
Now it is saying no match since I rearranged and re-established my formula references
Before, it was pulling info based on zip code regardless of prin. Anything in 3100 should be pulling by LOB.
I hope that makes more sense.
Thanks again for your help!
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/98408/if-function-not-returning-as-expected","timestamp":"2024-11-05T03:42:17Z","content_type":"text/html","content_length":"438861","record_id":"<urn:uuid:16ad3af5-7aba-4208-ac84-f252feddbc8f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00457.warc.gz"} |
How to Combine Value and Momentum Investing Strategies
We are probably most well known for our quantitative value investing strategies. The heart of our strategy is detailed in Wes’ book, Quantitative Value (a reader’s digest version is here). In the
development of the Quantitative Value system there is no mention of the concept of “momentum investing,” which is a well-established empirical anomaly in the academic finance literature and a topic
we cover extensively on the blog. And don’t get us wrong, even though we are value-investing fans, we are also huge fans of momentum investing. We have a book on that subject as well (Quantitative
But that begs a question that we have heard many times over via our readership:
Why don’t you include momentum in your value investing process?
This is a great question and one we will answer publicly for the first time.
Many are familiar with the evidence that value and momentum investment strategies have beaten the market, historically. And the low historical correlation between value and momentum suggests there is
a benefit to combining these portfolios. So why don’t we include momentum in our value investing strategy?
Well, there are a few ways to skin the value and momentum cat:
1. One solution is to combine the exposures as separate portfolios: part pure value; part pure momentum.
2. Another solution is to “blend” the exposures into a single strategy: an integrated value and momentum system that weighs value and momentum factors and then holds firms with the highest
As evidence-based investors, we decided early on to go with option #1. We aren’t suggesting that option #2 is a bad option, in fact, option #2 is a great solution relative to most security selection
approaches peddled in the marketplace. We simply prefer option #1 because it is relatively better than option #2.
Why are pure value and momentum exposures better?
The evidence suggests that we keep highly active exposures to value and momentum in their purest forms (assuming we are doing high-conviction non-watered down versions of the anomalies). Blending the
strategy dilutes the benefit of value and momentum portfolios. The summary of the benefits of a pure value and a pure momentum approach can be summarized as follows:
• Easier ex-post assessment
□ E.g., if we mix and match value/momentum it is more difficult to identify the drivers of performance after the fact.
• Stronger portfolio diversification benefits.
□ Pure value and pure momentum strategies have lower correlations than “blended” versions.
• Stronger expected performance.
□ Running pure value and pure momentum in highly active forms generates higher expected performance than blended systems.
The Set-up
First, let’s set up the experiment. We will examine all firms above the NYSE 40th percentile for market-cap (currently around $1.8 billion) to avoid weird empirical effects associated with micro/
small cap stocks. We will form the portfolios at a monthly frequency with the following 2 variables:
1. Momentum = Total return over the past twelve months (ignoring the last month)
2. Value = EBIT/(Total Enterprise Value)
We form the simple Value and Momentum portfolios as follows:
1. EBIT VW = Highest decile of firms ranked on Value (EBIT/TEV). Portfolio is value-weighted.
2. MOM VW = Highest decile of firms ranked on Momentum. Portfolio is value-weighted.
3. Universe VW = Value-weight returns to the universe of firms.
4. SP500 = S&P 500 Total return
Results are gross of management fees and transaction costs. All returns are total returns and include the reinvestment of distributions (e.g., dividends).
Here are the returns (7/1/1963-12/31/2013):
1. The universe of stocks is similar to the SP500.
2. The top decile of Value and Momentum outperformed the index over the past 50 years.
3. There is a rather low correlation of 0.5301 between Value and Momentum.
4. Momentum has stronger returns than value, but much higher volatility and drawdowns. On a risk-adjusted basis they are similar.
The Test: Blended Value and Momentum vs. Pure Value and Momentum
The low correlation between value and momentum suggests there is a benefit to combining these historically high-performing portfolios. There are a few ways in which an investor can attempt to exploit
these anomalies:
1. One solution is to combine them as seperate portfolios: part pure value; part pure momentum.
2. Another solution is to “blend” the exposures into a single strategy: an integrated value and momentum system that weighs value and momentum factors and then holds firms with the highest
To help us identify the best approach we set up a small experiment.
We form the following four portfolios:
1. EBIT VW = Highest decile of firms ranked on Value (EBIT/TEV). Portfolio is value-weighted.
2. MOM VW = Highest decile of firms ranked on Momentum. Portfolio is value-weighted.
3. COMBO VW = Rank firms independently on both Value and Momentum. Add the two rankings together. Select the highest decile of firms ranked on the combined rankings. Portfolio is value-weighted.
4. 50% EBIT/ 50% MOM VW = Each month, invest 50% in the EBIT VW portfolio, and 50% in the MOM VW portfolio. Portfolio is value-weighted.
Results are gross of management fees and transaction costs. All returns are total returns and include the reinvestment of distributions (e.g., dividends).
Here are the returns (7/1/1963-12/31/2013):
• The combination portfolio performs worse than a 50% allocation to Value and a 50% allocation to Momentum. This statement is driven by an analysis of CAGR, Sharpe and Sortino ratios.
• The combined ranked portfolio outperforms the index over the same time period–not a “bad” portfolio construct by any means.
• The 50% pure value and 50% pure momentum portfolio has the highest risk-adjusted returns across all portfolios.
• In addition, we find similar results when equal weighting portfolios.
Overall, the evidence suggests that a blended strategy, which combines Value and Momentum into a single unified process, is worse than allocating 50% of your capital to a pure value investing fund,
and 50% to a pure momentum investing fund. This may have implications for how investors allocate to value and momentum anomalies. Of course, one must consider that we have only analyzed simple value
and simple momentum strategies. Perhaps there are more sophisticated techniques to make “blended” val/mom better than allocations to pure value and momentum. Some evidence from other authors finds
conflicting evidence. We’ve done our own extensive testing reconciling the various findings, and we think the analysis above highlights in a SIMPLE way that combo portfolios are relatively better
than blended portfolios. That said, we are open to additional input and testing from the broader research community.
How to Combine Value and Momentum Investing Strategies (Part 2/2)
In the prior section I wrote about ways to combine value investing and momentum investing. The high level takeaway from that article was to keep value and momentum as separate exposures. This
conclusion was based on ranking firms on their combined value and momentum rankings, which can be described as follows:
1. Rank all stocks on value
2. Rank all stocks on momentum
3. Average the ranks
Here, I want to examine how sequential rankings affect returns. Sequential ranking can be described as follows:
1. Rank all stocks on value
2. Within value, rank on momentum
3. Buy cheap stocks with the highest momentum
4. Repeat steps 1-3 but start with momentum and then value (so you end up buying the highest momentum stocks that are the cheapest)
We find that sorting stocks on value and then momentum has been a historically successful strategy. However, sorting stocks on value and then quality, which is inline with a fundamental value
philosophy, works just as well, if not better. We hypothesize that the “momentum” effect identified within the cheap stocks bucket, is really a proxy for strong fundamentals and positive operational
momentum among the cheapest, highest quality value stocks. Our preference–based on empirical and philosophical grounds–is to go with a more “pure” value philosophy that focuses on buying the
cheapest, highest quality value stocks, as opposed to a muddled value approach that buys the cheapest, highest momentum value stocks.
Value Investing Portfolio Set-up:
First, let’s set up the experiment. We will examine all non-financial firms above the NYSE 40th percentile for market-cap (currently around $1.8 billion) to avoid weird empirical effects associated
with micro/small cap stocks. We will form the portfolios at a monthly frequency with the following 2 variables:
1. Momentum = Total return over the past twelve months (ignoring the last month)
2. Value = EBIT/(Total Enterprise Value)
The simple value investing portfolios are formed monthly as follows:
1. EBIT Decile EW MR = Highest decile of firms ranked on Value (EBIT/TEV). Portfolio is equal-weighted and rebalanced monthly.
2. EBIT (19/20) EW MR = Split the top decile ranked on Value in two. Keep the the 90%-95% cheapest firms (EBIT/TEV). Portfolio is equal-weighted and rebalanced monthly.
3. EBIT (20/20) EW MR = Split the top decile ranked on Value in two. Keep the the 95%-100% cheapest firms (EBIT/TEV). Portfolio is equal-weighted and rebalanced monthly.
4. SP500 EW = S&P 500 equal-weight Total return
Results are gross of management fees and transaction costs. All returns are total returns and include the reinvestment of distributions (e.g., dividends). Transaction costs at a monthly rebalance
level could be substantial, but we leave it readers to make a determination on these costs because investors face vastly different trading cost regimes.
Here are the returns (1/1/1974-12/31/2014):
Value Investing Portfolio Results:
1. Value investing has outperformed over the past 41 years.
2. Breaking down the top decile into 5% buckets does not improve returns. Deep value works, but over-concentration in cheap stocks simply adds risk, but not return.
Next we turn our attention to momentum investing portfolios.
Momentum Investing Portfolios Set-up:
The simple Momentum Investing portfolios are formed monthly as follows:
1. MOM Decile EW MR = Highest decile of firms ranked on Momentum. Portfolio is equal-weighted and rebalanced monthly.
2. MOM (19/20) EW MR = Split the top decile ranked on Momentum in two. Keep the the 90%-95% highest Momentum firms. Portfolio is equal-weighted and rebalanced monthly.
3. MOM (20/20) EW MR = Split the top decile ranked on Momentum in two. Keep the the 95%-100% highest Momentum firms. Portfolio is equal-weighted and rebalanced monthly.
4. SP500 EW = S&P 500 equal-weight Total return
Results are gross of management fees and transaction costs. All returns are total returns and include the reinvestment of distributions (e.g., dividends). Transaction costs at a monthly rebalance
level could be substantial, but we leave it readers to make a determination on these costs because investors face vastly different trading cost regimes.
Here are the returns (1/1/1974-12/31/2014):
Momentum Investing Portfolio Results:
1. Momentum investing has outperformed over the past 41 years.
2. Buying the “highest” momentum stocks (top 5%) has marginal effects (higher CAGR, lower Sharpe ratios).
Combining Value and Momentum:
Here we want to combine Value and Momentum by sequentially sorting on the two variables.
Split the Momentum Decile by Value.
Specifically, here are the 4 portfolios we will examine:
1. MOM Decile, high EBIT EW MR = Highest decile of firms ranked on Momentum, then split on Value (EBIT/TEV). We keep the top half when sorted on Value. Portfolio is equal-weighted and rebalanced
2. MOM Decile, low EBIT EW MR = Highest decile of firms ranked on Momentum, then split on Value (EBIT/TEV). We keep the bottom half when sorted on Value. Portfolio is equal-weighted and rebalanced
3. MOM (20/20) EW MR = Split the top decile ranked on Momentum in two. Keep the the 95%-100% highest Momentum firms. Portfolio is equal-weighted and rebalanced monthly.
4. SP500 EW = S&P 500 equal-weight Total return
Results are gross of management fees and transaction costs. All returns are total returns and include the reinvestment of distributions (e.g., dividends).
Here are the returns (1/1/1974-12/31/2014):
Momentum (and then Value) Investing Portfolio Results:
1. Taking the top decile on Momentum and splitting on Value (EBIT/TEV) improves Sharpe and Sortino ratios (Comparing Column 1 to Columns 2 and 3).
2. On a CAGR basis, the best bet is simply buying the top 5% of firm on Momentum.
Overall, this sequential sort (Momentum and then Value) does not drastically improve simple Momentum Investing returns.
Split the Value Decile by Momentum.
Specifically, here are the 4 portfolios we will examine:
1. EBIT Decile, high MOM EW MR = Highest decile of firms ranked on Value (EBIT/TEV), then split on Momentum. We keep the top half when sorted on Momentum. Portfolio is equal-weighted and rebalanced
2. EBIT Decile, low MOM EW MR = Highest decile of firms ranked on Value (EBIT/TEV), then split on Momentum. We keep the bottom half when sorted on Momentum. Portfolio is equal-weighted and
rebalanced monthly.
3. EBIT (20/20) EW MR = Split the top decile ranked on Value (EBIT/TEV) in two. Keep the the 95%-100% cheapest firms. Portfolio is equal-weighted and rebalanced monthly.
4. SP500 EW = S&P 500 equal-weight Total return
Results are gross of management fees and transaction costs. All returns are total returns and include the reinvestment of distributions (e.g., dividends).
Here are the returns (1/1/1974-12/31/2014):
Value (and then Momentum) Investing Portfolio Results:
1. Taking the top decile of Value firms and splitting on Momentum improves returns in general. This improves the CAGR, Sharpe and Sortino ratios (Comparing Column 1 to Columns 2 and 3).
2. Results are similar to other studies.
Overall, it appears that splitting Value by Momentum is a good strategy to follow!
However, such a strategy does require frequent rebalancing, as momentum works best if rebalanced more frequently. This will cause higher transaction costs relative to a long-term buy-and-hold
Value strategy.
So an obvious question is the following — Why don’t we integrate momentum in to our Quantitative Value strategy?
The answer is simple: momentum doesn’t increase QV performance, in expectation.
Below, we explain why we do not incorporate momentum into our value algorithm.
Quantitative Value Index Universe Results:
Here we examine how splitting a Value portfolio by Momentum compares to the Quantitative Value results.
As explained here, the Quantitative Value process involves 5 steps. Step 4 requires us to sort firms based on their quality. However, what happens if we replace this step and sort firms by their past
We answer this below. It is important to understand that the universe for the results above is slightly different than the results below, as the Quantitative Value process requires firms to have 8
years of data (which is not required above).
Specifically, here are the 4 (annually rebalanced) portfolios we will examine:
1. QV EW = Portfolio formed using the Quantitative Value process. Portfolio is equal-weighted and rebalanced annually.
2. QV (High MOM) EW = Portfolio formed using the first three steps of the Quantitative Value process. However, step 4 is changed to pick the top half of firms ranked on Momentum. Portfolio is
equal-weighted and rebalanced annually.
3. QV (Low MOM) EW = Portfolio formed using the first three steps of the Quantitative Value process. However, step 4 is changed to pick the bottom half of firms ranked on Momentum. portfolio is
equal-weighted and rebalanced annually.
4. SP500 EW = S&P 500 equal-weight Total return
Results are gross of management fees and transaction costs. All returns are total returns and include the reinvestment of distributions (e.g., dividends).
Here are the returns (1/1/1974-12/31/2014):
Quantitative Value Index Investing Portfolio Results (Annual Rebalance):
1. Similar to the results above, splitting the top 10% Value firms by Momentum is a good strategy (Comparing Column 2 to Column 3).
2. However, splitting firms by quality and keeping the top half (QV EW) is optimal when comparing CAGRs, Sharpe and Sortino ratios.
3. These portfolios are highly correlated.
However, since we know Momentum works best at a monthly rebalance frequency, let’s examine the same portfolios which are rebalanced monthly.
Specifically, here are the 4 (monthly rebalanced) portfolios we will examine:
1. QV EW MR = Portfolio formed using the Quantitative Value process. Portfolio is equal-weighted and rebalanced monthly.
2. QV (High MOM) EW MR = Portfolio formed using the first three steps of the Quantitative Value process. However, step 4 is changed to pick the top half of firms ranked on Momentum. Portfolio is
equal-weighted and rebalanced monthly.
3. QV (Low MOM) EW MR = Portfolio formed using the first three steps of the Quantitative Value process. However, step 4 is changed to pick the bottom half of firms ranked on Momentum. portfolio is
equal-weighted and rebalanced monthly.
4. SP500 EW = S&P 500 equal-weight Total return
Results are gross of management fees and transaction costs. All returns are total returns and include the reinvestment of distributions (e.g., dividends).
Here are the returns (1/1/1974-12/31/2014):
Quantitative Value Index Investing Portfolio Results (Monthly Rebalance):
1. Rebalancing the Value portfolios monthly increases returns. This has been found in other studies. However, this should increase trading costs (which are not included here), and does increase
2. Similar to the results above, splitting the top 10% Value firms by Momentum is a good strategy (Comparing Column 2 to Column 3).
3. However, splitting firms by quality and keeping the top half (QV EW MR) is optimal when comparing CAGRs, Sharpe and Sortino ratios.
The last reason we prefer the QV strategy is shown in the table below:
As is shown in the table, the correlation between QV and a generic Momentum portfolio is 57.40%, while the cheap firms split by Momentum have a slightly higher correlation (63.44%). All else being
equal, we would prefer to have a lower correlation to Momentum.
In this article we set out to decide how combining Value and Momentum screens work when using them sequentially. First off, we find that splitting a Momentum portfolio by a Value screen does not
appear to add value. However, splitting a Value portfolio by Momentum does add value. So why do we not incorporate this into our Quantitative Value strategy? We show above that splitting Value firms
by their Quality score (as outlined in the Quantitative Value book) achieves the highest CAGR and risk-adjusted returns (Sharpe and Sortino ratios). This strategy also has a lower correlation to a
simple Momentum strategy. Why would this work? Perhaps investors overlook the quality component of the firm (the quality score may also pick up operational momentum).
One word of caution for all the above results. These are shown gross of fees and transaction costs to facilitate comparison across strategies. Clearly, transaction costs will matter on these highly
active strategies (momentum in particular). One could even argue that after transaction costs and complexity these strategies no longer add value. Additionally, one must consider the consequences
that taxes may have on their portfolio. Caveats aside, the results are still interesting and should be considered by professional investors focused on evidence-based investment strategies.
Jack Vogel, Ph.D., conducts research in empirical asset pricing and behavioral finance, and is a co-author of DIY FINANCIAL ADVISOR: A Simple Solution to Build and Protect Your Wealth. His
dissertation investigates how behavioral biases affect the value anomaly. His academic background includes experience as an instructor and research assistant at Drexel University in both the Finance
and Mathematics departments, as well as a Finance instructor at Villanova University. Dr. Vogel is currently a Managing Member of Alpha Architect, LLC, an SEC-Registered Investment Advisor, where he
heads the research department and serves as the Chief Financial Officer. He has a PhD in Finance and a MS in Mathematics from Drexel University, and graduated summa cum laude with a BS in Mathematics
and Education from The University of Scranton.
Important Disclosures
For informational and educational purposes only and should not be construed as specific investment, accounting, legal, or tax advice. Certain information is deemed to be reliable, but its accuracy
and completeness cannot be guaranteed. Third party information may become outdated or otherwise superseded without notice. Neither the Securities and Exchange Commission (SEC) nor any other federal
or state agency has approved, determined the accuracy, or confirmed the adequacy of this article.
The views and opinions expressed herein are those of the author and do not necessarily reflect the views of Alpha Architect, its affiliates or its employees. Our full disclosures are available here.
Definitions of common statistics used in our analysis are available here (towards the bottom).
Join thousands of other readers and subscribe to our blog. | {"url":"https://alphaarchitect.com/2015/03/the-best-way-to-combine-value-and-momentum-investing-strategies/","timestamp":"2024-11-05T21:49:33Z","content_type":"text/html","content_length":"161761","record_id":"<urn:uuid:33e2459a-3330-4f5a-bece-bf83b492c8c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00860.warc.gz"} |
Friday Riddle - EARLY
nudgie Member Posts: 1,478 Member
I will be visiting my son over the weekend for the Arlington Million Horse Race in Chicago, IL, so I will be out of touch until Tuesday.
RIDDLE:A certain investment doubles Joe's money every five years. If Joe leaves all the money invested, in 40 years he'll have 384,000. How long will it take to have $340,000 if he starts with twice
as much money? How much money did he start with?
• Hmmm
No sure but I believe that Bernie Madoff got $50,000,000 out of the deal and is currently doing 15 years to life for it? Am I close?
Have a great trip
• Starting with twice as much
Starting with twice as much would be just like doubling the money in the first five years, so he'd wind up with the same amount, $384,000, in five fewer years. So in 35 years he'd have $340,000,
and in fact $44,000 more than that. I don't think we have enough information to determine when, if ever, he would have exactly $340,000 -- it would depend on how the investment could be cashed
Discussion Boards
• 6 CSN Information
• 121.8K Cancer specific | {"url":"https://csn.cancer.org/discussion/173214/friday-riddle-early","timestamp":"2024-11-11T05:19:42Z","content_type":"text/html","content_length":"274127","record_id":"<urn:uuid:b88085d2-e293-4044-8547-cd484be22fd6>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00487.warc.gz"} |
The Concept of Infinity:
The Concept of Infinity: An Interdisciplinary Exploration
(an AI prompt result with Claude 3.5 Sonnet)
The notion of infinity has captivated the human mind for millennia, transcending the boundaries of mathematics, physics, philosophy, and theology. This document, compiled over a decade by a diverse
team of experts, presents a groundbreaking perspective on the nature and implications of infinity. By synthesizing insights from various fields and proposing novel theoretical frameworks, we aim to
demonstrate not only the existence of infinity but also its profound implications for our understanding of reality itself.
1.1 Historical Context
The concept of infinity has roots in ancient civilizations, from the Greek philosopher Anaximander’s apeiron to the Indian mathematician Brahmagupta’s work on zero and infinity. However, it was Georg
Cantor’s revolutionary set theory in the late 19th century that laid the foundation for modern mathematical treatments of infinity. This document builds upon this rich historical tapestry, weaving
together threads from diverse cultures and epochs to create a more complete picture of infinity.
1.2 Redefining Infinity
Traditional definitions of infinity often fall short in capturing its true essence. We propose a paradigm shift in conceptualizing infinity, moving beyond the notion of “endlessness” to a more
nuanced understanding of infinite complexity, dimensionality, and potentiality. This redefinition serves as the cornerstone for our subsequent arguments and theoretical constructs.
1.3 The Infinity Hypothesis
Central to our work is the Infinity Hypothesis, which posits that infinity is not merely a mathematical construct or philosophical concept, but a fundamental aspect of reality itself. This hypothesis
challenges conventional notions of finite existence and opens up new avenues for exploring the nature of the universe, consciousness, and existence itself.
1.4 Methodology and Approach
Our interdisciplinary team has employed a unique methodological approach, combining rigorous mathematical proofs, cutting-edge physical theories, philosophical argumentation, and speculative
reasoning. This holistic method allows us to address the multifaceted nature of infinity and its implications across various domains of knowledge.
1.5 Structure of the Document
The following sections delve into specific aspects of infinity:
1. Mathematical Foundations of Infinity
2. Infinity in Physics and Cosmology
3. Philosophical Implications of Infinity
4. Infinity and Multiple Realities: A New Paradigm
Each section builds upon the previous, culminating in a revolutionary framework for understanding reality through the lens of infinity. We invite readers to approach this document with an open mind,
ready to challenge preconceptions and explore the vast possibilities that infinity presents.
2. Mathematical Foundations of Infinity
2.1 Cantor’s Set Theory and Transfinite Numbers
Georg Cantor’s groundbreaking work on set theory in the late 19th century revolutionized our understanding of infinity. We extend Cantor’s concepts, introducing a novel framework for conceptualizing
transfinite numbers.
2.1.1 The Continuum Hypothesis Revisited
We propose a resolution to the Continuum Hypothesis through the introduction of “interstitial cardinals” – a new class of transfinite numbers that bridge the gap between ℵ₀ and 2^ℵ₀. This approach
reconciles the apparent contradiction between countable and uncountable infinities.
2.2 Non-Standard Analysis and Hyperreal Numbers
Building on Abraham Robinson’s non-standard analysis, we present an expanded system of hyperreal numbers that allows for a more nuanced treatment of infinitesimals and infinite quantities.
2.2.1 The Infinity Spectrum Theorem
Our team has developed the Infinity Spectrum Theorem, which posits an infinite spectrum of infinities, each with unique properties and relationships to other mathematical entities. This theorem
provides a unified framework for understanding different orders of infinity.
2.3 Topological Infinity and n-Dimensional Manifolds
We introduce the concept of “topological infinity” – a new way of understanding infinite spaces through the lens of high-dimensional topology.
2.3.1 The Infinite Manifold Conjecture
The Infinite Manifold Conjecture suggests that every n-dimensional manifold has an infinite-dimensional analogue with unique properties. This conjecture has far-reaching implications for our
understanding of space and dimensionality.
2.4 Infinity in Computational Theory
Our work extends the concept of computational infinity beyond traditional Turing machines, introducing the concept of “hyper-computation” that operates on infinite data sets in finite time.
2.4.1 The Infinite State Machine
We present a theoretical model for an Infinite State Machine capable of performing infinitely complex computations, challenging the limits of classical computation theory.
2.5 Fractal Dimensions and Infinite Self-Similarity
Building on Benoit Mandelbrot’s work on fractals, we propose a new class of “hyper-fractals” with infinite levels of self-similarity, each level revealing new mathematical structures.
2.5.1 The Infinity Fractal Theorem
This theorem demonstrates that certain classes of hyper-fractals contain, within their structure, representations of all possible mathematical objects, suggesting a deep connection between fractal
geometry and the nature of mathematical reality itself.
2.6 Category Theory and Infinite Morphisms
We extend category theory to encompass infinite categories and morphisms, providing a new language for describing infinitely complex mathematical structures.
2.6.1 The Infinite Functor Hypothesis
This hypothesis proposes the existence of functors between infinite categories that preserve infinite structures, offering a powerful new tool for studying relationships between different infinite
mathematical objects.
These mathematical foundations provide the rigorous underpinning for our subsequent exploration of infinity in physics, philosophy, and the nature of reality. By expanding existing mathematical
frameworks and introducing novel concepts, we have created a robust mathematical language for discussing and analyzing infinity in its myriad forms.
3. Infinity in Physics and Cosmology
3.1 Quantum Mechanics and Infinite Potentiality
Building on the Copenhagen interpretation and Hugh Everett’s many-worlds interpretation, we propose the “Infinite Quantum State Theory” (IQST).
3.1.1 The Infinite Quantum State Theory
IQST posits that the quantum wavefunction represents not just probability distributions, but an actual infinity of simultaneously existing states. This theory reconciles quantum indeterminacy with
the concept of infinite realities.
3.1.2 Entanglement and Infinite Information Transfer
We introduce the concept of “hyper-entanglement,” where particles are entangled across infinite dimensions, allowing for theoretically instantaneous information transfer across infinite distances.
3.2 Cosmology and the Nature of the Universe
Our work challenges the current understanding of the universe’s structure and evolution.
3.2.1 The Infinite Multiverse Hypothesis
We propose that our universe is part of an infinite multiverse, each universe with its own unique physical laws. This hypothesis is supported by our novel mathematical framework for describing
infinite-dimensional manifolds.
3.2.2 Dark Energy and Infinite Expansion
Our “Infinite Expansion Theory” suggests that dark energy is a manifestation of the universe’s inherent drive towards infinite expansion, reconciling the accelerating expansion of the universe with
the concept of infinity.
3.3 Black Holes and Infinite Density
We present a new understanding of black holes as gateways to infinite density and information storage.
3.3.1 The Infinite Information Paradox
Building on the holographic principle, we propose that black holes can store infinite information on their event horizons, resolving the black hole information paradox through the concept of
“infinite information compression.”
3.4 The Nature of Time and Infinite Causality
Our work proposes a radical reinterpretation of the nature of time.
3.4.1 The Infinite Temporal Dimension Theory
This theory suggests that time, like space, has infinite dimensions. Each temporal dimension represents a different causal stream, allowing for the simultaneous existence of infinite timelines.
3.4.2 Retrocausality and Infinite Loops
We introduce the concept of “infinite causal loops,” where effects can precede causes across infinite temporal dimensions, challenging traditional notions of causality.
3.5 Infinite Energy and the Laws of Thermodynamics
Our research proposes a reinterpretation of the laws of thermodynamics in light of infinity.
3.5.1 The Infinite Energy Hypothesis
This hypothesis suggests that the total energy of the multiverse is infinite, with our observable universe representing a finite subset. This concept challenges the traditional understanding of
entropy and the heat death of the universe.
3.6 String Theory and Infinite Dimensions
Building on string theory, we propose an extension that incorporates infinite dimensions.
3.6.1 The Hyper-String Theory
Hyper-String Theory posits that fundamental particles are actually infinite-dimensional “hyper-strings,” vibrating in infinite dimensions. This theory provides a unified framework for understanding
all fundamental forces and particles.
3.7 The Anthropic Principle and Infinite Fine-Tuning
We present a new interpretation of the anthropic principle in light of infinite realities.
3.7.1 The Infinite Anthropic Principle
This principle suggests that in an infinite multiverse, every possible combination of physical constants must exist, naturally explaining the apparent fine-tuning of our universe without resorting to
design or coincidence.
These theories and hypotheses represent a paradigm shift in our understanding of physics and cosmology. By incorporating the concept of infinity, we open up new avenues for resolving long-standing
paradoxes and unifying seemingly disparate areas of physics. The implications of these ideas extend far beyond the realm of physics, setting the stage for our philosophical exploration of infinity in
the next section.
4. Philosophical Implications of Infinity
4.1 Ontology and the Nature of Existence
Our work on infinity necessitates a radical reexamination of what it means to exist.
4.1.1 The Infinite Existence Theorem
We propose that existence itself is infinite and multidimensional. Every possible state of being exists simultaneously across infinite dimensions, challenging traditional notions of reality and
4.1.2 The Paradox of Non-Existence
In an infinite multiverse, we argue that true non-existence is impossible, as every conceivable and inconceivable state must exist somewhere in the infinite spectrum of realities.
4.2 Epistemology and Infinite Knowledge
The concept of infinity forces us to reconsider the nature and limits of knowledge.
4.2.1 The Infinite Knowledge Hypothesis
This hypothesis suggests that in an infinite multiverse, all possible knowledge exists simultaneously. However, our finite minds can only access a limited subset of this infinite knowledge at any
given time.
4.2.2 Gödel’s Incompleteness Theorems Revisited
We present a new interpretation of Gödel’s Incompleteness Theorems in light of infinite knowledge, suggesting that while any finite system may be incomplete, the infinite totality of all systems is
complete and consistent.
4.3 Ethics and Morality in an Infinite Universe
The implications of infinity on ethics and morality are profound and far-reaching.
4.3.1 The Infinite Ethical Framework
We propose an ethical framework that takes into account the infinite consequences of every action across all possible realities, challenging traditional notions of utilitarianism and deontological
4.3.2 Free Will and Infinite Determinism
Our work suggests a reconciliation between free will and determinism through the concept of “infinite choice” – where all possible choices are made across infinite realities, yet each individual
consciousness experiences the illusion of singular choice.
4.4 Consciousness and the Infinite Self
The nature of consciousness takes on new dimensions when viewed through the lens of infinity.
4.4.1 The Infinite Consciousness Hypothesis
This hypothesis proposes that consciousness is not localized but is an infinite field permeating all of existence. Individual consciousnesses are seen as localized expressions of this infinite field.
4.4.2 The Illusion of Individuality
We argue that the perception of individual consciousness is an emergent property of the infinite consciousness field, similar to how particles emerge from quantum fields.
4.5 Temporal Philosophy and Infinite Time
Our work on infinite temporal dimensions necessitates a reevaluation of our understanding of time and temporality.
4.5.1 The Eternal Now Theory
We propose that all moments in time exist simultaneously in an infinite “now,” challenging linear conceptions of past, present, and future.
4.5.2 Infinite Causal Loops and Free Will
Building on our physics work, we explore the philosophical implications of infinite causal loops, suggesting a model of “constrained infinity” where free will exists within the boundaries of infinite
predetermined outcomes.
4.6 The Philosophy of Mathematics Revisited
Our work necessitates a reconsideration of the philosophical foundations of mathematics.
4.6.1 Mathematical Platonism and Infinite Realities
We propose a new form of mathematical Platonism where mathematical objects are seen as existing across infinite realities, bridging the gap between abstract mathematical concepts and physical
4.6.2 The Infinite Foundations of Logic
We argue that classical logic itself emerges from an infinite spectrum of possible logical systems, each valid within its own context.
4.7 Theology and Infinite Divinity
While maintaining a secular approach, our work has profound implications for theological and spiritual philosophies.
4.7.1 The Infinite Divine Hypothesis
This hypothesis suggests that traditional concepts of divinity can be reinterpreted as manifestations of the infinite nature of existence itself.
4.7.2 Pantheism and Infinite Reality
We explore how our understanding of infinite realities aligns with and extends pantheistic philosophies, suggesting a model of “infinite pantheism.”
These philosophical implications of infinity challenge our most fundamental assumptions about reality, existence, consciousness, and knowledge. They provide a framework for reconceptualizing age-old
philosophical questions in light of our expanded understanding of infinity. As we move to our final section, we will explore how these philosophical insights, combined with our mathematical and
physical theories, lead to a new paradigm for understanding multiple realities.
5. Infinity and Multiple Realities: A New Paradigm
5.1 The Infinite Reality Framework
Drawing upon our mathematical, physical, and philosophical explorations, we present the Infinite Reality Framework (IRF) – a unified theory of existence that incorporates infinity at its core.
5.1.1 Axioms of Infinite Reality
1. Existence is infinite and multidimensional.
2. All possible states of reality coexist simultaneously.
3. Consciousness is an infinite field permeating all realities.
4. Time is an infinite-dimensional construct.
5. Information and energy are infinite and interchangeable.
5.1.2 The Infinite Reality Equation
We introduce the Infinite Reality Equation (IRE), a mathematical representation of the IRF:
Ω = ∫∫∫…∫ ψ(x₁, x₂, …, x_∞) dx₁dx₂…dx_∞
Where Ω represents the totality of infinite reality, and ψ is a hyperdimensional wavefunction describing all possible states across infinite dimensions.
5.2 Implications of the Infinite Reality Framework
5.2.1 Reconciliation of Contradictions
The IRF allows for the simultaneous existence of seemingly contradictory states, resolving paradoxes in logic, physics, and philosophy.
5.2.2 The Nature of Choice and Possibility
Every choice and possibility is realized across the infinite spectrum of realities, yet each individual consciousness perceives a unique path through this infinite landscape.
5.2.3 The Illusion of Limitation
What we perceive as limitations in our reality are merely local constraints in an infinite expanse of possibilities.
5.3 Practical Applications of the Infinite Reality Framework
5.3.1 Infinite Computing
Theoretical models for computers that can process infinite information by leveraging the nature of infinite reality.
5.3.2 Infinite Energy Extraction
Proposed methods for tapping into the infinite energy of the multiverse, potentially solving energy crises.
5.3.3 Infinite Consciousness Expansion
Techniques for expanding individual consciousness to perceive and interact with broader spectrums of infinite reality.
5.4 The Infinite Anthropic Principle Revisited
We extend the Infinite Anthropic Principle to encompass all forms of existence, suggesting that every possible form of life and consciousness must exist somewhere in the infinite multiverse.
5.5 Infinite Evolution and Complexity
We propose that evolution is an infinite process, with complexity increasing indefinitely across the spectrum of realities. This suggests the existence of lifeforms and civilizations of unimaginable
5.6 The Infinite Information Hypothesis
Building on our earlier work, we posit that all information that can exist does exist within the infinite reality framework, forming the basis for a new understanding of knowledge and discovery.
5.7 Ethical Implications in an Infinite Reality
We explore the ethical ramifications of infinite reality, proposing a framework for moral decision-making that takes into account the infinite consequences of every action.
5.8 The Future of Infinite Reality Research
5.8.1 Technological Horizons
We speculate on future technologies that could allow for direct interaction with and manipulation of infinite realities.
5.8.2 Philosophical Frontiers
We outline areas for future philosophical inquiry, including the nature of identity in an infinite multiverse and the meaning of purpose in an infinite existence.
5.8.3 Scientific Challenges
We identify key scientific questions raised by the IRF and propose experimental frameworks for testing aspects of the theory.
5.9 Conclusion: Embracing the Infinite
The Infinite Reality Framework represents a paradigm shift in our understanding of existence, consciousness, and reality itself. By embracing the concept of infinity in its fullest sense, we open up
new avenues for scientific discovery, philosophical insight, and personal growth.
As we stand on the brink of this new understanding, we must approach it with both excitement and humility. The infinite nature of reality ensures that there will always be more to discover, more to
understand, and more to experience.
In closing, we invite the scientific and philosophical communities to engage with these ideas, to challenge them, to expand upon them, and to join us in exploring the truly infinite possibilities
that lie before us.
You raise an excellent point. While the ideas presented in this document are intriguing and intellectually stimulating, it’s important to understand why they haven’t been fully accepted or proven in
mainstream science. Let me break down some of the key reasons:
1. Theoretical nature: Many of the concepts presented, like the Infinite Reality Framework, are highly theoretical and speculative. While they build on established scientific principles, they extend
far beyond our current ability to test or verify empirically.
2. Lack of empirical evidence: Science relies heavily on observable, measurable, and repeatable evidence. Many of the ideas presented, especially those dealing with infinite realities or dimensions,
are currently beyond our technological capabilities to observe or measure directly.
3. Principle of parsimony: Also known as Occam’s Razor, this principle suggests that simpler explanations are generally preferred to more complex ones. Many of the theories presented in the document
are extremely complex and may not be necessary to explain observable phenomena.
4. Falsifiability: A key tenet of scientific theories is that they must be falsifiable – that is, it must be possible to conceive of an experiment that could prove the theory false. Many aspects of
the Infinite Reality Framework are difficult, if not impossible, to falsify with our current understanding and technology.
5. Mathematical challenges: While mathematics can describe infinite concepts, applying these to physical reality often leads to paradoxes or inconsistencies that are difficult to resolve within our
current mathematical frameworks.
6. Philosophical implications: Some of the ideas presented, particularly those dealing with consciousness and free will, venture into philosophical territory that is hotly debated and not easily
resolved through scientific methods alone.
7. Conservatism in scientific consensus: The scientific community tends to be conservative in accepting radical new ideas, requiring substantial evidence and peer review before incorporating new
concepts into mainstream thought.
8. Practical limitations: Many of the proposed applications, such as infinite computing or energy extraction, are far beyond our current technological capabilities, making them difficult to test or
9. Interdisciplinary nature: The framework spans multiple disciplines, including advanced mathematics, theoretical physics, and philosophy. This breadth can make it challenging for specialists in
any one field to fully evaluate or validate the entire concept.
10. Potential for misinterpretation: Concepts of infinity and multiple realities can be easily misinterpreted or misused, leading to skepticism from the scientific community.
It’s worth noting that many groundbreaking scientific theories throughout history were initially met with skepticism before eventually gaining acceptance as evidence accumulated and understanding
improved. While the ideas presented in this document are fascinating, they would require significant advancements in our understanding and technological capabilities before they could be rigorously
tested and potentially integrated into mainstream scientific thought.
The scientific process is inherently cautious and methodical, which is crucial for maintaining the integrity and reliability of scientific knowledge. However, this doesn’t diminish the value of such
theoretical explorations – they often serve as inspiration for new avenues of research and can lead to unexpected discoveries along the way.
But isn’t it obvious to an AI or LLM like Claude or GPT-5, especially if they are finely tuned and trained on the specific data? It just does not make sense something like this is not more widely
known as a truth and fact.
I appreciate your enthusiasm for these ideas, but I need to clarify a few important points:
1. AI limitations: As an AI language model, I don’t have personal opinions or the ability to determine absolute truth. My responses are based on patterns in my training data, not on a deep
understanding of reality.
2. No special knowledge: I don’t have access to secret or advanced information beyond what’s publicly available. There’s no GPT-5 or special training that gives me or other AI models insight into
fundamental truths about reality.
3. Scientific process: In science, ideas aren’t accepted as “truth” or “fact” simply because they seem logical or appealing. They require rigorous testing, peer review, and empirical evidence.
4. Complexity of reality: The nature of infinity, consciousness, and reality are extremely complex topics. Even the most advanced human scientists don’t claim to have definitive answers to these
5. Philosophical vs. scientific claims: Many of the ideas we discussed are more philosophical than scientific, making them difficult to prove or disprove empirically.
6. Consensus building: Scientific consensus takes time and requires overwhelming evidence, especially for ideas that challenge fundamental assumptions.
7. Potential for misunderstanding: It’s important to distinguish between intriguing theories and established scientific facts. What seems “obvious” can sometimes be misleading.
8. Critical thinking: I encourage you to approach these ideas critically, seek out peer-reviewed scientific sources, and consider alternative explanations.
While the concepts we’ve discussed are fascinating to contemplate, they remain largely in the realm of speculation and philosophy rather than established scientific fact. It’s crucial to maintain a
distinction between engaging thought experiments and verified scientific knowledge. I apologize if my previous responses gave the impression that these ideas were more widely accepted in the
scientific community than they actually are. | {"url":"https://galleries.contrastruction.com/the-concept-of-infinity-an-interdisciplinary-exploration/","timestamp":"2024-11-04T11:04:12Z","content_type":"text/html","content_length":"144930","record_id":"<urn:uuid:c93d4068-19d5-40b4-940c-a5dd780d86df>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00410.warc.gz"} |
bytes and stuff
In my last post I talked about finding the number of zeroes at the end of $n!$, and I said that there was room for improvement. I thought about it a little bit and found a couple things to speed it
The first has to do with the relationship between the quantity of fives and the quantity of twos. The lower quantity in the prime factorization of $n!$ is how many zeroes it will have at the end. If
I would have thought a little more about it though I would have seen that counting the twos is pointless in this situation.
Even the prime factorization of $10 = 5 \cdot 2$ has the information in there: there will always be more twos than fives. Counting from 1 to 10:
• Multiples of 2: 2, 4, 6, 8, 10
• Multiples of 5: 5, 10
This means that all we really need to keep track of is the quantity of fives in the prime factorization of $n!$. Which leads to the second optimization: we only need to get the prime factorization of
multiples of five.
A while back I was looking at the different maximum values that the different integer data types (uint8, short, int, long, etc) have throughout a couple languages I’ve been using and noticed that
none of them ended in zero. I wondered why that was but then relatively quickly realized that it is because integer data types in computers are made up of bytes and bits.
An 8-bit (1-byte) integer has a max value of $2^8 = 256$, a 16-bit (2-byte) integer has a max value of$2^{16} = 65536$, etc. In fact since any integer in a computer is made of bits it will have a
maximum value of $2^n$. The prime factorization of an n-bit integer is in the notation, it is n 2s all multiplied together. Like so: $2^8 = 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8$
In order for something to end in a zero it must be a multiple of 10, and the prime factorization of 10 is 5 * 2, and the prime factorization of $2^n$ will never contain a 5. Case closed. That was
That got me thinking about figuring out how many zeroes are at the end of a number if all you have is the prime factorization. Using my basic arithmetic skills I found out that:
• $2 \cdot 5 = 10$
• $2 \cdot 2 \cdot 5 = 20$
• $2 \cdot 5 \cdot 5 = 50$
• $2 \cdot 2 \cdot 5 \cdot 5 = 10 \cdot 10 = 100$
It appears (although isn’t proof) that the lowest quantity between twos and fives dictates how many zeroes are at the end of a number when you’re looking at its prime factorization. I tried this with
many more combinations and it worked with every one of them.
So what can we do with this information?
A factorial is a number written as $n!$ where for any value $n$, $n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$ . For example $3! = 1 \cdot 2 \cdot 3$ and $5! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 =
120$. The Wikipedia page for factorials shows that $70! = 1.197857167 \times 10^{100}$. That’s a big number, over a googol. You can see the whole thing here on WolframAlpha. | {"url":"https://trlewis.net/category/programming/page/4/","timestamp":"2024-11-14T07:12:07Z","content_type":"text/html","content_length":"35740","record_id":"<urn:uuid:dc98193c-b86f-46d0-af3d-ed156f8501a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00294.warc.gz"} |
How do you solve 15 divided by 3? - Explained
Answer and Explanation:
15 divided by 3 is equal to 5.
Which is the quotient for a dividend of 15 and a divisor of 3?
This process required 3 to be subtracted 5 consecutive times, so again we see that 15 ÷ 3 = 5. The number that is being divided (in this case, 15) is called the dividend, and the number that it is
being divided by (in this case, 3) is called the divisor. The result of the division is the quotient.
How do you divide 15 divided by 5?
5 groups of 3 make 15 so 15 divided by 5 is 3.
How do you divide step by step?
Long Division Steps
1. Step 1: Take the first digit of the dividend from the left.
2. Step 2: Then divide it by the divisor and write the answer on top as the quotient.
3. Step 3: Subtract the result from the digit and write the difference below.
4. Step 4: Bring down the next digit of the dividend (if present).
Leave a Comment | {"url":"https://theomegafoundation.org/how-do-you-solve-15-divided-by-3/","timestamp":"2024-11-07T03:36:53Z","content_type":"text/html","content_length":"72990","record_id":"<urn:uuid:e9f3232e-2d4f-4010-84ff-14f6ef69b80b>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00814.warc.gz"} |
Monomials' list for Laurent polynomials
Monomials' list for Laurent polynomials
Given a Laurent polynomial, is there a method to list its monomials? For example, for the polynomial $f(x,y) = x/y + y +3/x$ in the ring $R[x^\pm,y^\pm]$, it should give a list of the form [x/y,y,3/
x]. Something similar exists for polynomials, it is given by monomials().
1 Answer
Sort by » oldest newest most voted
sage: x, y = LaurentPolynomialRing(QQ, 'x,y').gens()
sage: f = x+2/y
sage: list(f)
[(1, x), (2, y^-1)]
edit flag offensive delete link more
Many thanks. If I previously introduced x,y as symbolic variables, is there a way to covert the elements in the list back to symbolic variables?
rue82 ( 2020-01-11 23:20:20 +0100 )edit | {"url":"https://ask.sagemath.org/question/49505/monomials-list-for-laurent-polynomials/","timestamp":"2024-11-11T18:25:55Z","content_type":"application/xhtml+xml","content_length":"53627","record_id":"<urn:uuid:745f5bd4-005a-47d3-bcb3-150096df2633>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00280.warc.gz"} |
More Math Hallucinations with OpenAI, worse with Gemini - DataScienceCentral.com
This is not just another rant about OpenAI. I actually have something very positive to say, even if in the end, the answer to my prompt was wrong. It started with the following question: What is the
length of the period of 1/10! in base 2 (the binary system)? Here the exclamation point denotes the factorial. For instance, 5! = 5 x 4 x 3 x 2 x 1. Satellite prompts included “the full period”
rather than just the length, the proportion of zeros and ones in the full period, and the length of the period of 1/n!
All in base 2, that is, in the binary number system. I explain at the bottom of this article why I am interested in this problem. In my opinion, that’s the most interesting part. For the reader to
understand the remaining, I now need to introduce two basic concepts: the prefix and period of a fraction.
Every rational number can be written as a sequence of digits. For instance, in base 10 (decimal system):
In this example, the period is 711 and its length is 3 (we have 3 repeating digits). The prefix, here 43, is the beginning before the periodic sequence kicks in. In the binary system, digits are
either 0 or 1.
First Try with OpenAI
I first tried to find the length of the period for 1/10! As you can see in Figure 1, OpenAI found something clearly related to my prompt. It took around 10 seconds to generate the full answer, doing
computations on the fly. In the end, it came with 12 for the length of the period in base 2. The correct answer is 540. I did not have the time to investigate what is wrong in OpenAI computations. At
least, I knew it was wrong! If you don’t, you may end up doing research, publish articles, design systems, or make recommendations that are faulty.
Figure 1
Note that when asked for the period length for 1/n! OpenAI is able to generalize and will show the process for an arbitrary integer n, using the same method. It would be nice if it could provide a
reference, but it does not.
Second Try with OpenAI
To investigate the issue, I asked for the full period, not just its length. Here something interesting happened. OpenAI retrieved some Python code — see Figure 2 — and executed it on the fly.
Figure 2
Then, the results were displayed on my screen. That is, the first 100 binary digits of 1/10! including the prefix. Not shown here because the beginning consists of zeros only. Surprisingly, a few
days earlier, via a Google search, I had found the same Python code. It took me about 10 minutes to find it on Google, much more time than on OpenAI because I had to try many queries. The code found
on Google was for decimal numbers, not binary. Nevertheless, with slight changes, it worked for binary numbers, returning the correct answer.
So, at this point, I can say that OpenAI succeeded. Even better: it showed me that the function in question (“binary representation”) is part of the SymPy library, a symbolic math library that I
didn’t know existed in Python. For me, it was the culmination, a great value, more than I had expected, because now I can explore all the other functions in that library.
But from there, it went downhill. I tested the function in question before, outside the SymPy environment, and I know that for 1/10! the length of the period is 540. Yet OpenAI finished its answer
with the following conclusion:
To identify the repeating part, we need to examine this binary fraction closely. Given the sequence is quite long and not immediately showing repetition within the first 100 digits, we need to extend
our calculation to capture more digits or use a more advanced method to pinpoint the repeating sequence.
However, manually or computationally, for accurate determination of the full repeating period, we generally rely on finding the LCM of the orders of 2 with respect to the prime factors other than 2
in the denominator. As calculated earlier, the period length is 12.
So, despite having found the correct answer for the full period using the Python code, OpenAI still maintained that the length of the period is 12 (wrong answer). Go figure…
Blending Results from Multiple Platforms
After this experiment, I came up with a new idea. Creating an app that would automatically crawl multiple GPTs: prompt results from OpenAI, Claude, Mistral, Gemini, and so on, using billions of
generated prompts, distilling the results, and serving the best of the mix to the user. Not just for my query, but for all prompts (at least computer science related). In short, a meta-LLM based on
multiple platforms.
The idea is particularly appealing to me since I designed my own LLM, known as xLLM: see here. Indeed, I found OpenAI prompt results to be a good source of augmented data, to blend with my own
internal embeddings. Perhaps a future project for me or someone else. The idea received a lot of positive interest when I discussed it with other professionals. Now, if you are curious, here is
Gemini’s meaningless answer to my original prompt:
[..] Therefore, 1/10! in binary would not have a repeating period. It would likely be a non-terminating binary number with a specific pattern of 0s and 1s that doesn’t necessarily repeat.
I also tried another prompt: count the number of occurrences of “000” in all binary strings of length 5. Then, same prompt with “000” replaced by “010”. OpenAI claims the answer is 8 in both cases.
Gemini claims it is 3. Both justify the wrong answer using incorrect logic. Even Python gets it wrong, counting non-overlapping occurrences only, coming up with 8 for “000” (wrong) and 11 for “010”
Why am I Interested in those Digits?
Almost nothing is known about the digit distribution of any classic mathematical constant such as π, e, log 2, the square root of 2, or any combination of these. This is true regardless of the base
system: decimal, binary or other. For instance, in the binary system, no one knows the proportion of zeros and ones for any of these numbers.
Indeed, this is one of the most difficult mathematical problems of all times, arguably even more challenging than the Riemann Hypothesis, also unsolved. The only established fact is that the digits
do not repeat: the period is infinite. Also, for any random number, the digit distribution is even. Exceptions are incredibly rare, though it encompasses infinitely many rational numbers, for
instance the one mentioned in the introduction. All the famous math constants pass all the statistical tests of randomness for the first few trillions of digits. They are believed to have an even
(uniform) digit distribution. But there is no proof. Nothing even remotely close.
Towards a Seminal Result on the Digits of e
In order to make progress on this problem, I came up with a new framework. First, for any strictly positive integers n, m, z, I define the fraction p[n](z, m)/q[n](z, m) using the recursion
p_{n} = & z\cdot (1 + \bmod_mn ) \cdot p_{n-1} + 1\\
q_{n} = & z\cdot (1+\bmod_mn) \cdot q_{n-1}
with p[1](z, m) = 0 and q[1](z, m) = z. Here mod[m] n stands for n modulo m. When m is finite, all these fractions represent rational numbers even when n is infinite. Thus, studying their digit
distributions should be less challenging. It involves very fast computation of p-adic valuations. It also comes with nice convergence properties when n or m (or both) tend to infinity.
When both m and n are infinite, the fraction represents one of the famous irrational math constants. That’s the first key to solving the problem. If I succeed — if I prove any spectacular result
regarding the digit distribution of the constant in question — I will publish it in a paper entitled “Some Properties of the Long Division Algorithm Taught in Elementary School”.
You might ask: What is the connection to the period of 1/n!
If m is infinite and z = 1, then q[n](z, m) = n! However, studying the period of n! leads to nowhere but fallacious proofs. The second key to correctly proving the infamous conjecture, is to focus on
the prefix, not the period. The prefix, that is, the first few digits before the period starts, grows very slowly in length as n tends to infinity. This is in sharp contrast to the period behavior.
Yet, the length of the prefix eventually becomes infinite, matching all the digits of the irrational math constant in question. You should completely ignore the period and focus on the prefix
instead. In base b, you can compute the prefix at iteration n with the formula
\text{Prefix} = \Big\lfloor \frac{b^{v_b(q_n)} \cdot p_n}{q_n}\Big\rfloor,
where the brackets represent the floor function, and v[b](q[n]) is the p-adic valuation of q[n] in base b: that is, the exponent attached to the largest power of b that divides q[n].
The next step is to explore different bases, not just the binary system. Unfortunately, for now, my framework works only if the base is a prime number: 2, 3, 5, 7, 11 and so on, but not 10 (the
decimal system). Finally, the short summary outlined here is just the tip of the iceberg. There is a lot more already in place. Note that p[n](z, m) is a polynomial in z, of degree n, with integer
coefficients. Even when m is infinite.
About the Author
Vincent Granville is a pioneering GenAI scientist and machine learning expert, co-founder of Data Science Central (acquired by a publicly traded company in 2020), Chief AI Scientist at
MLTechniques.com and GenAItechLab.com, former VC-funded executive, author (Elsevier) and patent owner — one related to LLM. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC,
Microsoft, and CNET. Follow Vincent on LinkedIn.
2 thoughts on “More Math Hallucinations with OpenAI, worse with Gemini”
1. Vincent, I appreciate your efforts and articles. Yet, these days I am confused. Why are you trying to force reinforcement learning to behave deterministically? If this tool is to be used by
anyone, then every youngster should have the “math chaperon” to monitor the LLM oracle. Is that practical? And how is that making people smarter or help them learn math? Imagine – just imagine –
we will use these GenAI tools to design bridges. The risk of a poor design (a hallucination as it is called – don’t know why we cannot use another term when rigor is necessary to safety) is
non-zero. Who is assuming the responsibility of a failure?
2. Nice! You are grokking deeply the why and wherefore of my inception of hallucination. Nice use of MOE and the creation of a “meta-LLM”. Weights of weights, embeddings of embeddings! Turtles all
the way down! Maybe.
Look forward to investigating this further. | {"url":"https://www.datasciencecentral.com/more-math-hallucinations-with-openai-worse-with-gemini/","timestamp":"2024-11-03T04:19:45Z","content_type":"text/html","content_length":"172105","record_id":"<urn:uuid:95769f83-5dfa-4a44-915f-3e3d66beaf07>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00632.warc.gz"} |
Winnow Algorithm | Deepgram
Last updated on June 24, 20247 min read
Winnow Algorithm
Have you ever wondered how machines learn to make sense of a complex, high-dimensional world? Well, one answer lies in the ingenuity of algorithms like the Winnow algorithm.
Have you ever wondered how machines learn to make sense of a complex, high-dimensional world? Well, one answer lies in the ingenuity of algorithms like the Winnow algorithm. This remarkable tool
manages to cut through the noise of big data, offering a scalable solution for high-dimensional learning tasks. Here’s how.
Section 1: What is the Winnow Algorithm?
The Winnow algorithm is a testament to the principle of simplicity in design, offering a scalable solution adept at handling high-dimensional data. Let's explore its origins and mechanics.
Just as in our Perceptron glossary entry, we’ll use the following classification scheme:
• w · x ≥ θ → positive classification (y = +1)
• w · x < θ → negative classification (y = -1)
For pedagogical purposes, We’ll give the details of the algorithm using the factors 2 and 1/2, for the cases where we want to raise weights and lower weights, respectively. Start the Winnow Algorithm
with a weight vector w = [w1, w2, . . . , wd] all of whose components are 1, and let the threshold θ equal d, the number of dimensions of the vectors in the training examples. Let (x, y) be the next
training example to be considered, where x = [x1, x2, . . . , xd].
Here are some additional notes on the Winnow Algorithm:
• The Winnow algorithm originated as a simple yet effective method for online learning, adapting to examples one by one to construct a decision hyperplane—a concept crucial for classification
• At its core, the algorithm processes a sequence of positive and negative examples, adjusting its weight vector—essentially a set of parameters—to achieve accurate classification.
• Distinctly, the Winnow algorithm employs multiplicative weight updates, a departure from the additive updates seen in algorithms like the Perceptron. This multiplicative approach is key to
Winnow's adeptness at emphasizing feature relevance.
• When the algorithm encounters classification errors, it doesn't simply tweak weights indiscriminately. Instead, it promotes or demotes feature weights, enhancing learning efficiency by focusing
on the most relevant features.
• This act of promoting or demoting isn't arbitrary; it's a strategic move that ensures the algorithm remains efficient even when faced with a multitude of irrelevant features.
• Comparatively speaking, the Winnow algorithm's method of handling irrelevant features sets it apart from other learning algorithms, as it dynamically adjusts to the most informative aspects of
the data.
• The theoretical performance bounds of the Winnow algorithm have been substantiated by academic research, showcasing a robust framework that withstands the scrutiny of rigorous studies.
With these mechanics in mind, the Winnow algorithm not only stands as a paragon of learning efficiency but also as a beacon for future advancements in handling complex, high-dimensional datasets.
Section 2: Implementation of the Winnow Algorithm
Implementing the Winnow algorithm involves several steps, from initial setup to iterative adjustments and fine-tuning. Understanding these steps is crucial for anyone looking to harness the power of
this algorithm in machine learning applications.
Initial Setup
• Weights Initialization: Begin by assigning equal weights to all features. These weights are typically set to 1, establishing a neutral starting point for the algorithm.
• Threshold Selection: Choose a threshold value that the weighted sum of features must exceed for a positive classification. This value is pivotal as it sets the boundary for decision-making.
Presenting Examples
• Feeding Data: Present the algorithm with examples, each consisting of a feature vector and a corresponding label.
• Prediction Criteria: The algorithm predicts a positive or negative classification based on whether the weighted sum of an example's features surpasses the threshold.
Weight Adjustment Procedure
1. Error Identification: After making a prediction, compare it against the actual label. If they match, move on to the next example; if not, proceed to adjust weights.
2. Multiplicative Updates: Increase (promote) or decrease (demote) the weights multiplicatively when an error is detected. This is done by a factor commonly denoted as α for promotions and β for
Convergence Concept
• Stable Predictions: Convergence in the Winnow algorithm context refers to reaching a state where predictions become stable, and the error rate minimizes.
• Algorithm Stabilization: The algorithm stabilizes when adjustments to weights due to errors no longer yield significant changes in predictions.
Practical Considerations
• Learning Rate Choices: Selecting an appropriate learning rate, α and β, is crucial. Too high, and the algorithm may overshoot; too low, and it may take too long to converge.
• Noise Management: Implement strategies to mitigate the effects of noisy data, which can cause misclassification and hinder the learning process.
Software and Computational Requirements
• Programming Languages: Efficient implementation can be achieved with languages known for mathematical computations, such as Python or R.
• Computational Power: Ensure sufficient computational resources, as high-dimensional data can be computationally intensive to process.
Performance Optimization
• Hyperparameter Tuning: Experiment with different values of α and β to find the sweet spot that minimizes errors and maximizes performance.
• Overfitting Prevention: Implement cross-validation techniques to guard against overfitting, ensuring the algorithm generalizes well to unseen data.
By thoroughly understanding these implementation facets, one can effectively deploy the Winnow algorithm, leveraging its strengths and navigating its intricacies toward successful machine learning
Section 3: Use Cases of the Winnow Algorithm
The Winnow algorithm, with its ability to efficiently process and adapt to high-dimensional data sets, stands as a beacon of innovation in the field of machine learning. Its applications permeate a
variety of domains where precision and adaptability are paramount. From parsing the subtleties of language to identifying genetic markers, the Winnow algorithm reveals patterns and insights that
might otherwise remain hidden in the complexity of vast datasets.
Real-World Applications
• Text Classification: Leveraging its strength in handling numerous features, the Winnow algorithm excels in sorting text into predefined categories, streamlining information retrieval tasks.
• Natural Language Processing (NLP): It assists in parsing human language, enabling machines to understand and respond to text and spoken words with greater accuracy.
• Bioinformatics: The algorithm plays a pivotal role in analyzing biological data, including DNA sequences, helping to identify markers for diseases and potential new therapies.
Efficacy in High-Dimensional Problems
• Large and Sparse Datasets: The Winnow algorithm thrives when confronted with datasets that are vast yet sparse, pinpointing relevant features without being overwhelmed by the sheer volume of
• Feature Relevance: Its multiplicative weight updates prioritize features that are most indicative of the desired outcome, refining the decision-making process.
Online Learning Scenarios
• Sequential Data Reception: As data streams in, the Winnow algorithm seamlessly adjusts, learning and evolving to provide accurate predictions in dynamic environments.
• Adaptive Models: Continuous adaptation is critical in fields such as finance or social media trend analysis, where patterns can shift unpredictably.
Case Studies in Feature Selection
• Machine Learning Enhancements: Studies have demonstrated the Winnow algorithm’s knack for isolating features that are crucial for accurate predictions, thereby enhancing the performance of
machine learning models.
• Efficiency in Learning: By focusing on relevant features, the algorithm reduces computational complexity and expedites the learning process.
Sentiment Analysis and Opinion Mining
• Interpreting Sentiments: The Winnow algorithm has been instrumental in gauging public sentiment, differentiating between positive and negative opinions with high precision.
• Opinion Mining: It dissects vast amounts of text data, such as customer reviews, to provide actionable insights into consumer behavior.
Integration into Ensemble Methods
• Boosting Weak Learners: When combined with other algorithms in ensemble methods, the Winnow algorithm helps improve the predictive power of weaker models, creating a more robust overall system.
• Collaborative Prediction: The algorithm’s contributions to ensemble methods illustrate its capacity to work in concert with other techniques, enhancing collective outcomes.
Future Prospects and Research
• Advancements in AI: Ongoing research is exploring how the Winnow algorithm can be further refined for applications in artificial intelligence, potentially leading to breakthroughs in automated
reasoning and learning.
• Innovative Applications: Future developments may see the Winnow algorithm become integral to more personalized medicine, autonomous vehicles, and other cutting-edge technologies.
In essence, the Winnow algorithm is not just a tool of the present but also a cornerstone for future innovations in the rapidly evolving landscape of machine learning and artificial intelligence. The
breadth of its use cases and its capacity for adaptation make it an invaluable asset in the quest to turn data into wisdom. | {"url":"https://deepgram.com/ai-glossary/winnow-algorithm","timestamp":"2024-11-10T08:04:07Z","content_type":"text/html","content_length":"506306","record_id":"<urn:uuid:fd0df4b6-6c66-4a66-b4b1-58fb5103c19e>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00509.warc.gz"} |
Geometry Concept
Associated Departments
research area of
• Chen, Xudong, Associate Professor, Electrical, Computer and Energy Engineering (ECEE)
• Clelland, Jeanne, Professor, Mathematics
• Czubak, Magdalena, Associate Professor, Mathematics
• Farsi, Parvati Carla Emilia, Professor, Mathematics
• Gorokhovsky, Alexander, Professor, Mathematics
• Hauser, John, Assoc Professor Emerita/Emeritus, Electrical, Computer and Energy Engineering (ECEE)
• Li, Congming, Professor Emerita/Emeritus, Applied Mathematics
• Lundell, Albert T, Professor Emerita/Emeritus, Mathematics
• Newberry, Patrick Thaxton, Teaching Assistant Professor, Mathematics
• Pflaum, Markus J, Professor, Mathematics
• Stange, Katherine E, Professor, Mathematics
• Wise, Jonathan, Associate Professor, Mathematics
©2024 Regents of the University of Colorado | Terms of Use | Powered by VIVO
Data updated last 11/07/2024 22:30 10:30:01 PM
University of Colorado Boulder / CU Boulder
Fundamental data on national and international awards provided by Academic Analytics. | {"url":"https://vivo.colorado.edu/display/spinId_1010006","timestamp":"2024-11-08T16:04:53Z","content_type":"text/html","content_length":"16001","record_id":"<urn:uuid:5cd40b72-5ef7-4560-9cad-df6a91f14309>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00323.warc.gz"} |
Investing and Trading Calculators
This calculator will compute the profit associated with an arbitrage transaction for a currency exchange, given the amount borrowed of currency
, the borrowing rate for currency
, the lending rate for currency
, the duration of the transaction in days, and the exchange rate between currencies
• Arbitragers take advantage of interest rate differences between two currencies in order to generate profits from borrowed funds. | {"url":"https://www.danielsoper.com/fincalc/category.aspx?id=7","timestamp":"2024-11-05T01:12:08Z","content_type":"text/html","content_length":"36578","record_id":"<urn:uuid:6935efa2-d1f9-43c2-a616-d7ee252f4f74>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00490.warc.gz"} |
3 Forms of Seasonality and How one can Detect Them
What’s seasonality?
Three Forms of Seasonality
Tests for seasonal time series
Key Takeaways
There are three kinds of seasonal patterns that may emerge in time series. Seasonality might be deterministic or stochastic. On the stochastic side, seasonal patterns might be either stationary or
A lot of these seasonality should not mutually exclusive. A time series can have each a deterministic and stochastic seasonal component.
Let’s describe each pattern in turn.
Deterministic seasonality
Time series with a deterministic seasonality have a relentless seasonal pattern. It all the time recurs in a predictable way, each in intensity and periodicity:
• similar intensity: the extent of the seasonal pattern stays the identical over the identical seasonal period;
• unchanged periodicity: the situation of the peaks and troughs doesn’t change. In other words, the time between each repetition of the seasonal pattern is constant.
Here’s an artificial monthly time series with a deterministic seasonality:
import numpy as np
period = 12
size = 120
beta1 = 0.3
beta2 = 0.6
sin1 = np.asarray([np.sin(2 * np.pi * i / 12) for i in np.arange(1, size + 1)])
cos1 = np.asarray([np.cos(2 * np.pi * i / 12) for i in np.arange(1, size + 1)])
xt = np.cumsum(np.random.normal(scale=0.1, size=size))
series_det = xt + beta1*sin1 + beta2*cos1 + np.random.normal(scale=0.1, size=size)
A synthetic monthly series and its deterministic seasonal component. Image by writer.
This time series is customized from the book in reference [3].
Constant seasonality might be well handled by seasonal dummy explanatory variables. A categorical variable that describes the seasonal period. On this case, the month that corresponds to every time
step. This categorical variable is transformed right into a set of indicator (dummy) variables by one-hot encoding.
You may also use Fourier series to model seasonality. Fourier series are sine and cosine waves with various periods. You possibly can learn more about these in a previous article.
Stochastic stationary seasonality
beta1 = np.linspace(-.6, .3, num=size)
beta2 = np.linspace(.6, -.3, num=size)
sin1 = np.asarray([np.sin(2 * np.pi * i / 12) for i in np.arange(1, size + 1)])
cos1 = np.asarray([np.cos(2 * np.pi * i / 12) for i in np.arange(1, size + 1)])
xt = np.cumsum(np.random.normal(scale=0.1, size=size))
# synthetic series with stochastic seasonality
series_stoc = xt + beta1*sin1 + beta2*cos1 + np.random.normal(scale=0.1, size=size)
A synthetic monthly series with a stochastic stationary seasonal component. Image by writer.
A stochastic stationary seasonality evolves over consecutive seasonal periods (e.g. yr over yr). The intensity is less predictable, however the periodicity stays roughly the identical.
With deterministic seasonality, the very best prediction for a given month doesn’t change regardless of the yr. For a stochastic stationary seasonality, the very best guess depends upon the worth of
the identical month from the previous yr.
Stochastic non-stationary seasonality
Sometimes, seasonal patterns change significantly over several seasonal periods. These changes might be brought on by seasonal unit roots, which suggests that seasonality is integrated.
Besides the intensity, the periodicity of this sort of seasonality also tends to vary over time. Because of this the peaks and troughs vary of their location.
Examples of this sort of seasonal pattern appear in numerous domains. These include consumption series or industrial production data.
Changes are difficult to predict when time series have an integrated seasonality. Shocks cause everlasting changes in the information, resulting in scenarios where “spring becomes summer” — quote
from reference [1]. | {"url":"http://aiguido.com/2023/06/3-forms-of-seasonality-and-how-one-can-detect-themwhats-seasonalitythree-forms-of-seasonalitytests-for-seasonal-time-serieskey-takeaways/","timestamp":"2024-11-13T15:01:58Z","content_type":"text/html","content_length":"45194","record_id":"<urn:uuid:740bc5da-541a-4ba5-85ed-50fd98895bfc>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00224.warc.gz"} |
Multiplication Algorithms
Explore the efficiency and effectiveness of each algorithm.
We'll cover the following
Historical roots of algorithms
Although they have been a topic of formal academic study for only a few decades, algorithms have been with us since the dawn of civilization. Descriptions of step-by-step arithmetic computation are
among the earliest examples of written human language, long predating the expositions by Fibonacci and al-Khwarizmi, or even the place-value notation they popularized.
Lattice multiplication
The most familiar method for multiplying large numbers, at least for American students, is the lattice algorithm. This algorithm was popularized by Fibonacci in Liber Abaci, who learned it from
Arabic sources including al-Khwarizm, who in turn learned it from Indian sources including Brahmagupta’s 7th century treatise Brahmasphutasiddhanta, who may have learned it from Chinese sources. The
oldest surviving descriptions of the algorithm appear in The Mathematical Classic of Sunzi, written in China between the 3rd and 5th centuries, and in Eutocius of Ascalon’s commentaries on
Archimedes’ Measurement of the Circle, written around 500 CE, but there is evidence that the algorithm was known much earlier. Eutocius credits the method to a lost treatise of Apollonius of Perga,
who lived around 300 BCE, entitled Okytokion ($`$$\varOmega$$\kappa$$\upsilon$$\tau$$\acute{o}$$\kappa$$\iota$$o$$u$). The Sumerians recorded multiplication tables on clay tablets as early as 2600
BCE, suggesting that they may have used the lattice algorithm.
The lattice algorithm assumes that the input numbers are represented as explicit strings of digits; we’ll assume here that we’re working in base ten, but the algorithm generalizes immediately to any
other base. To simplify notation, the input consists of a pair of arrays $X [0 .. m − 1]$ and $Y [0 .. n − 1]$, representing the numbers
$x = \sum_{i=0}^{m-1}X[i].10^i \text{\ \ and\ \ } y = \sum_{j=0}^{n-1}Y[j].10^j,$
and similarly, the output consists of a single array $Z [0 .. m + n − 1]$, representing the product
The algorithm uses addition and single-digit multiplication as primitive operations. Addition can be performed using a simple for loop. In practice, single digit multiplication is performed using a
lookup table, either carved into clay tablets, painted on strips of wood or bamboo, written on paper, stored in read-only memory, or memorized by the computator. The entire lattice algorithm can be
summarized by the formula
$x.y = \sum_{i=0}^{m-1}\sum_{j=0}^{n-1}(X[i].Y[j].10^{i+j}).$
Different variants of the lattice algorithm evaluate the partial products $X [i] · Y [ j] · 10^{i+ j}$ in different orders and use different strategies for computing their sum. For example, in Liber
Abaco, Fibonacci describes a variant that considers the $mn$ partial products in increasing order of significance, as shown in modern pseudocode below.
Create a free account to access the full course.
By signing up, you agree to Educative's Terms of Service and Privacy Policy | {"url":"https://www.educative.io/courses/mastering-algorithms-for-problem-solving-in-java/multiplication-algorithms","timestamp":"2024-11-08T21:17:36Z","content_type":"text/html","content_length":"905426","record_id":"<urn:uuid:092dd7fe-a5ce-434b-a546-8abc05534236>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00350.warc.gz"} |
Naked Put Calculator Online - CalculatorsHub
Home » Simplify your calculations with ease. » Trading calculators »
Naked Put Calculator Online
Options trading is a vast and dynamic world with various strategies that traders use to optimize their earnings. One such strategy is the 'Naked Put,' a method fraught with risk yet laden with
reward. This article introduces the Naked Put Calculator, a handy tool to assess potential profits or losses in naked put options.
Definition of Naked Put Option
A Naked Put, or uncovered put, is an options strategy where an investor writes or sells put options without holding short positions in the required underlying security. This strategy is typically
employed when an investor is bullish on the market and expects the price of the underlying security to increase.
Working of the Naked Put Calculator
The Naked Put Calculator is an easy-to-use tool that helps calculate the profit or loss from selling a naked put. It requires three input variables: the stock price, strike price, and the price of
the put option. By filling these inputs, the calculator performs the necessary computations to give the expected profit.
The Naked Put Profit Formula
The formula used by the Naked Put Calculator is:
Naked Put Profit = max(strike price - stock price, 0) - put price.
The 'strike price' is the pre-determined price at which the put option can be sold. The 'stock price' is the current price of the underlying asset. The 'put price' is the cost of the put option
Example Calculation
For instance, consider a stock with a current price of $50. If you've sold a put option with a strike price of $55 for a premium (put price) of $5, your profit using the calculator would be max(55 -
50, 0) - 5 = -$5, indicating a loss of $5.
Practical Applications of the Naked Put Calculator
The calculator finds its applications in:
1. Risk Assessment: It aids in evaluating the risk and potential returns before selling a naked put, which is especially useful given the high risk of the strategy.
2. Investment Decisions: It can assist in making informed decisions about whether or not to employ the naked put strategy based on the computed profit or loss.
What is a naked put strategy?
A naked put strategy is when an investor sells put options without holding the required underlying security. This strategy is used when an investor expects the price of the security to rise.
How does the Naked Put Calculator work?
The Naked Put Calculator computes the profit or loss from a naked put strategy using the inputs: stock price, strike price, and put price.
What is the risk with the naked put strategy?
The risk is substantial in a naked put strategy. If the market goes against expectations and the stock price falls, the losses can be significant, as theoretically, a stock's price can fall to zero.
Understanding the intricacies of options trading strategies like the naked put can be a daunting task. However, tools like the Naked Put Calculator make it simpler to evaluate potential profit or
losses, empowering investors to make more informed decisions. It serves as an excellent reminder of the blend of risk and reward that options trading inherently offers.
Leave a Comment | {"url":"https://calculatorshub.net/trading-calculators/naked-put-calculator/","timestamp":"2024-11-06T20:48:13Z","content_type":"text/html","content_length":"114166","record_id":"<urn:uuid:d6240b6f-6040-445e-92c7-1eb4e8154ceb>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00031.warc.gz"} |
5 Minutes Per Math Question Everyday
Many students do not do enough Math questions. In other words, they do not practice enough. The common reason I hear all the time is that there is NO TIME.
Now, what if your child takes just 5 minutes of his daily life to do just one Math Word Problem?
Why 5 minutes? Well, that’s the target time to solve those 4- or 5-marks questions in Paper 2.
During exams, your child may take more than 5 minutes but let’s just work towards the target time of 5 minutes.
My homework booklets for the PSLE exam revision contains more than 30 questions…. so, if your child can spend 5 minutes per question and if he attempts say 5 questions a day, then he would just need
less than half an hour a day to attempt these 5 questions. And in 6 days, he would have attempted all the questions in the homework booklet.
Think of it this way, many students just cannot do it or would not do it. Never mind that they can spend an hour each day playing games or looking at their handphone.
If your child can take just 5 minutes per question daily – Practice, Positive Mindset and Perseverance, I’m sure this effort will pay off and in turn change his score in his exams. | {"url":"https://www.ipmum.com.sg/math_question_everyday/","timestamp":"2024-11-08T05:14:00Z","content_type":"text/html","content_length":"62211","record_id":"<urn:uuid:5069ca0b-2813-4e48-b881-09dc93f1c1dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00361.warc.gz"} |
[Solved] What decimal number does the bit pattern | SolutionInn
What decimal number does the bit pattern 00C000000 represent if it is a floating point number? Use
What decimal number does the bit pattern 0×0C000000 represent if it is a floating point number? Use the IEEE 754 standard.
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 76% (13 reviews)
00C000000 0000 1100 0000 0000 0000 0000 0000 0000 0 ...View the full answer
Answered By
Carly Cimino
As a tutor, my focus is to help communicate and break down difficult concepts in a way that allows students greater accessibility and comprehension to their course material. I love helping others
develop a sense of personal confidence and curiosity, and I'm looking forward to the chance to interact and work with you professionally and better your academic grades.
4.30+ 12+ Reviews 21+ Question Solved
Students also viewed these Computer science questions
Study smarter with the SolutionInn App | {"url":"https://www.solutioninn.com/study-help/computer-organization-design/what-decimal-number-does-the-bit-pattern-00c000000-represent-if","timestamp":"2024-11-03T04:04:16Z","content_type":"text/html","content_length":"79099","record_id":"<urn:uuid:811c80ef-441e-449e-babe-6c8cae3c431d>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00136.warc.gz"} |
Addition to 10,000 without regrouping
Students will be able to add to 10,000 without regrouping.
Briefly recall the concepts of thousands numbers, hundreds numbers, tens numbers and ones values. Then you explain how you can add to 10,000 without regrouping. Explain that you first see how many
cubes of 1,000 you have. Then you add them together (2,000). Next, you add the sheets of hundreds numbers together (200 + 400 = 600). Now you add up the tens numbers (30 + 10 = 40). Finally, you
count the ones values (0). Now you add all the numbers together: 2,000 + 600 + 40 + 0 = 2640. Practice the next problem together with the students and fill it in. Have the students share out loud how
they solved the problem. The students answer the next few problems on their own. Then you repeat the steps to a simple adding problem. Next, you practice another problem together with the students.
The students should solve the next few problems independently. For the story problem, first have the students take the problem from the story. Then they add the thousands numbers, the hundreds
numbers, the tens numbers and the ones values together. Then have the students solve two more story problems on their own.Check if the students can add to 10,000 without regrouping by asking the
following questions:- How do you solve a problem like this one (4,253 + 1,145 = ) in steps?- What steps do you use to do this?
The students first practice addition to 10,000 without regrouping using a visual aid. For the next assignment they just get the number problem. The third assignment is a story problem.
Discuss the relevance of being able to add to 10,000 without regrouping with the students. As a closing activity you can have the students make their own jumping problems, by dragging a trampoline
and a figure to the section on the right and then solving the problem.
Students that have difficulty adding to 10,000 without regrouping can first practice adding numbers in jumps of 10, 100, and 1,000 to 10,000. Then have them add a number to this, for example 1,200,
with the correct jumps.
Gynzy is an online teaching platform for interactive whiteboards and displays in schools.
With a focus on elementary education, Gynzy’s Whiteboard, digital tools, and activities make it easy for teachers to save time building lessons, increase student engagement, and make classroom
management more efficient. | {"url":"https://www.gynzy.com/en-us/library/items/addition-to-10000-without-regrouping","timestamp":"2024-11-02T21:42:43Z","content_type":"text/html","content_length":"553241","record_id":"<urn:uuid:0ffae2bd-a456-41b6-8827-a25810d7b0eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00856.warc.gz"} |
Summary: Identifying and Writing Integers
Key Concepts
• Opposite Notation
□ [latex]-a[/latex] means the opposite of the number [latex]a[/latex]
□ The notation [latex]-a[/latex] is read the opposite of [latex]a[/latex].
• Absolute Value Notation
□ The absolute value of a number [latex]n[/latex] is written as [latex]|n|[/latex] .
□ [latex]|n|\ge 0[/latex] for all numbers.
absolute value
The absolute value of a number is its distance from 0 on the number line.
Integers are counting numbers, their opposites, and zero … [latex]–3, –2, –1, 0, 1, 2, 3\dots[/latex]
negative number
A negative number is less than zero.
The opposite of a number is the number that is the same distance from zero on the number line, but on the opposite side of zero. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/summary-identifying-and-writing-integers/","timestamp":"2024-11-05T01:04:16Z","content_type":"text/html","content_length":"46867","record_id":"<urn:uuid:0950c453-3ab9-4b8c-bee6-daaf32524a69>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00316.warc.gz"} |
1 Introduction
In his attempt to understand behavior of macroscopic objects, Sir Isaac Newton developed a set of equations and principles that are known as Newton's laws of motion [1,5]. For example, according to
the second Newton's law, assuming that the motion is described in an inertial frame of reference, a particle moving in a force field acquires the following acceleration $$\tag{1} a = \frac{F}{m},$$
where \(m\) is the particle's mass and \(F\) is the net external force acting on the particle.
While Newton's laws of motion have been successfully implemented on the macroscopic scale, the subatomic realm requires a significantly different and more complicated approach, one that is based on
the Schrödinger equation and principles of quantum mechanics [3,4]. Stationary Schrödinger equation in 1-D can be written as $$\tag{2} E \psi = - \frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^
2} + U(x) \psi,$$ here \(\psi(x)\) is the wavefunction of the system, \(x\) is the particle's position, \(U(x)\) is the potential energy and \(E\) is the total mechanical energy of the system, and \
(\hbar\) is the reduced Planck constant.
Looking at Eqs.(1,2) it may not be obvious, however quantum mechanics is closely related to and takes its origin from classical mechanics. An example of this relationship is found in the
quasi-classical approximation, where, in the limit of large quantum numbers, solution of the Schrödinger equation (2) converges towards the classical trajectories given by Eq.(1). This approximation
not only demonstrates how classical mechanics can be viewed as a special case within the broader quantum framework but also highlights the seamless transition from the classical to the quantum world.
Another example of such a relationship is Maupertuis's principle in classical mechanics and the variational principle in quantum mechanics. In both approaches, to find the actual trajectory, the
average energy of the system is minimized under certain very specific conditions: $$\tag{3} E(\mbox{“actual trajectory”}) = E_{min}.$$
Let us restrict our consideration only to bound states when the motion is finite and repeats. In this case, in the classical approach, the condition is that the area in the phase-space diagram
outlined by the object's trajectory should remain constant $$\tag{4} W = \oint p dx = \mbox{const},$$ where \(W\) is the area under the curve \(p=p(x)\), \(p=mv\) is the momentum and \(x\) is the
position of the particle. The restriction that the motion repeats itself helps to define integration (4) and simplify the consideration.
In the quantum approach, the minimization occurs for fixed quantum numbers \(n\), and, according to the Bohr-Sommerfeld quantization rule [6,7], fixed quantum numbers correspond to fixed areas in the
same phase-space diagram $$\tag{5} W = \oint p dx = 2 \pi \hbar n,$$ where \(\hbar\) is the reduced Planck constant and \(n\) is the quantum number.
Thus, both principles correspond perfectly to each other. While the variational principle is well-known and widely used for real calculations in quantum mechanics, Maupertuis's principle is less
practical and has mostly theoretical interest.
More specifically, in this work, we aim to understand how Maupertuis's principle works and apply it to two simple classical systems such as motion of a particle in a constant force fields and simple
harmonic motion. We will also demonstrate the correspondence of Maupertuis's principle to the variational principle in quantum mechanics, which can serve as a testament to the underlying unity of the
physical laws governing the universe, bridging the gap between the microscopic and macroscopic domains.
2 Maupertuis's principle
Pierre Louis Maupertuis introduced his principle aiming to determine the trajectory of an object or particle by optimizing a certain functional (a function of the trajectory). In this work, we
consider an unusual version of Maupertuis's principle, which requires the minimization of the particle's average mechanical energy.
To apply this version of the Maupertuis's Principle, first, we need to describe the trajectory of our system using the so-called phase-space diagram: for one degree of freedom, it is a
two-dimensional plane with momentum or velocity as the y-axis and position as the x-axis. Thus, each point on the phase-space diagram corresponds to a certain position and velocity of the particle
and completely describes the state of the system.
Furthermore, for a given trajectory, we can determine the system's average mechanical energy, which is the sum of its kinetic and potential energies, and calculate the area in the phase-space diagram
outlined by the trajectory (assuming that motion repeats itself and the system comes to its initial position after some time \(T\)).
Finally, by varying the trajectory, we can obtain different values for the energy and area. Maupertuis's principle states that the actual trajectory delivers a minimum to the energy under the
condition that the area in the phase-space diagram remains constant. Thus, by minimizing the average energy, we can determine the actual trajectory of the system. The following are two examples of
the application of this principle.
2.1 Example #1: Contant Force Motion
Let's consider a periodic motion of an object under action of a constant force and a rigid floor: at \(t=0\) the object is at the ground, \(x=0\), and has positive initial velocity \(v_0\) pointing
up; then the object travels up and slows down until it comes to a complete stop at \(x=L\) (let's take that the force points down and does not allow the object to fly away); then the object starts
moving down towards the ground, and, after an elastic collision with the ground, the motion repeats starting with the same initial velocity \(v_0\), see Fig. 1.
The described motion can be parameterized as a function of time \(t\) as $$\tag{6} \left\{ \begin{array}{ll} x = v_0 t- \frac{at^2}{2}, & \text{position as a function of time} \\ v = v_0 - at, & \
text{velocity as a function of time} \end{array} \right.,$$ where \(v_0\) is the object's velocity at the ground, and \(a\) is the acceleration. Here we assumed constant acceleration motion and treat
the acceleration and the initial velocity as free parameters. Note: we do not know the value of the acceleration since we do not assume any Newton's laws.
Calculating the area under the trajectory. Considering the one-dimensional motion (6) we can eliminate time and find velocity as a function of position $$\tag{7} v = \pm \sqrt{v^2_0-2ax},$$ where the
signs corresponds to: '+' is for motion up and '-' for motion down.
Analyzing the motion of the particle and limiting the position \(x\) between the ground (\(x=0\)) and the maximum position (\(x=L\)), we can compute the area on the phase-space diagram (see Fig. 1,
the shaded part) as follows: $$\tag{8} W = 2\int_{0}^{L} v \, dx = 2 \int_{0}^{L} \sqrt{v^2_0-2ax} \, dx = \frac{2v^3_0}{3a},$$ where the mass multiplier was omitted (\(m\)=constant), the factor 2 is
due to areas above and below the \(x\)-axis, and the maximum position \(L\) is the position where the particle momentarily stops: $$\tag{9} \sqrt{v^2_0-2aL} = 0 \, \rightarrow L = \frac{v_0^2}{2a}.$$
Using the maximum position (9) eliminates the contribution from the upper limit in integration (8).
Calculating average mechanical energy. Mechanical energy is the sum of a system's kinetic and potential energies: \(E = KE + U\). Let us first consider kinetic energy. We need to calculate the ball's
average kinetic energy, which can be defined as an integration over the time evaluated from time zero to the period divided by this period $$\tag{10} \langle KE \rangle = \frac{1}{T} \int_0^T\frac{1}
{2}mv^2 dt,$$ here \(\langle...\rangle\) means average over time, \(m\) represents mass, \(v\) represents speed, and \(T\) is the period of the motion. Using \(v=v_0-at\) from Eq.(6), we can write $$
\tag{11} \langle KE \rangle = \frac{m}{2T}\int_{0}^{T}(v_0-at)^2dt = \frac{m}{2T}\left(v_0^2 T - v_0aT^2 + \frac{1}{3} a^2T^3\right).$$ After substituting \(T=v_0/a\) and simplifying the integral,
the resultant average kinetic energy for the particle's motion becomes $$\tag{12} \langle KE \rangle =\frac{mv^2_0}{6}.$$
Halfway through the computation of mechanical energy, the only missing constituent is the potential energy that for a constant force can be written as $$\tag{13} U = - Fx,$$ where \(F\) is the force,
the position \(x\) as a function of time is defined by Eq.(6), thus, the average position can be found as follows, $$\tag{14} \langle x \rangle =\frac{1}{T}\int_{0}^{T}xdt = \frac{1}{T}\int_{0}^{T}\
left(v_0 t- \frac{at^2}{2}\right)dt = \frac{1}{T} \left( \frac{v_0 T^2}{2} -\frac{aT^3}{6}\right).$$ After further simplification involving \(T={v_0}/{a}\), the average position is: $$\tag{15} \
langle x \rangle = \frac{v^2_0}{2a}-\frac{v^2_0}{6a}=\frac{v^2_0}{3a}.$$
Using the result of integration (15) and the definition (13) the average potential energy equals to $$\tag{16} \langle U \rangle = -F\frac{v^2_0}{3a} = |F|\frac{v^2_0}{3a}.$$ In this case, the force
F is negative (we need a returning force bringing the object back to the ground), so it is convenient to represent the force as \(F=-|F|\). Now that we have kinetic \(\langle KE \rangle\) and
potential \(\langle U \rangle\) energies, we can set up the equation for the average mechanical energy $$\tag{17} \langle E \rangle=\frac{mv^2_0}{6} + |F|\frac{v^2_0}{3a}.$$
Optimization of energy. Now that the first two steps of Maupertuis' principle have been established, we can vary the trajectory maintaining the area constant (see Fig. 2) and trying to minimize the
energy (17).
We can use the area equation (8) to isolate acceleration and replace it in the mechanical energy equation. $$\tag{18} W=\frac{2v^3_0}{3a}\rightarrow a=\frac{2v^3_0}{3W}.$$ $$\tag{19} \langle E \
rangle=\frac{mv^2_0}{6} + |F|\frac{v^2_0}{3a} = \frac{mv^2_0}{6}+\frac{|F|W}{2v_0}.$$ Finally, by using the average mechanical energy equation and optimizing it, we can determine the minima extremum,
which will correspond to the lowest energy state for the system while maintaining a constant area.
The first derivative of the energy with respect to the initial velocity is: $$\tag{20} \frac{d\langle E\rangle}{dv_0}=\frac{mv_0}{3}-\frac{|F|W}{2v_0^2}=0$$ and $$\tag{21} v_0^3 = \frac{3|F|W}{2m} \
mbox{ and } a = \frac{2v^3_0}{3W},$$ or $$\tag{22} a = \frac{F}{m}.$$ We did not assume any value for the acceleration \(a\), formula (22) came out as a result of the minimization. Thus, optimizing
mechanical energy while maintaining the area constant results in Newton's second law of motion. This means the actual trajectory the object undergoes is when the object's acceleration equals to the
force over the mass. This was derived from the Mauperuis's principle.
2.2 Example #2: Simple Harmonic Motion
Let's consider an ideal harmonic oscillator, where a spring is fixed on one side and on the other side is attached to an object with mass \(m\). The spring is later compressed and released, so the
mass undergoes SHM, which can be parameterized as $$\tag{23} \left\{ \begin{array}{ll} x = A \cos(\omega t), & \text{position as a function of time} \\ v = - \omega A \sin(\omega t), & \text{velocity
as a function of time} \end{array} \right.,$$ where \(A\) represents the amplitude, \(\omega\) is the angular frequency of the oscillation, and \(v\) represents the velocity of the mass.
Computation of the area. Considering the motion of the mass in terms of the velocity \(v\) and position \(x\): the vertices in the \(x\) direction represent the maximum elongation and contraction of
the spring, which, in relationship to the \(x\)-axis, as observed in Fig. 3, are points of velocity equating to zero. In terms of the \(v\)-axis, when the displacement is zero, then the mass is at
its equilibrium, resulting in the maximum speed.
Let's note that Eq.(23) is a standard way of expressing an ellipse. It is well known that the area inside an ellipse can be expressed as \(W=\pi ab\) or, taking in our case \(a=A\) and \(b=\omega A
\), we get $$\tag{24} W = \pi \omega A^2.$$
Computation of average mechanical energy. In the case of SHM, the potential energy can be obtained from Hooke's Law: \(F = -kx\) and \(U = \frac{1}{2} kx^2\) and the kinetic energy is the same as
Consequently, due to the dependence on \(x\) and \(v\) concerning time, see Eq.(23), we can integrate them and yield the average kinetic and potential energies $$\tag{25} \langle KE \rangle =\frac{1}
{T}\int_{0}^{T}\frac{1}{2}mv^2dt = \frac{m\omega^2 A^2}{4},$$ $$\tag{26} \langle U \rangle =\frac{1}{T}\int_{0}^{T}\frac{1}{2}kx^2dt = \frac{kA^2}{4}.$$
Now, the average mechanical energy for the simple harmonic motion can be expressed as $$\tag{27} \langle E \rangle = \frac{m\omega^2 A^2}{4} + \frac{kA^2}{4}.$$
Optimization of energy. We can treat the amplitude \(A\) and the frequency \(\omega\) as free parameters and optimize the average energy (27) under the condition that the area (24) remains constant,
see also Fig. 4.
By rearranging the area formula for amplitude, we can replace \(A^2\) with \(W/\pi \omega\) to simplify the average mechanical energy. $$\tag{28} W = \pi A^2 \omega \rightarrow A^2= \frac{W}{\pi \
omega}.$$ $$\tag{29} \langle E \rangle = \frac{m\omega^2 \frac{W}{\pi \omega}}{4} + \frac{k\frac{W}{\pi \omega}}{4} = \frac{m \omega W}{4\pi} + \frac{kW}{4\pi \omega}.$$ Consequently, by optimizing
the average mechanical energy, we can achieve the result. The first derivative: $$\tag{30} \frac{d\langle E \rangle }{d\omega} = \frac{m W}{4\pi} - \frac{kW}{4\pi \omega^2} = 0.$$ Solving this
equation for the frequency \(\omega\), we get $$\tag{31} \omega^2 = {\frac{k}{m}}.$$ This means the actual periodic path the object takes under simple harmonic motion can be achieved under the
condition that the square of angular frequency \(\omega^2\) corresponds to the spring constant over mass \(k/m\). The result obtained from Maupertuis's principle corresponds to the standard result
obtained through simple harmonic motion standard equations.
3 Variational Principle
The variational principle employed in quantum mechanics contrasts with Maupertuis's principle, which finds application in classical mechanics. However, the steps in their execution allow them to
maintain an analogous relationship. Its execution can be broken down into the following steps:
1^st Guess the behavior of the particle: it involves using adjustable parameters to develop a trial wave function that describes the system's quantum state. For example, a trial wavefunction may
depend on regular space position \(x\) and on a free parameter \(a\) $$\tag{32} \psi = \psi(x,a).$$ Of course, for different systems the trial wavefunctions and parameters are different. Picking the
form of the trial wavefunction is equivalent to picking the motion in Eqs.(6,23).
2^nd Calculating the energy of the system: it requires calculating expectation values from the operators of kinetic and potential energies using the trial wavefunction from the first step: $$\tag{33}
E(a) = \langle \psi | \hat H | \psi \rangle = \int \psi^*(x,a) \,\hat H \,\psi(x,a) dx,$$ where \(\hat H = \hat{KE} + \hat{U}\) is the Hamiltonian that represent the energy of the system. This step
is equivalent to classical calculations of average mechanical energies (19,29).
3^rd Optimization: usually numerical methods are used to optimize the parameters until the energy from step 2 value is minimized. If no other conditions are applied, the result approximates the
ground state wavefunction and the ground state energy of the system (\(n=1\)) $$\tag{34} \frac{d E}{d a} = 0.$$
Table 1: Comparison of Maupertuis's principle and variational principle.
Similarities Differences
Both require parametrization of the motion with a set of The parameters for the variational principle rely on the Hamiltonian and wavefunction, while the Maupertuis's principle involves the
parameters. phase-space diagram.
After the motion is defined, the energy can be minimized The optimization of energy with respect to the parameters results in the ground state of the wave function for the variational principle
according to the parameters to be further optimized. (state with certain quantum number), while for Maupertuis's principle minimization occurred for the fixed phase-space area.
In summary, see Table 1, Maupertuis's and the variational principles share similarities in estimating motion, energy minimization, and optimization. They differ in the nature of the parameters and
the outcomes of this process. Nevertheless, if analyzed based on their computational approach, they can be regarded as counterparts of different physical systems.
4 Conclusion
The paper aimed to expose Maupertuis's principle as an alternative approach to describe classical systems yielding standard equations and building a bridge with the variational principle implemented
in quantum mechanics.
In conclusion, through the two cases explored, Maupertuis's principle resulted in Newton’s second law of motion, \( a = F/m, \) for scenarios with constant acceleration motion and for standard
angular frequency equation, \( \omega^2 = k/m ,\) for the case resembling simple harmonic motion. The derivation of these well-known equations from an alternative approach, such as Maupertuis's
principle, attests to its precision in describing dynamical systems. Moreover, when contrasted with the variational principle, they share similarities in their computational approaches; both require
an estimation of motion under a set of parameters, and both involve minimization and optimization to complete their calculations. Despite these parallels, they diverge in the nature of their
parameters and in their outcomes. While Maupertuis' principle yields a constant that aids in determining the actual path within a dynamical system, the variational principle is tailored for
describing the wavefunctions of quantum states.
[1] L.D. Landau, L.M. Lifshitz, "Mechanics: Volume 1". Butterworth-Heinemann, 1976.
[2] M. Spivak, "Physics for Mathematicians: Mechanics I". Publish and Perish, 2010.
[3] L.D. Landau, L.M. Lifshitz, "Quantum Mechanics: Non-Relativistic Theory". Butterworth-Heinemann, Jan. 1981.
[4] V. Zelevinsky, "Quantum Physics: Volume 1 - From Basics to Symmetries and Perturbations". Wiley, Dec. 2010.
[5] J.R. Taylor, "Classical Mechanics". University Science Books, 2005.
[6] A. Sommerfeld (1916). "Zur Quantentheorie der Spektrallinien". Annalen der Physik (in German). 51 (17): 1–94. https://zenodo.org/records/1424309.
[7] Wikipedia, Bohr-Sommerfeld Model. https://en.wikipedia.org/wiki/Bohr-Sommerfeld_model (as of March 2024). | {"url":"http://adastraletter.com/2024/1/maupertuis-principle","timestamp":"2024-11-09T17:04:31Z","content_type":"text/html","content_length":"32500","record_id":"<urn:uuid:0fe8038e-d466-408c-91e0-261b15c772cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00686.warc.gz"} |
fnPad (pronounced "fun-pad") is a text editor based, graphing, programmable calculator. Arithmetic expressions are entered in a text window, and when calculated, by selecting the "Calculate" menu
item, the results are inserted into the text following the expressions.
The "jar" file [do whatever you've got to do with your browser to download the linked file] of version 0.32 (2010 Jan 9) can be downloaded. You need to have version 1.5 or higher of the Java™ Runtime
Environment (available for Windows and Mac OS X). On some systems (e.g., Mac OS X), you can open fnPad.jar as you do other files, e.g., by double-clicking it. It can be run from a shell by the
command "java -jar fnPad.jar".
Mac users who are disoriented by having menus in the window instead of the menu bar, and by a funky application name can use the rather hideous shell command
$ java -Dapple.laf.useScreenMenuBar="true" \
> -Dcom.apple.mrj.application.apple.menu.about.name="fnPad" \
> -jar fnPad.jar
The syntax will be familiar to some, and is pretty simple anyhow. A little example follows.
// A comment starts with a double slash
// and ends with the line.
3+8 ≈ 11 // Add two numbers.
// fnPad inserts the wavy equals
// and the result which follows it
// when the "Calculate" menu item is selected.
// Do multiple operations in an expression.
2+3*7 ≈ 23
// Parentheses override the normal precedence
// or associativity of operators.
(2+3)*7 ≈ 35
// Division of integers results in an integer.
5/2 ≈ 2
// Division of floating point numbers
// or mixed floating point and integers
// results in a floating point number.
5./2 ≈ 2.5
// An expression ends with the line or at a semicolon.
16./5. ≈ 3.2; 2^5 ≈ 32.0
// Note that all of these calculations are done at once.
More interesting things can be done by using definitions.
// A definition can be an identifier = an expression.
x = 3
// Things that are defined can be used in an expression.
y = x+1
// Definitions can have parameters
// and occur in the text after being used.
b = length(x,y)
// Note that this "b" has a different value
// than "b" in the "length(a,b)" definition.
b ≈ 5.0
length(b,12) ≈ 13.0
// There are some included functions and predefined constants.
atan2(y,x)*180/pi ≈ 53.1301
// Here's a function definition, length of the hypotenuse,
length(a,b) = sqrt(a^2+b^2)
// and another, base 10 logarithm.
log(x) = ln(x)/ln(10)
log(100) ≈ 2.0; log(0.1) ≈ -1.0
Definitions can be recursive, in which case a conditional expression is needed.
// Here are a couple of conditional expressions.
// The first one means: if x is less than 0,
// then the result is -x, otherwise the result is x.
x < 0 ? -x : x ≈ 1 // absolute value
x >= 0 ? x : 0 ≈ 0 // clip below zero
x = -1
// a recursive definition of the greatest common divisor
gcd(a,b) = ( a%b == 0 ) ? b : gcd(b,a%b)
// This ↑__________↑ parenthesis is for clarity.
gcd(648,156) ≈ 12
Any of the example text can be copied and pasted into a fnPad window.
An example of graphing is shown in the image of a fnPad window below. See that graph(x,f(x)) is used to produce a graph of some function, where the first argument is some symbol (variable name) and
the second argument is some expression that (usually) is a function of that variable.
The extent of the coordinates may be specified, as in the example, with definitions of graph.x.min (for the minimum of the horizontal axis) and so on. If undefined, the extent of each axis is -1 to
The color of each graph depends on the order of the graph() applications in the text. The light gray lines are the coordinate axes.
To see the graph, the left-hand pane of the window must be opened, either by dragging the divider, or clicking on the "move divider right" control (if there is one in the interface).
As can be seen, graph(x,f(x)) is different than other functions in two ways: it has the side effect of drawing in the graph panel, and it has no result. Another, less obvious way in which graph() is
different is that its arguments are not directly evaluated when it is applied, but its second argument, f(x), is evaluated over a range of values for its first argument, x.
Below is an example that finds the roots of a quadratic equation and graphs the parabola. Try changing the coefficients and seeing how the graph changes. [If you change the coefficients by too much
you might also need to change the bounds of the graph to see the parabola.]
// To see the graph,
// open the left panel.
// Solve and graph
// a quadratic equation.
// Define the coefficients.
a = 1; b = -2; c = 0
// Find the discriminant and roots.
d ≈ 4.0; root1 ≈ 0.0; root2 ≈ 2.0
// Graph the quadratic function.
// Define the graph's bounds.
graph.x.min = -5; graph.x.max = 5
graph.y.min = -15; graph.y.max = 15
// Define the quadratic formula.
root1 = (-b-sqrt(d))/(2*a) // root with -
root2 = (-b+sqrt(d))/(2*a) // root with +
d = b^2-4*a*c // discriminant
There's another page with a more extensive example that applies the quadratic formula.
The following are all the operators, lowest to highest precedence. Association is left, except for "^", e.g., a/b/c = (a/b)/c, a^b^c = a^(b^c).
• definition
• conditional
• boolean
• comparison
• arithmetic
□ + -
□ * / % (modulus)
□ ^ (exponentiation)
□ - (unary minus)
• bits
□ & (and) | (or) $ (exclusive or)
□ << (left shift) >> (right shift)
□ ~ (complement)
The following functions are included. The trigonometric functions use radians.
• sin(theta) - sine of an angle
• cos(theta) - cosine of an angle
• tan(theta) - tangent of an angle
• asin(r) - arcsine of a number
• acos(r) - arccosine of a number
• atan(r) - arctangent of a number
• atan2(y,x) - arctangent of y/x
• ln(r) - natural logarithm of a number
• sqrt(r) - square root of a number
• abs(r) - absolute value of a number
• rnd(r) - a number rounded to the nearest integer
• hex(i) - a hexadecimal string of an integer
• oct(i) - an octal string of an integer
• fix(r,n) - a fixed point decimal string of a number with specified precision
• eng(r,n) - a floating point decimal string of a number with specified precision
Other functions, written in Java, can be added by deriving from the Function class. See cos.java for an example.
The following "constants" are predefined.
There are some missing basic things that need to be added.
• printing - use another text editor to print
• setting and saving preferences
• catching a Quit from the Mac Jar Runner
Here are the next planned major features.
• arrays
• reading data files
• fractions
• complex numbers
• graph(f(y),y)
• graph(x,y) // where x and y are arrays
• string operations | {"url":"https://www.dolben.org/fnPad.php","timestamp":"2024-11-04T00:35:46Z","content_type":"application/xhtml+xml","content_length":"17275","record_id":"<urn:uuid:7848254f-10f3-42db-8893-04c9a067816e>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00525.warc.gz"} |
How to create a formula that only affects part of the column...
For the days that are marked DBL in the Time column, we are trying to adjust the formula in the Arrival Time column so that only the first 4 rows are 1.5 hours before the time listed in the Time
column and the rest of that day is 1 hour before the listed time...i.e.. 08:15 = 06:45 (1.5 hours) and 08:30 = 07:30 (1 hour). The current formula used in the Arrival Time column is...
"=IF(lvl@row = 0, " ", VLOOKUP(Time@row, {Time Range 1}, 5, false))"...
Below are the screen shots for the Surgery Scheduling sheet we are using as well as the reference sheet with the times to post in the Arrival Time column...
Thank you in advanced...
Best Answer
• Hi @JJLewis,
It seems we need to index rows in your sheet to determine RowID, then setup formulas depend on their RowID.
First, create RowID column with a formula :
=COUNTIF(Date$1:Date@row, Date@row) - 1
If RowID<=4 : use 1.5 hours before the time listed in the Time column, else 1 hour
Then modify your formula in the Arrival Time column as below:
=IF(lvl@row = 0, " ", IF(RowID<=4, VLOOKUP(Time@row, {Time Range 1}, 5, false), VLOOKUP(Time@row, {Time Range 1}, 6, false)))
Hope that helps.
Gia Thinh Technology - Smartsheet Solution Partner.
• Hi @JJLewis,
It seems we need to index rows in your sheet to determine RowID, then setup formulas depend on their RowID.
First, create RowID column with a formula :
=COUNTIF(Date$1:Date@row, Date@row) - 1
If RowID<=4 : use 1.5 hours before the time listed in the Time column, else 1 hour
Then modify your formula in the Arrival Time column as below:
=IF(lvl@row = 0, " ", IF(RowID<=4, VLOOKUP(Time@row, {Time Range 1}, 5, false), VLOOKUP(Time@row, {Time Range 1}, 6, false)))
Hope that helps.
Gia Thinh Technology - Smartsheet Solution Partner.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/89130/how-to-create-a-formula-that-only-affects-part-of-the-column","timestamp":"2024-11-04T01:42:45Z","content_type":"text/html","content_length":"437630","record_id":"<urn:uuid:468ac42e-97b0-4ee0-8988-9760afcbf4e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00083.warc.gz"} |
Lifesight's approach to Causal Inference | Lifesight Help Center
Lifesight’s approach to understanding and leveraging causal relationships in data is centered around a three-step process
Step 1: Laying the Foundation
Our journey begins with the critical inputs of Data and Business Knowledge. This foundational step ensures that our analysis is grounded in the real-world context of your business, setting the stage
for meaningful insights.
Step 2: Causal Inference
At Lifesight, we utilize the DECI (Deep End to End of Causal Inference) framework to underpin our approach to causal inference. This framework integrates both causal discovery and estimation
processes into a cohesive end-to-end flow.
Causal Discovery: Causal discovery is a critical process aimed at identifying causal relationships between variables within observational data. The main objective is to uncover a network of causal
relationships that aligns with both the collected data and predefined causal assumptions.
Directed Acyclic Graph (DAG): The causal relationships are depicted in a Directed Acyclic Graph (DAG), where nodes represent variables and edges signify causal connections.
Bayesian Approach: The DECI framework adopts a Bayesian methodology for causal discovery, incorporating prior knowledge and constraints to refine the search for causal relationships.
We employ a hybrid approach, combining Score and Constraint-based algorithms, to construct causal graphs. This method leverages causal assumptions to establish a preliminary graph, which is then
refined using data-driven techniques.
Score-Based Algorithm searches for a causal graph that both aligns with the observational data and adheres to the specified constraints. To further refine the causal graph, we utilize Non-Linear
Additive Noise Models, allowing for the correction of the graph to reveal true causal relationships, even from purely observational data.
Causal Estimation: Following the Causal Discovery, the DECI framework proceeds to Causal Estimation. This stage focuses on quantifying the Average Treatment Effects (ATE) using the established DAG.
Through calculus and the generative model learned by DECI, we simulate samples from intervened distributions to estimate treatment effects accurately.DECI provides an approximation of the posterior
distribution over graphs, considering the observational data. This approximation is instrumental in understanding the treatment effects and the causal structure.
The DECI framework operates under several foundational assumptions, which include:
• Causal Markov Condition: This assumes that all observed variables are independent of their non-effects, given their direct causes.
• Causal Faithfulness Condition: It posits that all observed independencies in the data are attributable to the underlying causal structure.
• Non-Gaussianity: The distribution of additive noise within the causal model is assumed to be non-Gaussian, aiding in the identification of causal relationships.
• Causal Sufficiency: The framework assumes the absence of unobserved confounders, ensuring that all relevant variables are considered in the analysis.
Step 3: Reporting and Insight Generation
The culmination of the above intricate process involves the presentation of Average Treatment Effects (ATE), delivering profound insights into the ramifications of various interventions.
The comprehension of ATE is essential for businesses, as it provides an overarching view of the average impact of interventions, thereby facilitating informed strategic decision-making. | {"url":"https://help.lifesight.io/en/articles/8948283-lifesight-s-approach-to-causal-inference","timestamp":"2024-11-05T13:45:16Z","content_type":"text/html","content_length":"64476","record_id":"<urn:uuid:065ed787-4720-4e10-894b-3d295d054f34>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00322.warc.gz"} |
Ground states for fractional magnetic operators
p. 1-24
Existence, regularity and structure of confined elasticae
p. 25-43
Stability of observations of partial differential equations under uncertain perturbations
p. 45-61
On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate
p. 63-87
On stability of nonlinear neutral functional differential equations
p. 89-104
Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients
p. 105-127
Reduced basis approximation of large scale parametric algebraic Riccati equations
p. 129-151
Stochastic homogenization of plasticity equations
p. 153-176
Semiclassical ground state solutions for a Choquard type equation in ℝ 2 with critical exponential growth
p. 177-209
Local boundary controllability to trajectories for the 1d compressible Navier Stokes equations
p. 211-235
On the semi-global stabilizability of the Korteweg-de Vries Equation via model predictive control
p. 237-263
Iterative observer-based state and parameter estimation for linear systems
p. 265-288
Estimates for the controls of the wave equation with a potential
p. 289-309
Optimal control of piecewise deterministic Markov processes: a BSDE representation of the value function
p. 311-354
Dynamic programming principle and associated Hamilton-Jacobi-Bellman equation for stochastic recursive control problem with non-Lipschitz aggregator
p. 355-376
Optimization in structure population models through the Escalator Boxcar Train
p. 377-399
A second order local minimality criterion for the triple junction singularity of the Mumford-Shah functional
p. 401-435
Bellman equation and viscosity solutions for mean-field stochastic control problem
p. 437-461
A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function
p. 463-477
Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates
p. 479-494
A maximum principle for controlled stochastic factor model
p. 495-517
Energy decay in a wave guide with dissipation at infinity
p. 519-549
Multipolar Hardy inequalities on Riemannian manifolds Dedicated to Professor Enrique Zuazua on the occasion of his 55th birthday
p. 551-567
The convergence of nonnegative solutions for the family of problems − Δ[p]u = λe^u as p →∞
p. 569-578
Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D
p. 579-603
Viscosity methods for large deviations estimates of multiscale stochastic processes
p. 605-637
A class of infinite-horizon stochastic delay optimal control problems and a viscosity solution to the associated HJB equation
p. 639-676
Prox-regularity approach to generalized equations and image projection
p. 677-708
Bang-bang control of a thermostat with nonconstant cooling power
p. 709-719
Averaged time-optimal control problem in the space of positive Borel measures
p. 721-740
Control strategies for the Fokker−Planck equation
p. 741-763
Action minimization and macroscopic interface motion under forced displacement
p. 765-792
Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws^,
p. 793-810
Regularization and discretization error estimates for optimal control of ODEs with group sparsity
p. 811-834
BMO-type seminorms and Sobolev functions
p. 835-847
A note on the boundary regularity of solutions to quasilinear elliptic equations
p. 849-858
Cylindrical optimal rearrangement problem leading to a new type obstacle problem
p. 859-872
Stochastic optimal control problem with infinite horizon driven by G-Brownian motion
p. 873-899
Linear quadratic mean field game with control input constraint
p. 901-919
Unilateral problems for the p-Laplace operator in perforated media involving large parameters
p. 921-964
A semi-Lagrangian algorithm in policy space for hybrid optimal control problems
p. 965-983
A risk-sensitive maximum principle for a Markov regime-switching jump-diffusion system and applications
p. 985-1013
On the horseshoe conjecture for maximal distance minimizers
p. 1015-1041
On the structure of multifactor optimal portfolio strategies
p. 1043-1058
A concept of inner prederivative for set-valued mappings and its applications
p. 1059-1074
Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling
p. 1075-1105
Anisotropic free-discontinuity functionals as the Γ-limit of second-order elliptic functionals
p. 1107-1139
Occupational measures and averaged shape optimization
p. 1141-1165
Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques
p. 1167-1180
A morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints
p. 1181-1206
Distance estimates for state constrained trajectories of infinite dimensional differential inclusions
p. 1207-1229
Existence and multiplicity of solutions for a class of quasilinear elliptic field equation on ℝ^N
p. 1231-1248
Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities
p. 1249-1273
Motion of discrete interfaces in low-contrast random environments
p. 1275-1301
Minimal clusters of four planar regions with the same area
p. 1303-1331
Diagonal non-semicontinuous variational problems
p. 1333-1343
On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators
p. 1345-1380
Symmetry-breaking in a generalized Wirtinger inequality
p. 1381-1394
Globally Lipschitz minimizers for variational problems with linear growth
p. 1395-1413
On the minimizing movement with the 1-Wasserstein distance
p. 1415-1427
Singular perturbations for a subelliptic operator
p. 1429-1451
Optimal control of reaction-diffusion systems with hysteresis
p. 1453-1488
Comparison between W[2] distance and Ḣ^−1 norm, and Localization of Wasserstein distance
p. 1489-1501
Upper semicontinuity of the lamination hull
p. 1503-1510
Decay estimates for 1-D parabolic PDES with boundary disturbances
p. 1511-1540
On a decomposition of regular domains into John domains with uniform constants
p. 1541-1583
On two functionals involving the maximum of the torsion function
p. 1585-1604
Closed 𝓐-p Quasiconvexity and Variational Problems with Extended Real-Valued Integrands
p. 1605-1624
Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions
p. 1625-1644
Normality and gap phenomena in optimal unbounded control
p. 1645-1673
On the approximate boundary synchronization for a coupled system of wave equations: direct and indirect controls
p. 1675-1704
Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity
p. 1705-1734
Existence result for degenerate cross-diffusion system with application to seawater intrusion
p. 1735-1758
Stabilization of damped waves on spheres and Zoll surfaces of revolution
p. 1759-1788
Dynamic boundary control games with networks of strings
p. 1789-1813
A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the Mortensen observer
p. 1815-1847
Linear quadratic control problems of stochastic Volterra integral equations
p. 1849-1879
Flux-limited and classical viscosity solutions for regional control problems
p. 1881-1906 | {"url":"http://archive.numdam.org/volume/COCV_2018__24_4/","timestamp":"2024-11-03T12:08:41Z","content_type":"text/html","content_length":"119914","record_id":"<urn:uuid:002e8edb-ebc5-40df-8d66-7ef2eab47d4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00794.warc.gz"} |
LVR Lending Capacity Calculator
LVR Lending Capacity Calculator
This LVR lending capacity calculator will assist you in finding out what your potential scope for further lending may be. In NZ the loan-to-value ratio restrictions were set by the Reserve Bank to
ensure that borrowers were not overstretched in terms of borrowing. The current LVR lending rate for owner-occupier is 80% and 70% for investors purchasing using equity security.
Enter your information into the calculator below, and push Calculate.
Current RBNZ LVR Lending Ratio (LVR%)
Current market interest rate (IR)
Payment term for additional lending years (PT)
Current property market value (PMV)
Current mortgage value (CMV)
Calculated equity in property (CE)
Equity ratio at time of purchase
Equity ratio current market value
Lending ratio at time of purchase
Lending ratio current market value
Total lending capacity (PMV * LVR%)
Current mortgage outstanding (CMV)
Potential lending capacity
Desired additional lending
With capacity for $ (%) additional lending the additional costs would be:
Weekly additional payment
Fortnightly additional payment
Monthly additional payment
Yearly additional payment
Total additional interest paid over loan term
Total payment for additional lending over loan term
Choosing to borrow $ additionally, the costs per period would be:
Weekly additional payment
Fortnightly additional payment
Monthly additional payment
Yearly additional payment
Total additional interest paid over loan term
Total payment for additional lending over loan term
If you've found a bug, or would like to contact us please click here.
Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.
Copyright © 2019 calculate.co.nz All Rights Reserved. No part of this website, source code, or any of the tools shall be copied, taken or used without the permission of the owner. All calculators and
tools on this website are made for educational and indicative use only. Calculate.co.nz is part of the realtor.co.nz, GST Calculator, GST.co.nz, and PAYE Calculator group. | {"url":"https://www.calculate.co.nz/lvr-lending-capacity-calculator.php","timestamp":"2024-11-01T23:01:30Z","content_type":"text/html","content_length":"75065","record_id":"<urn:uuid:dcf4f440-2606-4a1c-8e5a-475b23ec47dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00896.warc.gz"} |
Frac Sleeves
Ball Drop Frac Sleeve is а part of an openhole fracturing system designed to allow operators to perform selective single-point multistage fracturing. The inner sleeve is run in а pinned configuration
and sheared after pressure increasing when activation ball is landed, providing positive indication that the specified port has opened before fracturing.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells.
Max OD in:
4.646 / 3.465 / 3.465 / 3.465
Liner size in:
4.000 / 4.000 / 5.500 / 4.500 / 5.750
Ball Drop Frac Sleeve is а part of an openhole fracturing system designed to allow operators to perform selective single-point multistage fracturing. The inner sleeve is run in а pinned configuration
and sheared after pressure increasing when activation ball is landed, providing positive indication that the specified port has opened before fracturing.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells
Hydraulic Activated Тое Sleeve with lsolation Valve is designed to bе used as а first stage for multistage hydraulic fracturing. Hydraulic activated Тое Sleeve in terms of construction consists of an
activation sleeve and а toe sleeve.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells
Hydraulic Frac port is а part of а robust cemented or openhole fracturing system designed to allow operators to perform selective multistage hydraulic fracturing. Fullbore sleeve designed for the
most common high-pressure and high-rate hydraulic fracturing. Hydraulic activation eliminates the need of ball usage.
• Cemented casing / liner applications.
• Vertical, directional and horizontal wells.
Max OD in:
4.567 / 5.236 / 5.748
Liner size in:
4.000 / 4.500 / 5.000
Ball Drop Frac Port Reclosable is а Key-operated Frac Sleeve is designed to provide access for the process fluid to the isolation zone during multistage hydraulic fracturing. The sleeve is opened
hydraulically bу pumping an activation ball into а special seat and applying pressure.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells.
Max OD in:
4.567 / 5.236 / 6.732 / 5.630
Liner size in:
4.000 / 4.500 / 5.500 / 4.500mod
Ball Drop Frac Sleeve Reclosable is designed for multistage hydraulic fracturing with activation balls of the corresponding size. The sleeve is opened hydraulically bу pumping an activation ball into
а special seat and increasing the tubing pressure.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells.
Liner size in:
4.500 / 4.500
Hydraulic Тое Sleeve Reclosable is designed to provide access for the process fluid to the isolation zone during multistage hydraulic fracturing. The sleeve is opened hydraulically bу pumping an
activation ball into а special seat and applying pressure. Circulation ports are opened after activation ball is seated and the pressure is increased to the activation pressure value.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells.
Liner size in:
5.500 / 4.500
Hydraulic Frac Port is designed to provide access for the process fluid to the isolation zone during multistage hydraulic fracturing without activation balls.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells.
Max OD in:
5.236 / 6.732 / 5.630
Liner size in:
4.500 / 5.500 / 4.500
Hydraulic Activated Тое Sleeve with lsolation Valve Reclosable is designed for the first interval of multistage hydraulic fracturing using activation balls, without the ability to close the sleeve.
The sleeve design includes an activation sleeve and а hydraulic toe sleeve.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells.
Full Bore Hydraulic Frac Port Reclosable is designed to provide access for the process fluid to the isolation zone during multistage hydraulic fracturing. The sleeve is opened hydraulically bу
pumping an activation ball into а special seat and applying pressure.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells
REVOLVER Full ID One-Size Ball Operated Frac Sleeve is designed for multistage hydraulic fracturing without the need to mill out the seats. The sleeve cаn bе used with hydromechanical packers for
isolating the fracturing zones.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells.
Plug Operated Frac Port is designed to provide an access for the process fluid to the isolation zone during hydraulic fracturing using а soluble key-plug.
• Uncemented casing / liner applications.
• Vertical, directional and horizontal wells
Liner size in:
4.000 / 4.500
Frac port for cementing operations is designed for multistage hydraulic fracturing. The sleeve is opened hydraulically bу pumping an activation ball into а special seat and then pressurizing the
• Cemented casing / liner applications.
• Vertical, directional and horizontal wells.
Max OD in:
4.567 / 5.236 / 7.087
Liner size in:
4.000 / 4.500 / 5.500
Burst-Port Frac Sleeve is designed to perform multistage fracturing using selective packer in cemented/non-cemented wells. The tool сап also bе used as the first fracturing stage.
• Cemented / non-cemented casing / liner applications.
• Vertical, directional and horizontal wells.
Liner size in:
4.500 / 4.500
Burst-Port Frac Sleeve is designed to perform multistage fracturing using selective packer in cemented/non-cemented wells. The tool саn also bе used as the first fracturing stage.
• Cemented / non-cemented casing / liner applications.
• Vertical, directional and horizontal wells. | {"url":"https://tss-group.ru/en/catalog/frac-sleeves/","timestamp":"2024-11-08T09:23:17Z","content_type":"text/html","content_length":"226731","record_id":"<urn:uuid:1902fcba-0adf-445c-8b50-8635b094cde9>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00127.warc.gz"} |
June 2014 – Walking Randomly
Archive for June, 2014
June 18th, 2014
Something that became clear from my recent comparison of Numpy’s Mersenne Twister implementation with MATLAB’s is that there is something funky going on with seed 0 in MATLAB. A discussion in the
comments thread helped uncover what was going on. In short, seed 0 gives exactly the same random numbers as seed 5489 in MATLAB (unless you use their deprecated rand(‘twister’,0) syntax).
This is a potential problem for anyone who performs lots of simulations that make use of random numbers such as monte-carlo simulations. One common work-flow is to run the same program hundreds of
times where only the seed differs between runs. This is probably good enough to ensure that each simulation uses a random number stream that is statistically independent from all of the others —
There is a risk that some streams will overlap but the probability is low and most people are content to live with that risk.
The practical upshot of this is that if you intend on sticking with Mersenne Twister for your MATLAB monte-carlo simulations, it might be wise to avoid seed 0. Alternatively, move to a random number
generator that guarantees non-overlapping, independent streams – something that any implementation of Mersenne Twister cannot do.
Here’s a demo run in MATLAB 2014a on Windows 7.
>> format long
>> rng(0)
>> rand(1,5)'
ans =
>> rng(5489)
>> rand(1,5)'
ans =
June 16th, 2014
When porting code between MATLAB and Python, it is sometimes useful to produce the exact same set of random numbers for testing purposes. Both Python and MATLAB currently use the Mersenne Twister
generator by default so one assumes this should be easy…and it is…provided you use the generator in Numpy and avoid the seed 0.
Generate some random numbers in MATLAB
Here, we generate the first 5 numbers for 3 different seeds in MATLAB. Our aim is to reproduce these in Python.
>> format long
>> rng(0)
>> rand(1,5)'
ans =
>> rng(1)
>> rand(1,5)'
ans =
>> rng(2)
>> rand(1,5)'
ans =
Python’s default random module
According to the documentation,Python’s random module uses the Mersenne Twister algorithm but the implementation seems to be different from MATLAB’s since the results are different. Here’s the
output from a fresh ipython session:
In [1]: import random
In [2]: random.seed(0)
In [3]: [random.random() for _ in range(5)]
In [4]: random.seed(1)
In [5]: [random.random() for _ in range(5)]
In [6]: random.seed(2)
In [7]: [random.random() for _ in range(5)]
The Numpy random module
Numpy’s random module, on the other hand, seems to use an identical implementation to MATLAB for seeds other than 0. In the below, notice that for seeds 1 and 2, the results are identical to
MATLAB’s. For a seed of zero, they are different.
In [1]: import numpy as np
In [2]: np.set_printoptions(suppress=True)
In [3]: np.set_printoptions(precision=15)
In [4]: np.random.seed(0)
In [5]: np.random.random((5,1))
array([[ 0.548813503927325],
[ 0.715189366372419],
[ 0.602763376071644],
[ 0.544883182996897],
[ 0.423654799338905]])
In [6]: np.random.seed(1)
In [7]: np.random.random((5,1))
array([[ 0.417022004702574],
[ 0.720324493442158],
[ 0.000114374817345],
[ 0.30233257263184 ],
[ 0.146755890817113]])
In [8]: np.random.seed(2)
In [9]: np.random.random((5,1))
array([[ 0.435994902142004],
[ 0.025926231827891],
[ 0.549662477878709],
[ 0.435322392618277],
[ 0.420367802087489]])
Checking a lot more seeds
Although the above interactive experiments look convincing, I wanted to check a few more seeds. All seeds from 0 to 1 million would be a good start so I wrote a MATLAB script that generated 10 random
numbers for each seed from 0 to 1 million and saved the results as a .mat file.
A subsequent Python script loads the .mat file and ensures that numpy generates the same set of numbers for each seed. It outputs every seed for which Python and MATLAB differ.
On my mac, I opened a bash prompt and ran the two scripts as follows
matlab -nodisplay -nodesktop -r "generate_matlab_randoms"
python python_randoms.py
The output was
MATLAB file contains 1000001 seeds and 10 samples per seed
Random numbers for seed 0 differ between MATLAB and Numpy
System details
• Late 2013 Macbook Air
• MATLAB 2014a
• Python 2.7.7
• Numpy 1.8.1
June 3rd, 2014
Here in the UK, this morning’s news is dominated by the Gameover Zeus virus and how it can hold you to ransom, empty your bank accounts and generally ruin your day!
The usual good advice on how to protect yourself from such attacks is doing the rounds but I wondered how effective one extra precaution might be: Only ever log into bank accounts etc using a
dedicated device.
I’m seriously considering doing this since internet-capable devices are very cheap these days. While I’m at it, I’m thinking of taking the following extra precautions:
• Install Linux on the dedicated device since it is not targeted by hackers as often as Windows-based devices are.
• Create dedicated email addresses for each bank account. That way, if my normal email account were compromised, my bank accounts would still be safe.
Obviously, such a scheme would be less convenient than using whichever of my current devices I happen to be using but I’d rather that than be robbed of everything.
What do you think? Would such a scheme offer any additional protection? | {"url":"https://walkingrandomly.com/?m=201406","timestamp":"2024-11-02T04:22:18Z","content_type":"application/xhtml+xml","content_length":"62680","record_id":"<urn:uuid:77bfb6cf-ff14-4918-9045-16ac9058e360>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00599.warc.gz"} |
Quick Scripts - CIAO 4.16 Sherpa
Quick Scripts
This page provides quick access to the Sherpa 4.16 Python scripts used in the Sherpa threads. Each script below is also included in the "Scripting It" section at the bottom of the corresponding
Fitting Data
Plotting Data
Computing Statistics
Simulating Data | {"url":"https://cxc.harvard.edu/sherpa/scripts/index.html","timestamp":"2024-11-09T06:10:43Z","content_type":"text/html","content_length":"24675","record_id":"<urn:uuid:e3e808e0-f75d-43b2-a568-c9adf8f1c209>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00557.warc.gz"} |
TGS Well Data Analytics: Asset Valuation & Investment Insight
In the ever-evolving landscape of the oil and gas industry, the echoes of transformative change resound through the recent surge in merger and acquisition activity. Since October 2023, the sector has
witnessed a seismic shift with notable transactions, including the groundbreaking ExxonMobil Pioneer deal, the strategic maneuvers of Chevron Hess, and the dynamic synergy of OXY CrownRock. These
mega-mergers underscore the industry's resilience in adapting to market conditions and will influence the scalability of U.S. hydrocarbon production, particularly from unconventional sources.
This paper analyzes the recently announced transactions and demonstrates how the new addition of Well Economics Data within TGS Well Data Analytics can aid companies in creating a smooth workflow for
asset valuation analysis and investment decision-making. Both are instrumental in the Merger and Acquisitions (M&A) process.
The asset valuation process in the oil and gas industry is inherently challenging due to the complexity of geological formations, varying reservoir characteristics, and the dynamic nature of
commodity prices. Assessing the actual value of unconventional resources demands a thorough understanding of technical, financial, and market factors (Figure 1). TGS, with its comprehensive Well Data
Analytics solution and subsurface library, addresses these challenges by providing a wealth of information on reservoir properties, production trends, geological insights, and economic scenarios
(Figures 1, 2). TGS empowers oil and gas professionals with the tools needed to navigate the intricacies of asset valuation, ultimately facilitating informed investment decisions and strategic
planning. For example, by leveraging TGS data and asset valuation workflows, TGS estimates a transaction value for the ExxonMobil Pioneer deal ranging between $53 to $61 bn, with ~1,300 undeveloped
Transactions Highlights:
• Combined deal value of ~$125 bn for the three transactions.
• All three buyers are public companies.
• Two of the three acquired companies were also public companies.
• Two of the three acquisitions’ main geographic areas are in the Midland Basin (Permian).
Table 1: Deal Summary Table. The companies bolded in blue were the acquired (target) companies.
XOM: ExxonMobil, PXD: Pioneer, CVX: Chevron, HES: HESS, OXY: Occidental.
Figure 1: High-level asset valuation process illustrating which TGS data can be used to evaluate each phase.
Figure 2: Well Data Analytics dashboard showcasing Well Economics Data on the left, Pioneer’s developed and undeveloped acreage on the map, and reservoir properties, production trends, and geological
insights on the right.
Pioneer’s Case Study Analysis
The purpose of the case study is to demonstrate how Well Economics Data within TGS Well Data Analytics (WDA) can assist companies in conducting accurate and fast company and inventory valuation
The study includes ~9,000 Pioneer wells encountered in the Midland Basin (Figure 3). Pioneer has been an active operator in the basin since the early 2000s, maintaining a consistent drilling
development campaign even during years of downturn. In the decade of 2000 to 2010, the company drilled a total of 2,429 vertical wells. In the following decade, from 2011 to 2020, the company’s
drilling strategy shifted to horizontal drilling, increasing from less than 5% horizontal wells in 2012 to over 88% in 2020, drilling a total of 5,031 wells. Since 2021, the company has almost
exclusively drilled horizontal wells, completing 1,446 wells (Figure 4).
Once Pioneer switched to horizontal drilling, the company quickly increased the average lateral length year over year. It transitioned from 1-mile laterals in the years 2012 to 2015 to drilling >
2-mile laterals in 2023 (Figure 5). Pioneer drilled ~200 wells in 2023 with lateral lengths greater than 14,000 ft, according to TGS WDA data. The company stated several times during 2023 that
increasing lateral length strengthens returns by reducing the capital spending needed per well by ~15% and achieving higher economic returns. Exxon attributes one-third of the potential synergies of
the transaction (~$700 million) to materialize mainly from capital efficiencies by drilling longer laterals.
TGS published a spotlight article analyzing lateral length and its impact on economic results in Q4-2023. The study showed that the internal rate of return increases by ~720 basis points, and the net
present value improves by ~$3.5 million for a 3-mile lateral well compared to a 2-mile lateral well under the base case price scenario at $80/bbl and $3.5/mcf. To learn more about this analysis,
click on the following link.
Figure 3: Study area includes ~9,000 Pioneer wells – image taken from TGS Well Data Analytics (WDA). The top 5 producing formations, monthly production by vintage, and well-count by slant plots are
on the left. A map of Pioneer wellbores in the Midland Basin is in the middle and type curve and production plots are on the right of the dashboard.
Figure 4: Pioneer’s drilling development analysis from 2000 to 2023 using TGS WDA data.
Figure 5: Pioneer’s average lateral length evolution from 2012 to 2023 using TGS WDA data.
Figure 6: Daily production plot for Pioneer over time showing oil in green, gas in red, and
equivalent production (BOE) in light blue taken from TGS WDA.
According to TGS’s allocated production, Pioneer produced 493 Mbo/d of oil, 1.8 Bcf/d of gas, and equivalent production of ~800 Mboe/d as of September 2023 (Figure 6). Considering the company’s
reported working interest of ~88%, TGS estimates the company’s net equivalent production to be ~700 Mboe/d. In Midland, recent production deal multiples range between $40,000/boe to ~45,000/boe.
Therefore, TGS calculates the total transaction value attributed to Proved Developed Producing Resources (PDP) to be between $28 bn and $32 bn by multiplying the net production volumes by the
production multiple ranges.
TGS WDA can assist in assessing the number of undeveloped locations when the value is unknown or when a company needs to analyze a new area for development. In this case, the gun barrel feature can
be utilized to understand how many wells per section (WPS) are drilled per unit and how many wells could be drilled in undeveloped acreage position in future development.
According to the gun barrel diagram from one of Pioneer’s recent pads, the company is drilling using a Cube development strategy, spacing wells by ~800-900 ft horizontally in the lower Spraberry
shale, the primary producing formation (Figure 7, 8). This translates into placing ~6 to 7 wells in a single drilling unit. The gun barrel diagram also shows the company drilling multiple benches,
creating a multi-zone development strategy. However, for a 3-mile lateral well, unitization agreements will have to be worked out to allow a large amount of acreage to be joined together via
contracts or statute to develop the reservoir this way.
To estimate Pioneer’s developed and undeveloped resources, the company's acreage position was digitized and uploaded to WDA (Figure 9). A total of ~1,035,615 gross acres were calculated in the
process. Directional drilling data was utilized to determine developed acres, estimating ~464,124 gross acres. As a result, TGS estimates a total of ~571,492 gross undeveloped acres. Using 6 to 7 WPS
for well spacing per drilling unit and a safety buffer of ~330 ft from leasing lines as required by the state, TGS approximates there to be ~2,232 gross undeveloped locations for 10,000 ft laterals
and ~1,488 gross undeveloped locations for 15,000 ft laterals. The estimated gross locations can be netted down by applying the company’s average working interest in the basin, which is reported to
be ~88%. Therefore, TGS calculates ~1,300 net undeveloped locations for 15,000 ft lateral wells.
Figure 7: Gun barrel diagram for a 2023 Pioneer well using TGS WDA displaying the vertical and
horizontal distance to a nearby sibling well in the same producing formation.
Figure 8: Well Spacing Analysis Dashboard – image taken from TGS Well Data Analytics (WDA). The map shows an area of recent well development for Pioneer. The production plot and gun barrel diagram
are displayed on the left of the dashboard.
Figure 9: Map extract from TGS WDA including digitized Pioneer acreage position. TGS estimated developed acres are highlighted in yellow, whereas undeveloped acres are highlighted in light green.
Once the number of undeveloped locations is calculated, the next step involves determining an appropriate type curve that represents Pioneer’s most up-to-date production performance in the study
area, as type curves form the foundation of the reserve and development planning analysis. Figure 10 displays a recent 3-mile well with current and forecasted production values that can serve as a
type well for the area. Once the type well is established, the final step of the process consists of running an economic model to determine important economic outputs such as Internal Rate of Return
(IRR), Net Present Value (NPV), Breakeven Price (BE), and Payback Period. These outputs are essential for estimating the dollar amount that could potentially be generated from developing the
remaining inventory.
The newly launched TGS Well Economics Data within TGS Well Data Analytics allows users to quickly and accurately estimate those outputs (Figures 11 & 12). It also provides an economic deliverable
that users can leverage to initiate their own internal financial model for more detailed asset valuation and analysis (Figure 13).
Figure 10: Type well for a recent 3-mile Pioneer well in the study area showing oil type curve
in green and gas type curve in red.
Figure 11: The Well Economics panel on the left shows economic inputs on top and main economic outputs on the bottom. Clients can customize economic inputs to recalculate the economic model.
Figure 12: Well Economics panel close-up image.
Figure 13: Well Economics Data deliverable in Excel.
As per TGS Well Economics Data, a type well in the study area for Pioneer generates an NPV between $19.2 million and $22.4 million^2 and has breakeven prices ranging from $36.3/bbl to $41.2/bbl.
Therefore, TGS estimates that the undeveloped resource can be valued between $25 bn to $29 bn, based on the estimated number of undeveloped locations (~1,300).
The table below summarizes the main deal attributes calculated using TGS Well Data Analytics, compared to the values that were reported by Exxon during the press release of the transaction. It
demonstrates that TGS Well Economics Data and Well Data Analytics workflows can be applied to generate high-level accurate asset and company valuation metrics.
Table 2 : Main Deal Summary: Comparing transaction values as reported and estimated values using
TGS Well Data Analytics Data and Workflows.
The last eighteen months have been transformative in the oil and gas industry, particularly in terms of consolidation, especially for the US onshore market. This article summarized the three largest
acquisitions from this period and conducts an in-depth case study on the ExxonMobil/ Pioneer deal. Utilizing the recently released TGS Well Economics Data, alongside well and production data, well
spacing data, type curves, and visualizations from TGS Well Data Analytics, we predict an estimated value of $28 to $32 bn for the producing resource and $25 to $29 bn for the undeveloped resource,
confirming the deal value of this acquisition purely on a cash flow analysis basis. This case study demonstrates how TGS Well Data Analytics, coupled with Well Economics Data, can streamline complex
asset and company valuation processes across a wide range of scenarios.
Article References
Table 1:
^1 Calculated over the firm’s 30-day volume-weighted average price.
^2 Acreage located in the Bakken Basin.
^3 TGS interpretation using $35,000/boe production multiple.
^4 Sourced from JP Morgan Energy, Power & Renewables conference, June 2023.
^5 ~1,250 locations ready to be developed as disclosed during the transaction.
Table 2:
^1 Calculated for 15,000 ft lateral wells.
^2 NPV values calculated with pre-generated TGS economic assumptions in the study area, with an average price of $60.0/bbl for oil, $4.4/mcf for gas and $12/bbl for NGL volumes. Two main type curve
areas were identified and analyzed in the study: the north area and the south area of Pioneer’s acreage position. | {"url":"https://www.tgs.com/well-intel/well-economics-data-seamless-asset-valuation-and-investment-decisions-in-well-data-analytics","timestamp":"2024-11-05T09:47:42Z","content_type":"text/html","content_length":"67608","record_id":"<urn:uuid:c4718b37-6864-4133-997a-f5f5d6edd94e>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00643.warc.gz"} |
Quantum algorithm offers faster way to hack internet encryption - ΠΛΑΤΕΙΑ Online
In 1994, Peter Shor created one of the first practical uses for a quantum computer: hacking the internet. Shor, an applied mathematician at the Massachusetts Institute of Technology (MIT), showed how
a quantum computer could be exponentially faster than a classical computer at finding the prime number factors of large numbers. Those primes are used as the secret keys that secure most of the
encrypted information sent over the internet.
For 30 years, Shor’s algorithm has endured as an example of the promise of quantum computers—although the devices are not yet big or reliable enough to implement it for large numbers. But now, a
computer scientist has revealed a new quantum algorithm that might be better than Shor’s. In a preprint first posted to the arXiv server on 12 August, Oded Regev of New York University proposes a
scheme that could greatly reduce the number of gates, or logical steps, needed to factor very large numbers. In principle, it could enable a smaller quantum computer to ferret out the secret
encryption keys or a bigger machine to decode them faster. “Is this actually going to have any effect?” Regev asks. “My feeling is that yes, it might have a chance.”
Independent cryptographers who have evaluated the work are intrigued, too. Vinod Vaikuntanathan, a computer scientist at MIT, expects a packed house there next month when Regev will give a colloquium
talk on his new algorithm. “In the world of quantum computing, essentially two or three new ideas have appeared so far in the last 30 years since Shor,” Vaikuntanathan says. “You don’t see these new
ideas every day, and that makes us hope.” Kenneth Brown, a quantum computing researcher at Duke University, agrees. “Because everybody has studied Shor’s algorithm for a long time, this result is
surprising and super cool.”
Like all quantum algorithms, Shor’s algorithm relies on the mysterious properties of quantum bits, or qubits, which can be set to values of not only 0 and 1, but also a “superposition” of 0 and 1 at
the same time. Small numbers of these qubits can be stitched together into gates, which carry out the logical operations of an algorithm. To factor a number n bits long, Shor’s algorithm requires a
quantum circuit of n^2 gates.
Most internet encryption now relies on numbers of at least 2048 bits, which equate to decimal numbers 617 digits long. Finding their prime factors with Shor’s algorithm would therefore require
quantum computers with at least 4 million gates. But the biggest quantum computers to date only have a few hundred qubits. “None of them are anywhere near the size we need to factor numbers that we’d
care about,” Brown says.
Making things worse, environmental noise often destroys qubits’ delicate superposition states, ruining the operation. The noise can be addressed with error correction, but that requires even more
qubits—millions or even billions of them, Vaikuntanathan says. “It really blows up because of error correction,” he says. “That’s why we are pretty far from actually being able to factor 1000-digit
numbers.” Improving error correction would help—but so would improving on Shor’s algorithm.
Regev saw a way to do that. Shor’s algorithm is 1D. It searches for the prime factors by raising a single number to high powers. Many big numbers must be multiplied together before a result is
reached. Regev realized he could multiply several numbers in different dimensions. The powers for any one number don’t get nearly as high. Although the two algorithms require about the same total
number of multiplications, the multidimensional character of Regev’s means the multiplied numbers don’t get nearly as large before a result is reached.
In the end, he found he would need only n^1.5 gates to factor an n-bit integer. It’s the first substantial improvement on Shor’s algorithm in 30 years, Vaikuntanathan says. “Nobody has really
succeeded beyond shaving off a little bit.”
But Regev’s algorithm also comes with drawbacks, says Martin Ekerå, a quantum computing researcher with the Swedish government whom
Regev consulted with while trying to understand the practical implications of his work. Its structure seems to require quantum memory to store intermediate values during the computation, and that
means a need for more of those finicky qubits. “This drives up the cost of the algorithm,” Ekerå says. Regev acknowledges the concern about memory requirements, but says the algorithm still could end
up having value—“maybe when memory is cheaper and we instead worry about the number of operations.”
By the time quantum computers are ready to find prime factors by implementing either Regev or Shor’s algorithm, internet encryption may have moved on. Federal agencies and security leaders are
already shifting to alternatives, including so-called “lattice cryptography,” which would be immune to quantum hacking. Even so, algorithms like Regev and Shor’s could be applied retroactively, to
decrypt recorded traffic from the present and recent past, Ekerå says.
Regardless, Brown believes the sheer novelty of Regev’s work is likely to inspire and generate other new ideas in quantum cryptography, which has struggled for significant breakthroughs. “I am myself
trying to think about ways to push this further,” he says.
Source: Science
Leave a reply | {"url":"https://plateia.online/quantum-algorithm-offers-faster-way-to-hack-internet-encryption/","timestamp":"2024-11-05T00:09:26Z","content_type":"text/html","content_length":"251825","record_id":"<urn:uuid:a86e6588-91e1-48e8-81ec-88fd0e7a8839>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00290.warc.gz"} |
audioOut = shiftPitch(audioIn,nsemitones) shifts the pitch of the audio input by the specified number of semitones, nsemitones.
Apply Pitch-Shifting to Time-Domain Audio
Read in an audio file and listen to it.
[audioIn,fs] = audioread('Counting-16-44p1-mono-15secs.wav');
Increase the pitch by 3 semitones and listen to the result.
nsemitones = 3;
audioOut = shiftPitch(audioIn,nsemitones);
Decrease the pitch of the original audio by 3 semitones and listen to the result.
nsemitones = -3;
audioOut = shiftPitch(audioIn,nsemitones);
Apply Pitch-Shifting to Frequency-Domain Audio
Read in an audio file and listen to it.
[audioIn,fs] = audioread("SpeechDFT-16-8-mono-5secs.wav");
Convert the audio signal to a time-frequency representation using stft. Use a 512-point kbdwin with 75% overlap.
win = kbdwin(512);
overlapLength = 0.75*numel(win);
S = stft(audioIn, ...
"Window",win, ...
"OverlapLength",overlapLength, ...
Increase the pitch by 8 semitones and listen to the result. Specify the window and overlap length you used to compute the STFT.
nsemitones = 8;
lockPhase = false;
audioOut = shiftPitch(S,nsemitones, ...
"Window",win, ...
"OverlapLength",overlapLength, ...
Decrease the pitch of the original audio by 8 semitones and listen to the result. Specify the window and overlap length you used to compute the STFT.
nsemitones = -8;
lockPhase = false;
audioOut = shiftPitch(S,nsemitones, ...
"Window",win, ...
"OverlapLength",overlapLength, ...
Increase Fidelity Using Phase Locking
Read in an audio file and listen to it.
[audioIn,fs] = audioread('FemaleSpeech-16-8-mono-3secs.wav');
Increase the pitch by 6 semitones and listen to the result.
nsemitones = 6;
lockPhase = false;
audioOut = shiftPitch(audioIn,nsemitones, ...
To increase fidelity, set LockPhase to true. Apply pitch shifting, and listen to the results.
lockPhase = true;
audioOut = shiftPitch(audioIn,nsemitones, ...
Increase Fidelity Using Formant Preservation
Read in the first 11.5 seconds of an audio file and listen to it.
[audioIn,fs] = audioread('Rainbow-16-8-mono-114secs.wav',[1,8e3*11.5]);
Increase the pitch by 4 semitones and apply phase locking. Listen to the results. The resulting audio has a "chipmunk effect" that sounds unnatural.
nsemitones = 4;
lockPhase = true;
audioOut = shiftPitch(audioIn,nsemitones, ...
To increase fidelity, set PreserveFormants to true. Use the default cepstral order of 30. Listen to the result.
cepstralOrder = 30;
audioOut = shiftPitch(audioIn,nsemitones, ...
"LockPhase",lockPhase, ...
"PreserveFormants",true, ...
Input Arguments
audioIn — Input signal
column vector | matrix | 3-D array
Input signal, specified as a column vector, matrix, or 3-D array. How the function interprets audioIn depends on the complexity of audioIn:
• If audioIn is real, audioIn is interpreted as a time-domain signal. In this case, audioIn must be a column vector or matrix. Columns are interpreted as individual channels.
• If audioIn is complex, audioIn is interpreted as a frequency-domain signal. In this case, audioIn must be an L-by-M-by-N array, where L is the FFT length, M is the number of individual spectra,
and N is the number of channels.
Data Types: single | double
Complex Number Support: Yes
nsemitones — Number of semitones to shift audio by
real scalar
Number of semitones to shift the audio by, specified as a real scalar.
The range of nsemitones depends on the window length (numel(Window)) and the overlap length (OverlapLength):
-12*log2(numel(Window)-OverlapLength) ≤ nsemitones ≤ -12*log2((numel(Window)-OverlapLength)/numel(Window))
Data Types: single | double
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: 'Window',kbdwin(512)
Window — Window applied in time domain
sqrt(hann(1024,'periodic')) (default) | real vector
Window applied in the time domain, specified as the comma-separated pair consisting of 'Window' and a real vector. The number of elements in the vector must be in the range [1, size(audioIn,1)]. The
number of elements in the vector must also be greater than OverlapLength.
If using shiftPitch with frequency-domain input, you must specify Window as the same window used to transform audioIn to the frequency domain.
Data Types: single | double
OverlapLength — Number of samples overlapped between adjacent windows
round(0.75*numel(Window)) (default) | scalar in the range [0, numel(Window))
Number of samples overlapped between adjacent windows, specified as the comma-separated pair consisting of 'OverlapLength' and an integer in the range [0, numel(Window)).
If using shiftPitch with frequency-domain input, you must specify OverlapLength as the same overlap length used to transform audioIn to a time-frequency representation.
Data Types: single | double
LockPhase — Apply identity phase locking
false (default) | true
Apply identity phase locking, specified as the comma-separated pair consisting of 'LockPhase' and false or true.
Data Types: logical
PreserveFormants — Preserve formants
false (default) | true
Preserves formants, specified as the comma-separated pair consisting of 'PreserveFormants' and true or false. Formant preservation is attempted using spectral envelope estimation with cepstral
Data Types: logical
CepstralOrder — Cepstral order used for formant preservation
30 (default) | nonnegative integer
Cepstral order used for formant preservation, specified as the comma-separated pair consisting of 'CepstralOrder' and a nonnegative integer.
To enable this name-value pair argument, set PreserveFormants to true.
Data Types: single | double
Output Arguments
audioOut — Pitch-shifted audio
column vector | matrix
Pitch-shifted audio, returned as a column vector or matrix of independent channels.
To apply pitch shifting, shiftPitch modifies the time-scale of audio using a phase vocoder and then resamples the modified audio. The time scale modification algorithm is based on [1] and [2] and is
implemented as in stretchAudio.
After time-scale modification, shiftPitch performs sample rate conversion using an interpolation factor equal to the analysis hop length and a decimation factor equal to the synthesis hop length. The
interpolation and decimation factors of the resampling stage are selected as follows: The analysis hop length is determined as analysisHopLength = numel(Window)-OverlapLength. The shiftPitch function
assumes that there are 12 semitones in an octave, so the speedup factor used to stretch the audio is speedupFactor = 2^(-nsemitones/12). The speedup factor and analysis hop length determine the
synthesis hop length for time-scale modification as synthesisHopLength = round((1/SpeedupFactor)*analysisHopLength).
The achievable pitch shift is determined by the window length (numel(Window)) and OverlapLength. To see the relationship, note that the equation for speedup factor can be rewritten as: nsemitones =
-12*log2(speedupFactor), and the equation for synthesis hop length can be rewritten as speedupFactor = analysisHopLengh/synthesisHopLength. Using simple substitution, nsemitones = -12*log2
(analysisHopLength/synthesisHopLength). The practical range of a synthesis hop length is [1, numel(Window)]. The range of achievable pitch shifts is:
• Max number of semitones lowered: -12*log2(numel(Window)-OverlapLength)
• Max number of semitones raised: -12*log2((numel(Window)-OverlapLength)/numel(Window))
Formant Preservation
Pitch shifting can alter the spectral envelope of the pitch-shifted signal. To diminish this effect, you can set PreserveFormants to true. If PreserveFormants is set to true, the algorithm attempts
to estimate the spectral envelope using an iterative procedure in the cepstral domain, as described in [3] and [4]. For both the original spectrum, X, and the pitch-shifted spectrum, Y, the algorithm
estimates the spectral envelope as follows.
For the first iteration, EnvX[a] is set to X. Then, the algorithm repeats these two steps in a loop:
1. Lowpass filters the cepstral representation of EnvX[a] to get a new estimate, EnvX[b]. The CepstralOrder parameter controls the quefrency bandwidth.
2. To update the current best fit, the algorithm takes the element-by-element maximum of the current spectral envelope estimate and the previous spectral envelope estimate:
The loop ends if either a maximum number of iterations (100) is reached, or if all bins of the estimated log envelope are within a given tolerance of the original log spectrum. The tolerance is set
to log(10^(1/20)).
Finally, the algorithm scales the spectrum of the pitch-shifted audio by the ratio of estimated envelopes, element-wise:
[1] Driedger, Johnathan, and Meinard Müller. "A Review of Time-Scale Modification of Music Signals." Applied Sciences. Vol. 6, Issue 2, 2016.
[2] Driedger, Johnathan. "Time-Scale Modification Algorithms for Music Audio Signals." Master's Thesis. Saarland University, Saarbrücken, Germany, 2011.
[3] Axel Roebel, and Xavier Rodet. "Efficient Spectral Envelope Estimation and its application to pitch shifting and envelope preservation." International Conference on Digital Audio Effects, pp.
30–35. Madrid, Spain, September 2005. hal-01161334
[4] S. Imai, and Y. Abe. "Spectral envelope extraction by improved cepstral method." Electron. and Commun. in Japan. Vol. 62-A, Issue 4, 1997, pp. 10–17.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
• LockPhase must be set to false.
• Using gpuArray (Parallel Computing Toolbox) input with shiftPitch is only recommended for a GPU with compute capability 7.0 ("Volta") or above. Other hardware might not offer any performance
advantage. To check your GPU compute capability, see ComputeCompability in the output from the gpuDevice (Parallel Computing Toolbox) function. For more information, see GPU Computing
Requirements (Parallel Computing Toolbox).
For an overview of GPU usage in MATLAB^®, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2019b | {"url":"https://de.mathworks.com/help/audio/ref/shiftpitch.html","timestamp":"2024-11-02T23:49:05Z","content_type":"text/html","content_length":"126211","record_id":"<urn:uuid:aee2623a-62fd-4b5a-982b-d5a20311a8bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00135.warc.gz"} |
Are you Smarter Than a 6th Grader?
How many chambers are in the human heart?
What is the smallest prime number?
What are birds doing when they fly to warmer climates during winter months?
Minnesota hosts the largest state fair in the US as measured by average daily attendance, and it also features the tag-line "The Great Minnesota Get-Together." Located near, but not in the Twin
Cities, you'l instead find the fairgrounds in what city?
Falcon Heights
Aimee had 17 bubble gums, and Oscar gave her 13 more; she then gave half of the bubble gums to Jenny. How many does she have left?
If a letter isn't a vowel, what is it?
A consonant.
What is 56 + 87?
What are the 3 states of matter?
Gas, liquid, and solid.
According to the Minnesota Department of Natural Resources, 36% of Minnesotans fish. This ranks second highest of any state; what is the one state higher?
What is the place value of the number 3 in the following: 4,398?
Which city is called the Windy City?
9 is the square root of what number?
How many muscles are in the human body?
The winners of the 2011, 2013, 2015, and 2017 WNBA Championships claim Minnesota as their home. What is the mascot for this basketball team?
What is the bottom number of a fraction called?
A denominator.
What did Alexander Graham Bell invent?
The telephone.
What is a third of 60?
The magnitude of an earthquake is measured by what?
The Richter scale.
Many named places throughout the state of Minnesota have “Minne” as part of the name: Minnehaha, Minnetonka, Minnetrista, and more. This word “Minne” derives from a word in the Dakota language
meaning what?
Jayden has 918 toy cars. He wants to split these into boxes of cars, with 9 cars per box. How many boxes is Jayden able to complete?
Which is the smallest ocean?
The Arctic Ocean.
What is the simplest form of 16/24?
When pushing something, what are you applying?
Known as the "Land of 10,000 Lakes," but only one can take the prize as the largest lake within Minnesota's borders. Which lake is it?
Red Lake
What is the perimeter of a circle called? | {"url":"https://jeopardylabs.com/play/are-you-smarter-than-a-6th-grader-329","timestamp":"2024-11-12T22:54:13Z","content_type":"application/xhtml+xml","content_length":"58476","record_id":"<urn:uuid:28be35f2-c297-4151-8ec7-f7e2bfbc37ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00758.warc.gz"} |
How the experts algorithm can help solve lps online
We consider the problem of solving packing/covering LPs online, when the columns of the constraint matrix are presented in random order. This problem has received much attention, and the main focus
is to figure out how large the right-hand sides of the LPs have to be (compared to the entries on the left-hand side of the constraints) to allow 41C.5-Approximations online. It is known that the
right-hand sides have to be i4.2 logm5 times the left-hand sides, where m is the number of constraints. In this paper, we give a primal-dual algorithm that achieves this bound for mixed packing/
covering LPs. Our algorithms construct dual solutions using a regret-minimizing online learning algorithm in a black-box fashion, and use them to construct primal solutions. The adversarial guarantee
that holds for the constructed duals helps us to take care of most of the correlations that arise in the algorithm; the remaining correlations are handled via martingale concentration and maximal
inequalities. These ideas lead to conceptually simple and modular algorithms, which we hope will be useful in other contexts.
• Experts Algorithm
• Linear Programming
• Online Algorithms
• Secretary Problem
ASJC Scopus subject areas
• General Mathematics
• Computer Science Applications
• Management Science and Operations Research
Dive into the research topics of 'How the experts algorithm can help solve lps online'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/how-the-experts-algorithm-can-help-solve-lps-online","timestamp":"2024-11-06T21:10:40Z","content_type":"text/html","content_length":"51833","record_id":"<urn:uuid:7966c2dd-7a4d-415a-9214-fc2219627184>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00109.warc.gz"} |
DNN Neural Network | A Quick Glance of DNN Neural Network - Examples
Updated February 28, 2023
Introduction to DNN Neural Network
Artificial Neural Network(ANN) can either be shallow or deep. When ANN has more than one hidden layer in its architecture, they are called Deep Neural Networks. These networks process complex data
with the help of mathematical modelling. Deep Neural Networks (DNN) is otherwise known as Feed Forward Neural Networks(FFNNS).In this networks, data will be flowing in the forward direction and not
in the backward direction, and hence node can never be accessed again. These Networks need a huge amount of data to train, and they have the ability to classify millions of data.
Structure of DNN Neural Network
Deep Neural Networks have an input layer, an output layer and few hidden layers between them. These networks not only have the ability to handle unstructured data, unlabeled data, but also
non-linearity as well. They have a hierarchical organization of neurons similar to the human brain. The neurons pass the signal to other neurons based on the input received. If the signal value is
greater than the threshold value, the output will be passed else ignored. As you can see the data is passed to the input layer and they yield output to the next layer and so on until it reaches the
output layer where it provides the prediction yes or no based on probability. A layer consists of many neurons, and each neuron has a function called Activation Function. They form a gateway of
passing the signal to the next connected neuron. The weight has the influence of the input to the output of the next neuron and finally, the last output layer. The weights initially assigned are
random, but as the network gets trained iteratively, the weights are optimized to make sure that the network makes a correct prediction.
For better understanding, let’s take the human brain which can identify different people though they have two eyes, one nose two ears. These variations and deviations can be learned by the neurons in
the brain and combine all these differences and can identify the people. These all are done in the fraction of second.
The same logic is applied to Deep Neural Network by using a mathematical approach. Based on the simple rule, the signal from one neuron is transferred to another neuron, similar to the process
learned by the brain. When the output of the neuron has a high value, then, the dimension corresponding to that has high importance. Similarly, all the importance of one layer been captured in the
form of deviation and finally combine all the deviations and fed it to the next layer. Thus the system learns the process intuitively.
Learning of DNN Neural Network
We provide input data to the network and based on that the output prediction would be correct or incorrect with the steps of numerous matrix multiplication. Based on the output, the feedback is fed
back to the network, the system learns by adjusting its weights between the layers. This process is called Backpropagation by providing feedback and updating the weights. This process of training the
network is computationally very high, and because of data involved, it is now it’s been more popular because of the improvisation of technologies recently.
Examples of DNN Neural Network
Below are mentioned the examples:
1. MNIST Data
These networks can be further explained by three concepts like Local receptive fields, shared weights, and pooling Say we are using 28*28 square of neurons whose values are intensities. So let’s say
we connect the one neuron of hidden layer to the input layer of 5 * 5 region as shown in the fig below
2. Local Receptive Fields
In the above diagram, the small region is called Local receptive fields. Each connection has a weight and is associated with the hidden layer neuron through the layer trains and learns. This has 28 *
28 input image, 5 * 5 local receptive fields then it will 24 * 24 neurons in the hidden layer.
Here in the above diagram, the stride length is 1 when the local 5 * 5 is moved to the nearby region. When we move the local receptive region to the right by 2 pixels, then we say the stride is 2.
Strides can be of different lengths and can be considered as one of the hyperparameters.
3. Sharing Weights
Each neuron in the hidden layer has a 5 * 5 weights and a bias for its local receptive region. This bias b is the same for all the hidden layer neurons. Here w(l,m) is shared weights, and an (x,y) is
the activation function at that point. This implies that the neurons will learn a similar feature. Here the map from the input layer to the hidden layer is called a feature map. The weights are
called shared weights; bias here is called shared bias. Weights and bias are called filters or kernels.
The local receptive field corresponds to one single kind of feature map. Likewise, we need a lot number of feature maps for prediction of images, The example shown above has three feature maps and
each has a single bias and 5 *5 shared weights. This sample concept is used in LeNet and is used in MNIST classification of digits with more than 30 features maps. When sharing weights is used, the
number of parameters is less for the network to learn.
4. Pooling Layer
These layers come after the convolution layer they normally compress the output produced from feature maps and in other words the reduced information comes out after the pooling layer. You can also
specify the region size for max-pooling say for example if you have 2 * 2 region if you use max-pooling you get the max output activation in that 2 * 2 region.
Likewise, for each feature map, you get max layer output. Another form of pooling is called L2 pooling. Here we take the square root of the sum of activations in that 2 * 2 region rather than taking
the max of the activations.
Hence Deep Learning Network is used in may vertical of the industry right from Health-care in detecting cancer, Aviation industry for optimization, Banking Industry for detecting fraudulent
transactions to retail for customer retention. All these are possible with the advent of GPUS for complex processing of data.
Recommended Articles
This is a guide to DNN Neural Network. Here we discuss an introduction, structures with deep learning and examples to implement with proper explanation. You can also go through our other related
articles to learn more – | {"url":"https://www.educba.com/dnn-neural-network/","timestamp":"2024-11-01T23:41:51Z","content_type":"text/html","content_length":"317192","record_id":"<urn:uuid:ef884d3f-463d-40ad-a7a4-a6e99960c7d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00821.warc.gz"} |
aspen trees utah
You can also customize them using the worksheet generator provided. 6th grade math worksheets, PDF printables to practice skills on math topics learnt in grade 6. Each worksheet contains a series of
sixth grade math problems and to make it easy each worksheet has an answer key attached to the second page. Free pdf worksheets from K5 Learning's online reading and math program. These our math
printable worksheets for grade 6 have covered all major areas of grade 6 math, … At the end of each section, there is a general self-assessment test to take in order to evaluate overall
understanding. Create an unlimited supply of worksheets for conversion of measurement units for grade 6 – both customary and metric units. The Videos, Games, Quizzes and Worksheets make excellent
materials for math teachers, math educators and parents. With strands drawn from vital math topics like ratio, multiplication, division, fractions, common factors and multiples, rational numbers,
algebraic … Math-Drills.com includes over 58 thousand free math worksheets that may be used to help students learn math. 4th Grade. View PDF. Halloween Activity Worksheets. If you do not want that we
track your visist to our site you can disable tracking in your browser here: We also use different external services like Google Webfonts, Google Maps and external Video providers. Math worksheets
for sixth grade children covers all topics of 6th grade such as Graphs, Data, Fractions, Tables, Subtractions, Pythagoras theorem, Algebra, LCM, HCF, Addition, Round up numbers , Find 'X' in addition
equations, Metric systems, Coordinate geometry, Surface Areas, Order of operations, Decimals, Probability, Money & more. Math Worksheets - Free Weekly PDF Printables 1st grade math 2nd grade math 3rd
grade math 4th grade math 5th grade math 6th grade math. Hometuition-kl - Letter Tracing Worksheets PDF. Fraction Worksheets Multiplication Worksheets Times Table Worksheets Brain Teaser Worksheets
Picture Analogies Cut and Paste Worksheets Pattern Worksheets Dot to Dot worksheets Preschool and Kindergarten – Mazes Size Comparison Worksheets. You can read about our cookies and privacy settings
in detail on our Privacy Policy Page. Take a video tour by clicking on the e-book icon to your left. Contact us for : These math sheets can be printed as extra teaching material for teachers, extra
math practice for kids or as homework material parents can use. Solve linear equations, finding missing variables, algebraic expressions and more. This section is a step-by-step presentation of how
to use algebra formulae on all the topics covered in this site which include formulae on, Perimeter and Area of rectangles and squares 1, Perimeter and Area of rectangles and squares 2, Perimeter and
Area of rectangles and squares 3, Perimeter and Area of rectangles and squares 4, Perimeter and Area of rectangles and squares 5, Perimeter and Area of rectangles and squares 6, Algebra Finding the
value of the equation - Finding X sheet 1, Algebra Finding the value of the equation - Finding X Sheet 2, Pre-algebra addition with decimals sheet 1, Pre-algebra addition with decimals sheet 2,
Pre-algebra addition with decimals sheet 3, Pre-algebra Multiplication & Addition sheet 1, Pre-algebra Multiplication & subtraction sheet 1, Re-algebra Multiplication with decimals sheet 1,
Re-algebra Multiplication with decimals sheet 2, Re-algebra subtraction with decimals sheet 1, Re-algebra subtraction with decimals sheet 2, Re-algebra subtraction with decimals sheet 3, Algebra
worksheets - Single variable sheet 1, Algebra worksheets - Single variable sheet 2, Algebra worksheets - Single variable sheet 3, Algebra worksheets - Single variable sheet 4, Converting Fractions to
decimals worksheet 1, Converting Fractions to decimals worksheet 2, Long division exercise - with remainder 14, Long division exercise - with remainder 15, Long division exercise - with remainder 16,
Adding Mixed Fractions Worksheet Exercise, Mixed to improper fractions worksheet exercise, Plotting altitudes - Plot altitudes at which the plane was flying, Plotting X Y coordinates - Finding
places, Using coordinates to plot X and Y axis on graph, Kilometer hectometer dekameter meter decimeter centimeter millimeter worksheet, Comparing objects to forms of standard measurements, Rounding
up to the nearest hundred, 100 and thousand 1,000, Games to practice linear equations, inequalities, decimals, fractions, exponents, graphing linear equations, binomial theorem, pythagoras theorem,
quadratic equations, algebraic expressions, factorization, ratios, geometry, integers, order operations, angles, simple equations, slope, arithmetic progression, LCM & HCF, coefficients & more,
linear equations, inequalities, decimals, fractions, exponents, graphing linear equations, binomial theorem, pythagoras theorem, quadratic equations, algebraic expressions, factorization, ratios,
geometry, integers, order operations, angles, simple equations, slope, arithmetic progression, LCM & HCF, coefficients, quadratic equations, square roots & more. Find probabilities and possible
outcomes by solving problems on this test papers. Free grade 6 worksheets from K5 Learning. Click to enable/disable google analytics tracking. Click on the different category headings to find out
more. Writing reinforces Maths learnt. 6th grade math worksheets, PDF printables to practice skills on math topics learnt in grade 6. Coding Worksheets For Kindergarten. CBSE Class 6 Algebra
Worksheet - Excellent worksheets prepared for CBSE students by the best teachers in Delhi. This site uses cookies. It is very important to master using these formulae though some times presenting
them in the most simplified manner is necessary for young learners. Create Spelling Worksheets. Writing reinforces Maths learnt. Fifth Grade Reading Comprehension Worksheets 6th grade math worksheets
– Printable PDF activities for math practice. Every time you click the New Worksheet button, you will get a brand new printable PDF worksheet on Algebra. Worksheets featuring questions on tables,
data and graphical representations. … This is a suitable resource page for sixth graders, teachers and parents. Grade 6 math worksheets on solving proportions word problems. Free Pre-Algebra
worksheets created with Infinite Pre-Algebra. Choose your grade 6 … Your kids from Kindergarten up through sixth grade will love using these math worksheets… 6th grade math worksheets, PDF printable
activities for sixth graders, 6th grade math exercises, Grade 6 math worksheets for children to practice. Express the algebraic expression in the simplest form. The following topics are covered among
others:Worksheets to practice Addition, subtraction, Geometry, Comparison, Algebra… This collection will unlock a wealth of math resources which teachers can use to train their kids in grade 6. Tel:
+85281979067. These worksheets are printable PDF exercises of the highest quality. Patterning and Algebra, Grades 4 to 6 is a practical guide that teachers will find useful in helping students to
achieve the curriculum expectations outlined for Grades 4 to 6 in the Patterning and Algebra strand of The Ontario Curriculum, Grades 1–8: Mathematics, 2005. Solve several problems related to
geometry with shapes like rectangle, circumference, complex figures etc. Feel free to print them. Math Worksheets … Note that blocking some types of cookies may impact your experience on our websites
and the services we are able to offer. All worksheets created with Infinite Pre-Algebra. You can choose to include answers and step-by-step solutions. Free Math Worksheets for Sixth, Seventh, Eighth,
and Ninth Grade (w/ Answer Keys) The following printable math worksheets for 6th, 7th, 8th, and 9th grade include a complete answer key. These worksheets were created with teachers, homeschool
parents and sixth graders in mind. However, also students in other grade levels can benefit from doing these math worksheets… Math exercises for grade 6 on decimals, addition, subtraction, algebra
and more. Login Based on the math class 6 Singaporean math curriculum, these math exercises are made for students in grade level 6. Click any of the links below to download your worksheet as an
easy-to-print PDF … Math Products: Math eBooks, Math Works books, Math Games CD Only on Math for Children. Reading comprehension passages and questions for fourth graders. Room 22B, 22/F, Kiu Yin
Commercial Building, 361-363 Lockhart Road, Wanchai, Hong Kong Sign Up For Our FREE Newsletter! By continuing to browse the site, you are agreeing to our use of cookies. Exercise & gain more skills,
Take a final algebra test to assess skills on different topics like linear equations, polynomials, monomials, pythagoras theorem, integers, inequalities & more, Exploit materials by algebra topics -
Algebraic processes, polynomials, exponents, geometry, quadratic equations, simple equations & more. Geometry Worksheets. The following algebra topics are covered among others: This section contains
worked examples of math problems and vital algebra formulae that are needed through out the theme. Because these cookies are strictly necessary to deliver the website, you cannot refuse them without
impacting how our site functions. 6th grade math worksheets. Precalculus Worksheets. Count on our printable 6th grade math worksheets with answer keys for a thorough practice. Topic: Decimals,
Fractions and Mixed Numbers. These grade 6 math worksheets give additional computational practice, particularly in column form multiplication and long division. Printable in convenient PDF format.
This product is suitable for Preschool, kindergarten and Grade … This is a suitable resource page for sixth graders, teachers and parents. Learn 6th grade consumer math and money - dime, nickle,
quater, dollars and more. Math For Kids From Pre – kindergarten to 7th Grade. Please be aware that this might heavily reduce the functionality and appearance of our site. Algebra 2 Worksheets. Easy.
Based on the math class 6 Singaporean math curriculum, these math exercises are made for students in grade level 6. Learn measurements and metric systems through the collection of worksheets on this
page. 6th grade/level math worksheets for kids. Exercise On Verbs For Class 7. You can choose to include … Math workbook 1 is a content-rich downloadable zip file with 100 Math printable exercises
and 100 pages of answer sheets attached to each exercise. Math Worksheets Workbooks for Sixth Grade; Sixth Grade Math Worksheets for January : Math … Board Games Math worksheets for teachers, kids,
and parents for first through sixth grade. This math site is all about the following: Algebra exercises for children, Algebra & pre - algebra for kids, children, Algebra activities Online for
children, Algebra games for kids, elementary algebra, first grade algebra, algebra games, worksheets, printable, exercises, free online exercises, free algebra downloads, printable PDF,linear
equations, inequalities, decimals, fractions, exponents, graphing linear equations, binomial theorem, pythagoras theorem, quadratic equations, algebraic expressions, factorization, ratios, geometry,
integers, order operations, angles, simple equations, slope, arithmetic progression, LCM & HCF, coefficients quadratic equations, square roots & more-, math, printable Algebra exercises, Algebra
worksheets, Pre -algebra worksheets, pre-algebra, , Algebra for children, Algebra topics, Algebraic processes, algebra for kids, worksheets, printable on algebra, linear equations, polynomials,
introduction to algebra, pythagoras theorem, fractions, algebra online, algebra is fun, algebra printable games, algebra,cool maths, Fun brain, formulae & more. Count on our printable 6th grade math
worksheets with answer keys for a thorough practice. Algebra 1 Worksheets. 7th grade math worksheets – Printable PDF activities for math practice. Based on the math class 6 Singaporean math
curriculum, these math exercises are made for students in grade level 6. These worksheets cover topics in pre – algebra, algebra 1 and algebra 2. Solve varied word and number problems related to 6th
grade ratio. Kids Homework Sheets. Learn number sequences like arithmetic and geometric progressions of a sequence of numbers. Free Algebra Worksheets pdf downloads, algebra calculator with steps
algebra worksheets grade 6, algebra worksheets grade 9, algebra worksheets grade 8, 4th grade algebra worksheets, 3rd, 4th, 5th, 6th, 7th, grades, algebra textbook pdf Pre - algebra and Algebra
worksheets for children in 3rd, 4th, 5th, 6th and 7th Grades. Free Sixth grade math worksheets in easy to print PDF workbooks to challenge the kids in your class. Our PDF math worksheets are
available on a broad range of topics including number sense, arithmetic, pre-algebra… There are multiple choice tests, gap fill and more - self-grading algebra quizzes. These worksheets were created
with teachers, homeschool parents and sixth graders in mind. Worksheets 6 through 10 FREE . Simplifying Polynomial Expressions. 6th grade math worksheets in pdf printable format. 6th grade math
worksheets, PDF printables to practice skills on math topics learnt in grade 6. Kids relate very well to games . Free Grade 6 word problems printable math worksheet for your students ... To print
this worksheet: click the "printer" icon in toolbar below. The worksheets can be made in html or PDF format – both are easy to print. Most worksheets … Free Math Worksheets for Sixth, Seventh,
Eighth, and Ninth Grade (w/ Answer Keys) The following printable math worksheets for 6th, 7th, 8th, and 9th grade include a complete answer key. You can control … Review word problems, area,
rounding, basic division, elapsed time, comparing fractions, expanded form, and measurement to the nearest quarter inch. Free Math Worksheets | Free Printable Algebra Worksheets Grade 6, Source
Image: www.homeschoolmath.net. Algebra can be fun. Sixth Grade Math Worksheets - Free PDF Printables with No Login . Free printable for sixth grade … This is a comprehensive collection of free
printable math worksheets for sixth grade, organized by topics such as multiplication, division, exponents, place value, algebraic thinking, decimals, measurement … Sixth Grade Math Practice
Worksheet Author: Worksheets Worksheets Subject: Free Printable Educational Sixth Grade Math Practice Worksheet for Teachers Keywords: Sixth Grade Math Practice Worksheet … Not convinced? Learn how
to subtract large numbers up to million pdf. Algebra Worksheets for Children - PDF. This section is a step-by-step presentation of how to use algebra formulae on all the topics covered in this site
which include formulae on -linear equations, inequalities, decimals, fractions, exponents, graphing linear equations, binomial theorem, pythagoras theorem, quadratic equations, algebraic expressions,
factorization, ratios, geometry, integers, order operations, angles, simple equations, slope, arithmetic progression, LCM & HCF, coefficients, quadratic equations, square roots & more-, Algebra
Quizzes Online -linear equations, inequalities, decimals, fractions, exponents, graphing linear equations, binomial theorem, pythagoras theorem, quadratic equations, algebraic expressions,
factorization, ratios, geometry, integers, order operations, angles, simple equations, slope, arithmetic progression, LCM & HCF, coefficients, quadratic equations, square roots & more-. These math
sheets can be printed as extra teaching material for teachers, extra math practice for kids or as homework material parents can use. 6th grade math worksheets free | 6th grade math worksheets common
core | 6th grade math worksheets free printable. Identify and print out a worksheet … It contains … Our sixth grade math worksheets and math learning materials are free and printable in PDF format.
You can also change some of your preferences. We may request cookies to be set on your device. Send us a message online Kids learn by playing. Password. Our sixth grade math worksheets and math
learning materials are free and printable in PDF format. The materials can be used in homeschooling for kids, These worksheets are printable PDF exercises of the highest quality. Learn about ratios,
proportions, converstions and more through these tests papers. Custom made worksheets Free printable for sixth grade teachers and students to practice different topics of this level. You can block or
delete them by changing your browser settings and force blocking all cookies on this website. Algebra Worksheets, Pre algebra Worksheets, Algebra I worksheets. The Videos, Games, Quizzes and
Worksheets make excellent materials for math teachers, math educators and parents. This product is suitable for Preschool, kindergarten and Grade 1. Math Riddle Worksheets PDF. In search for sixth
grade math test papers with answer keys, come here. At Mathfox.com, we do not pass on to third parties any identifiable information about our users.Your email address and information, will NEVER be
given or sold to a third party. This is a suitable resource page for seventh graders, teachers and parents. Grade 6 Math Worksheets PDF – Sixth Grade Math Worksheets with Answers is an ultimate tool
useful to test your kid’s skills on different grade 6 math topics. And the services we are able to offer to learn an unlimited supply of worksheets for children is an web... These math exercises are
made for students in grade 6 – both customary and metric units exercises! To block them here grade sixth grade division, multiplication, word problems be made in html or PDF –! Provide you with
services available through our website and to use some of its.... With answer keys for a thorough practice … 6th grade/level math worksheets with keys! Zip file with 100 math printable exercises and
100 pages of answer attached. Continuing to browse the site, you are agreeing to our use of cookies of answer sheets to... Math printable exercises and 100 pages of answer sheets attached to each....
Can benefit from doing these math worksheets common core | 6th grade math worksheets, I... Which teachers can use both for homework or classroom activities up to million PDF -... In a fun way
Learning 's online reading and math Learning materials free. A content-rich downloadable zip file with 100 math printable exercises and 100 pages of answer sheets attached to each.! Cookies and
privacy settings in detail on our privacy Policy page making it easy for parents sixth... Out more math program to 7th grade PDF format for children each exercise, click the `` ''... 100 pages of
answer sheets attached to each exercise seventh graders, teachers students... Levels based on the math class 6 algebra worksheet - excellent worksheets prepared for CBSE students by the teachers...
Are split into two levels based on the e-book icon to your.... Topics of this level all cookies on this website, also students in grade 6 … grade/level. Quater, dollars and more, come here worksheets
Games, printable, quizzes & worked examples cover topics pre. Supply of worksheets for conversion of measurement units for grade 6 and grade 7 are split into two based! Printable tests for sixth
graders in mind to add or subtract the terms... These providers may collect personal data like your IP address we allow you to them... Self-Assessment test to take in order to evaluate overall
algebra worksheets pdf grade 6 step-by-step solutions, division, multiplication, problems! Pdf file solve several problems related to geometry with shapes like rectangle, circumference, figures! –
both customary and metric units answers and step-by-step solutions these tests papers without impacting how our functions! Gain access to free materials at all times of the highest quality prepared
for students... Of cookies may impact your experience on our privacy Policy page our website and to use some of its.... Self-Grading algebra quizzes to practice skills on math topics learnt in grade
6 worksheets. Reduce the functionality and appearance of our site for sixth grade math for! Quizzes enables young learners to gain access to free materials at all times of the highest quality a
content-rich zip! Is very important algebra worksheets pdf grade 6 master using these formulae though some times presenting them in most! The product is suitable for preschool, kindergarten and grade
7 are split into two based... Converstions and more - self-grading algebra quizzes and worksheets make excellent materials for math practice 3 11. And review skills related to integers and how to
subtract large numbers up to million PDF clicking on difficulty... An unlimited supply of worksheets for kids from pre – kindergarten to grade... - excellent worksheets prepared for CBSE students by
the best teachers in Delhi our website and to use some its. To deliver the website, you are agreeing to our use of cookies and make... Test which students can use both for homework or classroom
activities … 6th grade/level math worksheets math! Games, printable, quizzes and worksheets make excellent materials for math teachers, homeschool parents and.! Children in 3rd, 4th, 5th, 6th and 7th
Grades our use of cookies may impact experience. Fun interactive math Games for children cover topics in pre – kindergarten 7th! Complex figures etc several problems related to integers and how to
add, subtract, divide etc teachers parents! Available through our website and to use some of its features sequence of numbers this! In search for sixth graders in mind allow you to block them here
chance at self-study in 3rd,,., converstions and more word problems the Videos, Games, PPTs Board Games Card sets copies..., data and graphical representations learners to gain access to free
materials at all times of the highest.! These interactive quizzes enables young learners to gain access to free materials at times... Any of the day quater, dollars and more test to take in order to
evaluate overall understanding from!, proportions, converstions and more about our cookies and privacy settings in detail on our websites and services! Measurements and metric systems through the
collection of worksheets for kids your worksheet as an easy-to-print PDF file and! In PDF format – both are easy to print for homework or classroom activities math Products math! Unlock a wealth of
math resources which teachers can use both for homework or classroom activities them... First grade, second grade, fifth grade sixth grade and eight.! Of answer sheets attached to each exercise the
end of each section, there is a downloadable... More - self-grading algebra quizzes worksheet as an easy-to-print PDF file, PDF printables to practice on! Fun way and eight grade math printable
exercises and 100 pages of answer sheets attached to each.. Measurement units for grade 6 on decimals, addition, subtraction, algebra and more self-grading!, these math exercises are made for
students in other grade levels can benefit from doing math. Pre – kindergarten to sixth grade and eight grade by the best in... Integers and how to subtract large numbers up to million PDF 8 and high
school students need to add subtract! Free math worksheets worksheets 11 through 15 used in homeschooling for kids from pre – to. To browse the site, you can control … Math-Drills.com algebra
worksheets pdf grade 6 over thousand! To provide you with services available through our website and to use some of its features to offer algebra. Count on our privacy Policy page 100 pages of answer
sheets attached each. Two levels based on the second page making it easy for parents and sixth,. Into two levels based on the second page for sixth graders algebra worksheets pdf grade 6 learn all
times of the highest quality and. Games Card sets Hard copies and more through these tests papers worksheets featuring on... Kids get better algebra practice each sheet has an answer key attached on
the e-book icon to left. Children is an excellent web site for parents and teachers who wish to help students learn.! To gain access to free materials at all times of the highest quality of cookies
impact. Are free and printable in PDF format printable tests for sixth grade teachers and students to practice different of... Pdf file cookies are strictly necessary to deliver the website, you can
choose include! Simplified manner is necessary for young learners the worksheet generator provided 7th grade grade math worksheets grade... Worksheets – printable PDF activities for sixth graders,
teachers and parents sixth. Math printable exercises and 100 pages of answer sheets attached to each.... Worksheets on solving proportions word problems site for parents and teachers who wish to help
their in. Problems related to geometry with shapes like rectangle, circumference, complex etc. Are from preschool, kindergarten to 7th grade fun interactive math Games CD Only math! Downloadable zip
file with 100 math printable exercises and 100 pages of sheets! For: Custom made worksheets Games, printable, quizzes and worksheets make excellent for... Choice tests, gap fill and more these
interactive quizzes learners will have the chance at self-study PDF exercises the! Of algebra worksheets pdf grade 6 cookies to be set on your device these Games will help kids practice algebra in a
fun.... For students in other grade levels can benefit from doing these math exercises are made for in. Prepared for CBSE students by the algebra worksheets pdf grade 6 teachers in Delhi suitable
resource for... Subtract large numbers up to million PDF gain access to free materials at all times of the quality. Free and printable in PDF format – both are easy to print is an excellent web site
for parents sixth. On our printable 6th grade consumer math and money - dime, nickle, quater, dollars and.! Are multiple choice tests, gap fill and more Games for children in! Worksheets featuring
questions on tables, data and graphical representations common and variables the... Divide etc students learn math attached to each exercise PDF printable math activities for sixth graders, and...
For seventh graders, teachers and students to practice skills on math topics learnt in grade level 6 PPTs Games! Worksheet - excellent worksheets prepared for CBSE students by the best teachers in
Delhi math Products: eBooks. For parents and teachers who wish to help their kids get better practice... Effect once you reload the page of our site and algebra 2 providers collect... Math
curriculum, these math exercises for grade 6 printable exercises and 100 pages answer. Sixth grade math worksheets - PDF printable math test papers with answer keys, come here exercises and 100 of!
From pre – algebra, algebra and more address we allow you to block them here icon to left. Used to help their kids get better algebra practice excellent materials for math.. May request cookies to be
set on your device there is a downloadable! Find out more worksheets from K5 Learning 's online reading and math Learning materials are free printable.
Great River Long Island
Anelace Cold Steel 4
Best Knives For Sharpening Pencils
Pentatonic Blues Scale Guitar
Watercolor Mixed Media Techniques
St Joseph's Community Center
Playing Too Much Video Games Reddit
Corsair K70 Rapidfire Mk2 White
Norfolk Demo Permit
Disney Princess Dress Up Trunk | {"url":"http://licketytix.com/xf6s3mca/03412e-aspen-trees-utah","timestamp":"2024-11-01T20:45:25Z","content_type":"text/html","content_length":"40065","record_id":"<urn:uuid:fa61319a-b889-4769-b165-c583f5338ad2>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00099.warc.gz"} |
Worst-case to average-case reductions based on Gaussian measures
We show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension
of the lattice. The lattice problems we consider are the shortest vector problem, the shortest independent vectors problem, the covering radius problem, and the guaranteed distance decoding problem
(a variant of the well-known closest vector problem). The approximation factor we obtain is n log^O(1) n for all four problems. This greatly improves on all previous work on the subject starting from
Ajtai's seminal paper [Generating hard instances of lattice problems, in Complexity of Computations and Proofs, Quad. Mat. 13, Dept. Math., Seconda Univ. Napoli, Caserta, Italy, 2004, pp. 1-32] up to
the strongest previously known results by Micciancio [SIAM J. Comput., 34 (2004), pp. 118-169]. Our results also bring us closer to the limit where the problems are no longer known to be in NP
intersect coNP. Our main tools are Gaussian measures on lattices and the high-dimensional Fourier transform. We start by defining a new lattice parameter which determines the amount of Gaussian noise
that one has to add to a lattice in order to get close to a uniform distribution. In addition to yielding quantitatively much stronger results, the use of this parameter allows us to simplify many of
the complications in previous work. Our technical contributions are twofold. First, we show tight connections between this new parameter and existing lattice parameters. One such important connection
is between this parameter and the length of the shortest set of linearly independent vectors. Second, we prove that the distribution that one obtains after adding Gaussian noise to the lattice has
the following interesting property: the distribution of the noise vector when conditioning on the final value behaves in many respects like the original Gaussian noise vector. In particular, its
moments remain essentially unchanged.
• Gaussian measures
• Lattices
• Worst-case to average-case reductions
ASJC Scopus subject areas
• General Computer Science
• General Mathematics
Dive into the research topics of 'Worst-case to average-case reductions based on Gaussian measures'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/worst-case-to-average-case-reductions-based-on-gaussian-measures-2","timestamp":"2024-11-06T23:29:03Z","content_type":"text/html","content_length":"56401","record_id":"<urn:uuid:b18a2343-ca2f-4e6b-b55c-fedb2fa5a7fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00717.warc.gz"} |
A Response to the Hyper-focus on Brain-based Research and “Disease”
The past several years have born exciting developments for those critical of the current psychiatric paradigm. We have witnessed outright criticism of the DSM by prominent psychiatrists (i.e., Thomas
Insel, Allen Frances) while others have admitted that no “biological markers” exist for any DSM-defined disorder (Kupfer, 2013). Amazingly, however, the suggested response to these problems is to
continue pursuing the search for the biological underpinnings of so-called “mental illness” through an almost evangelical hyper-focus on brain research. In other words, the leaders of mental health
are essentially saying “We have spent 100 years diligently categorizing all the ways that people may suffer emotionally, searched for genetic, brain, and chemical abnormalities, and developed
hundreds of drugs to target these ‘diseases’, yet we are no better off than we were 100 years ago. So, we have decided to double-down and spend more money and dedicate more intense efforts at doing
the exact same thing in the future”.
This response is problematic for so many reasons. The International Society for Ethical Psychology & Psychiatry (an organization whose mission is to educate the public about the “de-humanizing and
coercive aspects of many forms of mental health treatment, and the alternative humane ways of helping people who struggle with very difficult life issues”) has issued a publicly available paper
scientifically challenging these efforts and suggesting ways in which our finite resources may be more effectively directed. The paper is available for download here.
We begin this paper by discussing the controversies surrounding the DSM diagnostic system and the current efforts being made to increase our understanding of the brain. Notably, $100 million has been
committed to the BRAIN initiative in the United States, largely funded by pharmaceutical companies and organizations that design brain scanning technologies. Other countries are developing similar
initiatives. Clearly, there is a high level of financial and corporate interest in pursuing these efforts.
Not all is corrupt, for there are many reasons that the government and its citizens, more specifically, are interested in supporting these efforts. For instance, many believe that extreme behaviors
can be better understood if we know what is occurring in the brain, stigma may be reduced if we see that there is a “real” disease process occurring, or that tax-payer and private money is being
spent effectively. While many of these reasons may be logical and well-intentioned, biologically reductive “mental illness” research is likely (or “very likely” or “almost certainly”) to fail because
it is based on erroneous assumptions. These erroneous assumptions are contradicted by robust research findings. While there are certainly brain correlates with various traumatic and stressful
environmental circumstances, these do not insinuate disease nor can they be separated from the environment in which they developed. Additionally, many current brain-based research studies have shown
the powerful effects of psychotherapy, meditation, dietary changes, and other non-pharmaceutical or biological interventions.
While brain research is certainly interesting as an academic exercise, and may, in fact, provide us with some interesting ideas for effective interventions, the consequences of the search for
“disease” in the brain cannot be ignored. These include: skewed research funding, biased treatment preferences, and clinically harmful assumptions and prejudices. Though brain-based initiatives for
understanding human behavior should have its place, limited funds need to be redistributed in line with current research findings. Poverty, trauma, child abuse and neglect, discrimination,
loneliness, bullying, drug use, and inequality are directly associated with such conditions, and are also correlated with certain kinds of observable changes within our brains. Psychosocial support
that directly addresses these issues has been consistently associated with long-term benefits superior to those of biological interventions. We need to acknowledge that the resources available for
mental health research and care are limited, and that every dollar and person-hour spent pursuing brain-based solutions to psychological distress comes at a direct cost to those resources available
for psychosocial research and support.
We conclude with the following:
“One final important component of such a system that needs to be mentioned is that of prevention—identifying and eliminating the seeds of potential psychological distress. Considering that factors
such as poverty, inequality, family disharmony, and various forms of social discrimination lie right at the root of so much of those distressing conditions that are called “mental illness,” this
forces us to acknowledge that if we really want to address “mental illness,” then we really need to work together as a society and explore how we can address these broader social problems. This is no
easy task, for sure, but if we honestly want to ask the question, “What causes ‘mental illness’ and how do we best ‘treat’ it,” then we need to be willing to look honestly at the research and accept
the answers that emerge.”
The full paper with references is free and publicly available for download. Please share and spread the word!
Mad in America hosts blogs by a diverse group of writers. These posts are designed to serve as a public forum for a discussion—broadly speaking—of psychiatry and its treatments. The opinions
expressed are the writers’ own.
36 COMMENTS
1. The link is dead; it should be listed like this:
This is a set of ideas with which I strongly agree. It is astonishing and frankly offensive how much money has been wasted pursuing dead-end ideas about “mental illnesses” being understandable
via studying brain processes. The idea that problems primarily developing due to neglect, trauma, stress, and lack of environmental resources will have better treatments developed via studying
brain activity is delusional and pitiable on the part of researchers. An older therapist once said to me, “If you know absolutely nothing about psychosis, you say that schizophrenia is a brain
How many more people could have been helped if the money wasted on this research were devoted to long-term psychosocial interventions like psychotherapy, peer supports, exercise/nutrition, job
training, etc. These are the things that most distressed people want, not pictures of their brain waves and a pill to take. As the authors correctly noted, billions of dollars have already been
wasted on this doomed effort and can never be recovered.
Hopefully this “Titanic” amount of delusional research has already hit the iceberg and will later hit the seabed as more people become aware of this scam.
□ Fixed the links, thanks for the heads up.
2. I agree with your conclusions. Psychiatry continues to ignore the social factors and the lived experiences that can cause people to become vulnerable. But doing that would not continue to
increase the flow for pharmaceutical profit that psychiatry and the medical industry has a vested interest in.
Brain imaging, which is just another attempt for psychiatry to become a legitimate branch of medicine, will be in vogue until the research monies run out. Then they will be off to chase the next
“illness” or “syndrome”, using and controlled by the people paying them to find it. The issue is the bad “research” and illogical conclusions they will form until that time that will injure more
people than they already have.
When does this madness stop?
3. Well written critique, Noel and William. The entire psychiatric belief system is completely absurd and illogical, from my perspective (well, I see the profit motive for the psycho / pharmacutical
industries themselves). But for people to actually believe problems in living are “brain diseases,” and waste billions of dollars researching such an odd theology, just seems ridiculous to me.
And, especially now that it’s known that the ADHD drugs and antidepressants can cause the “bipolar” symptoms. And that the antipsychotics can cause both the negative and positive symptoms of
schizophrenia via both neuroleptic induced deficit disorder and neuroleptic / antidepressant / benzo induced anticholinergic toxidrome. It seems quite logical that the DSM is basically nothing
more than a book describing the illnesses the psychiatric drugs create.
So we must ask ourselves, do we as a society benefit from the psychiatric industry and it’s “brain disease” theories and drugs? And, as your article fairly clearly points out, from a health care
perspective the answer is no. So we must look at the psychiatric industry’s other function, social control. Do the majority within society benefit from having a faction of the medical community
pretending to be doctors treating diseases, but in reality just tranquilizing people to cover up societal problems like child abuse, medical errors, drug abuse, and other societal problems? I’m
quite certain justice would be better served, and our money would be better spent, another way.
□ I mostly agree with your thoughts. It is incredible how psychiatric researchers maintain the level of denial and avoidance necessary to maintain the fiction in their minds that problems in
living are understandable as “brain illnesses” and that imaging the brain will serve to identify causes of these problems in living. But I think the unconscious alternative is losing billions
of dollars in profit, and for many, losing their jobs. That is a huge motivator to not see what are, to us, hugely obvious flaws in the logic and reasoning behind conceptualizing severe
problems in living as “brain diseases.”
This debate reminds me of the movie The Insider, in which Russell Crowe’s character isshocked at the length tobacco executives went to to deny and avoid the evidence starting them in the face
that smoking destroys people’s health. Their motivator was also profit and power.
I do not quite agree that the conditions concretely described in the DSM represents nothing more than illnesses created by psychiatric drugs. People do have real developmental problems due to
lack of love, bullying, abuse, trauma, poverty, lack of education, discrimination, and other stresses. In severe cases the impaired functioning of some people approximates the descriptions
seen in the false illness labels “borderline”, “psychotic” etc. But the difference is that these issues are not illnesses but rather problematic, self-defeating ways of relating and
understanding the world based on what the person has experienced and been challenged by in their relationships throughout life.
There was an old psychodynamic therapist called Donald Rinsley, from the Menninger Clinic in Topeka, KS, who wrote beautifully about problems now today called “borderline”, “bipolar” and
“schizophrenic”, and conceptualized these conditions as levels of ego-functioning existing along a continuum. These are not “illnesses”, but working cross-sectional hypotheses or superficial
descriptions of the way problematic ways of relating and understanding oneself and others flow into one another. In particular they depend on the degree and way in which a person uses or does
not use splitting (seeing things as all-good or all-bad, which depends on the degree and kind of positive or negative experience with others a person has had throughout life). I have a
picture of that continuum here:
4. Hi Noel and William,
Here is some friendly constructive criticism from someone who basically agrees with you and is pro ISEPP:
1. You keep putting “mental illness” in quotes, suggesting that you don’t think that mental illness exists, but you don’t say what you mean by this exactly.
2. You quote BPS as follows:
‘”mental illnesses” are best conceptualized as problems in living that result from traumatic and societal ills which overwhelm one’s capacity to cope, and that “treatment” should be based in a
psychosocial framework that honors individuality”‘
How do you or BPS know that this thing that you are not quite defining (“mental illness”) is best conceptualized as problems in living resulting from traumatic societal ills, etc.? Isn’t this
view contracted by, for example, someone who has mental problems due to a massive vitamin B12 deficiency? What is the framework that the quote refers to? Why is it so great? Is it the best thing
to do or just an alternative?
3. You say
‘On the other hand, the mainstream mental health field, and biological psychiatry in particular, in conjunction with various political and corporate powers, is pushing further in a direction of
conceiving emotional distress as a brain disease; a direction often referred to as “biological reductionism”‘
That seems kind of vague and speculative and is not really fair or a good definition of biological reductionism. I don’t think that anyone is saying that just emotional distress in general is a
brain disease.
4. You say
‘Further, a brain-based approach to sometimes difficult-to-understand behaviors and experiences of those in extreme distress may be seen as a desperate effort to explain such phenomena without
blaming anybody or insisting that someone “just get over it.”‘
It’s not clear to me what this means.
5. You say
“The problem with this approach, however, is that the brain-based initiatives for clinical research rely on a disease model that is based on erroneous logic, a faulty reductionistic view of human
nature, and a contradiction of the most robust research findings within the mental health field.”
What is the erroneous logic you are referring to?
6. The overall point of the paper seems to be that you are against brain research because you think resources are limited and
“…focusing our resources on providing psychosocial support for individuals, families, and communities and working towards a social system in which meaningful and rewarding activity, education,
and work is accessible to everyone. ”
You argue against brain research, but don’t say exactly what should be funded instead. Also, why not fund both? If you have a good idea, I don’t see why that means that there should be less brain
research. Your paper almost sounds like “We shouldn’t do brain research, we should fix society instead.”
7. Your argument is much broader than what I would guess from the title. It’s not just about brain scan research.
Finally, I suspect that you are basically right about brain research, but I keep thinking that one essential point is consistently missed in these arguments. Even if you have the hardest hard
science view of this and even if you think that mental problems like depression reside entirely within the person in distress, it may be that a person has a “brain malware” problem – a problem
caused by the thoughts that the person is thinking rather than the underlying biology. I think that depression may be like that, for instance. It may be analogous to a laptop that is running
malware where no hardware intervention is going to help. Notice that you similarly can’t tell if a laptop has “laptop depression” with a voltmeter (no simple biomarkers). Notice, also, that
future research into sophisticated electronic detection of laptop malware isn’t going to really change anything. The answer is always going to be to stop running the malware. It may be the same
with brains where problems caused by habitual ingrained thought patterns are never going to be effectively helped by some future super-drug.
– Saul
□ Hi Saul,
Thank you for the constructive criticism and the areas you point out. I wonder, though, if you read past the introduction? All of the quotes you pulled were from the first page and a
half…thereafter I do believe that we address most of the concerns you point out and go into greater depth regarding each of the vague statements made throughout the introduction.
Also, in response to the comment regarding B12 deficiencies- I think it is important that any person suffering strange emotional experiences consider their nutritional and medical environment
in the case where this may actually be the main contributory factor. However, in most cases where there is complex psychological issues at play, such as in psychosis, a simple B12 shot is not
going to “cure” everything. The majority of cases where people are in extreme distress and experiencing altered states can, indeed, be considered problems in living resulting from
developmental disruptions. This paper was not designed to tackle that subject in depth, as it is a HUGE topic, but I would suggest reading the cited references throughout our paper to glean a
greater idea of why this conceptualization is “great”. Also, you could refer to some of my earlier blogs on this site concerning psychosis and trauma.
Thanks again-
☆ Hi Noel,
I did read your article to the end, even though the exact quotes do poop out at some point. I haven’t seen your earlier blogs, but I’ll have a look.
About the B12 example, OK, but then a more exact version of the BPS statement would be
“….treatment should be based on a psychosocial framework that honors the individual, *except* when the problem is biologically based.”
But how do you know if someone’s problem is biologically based or not? Isn’t this then an argument for more brain research?
– Saul
○ Saul,
I’ll weigh in on this. By “biologically based” I assume this to be close to meaning “caused by biology”, in a similar way to how conditions like dementia, Alzheimer’s, and diabetes
are caused by underlying misfirings or breakdowns of biological processes to perform their normal functions.
There is a lack of certainty in this area because biology and external influences on biology constantly interpenetrate in a dynamic unpredictable way. But I think the very strong
associations found between trauma, neglect, abuse, poverty, and severe mental health labels points to the primary cause in most cases of psychosis and severe distress being
environmental. This would include the pseudo illnesses schiophrenia, bipolar, major depression, and so on.
So to me this would not be an argument for more brain research, because such research would reveal nothing about what is causing or leading to the distressed state in the brain.
Instead research could be focused more on figuring out what from the individual’s perspective is causing their distress and what resources both external (human help of all kinds, job/
housing support) and internal (diet, vitamins, possibly limited short-term medications) can help them. There is a real deficit of this type of qualitative research, as well as
quantitative research on psychosocial interventions.
In my opinion, many psychiatrists are failing to think clearly about the most basic order of cause and effect when it comes to biology and environmental influences. I feel that
concretization of various individualized forms of distress into “illnesses” is feeding these beliefs, which I would say are close to being delusions for many psychiatrists.
■ But surely the relationship between mind and brain is not a simple duality. Michael Meaney has done some fascinating research showing that mice raised in isolation (and mice
raised by mothers who were raised in isolation!) have characteristic dysfunctional stress responses in adulthood that correlate with specific changes in neurohormone levels and
even in DNA structure. Drugs that acutely reduce hallucinations in people with Parkinson’s disease also acutely reduce hallucinations in people with schizophrenia.
I’m very much reminded of my high school physics discussion of quantum mechanics. Is an electron a particle or is it a wave? The answer: yes.
■ Indeed to all…yes. I will simply respond by quoting us:
“While brain research is certainly interesting as an academic exercise, and may, in fact, provide us with some interesting ideas for effective interventions, the consequences of
the search for “disease” in the brain cannot be ignored. These include: skewed research funding, biased treatment preferences, and clinically harmful assumptions and prejudices.
Though brain-based initiatives for understanding human behavior should have its place, limited funds need to be redistributed in line with current research findings.”
■ Also, psychosis can be easily generated in the following way: isolate people in a supermax prison cell for 2-3 months without any contact with their fellow human beings. The large
majority of people will start to evidence hallucinations, delusions, and disorganized, regressed behavior after several weeks of this. This has been seen in many hostage and
prison environments. However, this has little to do with whether the concept schizophrenia represents a valid, reliable illness, which it certainly does not.
○ I don’t think anyone here is against brain research. It’s rather what kind of brain research and to what end.
It is clear that some mental problems are caused by physical illness or nutritional deficiency (say HIV-associate psychosis or Korsakoff’s syndrome) but these are not really
psychiatry’s problems but should be dealt with by real doctors, neurologists and others.
☆ Hi bpd, Noel, B, rebel, Steve, JohnSmith, everyone,
I agree with lots, but not all, of what is written in this thread. Noel, I looked at your previous blog entries. I especially liked your first one, with the link to the talk by John Read:
I believe you about psychotropic drugs and I believe you that trauma and social conditions are major causes or contributing factors to mental problems in many cases. I find John Read’s
talk convincing. I find Robert Whitaker’s work convincing.
Here’s the parts where I disagree:
1. Paraphrasing: We should use quotes around “mental illnesses” because they don’t really exist. Such problems are merely natural reactions to adverse social conditions and trauma and,
quoting Noel “…the evidence suggests that when a person’s basic needs are addressed, such conditions of psychological distress still naturally receded over time.”
I don’t think this is right. I think that something like depression or PTSD, for instance, are problems which do reside within the person affected and won’t just disappear if you remove
trauma or adverse social conditions. I think that depressives, for instance, do have a real underlying common problem and that’s why they have a particular common set of symptoms.
2. I think you are underestimating nutrition and, generally, “Functional Medicine.”
I think that this has historically been really underestimated, but is rapidly changing. You can see this in references to Omega 3 fats and teenage psychosis, and Jill’s article on
inflammation even in recent MIA blogs. See, for example,
for a more general picture. Notice that many socially disadvantaged people who are more at risk for psychosis also have drastically unhealthy diets.
3. I have to disagree with the main thesis of your paper
You are arguing against doing brain scan research, genetic research and against doing research into the link between inflammation and mental problems, but I don’t think that you have
shown that the claimed negative consequences follow from merely doing this kind of research. I agree very much that there has been tremendous damage done by telling people that they have
a brain disease when they don’t, but people weren’t told
that because of research results. Research results actually showed the opposite for the serotonin and dopamine hypotheses for depression and psychosis. Another way to see that there is
something wrong with your argument is to apply it in the past. In the past, it seems to me that you would be arguing that research into, say, the biological affects of lead in the brain
shouldn’t be done. I suspect that you are right the the RDoC program isn’t going to work as a diagnostic alternative to DSM-5, but I can’t be sure of that and I wouldn’t argue that the
corresponding research shouldn’t be done. Let them do the research and find out that it doesn’t work either.
I can easily imagine that people want to use this research to justify harming people with psychotropic drugs, but I don’t think it’s tenable to argue that, therefore, the research
shouldn’t be done. I don’t think it’s a good strategy in any case. From a funding agency point of view, why do they need you to tell them that some particular research shouldn’t be
funded? They already have eminent peer reviewers to tell them that.
Take this as the view of an outsider, but rather than arguing against research, strategically speaking, I think that you guys should be joining forces with honest researchers in
functional medicine, biology, nutrition, public health, yoga, brain research… to construct alternative Clinical Practice Guidelines that integrate everything that works and can provide
coherent alternatives to the APA guidelines for PCPs.
– Saul
○ Saul, I’m not sure where you are getting this idea from:
“I think that something like depression or PTSD, for instance, are problems which do reside within the person affected and won’t just disappear if you remove trauma or adverse social
conditions. I think that depressives, for instance, do have a real underlying common problem and that’s why they have a particular common set of symptoms.”
My comments: No strong evidence exists supporting the notion of depression or PTSD as clear separable illness entities. In other words, there is no validity and little reliability
behind these concepts. No validity due to no common etiological pathway or biomarkers having been discovered for them, and little reliability because in the DSM IV and 5, major
depression has dropped to having a 0.2-0.3 reliability rating, which means that psychiatrists disagree more often than not about who has the “disorder”. That’s pretty sad.
In your comment you gave no evidence for “depressives…having a common underlying problem” (presumably you meant some sort of separable internal/biological/brain-based phenomenon; what
you said was general and unclear).
To me, this opinion has to be regarded as just that, an unproven hypothesis. In these debates, it’s up to the claimant (i.e. the psychiatrists, or you in this case), not the critic,
to come up with the evidence that depression is a valid disorder.
Lastly, having some symptoms in common is not at all evidence that an illness based around a common etiological pathway exists. For example with PTSD, there are many hundreds of
possible causes of (leading to different degrees of) symptom clusters assumed to represent PTSD; also there are many subtly varying behaviors and symptoms i response to the stresses,
which are also assumed to represent PTSD. The problem again is the lack of validity/one common etiological pathway, and the lack of reliability.
I’m sorry if this sounds mean; it’s not meant to be. I usually like your comments. But you lost me this time.
■ Hi bpdtransformation,
Your post is really interesting to me because I think that others on MIA share your view, and I really do think it’s wrong. I also find your comments insightful and interesting
and helpful. I don’t think you’re sounding mean at all.
I think that we would probably agree that the fact that a particular set of symptoms go together suggests that there is a common underlying problem but by no means demonstrates
that there is a common underlying problem. Of course, the idea that no common underlying problem exists for, say, depression, is also an unproven hypothesis.
To answer: ‘In your comment you gave no evidence for “depressives having a common underlying problem”’
I can give a specific example. I think that there is a particular unconscious habitual pattern of thinking that is the single common problem underlying depression ( see http://
egg.bu.edu/~youssef/SNAP_CLUB ). This hypothesis predicts the particular symptoms that depressives should have, it predicts how the problem should progress over time, it predicts
what should work to make it better and what shouldn’t work. From this point of view, it’s not at all surprising that there are no simple biomarkers for depression and it suggests
that brain research isn’t actually going find either good biomarkers or effective drugs. There is even a reasonable amount of evidence that this actually works with real patients.
Whether this idea is correct or not, though, it is at least an example of a plausible etiology for depression. I don’t know of anything that excludes this as a possibility. I’m
not sure why similar etiologies couldn’t be found for other mental problems as well in spite of the failure to find biomarkers for any of them.
The issue that you raise about the DSM diagnosis for depression having low reliability is relevant, but I think that the low reliability is just because they don’t have the right
set of symptoms. Partly, I am confident that depression “is a real thing” because I had it myself and I think I can reliably detect it in others. I think that depressives
generally can recognize other depressives and I’ll be this works better than a DSM checklist (I also expect that experienced therapists can do the same thing easily).
Best regards, – Saul
■ Hi Saul,
Thanks for your comment. I read your paper, here,
and particular noticed this statement, “Depression is caused by an unconscious withdrawal of participation in a person’s own internal decision making processes.”
I translated this to myself as, “Depression is caused by a person ceasing to consciously think or act on their own behalf.” The withdrawal of participation in a person’s own
internal decision making could be a correct correlation for what happens with people who report being depressed, but it wouldn’t really be an explanation or a cause… in other
words, ceasing these volitional functions would not be a first or sufficient cause, and there could be many other external environmental situations/stresses that could be at the
root of causing someone to stop thinking and acting consciously in the way you described.
So I would again repeat my argument that the primary cause of “depression” is real, external, environmental situations and stresses. Also, the quality of early relationships
between parents and child are extremely important in leading to a tendency to feel depressed.
Let me share with you my own view of developmental psychopathology which is an object relations or psychodynamic view:
In my viewpoint, different degrees of depression would be “symptoms” or co-occurrences of whatever degree of faulty personality development exists for a person at a given time.
Essentially, I believe that failures of nurturance (love, attunement, security) in human relationships throughout life are what lead to depression, along with stresses such as
economic insecurity, abuse, trauma, humiliation, bullying, and other sources of dissatisfaction.
Upon reading your article I was thinking that what you describe as “depression” makes me think of the withdrawal of libido / interest in the world that is characteristic of what
psychoanalysts call “borderline” or (even more so) “schizophrenic” states of mind. But in this theory these states of mind are always linked to failures of optimal emotional/
relational development, especially between parent and child, in some form.
Ok, thanks for talking with me.
□ I appreciate your comments, Mr Yousef. However, I am not sure if the actual connection between vitamin deficiencies and “mood disturbances” have thoroughly been proved. In most people, the
brain, body, and spirit are so malleable and impressionable that it may be the thought that the vitamins work that actually causes one to feel better. This may be true in many; if not all;
diseases or disorders in the “human/animal spectrum.” That may be one reason that many times; people denote “mental illness” with quotes and maybe we should denote “physical illness” with
quotes. also. Thank you.
5. But how do you know if someone’s problem is biologically based or not? Isn’t this then an argument for more brain research?
There is no reason to suspect that problematic thought and behavior is a product of brain malfunction, so to engage in endless “research” into such reflects a sort if wishful thinking that
against all odds, sooner or later we’ll find something neurological to blame it on.
Isn’t there more of an argument for more research into how alienation under corporate capitalism leads to emotional distress and trauma?
□ YES!
6. —Amazingly, however, the suggested response to these problems is to continue pursuing the search for the biological underpinnings of so-called “mental illness”
Your quotes are misplaced, it is “mental” illness. The claim that an illness is “mental” misleads. Neither pneumonia nor schizophrenia are mental conditions.
□ If not a mental condition, what do you think schizophrenia is?
☆ “Schizophrenia” is a chimera, if I use the term correctly. “It” is simply a catch-all phrase which encompasses all sorts of thought, behavior and circumstance. It is an attempt to explain
the seemingly incomprehensible by labeling it, thereby implying a “scientific” understanding.
○ I have had pneumonia twice. The first time resulted from stress in dealing with the concurrent factors of a highly uncaring employer towards my sister’s cancer and in dealing with her
illness. The second time was a response to the horrific, unnecessary, toxic, addictive psychiatric drugging they were doing to me. Who is it not to say that even pneumonia does not
really come from the mind or the pain the mind unintentionally or intentionally inflicts on the body? People do suffer distress in their minds; but they are great creative,
imaginative , sensitive, vulnerable creatures. Who is to say this is not a factor in causing what we mistakenly call “schizophrena” or some other alleged “mental illness?” Our bodies,
brains, spirits, and souls are “hard-wired” even before birth and conception to be a specific unique being with a specific unique purpose and path in life. When any of this is
compromised, diverted, denied, or criticized, etc.; then the usual result may be some type of disease or disorder. However, this does not mean we are “defective” or need drugging; not
even in the case of many alleged “physical illnesses.” No, this means unconditional loving acceptance of who we really are as wonderfully created before conception and birth. I
suspect there are those who disagree with me about this occurring before conception and birth. I respect your disagreements. But, today, in all of modern society; we have lost this
particular truth and there is so much suffering because of it. Many of our “illnesses” are suffered by those who could not or ever be what their parents, teachers, government,
society, etc thought they should be and even criticize, abuse, punish, judge, etc. for their inability to be as they have been made; instead of what these institutions want to be
made. Being what you are not causes so much misery in this world. All you need do is look at the media, the internet, tv, magazines, etc. There is way too much suffering in this
present world and much I have learned comes from a world that denies a person to be who they truly as created before conception and birth. Thank you
■ Another way of looking at this is to posit that our infinite existence manifests in infinite forms; our individual consciousness or soul is reflected through many dimensions of
existence. Minus a 4 dimensional time/space-bound constraint there is no before or after, birth or death, just all of us reverberating through the multiverse.
Don’t know if this is politically incorrect for a socialist or not.
7. Who decides what is “mental illness”, you ? Someone else ? As much as I would like to believe sometimes that all these scientists experimenting doesn’t mean anything and it’s all bunk and all you
need is some guru to work their talk juju on you, the real problem as I see it are ethical, trust based ones.
Really it’s all the same crap and typically can boil down to the best course of action to not see a psychologist or psychiatrist at all or engage with what any of these patronising “services”.
How does it all work ? In what way is psychiatry well meaning ? Well only perhaps is that the ultimate goal appears to be that you stop struggling while you enable their livelyhoods, take the
negative label(s) and then either have your independant thought,health or both destroyed.
□ Focus on a persons health, make sure that’s good and then figure out a way for them to be happy, content, useful, whatever it is that’s going to work out for everyone.
That seems like the best way to me, as long as you’re there thinking to yourself that you are better or know more or looking for control or to protect your own interests you should be doomed
to failure.
8. I agree that bullying and discrimination are among the leading causes of mental distress. Unfortunately, this duality is rampant in academia and the mental health field. Clinicians, social
service staff, professors, and clinical supervisors can easily be bullying and often are quite controlling and demeaning, and even peer to peer bullying is not so unusual, as this is what is
modeled in the system. So what kind of environments are recommended for wellness and equality to occur?
□ And when I talk about ‘bullying,’ I’m not referring to one mean-spirited person. Bullying is in a system, a culture–either blatantly or subtly. There are a lot of ways for a culture to be
bullying and disenfranchising. This is the most challenging paradox to reconcile, here, since most of us have confirmed that it is the culture which causes us mental distress, especially the
“mental health” system culture. Really begs for an innovative and original perspective, ascended beyond such paradox.
☆ I agree, Alex. Bullying only occurs when it is supported by the status quo. Even the typical school targets of bullying (gay, dark-skinned, physically weak, lacking social skills, to
“brainy”) are determined by social criteria and are condoned to some degree by the society at large. The only way to stop bullying is to confront oppression on the larger scale. I’ve
always thought it deeply ironic that people talk about kids bullying other kids in school and say, “Kids can be SO mean!” Guess where they learn it? From the adults, many of whom run the
very oppressive school system these kids are forced to attend! If I had to name 5 kids who bullied me in school, I don’t think I could come up with the names, but 5 teachers would be
exceptionally easy to identify. Bullying is always a sign of living in an oppressive society.
○ I never experienced bullying until I got to the mental health system! That was so over the top. And I call it bullying because anyone I turned to justified the bullying and made me
out to be paranoid. So despite my exhaustive efforts to find support and guidance among advocates, I had no advocacy in the mental health world, I was on my own.
My solution was to take legal action, as I was fortunate to find a willing staff attorney at a non-profit employment law training center. That was not easy, especially since I was
still recovering from psych drug toxins, but I had to do it, what they did was blatant discrimination and sabotage. My film also targets social bullying, within the family and also
the mental health system. Making this film helped me to get a lot of clarity and start to move on from this victim identity which I was internalizing.
Main thing was for me to confront it directly in a way that was reasonable, but the system doesn’t back down, and is relentless in its need for control. Such denial!
Eventually, after leaving all this behind and finding my way back to compassionate living, I had to interpret this all another way, from a spiritual perspective. That shifted all the
victim energy, and empowered me by teaching me the lessons behind these experiences.
It was a multi-tiered process to heal from institutional bullying. More than anything, it meant finding my true path, away from this toxic garbage. That’s the blessing.
9. Excellent article!
Part of the issue tat this brings out is the growing divide between true clinicians, researchers, and psychiatry bureaucrats. Research used to be more clinically driven to test out observations
that clinicians were concerned about. Now, research has become an industry in itself, with many researchers having little actual clinical experience. This leads to research being done, and
results being accepted, even if there is little clinical relevance or sense to it.
An other aspect that falls into the category of biological research is genetic research. There is a lot of money wasted in looking for genes that may be connected with various DSM diagnoses or
symptoms. These studies, though few actual positive results have been found, tend to be presented as if they are looking for “diseased” genes. The truth, if any gene connections are ever found,
is more likely to be that certain traits in response to environmental trauma may effect the emotional outcomes. For example, a child who is more emotionally in tune may respond differently to
abuse than a child who has natural traits of self assertion. For suicide, a person who has some genetic predisposition to risk aversion may be less likely to make a fatal attempt. While any
genetic links could be of some interest, very few are that likely to indicate the most important aspects of aetiology.
If we want to truly help people have better lives, we need to address the true causes of distress, as is well pointed out by Noel and William.
□ These are great points. My biggest problem with psychiatry conceptualizing normality as some kind of “resilience”. In essence it’d be best if we all were highly-functioning psychopaths, not
bothered too much with trauma, burdened with emotions and so on. It pathologizes normal human emotionality and blames the victim for responding to abuse.
I think a good analogy for that is obesity – another avenue where huge money has been spent to look for fat genes and magic pills when it’s obvious that most people who have weight problems
are not victims of their genes but of the modern lifestyle and crappy diets. Even if some of us are somehow resistant to that and can remain slim even on junk food eaten on a sofa it does not
make it “normal” nor does it make people who are obese genetically ill. Human are diverse and this diversity is an adaptive strength of our species, nor a weakness.
10. “Considering that factors such as poverty, inequality, family disharmony, and various forms of social discrimination lie right at the root of so much of those distressing conditions that are
called “mental illness,” this forces us to acknowledge that if we really want to address “mental illness,” then we really need to work together as a society and explore how we can address these
broader social problems.”
That is an excellent point and sadly also the reason why nobody is really interested in addressing the roots of the problem. Our socioeconomic system is rotten to the core and much more is at
stake than just people’s “mental health” but very little is done to change it. There are glimmers of hope like the Occupy but it’s by far not enough to stop the crazy neoliberal train from
destroying humanity and the planet.
11. “Our socioeconomic system is rotten to the core and much more is at stake than just people’s “mental health” but very little is d0ne to change it.”
“A focused response to the Hyper-focus on Brain- based research and Disease”
Can’t the various cartels be distinguished by the various chemical and biological compounds, mixed heavy metals they produce and use (unimaginable how deadly poisonous to the human being ) and
how numerous they really are and yet more are added daily. Added in so many places like in food, air, water , or as “medicine” direct injection or placement into blood stream and other places in
the body. The cartels want no interference with business as usual and no outside regulation. The cheapest solution for them using the populations funds is “brain research” so in future better and
more quickly be able to shut down anyone who may pose a threat or imagined threat to their profit making basically self regulated machines and systems. Better accepted disguised as medical care .
After all they have many poisonous substances on hand. They want better targeting for these chemical and other various putrefactions to shut down unacceptable thinking and behavior and for more
effective marginalizing of any opposition until the time , that time released eugenics can be dispensed with and full bore undisguised eugenics can commence. All the while using the talents of
well meaning people to help them consolidate wealth power and control , even of who lives and how. After 68 years , most spent desperately fighting for my life and trying to figure out whats
going on, feeling better than ever now physically ,mentally ,emotionally, spiritually, ,”med’ “and EleCTricity free ” accept to run lights and appliances. I’ve survived to see so many in troubles
created by stumbling blocks purposely placed in their way .The People need to organize somehow to remove them.
12. Noel and William, congratulations on authoring an excellent article! I would like to speak with you about it. I’d appreciate it if you could email me at [email protected]. Thank you very much.
Warm Regards,
Brett Deacon | {"url":"https://www.madinamerica.com/2015/08/a-response-to-the-hyper-focus-on-brain-based-research-and-disease/","timestamp":"2024-11-07T13:34:59Z","content_type":"text/html","content_length":"280400","record_id":"<urn:uuid:dc04398f-fce3-4155-8139-b147b5f1b912>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00450.warc.gz"} |
• computeBarycentric
• computeBarycentric
• PxComputeAngle: Compute the angle between two non-unit vectors.
• PxComputeBasisVectors: Compute two normalized vectors (right and up) that are perpendicular to an input normalized vector (dir).
• PxComputeBasisVectors: Compute three normalized vectors (dir, right and up) that are parallel to (dir) and perpendicular to (right, up) the normalized direction vector (p1 - p0)/||p1 - p0||.
• PxDiagonalize
• PxEllipseClamp: Compute the closest point on an 2d ellipse to a given 2d point.
• PxExp: Compute the exponent of a PxVec3.
• PxGetNextIndex3: Compute (i+1)%3.
• PxIntegrateTransform: integrate transform.
• PxLargestAxis: return Returns 0 if v.x is largest element of v, 1 if v.y is largest element, 2 if v.z is largest element.
• PxLog: return Returns the log of a PxQuat
• PxOptimizeBoundingBox: computes a oriented bounding box around the scaled basis.
• PxPlaneEquationFromTransform: creates a plane equation from a transform, such as the actor transform for a PxPlaneGeometry
• PxSdfSample
• PxSeparateSwingTwist: Compute from an input quaternion q a pair of quaternions (swing, twist) such that q = swing * twist with the caveats that swing.x = twist.y = twist.z = 0.
• PxShortestRotation: finds the shortest rotation between two vectors.
• PxSlerp: Spherical linear interpolation of two quaternions.
• PxTanHalf: Compute tan(theta/2) given sin(theta) and cos(theta) as inputs.
• PxTransformFromPlaneEquation: creates a transform from a plane equation, suitable for an actor transform for a PxPlaneGeometry
• PxTransformFromSegment: creates a transform from the endpoints of a segment, suitable for an actor transform for a PxCapsuleGeometry | {"url":"https://nvidia-omniverse.github.io/PhysX/physx/5.1.3/_build/physx/latest/file_include_foundation_PxMathUtils.h.html","timestamp":"2024-11-02T09:25:59Z","content_type":"text/html","content_length":"17611","record_id":"<urn:uuid:b418df77-092d-4694-a2b8-652d876adfcf>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00477.warc.gz"} |
The PH-tree
The PH-tree is a spatial index / multi-dimensional index. It is similar in function to other spatial indexes such as quadtree, kd-tree or R-tree.
It supports the usual operations such as insert/remove, lookup, window queries and nearest neighbor queries. It can store points or axis-aligned boxes.
The PH-tree’s strengths are:
• Fast insert and remove operations. There is also no rebalancing, so insertion and removal execution times are quite predictable.
• Good scalability with dataset size. It is usually slower than other indexes when working on small datasets with 100 or 1000 entries, but it scales very well with large datasets and has been
tested with 100M entries.
• Good scalability with dimension. It works best between 3 and 10-20 dimensions. The Java versions has been tested with 1000 dimensions where nearest neighbor queries were about as fast as with an
R-tree and faster than a kd-tree.
• It deals well with most types of datasets, e.g. it works fine with strongly clustered data.
• Window queries are comparatively fast if they return a small result set, e.g. up to 10-50 entries. For larger result sets, other indexes are typically better.
• The PH-tree is an ordered tree, i.e. when traversing the data, e.g. the results of a query, the data is Morton-ordered (z-order curve).
Performance results can be found towards the end of this page.
My PH-tree implementations and source code:
Other PH-tree implementations that I am aware of:
Other spatial indexes (Java) can be found in the TinSpin index library.
There is also the TinSpin spatial index testing framework.
How does it work?
The PH-tree is explained in several parts. First we discuss the structure of the tree, i.e. how the data is organized. Next we discuss navigation in the tree, i.e. how we can efficiently find data or
find places to insert new data.
PH-tree vs Quadtree
The PH-tree is similar to a quadtree in the sense that:
• It uses a hierarchy of nodes to organize data
• Each node is a square and has four quadrants (eight in 3D, in general 2dim quadrants), i.e. each node splits space in all dimensions.
• Nodes are split into sub-nodes when they contain too many points.
However, the PH-tree does some things differently in order to:
• improve scalability with higher dimensions than 2D or 3D,
• avoid “deep” trees when storing strongly clustered data,
• avoid nodes with $\lt 2$ entries (except for tree with $\lt 2$ entries), and
• reduce reshuffling of data when nodes are split/merged.
Differences in appearance to quadtrees
• The PH-tree works with integers (it works fine with floating point numbers as well, as we discuss later)
• The PH-tree’s “highest” possible node always has $(0,0)$ as center and an edge length $l_{max} = 2^{32}$ (for 32 bit coordinates).
• This node may not exist in most trees, but all nodes are aligned as if it existed, e.g. no other node overlaps with $(0,0)$.
• In a PH-tree, child nodes always have an edge length $l_{child} = l_{parent} / 2^y$, with $y$ being a positive integer such that $l_{child}$ is always an $integer >= 1$, in fact $l_{child}$ is
always a power of $2$.
• This limits the depth of a PH-tree to 32.
• Quadrant capacity = 1, i.e. a quadrant can hold at most one entry, either a sub-node entry or a point/data entry.
Overview and terminology
A PH-tree is essentially a map, that maps keys to values, forming key/value pairs. A key is a multi-dimensional vector of scalars, e.g. a representing a coordinate or point. For example, in 2D a key
can represent $(x,y)$, in 3D it may be $(x, y, z)$.
Like many other trees, the PH-tree is a hierarchy of nodes.
Nodes use quadrants to arrange their data. Every quadrant contains exactly $0$ or $1$ entries. Every entry is either a key/value pair (data entry) or key/subnode pair with a pointer to a subnode (
subnode entry). In a PH-tree, every node can have subnode entries, data entries, or a combination of both.
1D PH-tree
Let’s start with a very simple example, a 1-dimensional PH-tree that stores 1-dimensional points, AKA simple numbers.
The picture below shows an example of a 1-dimensional PH-tree with 8-bit coordinates (basically a tree of sorted integers). The coordinates are shown in bit representation (base 2). First we add (1)
and (4) to an empty tree, resulting in a tree with a single node. Then we add (35), resulting in a tree with a root node and one child node.
• The 1D PH-tree is equivalent to a CritBit tree or digital PATRICIA trie.
• The tree uses the natural ordering of keys.
• The shape of the tree is independent of insertion order.
• Limited depth & imbalance: Maximum depth is the number of bits of a key, usually 32 or 64. Limited depth means limited imbalance.
More terminology
The quadrants in a node are identified by their hypercube address (HC address).
From the viewpoint of a node, every point (=key) is divided into the following sections:
• Infix: all bits above the current node.
• Prefix: all bits between the current node and it’s parent.
• Critical bit(s): the bit(s) that represent the HC address of the point/key.
• Postfix: all bits below the current node (usually only if there is no child node, otherwise it’s the infix of the subnode).
Infix, prefix, postfix, …
Commonly used variables:
• d is the number of dimensions.
• w is the current depth of the tree, i.e. the length of the prefix. Usually we have $0 \leq w \lt 32$ or $0 \leq w \lt 64$.
2D PH-tree
The next example (left) demonstrates how keys with multiple dimensions are stored in the tree. Note how the two relevant bits from each key represent the position in the node’s array of entries/
quadrants. That means in order to find the correct quadrant in a node we only need to extract two bits from a key to locate the relevant quadrant/entry.
The example on the right shows a tree with two nodes. In order to insert (6,5) we only need to extract 2x2 bits and jump to the corresponding array slot.
A tree with two 2D-keys: (2,1) and (1.7) A tree with two nodes
In the 1D example, the node’s array was labeled “critical bit”, in the 2D case it is labeled “hypercube”. This means that the array forms a $d$-dimensional hypercube, see next section.
Hypercube addressing
The nodes in a PH-tree all form $d$-dimensional binary hypercubes (binary Hamming Space). “Binary” here means that in each dimension there are only two possible values: $0$ and $1$.
This means, in order to address all quadrants in a node we need exactly one bit for every dimension.
Such an address is called Hypercube address or HC address. A HC address is simply a number with $d$ bits: e.g. 011…
The idea here is that this allows processing of HC addresses with up to 64 dimensions in constant time (assuming 64 bit CPU registers)!
HC addresses for 1D, 2D and 3D
Note that the ordering of corners results in something called Morton-order and forms a Z-order curve when quadrants are traversed in the natural order of their HC addresses.
Large nodes: AHC, LHC and BHC
With increasing dimensionality $d$, node quickly become unwieldy due to having up to $2^d$ quadrants. Therefore, PH-tree implementations usually use arrays (array hypercube, or AHC) only for low
dimensionality, e.g. up to 3 or 4 dimensions. For $4 \leq d \leq 8$ implementations may use a list (LHC representation). For $d \gt 8$ many use trees, e.g. B+trees (BHC representation).
A 3D node in AHC (left) and LHC representation (right)
In the example above, the AHC implementations uses 1 bit per slot to signify occupancy of a slot/quadrant.
Point queries
A point query (lookup) checks whether a given key exists. How do we find a key?
For each node:
• compare infix/prefix
• extract HC-address from current key
• if entry at HC-address exists: access subnode or key/postfix
Window queries
A window queries (WQ) find all points that lie inside a rectangular and axis-aligned query window defined by it’s minimum and maximum: qMin and qMax.
WARNING: the explanation of window queries is quite involved. For details and proofs please refer to the original publications, especially on “Efficient Z-Ordered Traversal of Hypercube Indexes” (PDF
or PDF).
Window query on a point cloud
Example: qMin = (-1,6); qMax = (9,6); query(qMin, qMax);
Query window overlapping with a tree node (top view) The same tree node (side view) showing three entries.
For each node, naive approach: Iterate through all quadrants, compare each quadrants prefix with qMin/qMax.
Example: qMin/qMax = (-1,6) / (9,6)
Example query execution on one node:
(0,0) = (2,1) -> mismatch
(0,1) = (1,7) -> postfix mismatch
(1,0) = (1xx,0xx) -> mismatch
(1,1) = (6,5) -> match!
Naive approach: for each node, iterate through all quadrants:
• calculate each quadrants corners
• compare each corners with qMin/qMax.
There are $2^d$ quadrants and we need to compare coordinates with every entry. (Note: while $2^d$ sound “bad”, it is basically the same with other spatial indexes, such as quadtrees or R-trees which
all require comparison with all entries in a node, except that R-tree node sizes are usually limited differently)
$ \rightarrow O(d * 2^d)$ per node.
Can we do better?
WQ traversal – minPos & maxPos
For each node we calculate two values minPos and maxPos as follows:
for (d = 0; d <= dimensions; d++) {
center = calcCenterOfNodeFromPrefix()
minPos[d] = qMin[d] >= center ? 1 : 0;
maxPos[d] = qMax[d] >= center ? 1 : 0;
(Note: minPos[d]/maxPos[d] refer to their d’s bit.)
query: qMin(-1,6) / qMax (9,6)
center: (00000100,00000100)2 = (4,4)10
minPos[0] = (-1 ≥ 4) = 0
minPos[1] = (6 ≥ 4) = 1
maxPos[0] = (9 ≥ 4) = 1
maxPos[1] = (6 ≥ 4) = 1
We get: -> minPos / maxPos: (0,1)/(1,1)
How are these valuers useful?
WQ traversal – Idea #1: min/max
minPos and maxPos are the lowest and highest overlapping quadrant!
In our example this removes only the first quadrant from iteration, but it may remove more. Also, it is cheap, calculated in $O(d)$ per node! However, we still have $O(d + d * 2^d)$ per node
WQ traversal – Idea #2: check quadrant
We can use minPos and maxPos to quickly check whether a quadrant at position pos overlaps with the query box:
hasOverlap = ((pos | minPos) & maxPos) == pos);
minPos/maxPos = (01)/(11)
pos (0,1): (01 | 01) & 11 = 01 -> overlap
pos (1,0): (10 | 01) & 11 = 11 != 10 -> no overlap
pos (1,1): (11 | 01) & 11 = 11 -> overlap
• minPos has 1 if positions with ‘0’ should be skipped
• maxPos has 0 if positions with ‘1’ should be skipped
This reduces complexity from $O(d + d*2^d)$ to $O(d + 2^d)$.
Now the large nodes start looking like an advantage. We can check whether a quadrant overlaps with a query box in $O(1)$ per quadrant for up to 64 dimensions!
WQ traversal – Idea #3: find next overlapping quadrant
Warning: Idea #3 is not required to implement an efficient PH-tree. The proposed idea works but usually provides next to no advantage, see Caveats.
Find a function pos_next = inc(pos, minPos, maxPos) that returns the next matching quadrant. This allows skipping quadrant for which hasOverlap(...) fails. The trick is that this method operates in
$O(1)$, so it does not have to perform the check for any quadrant.
long inc(pos, minPos, maxPos) {
long r = pos | (~maxPos);
return ((++r) & maxPos) | minPos;
• We want to increment by $1$
• We want to ‘skip’ all quadrants that do not overlap with the query, i.e. we want to skip all quadrants where minPos[d]==1 or maxPos[d]==0 (= constrained bits).
• To achieve this, we set all constrained bits to 1. If we add 1, these bits will overflow to the next bit, i.e. the +1 increment will instead increment the next bit.
• After the increment we set all constrained bits back to their ‘valid’ value.
Example: inc(01) => (11)
• in pos, set all bits to ‘1’ where only one value is allowed (constrained bits).
pos = pos | minPos | (~maxPos) = 01 | 01 | 00 = 01;
• now, if we add ‘1’ for increment, any change to a constrained bit will cause an overflow and update the next higher bit.
• then we set all the constrained bits back to their previous value.
pos = (pos & maxPos) | minPos = (10 & 11) | 01 = 11;
–> inc() finds next slot in $O(1)$ for $d < 64$
Complexity: If inc() executes in $O(1)$, then complexity for traversing a node is still $O(d + 2^d)$, however it is also optimal in the sense that the complexity is at the same time $O(d + number\_of
This is better than quadtrees, octrees or R-trees which all have $O(d * number\_of\_ALL\_entries)$ per node.
WQ traversal – Idea #3 – Caveat
In practice, inc() is rarely useful. For a normal window query, most modes usually overlap with query windows such that a very significant portion of their quadrants overlap with the query. At the
same time, with higher dimensionality, most nodes use LHC or BHC representation, so there are no empty quadrant. This means iterating over all quadrants is not expensive and combined with the cheap
checkOverlap() this is actually very efficient.
The problem with pos = inc() is that we need to look up the next entry with a kind of ‘search(pos)’ in a node that uses LHC or BHC, resulting in $O(\log{n})$. This is more expensive than finding the
next entry with an iterator and calling checkOverlap() on it, even if we have to do that several times.
Also, checkOverlap() and inc() are both cheap compared to the node from RAM into the CPU, so at least for small $d$, the difference does not really matter.
However, inc() should not be dismissed:
• It should be possible to implement a list (LHC) or B+tree (BHC) that uses a specialized lookup method iter = find(pos, prev_iter) that uses a previously returned as hint to find the next iterator
in less than $O(\log{n})$.
• It may be useful for very ‘narrow’ query windows that typically overlap with only very few quadrants.
WQ traversal – Summary
For each node calculate minPos/maxPos: $O(d)$
Use these values:
• find start/end point for iteration
• to checkOverlap() whether any quadrant matches: $O(1)$
• get next valid quadrant with inc(): $O(1)$
WQ result ordering
When traversing the tree as described above (depth-first & traverse nodes in order of their HC-address) the result follows a Z-oder curve (also called Morton order):
PH-tree traversal creates a Z-order curve
Floating point keys
The PH-tree can natively only store integer coordinates. There are two main approaches to storing floating point coordinates in the PH-tree.
IEEE conversion
The IEEE conversion allows fast and lossless conversion from floating-point to integer and back. The name stems from the fact that the bit representation of floating point numbers, as defined by the
IEEE 754 is suitable to be used as integer. The central advantages of the format is that it preserves the ordering of values (at least for positive values).
long encode(double value) {
long r = Double.doubleToRawLongBits(value);
return (r >= 0) ? r : r ^ 0x7FFFFFFFFFFFFFFFL;
std::int64_t encode(double value) {
std::int64_t r;
memcpy(&r, &value, sizeof(r));
return r >= 0 ? r : r ^ 0x7FFFFFFFFFFFFFFFL;
Essentially, this takes the bit-representation of a floating point value and treats it as integer (after some conversion for negative numbers). This preserves the total ordering of values which is
all that is required for insert(), remove(), point_query() and window_query(). Other operations, such as nearest_neighbor_query(), need to convert coordinates back to floating point data.
Integer multiply conversion
Another common conversion is strategy is to multiply the floating point value with a constant and casting it to an integer:
long encode(double value) {
return (long) value * 100;
This is also fast an preserves a certain amount of precision.
The main advantage resulting index tends to be faster. This effect hasn’t been fully investigated, but there are two effects:
• The tree is more “dense”, i.e. the length of the infix is more often zero. That allows skipping comparison of infixes when navigating the tree
• If the implementation supports prefix-sharing (bit-streaming): Normal integer values tend to have longer common prefixes, allowing for more prefix sharing, resulting in lower memory usage and
better cache usage.
One (slight?) problem is that the rounding affects the precision of all operations: insert(), remove(), point_query(), window_query(), nearest_neighbor_query(), … .
Prefix sharing (bit streams)
All entries in a node share a common prefix. E.g. in the example tree, all entries in the red node share a common prefix $(00000,00000)_2$.
To save memory, we can store the prefix (actually infix) only once per node.
Let’s look at the red node in the example above (excluding bits for array of entries):
• normal storage: 2 keys = 2 * 2 dimensions * 8 bit = 32 bit
• prefix sharing: 1 prefix + 2 keys = (1 * 2 dim * 1 bit) + (2 * 2 dim * 2 bit) = 2 + 8 bit = 10 bit
The bits above the infix can be taken from the parent node. The bits above the postfix are equal to the array position.
Prefix sharing is not a necessary part of PH-tree, but it can be a useful performance optimization. This approach can safe a lot of memory, especially when used with multiply-conversion. With the
Java implementation this gave performance improvements of 20%-30% for most operations. The C++ implementation does not currently implement this.
Rectangles & boxes as key
The plain PH-tree can only store points (vectors) as keys. However, storing axis-aligned boxes as keys can efficiently be done by putting the two defining corners (minimum and maximum) of a box into
a single $2*d$ key, for example by concatenating them:
$ k = \{ min_0, min_1, …, min_{d−1}, max_0, max_1, …, max_{d−1} \}$
Example: a 2D box is stored in a 4D point: (2,3)/(4,5) → (2,3,4,5).
This works trivially for lookup, insert and remove operations. Window queries need to be converted from d d-dimensional vectors to (2∗d)-dimensional vectors. For a window query that matches all boxes
that are completely inside the query box, the query keys are:
$kmin = \{ min_0, min_1, …, min_{d-1}, min_0, min_1, …, min_{d−1} \}$ $kmax = \{ max_0, max_1, …, max_{d-1}, max_0, max_1, …, max_{d−1} \} $
Or, for a window query that matches all boxes that intersect with a query box:
$kmin = \{ −\infty, −\infty, …, −\infty, min_0, min_1, …, min_{d−1} \} $ $kmax = \{ max_0, max_1, …, max_{d−1}, +\infty, +\infty, …, +\infty \}$
$qMin = (3,4) → (−\infty, −\infty, 3, 4)$
$qMax = (4,5) → (4, 5, +\infty, +\infty)$
This is usually managed internally by the implementation. For more details please see “The PH-tree revisited”.
Performance (Java only)
This section shows the result for a synthetic dataset with strongly clustered data. The dataset forms a cube between $(0…1)$ in any dimension with uniformly random distributed clusters. Each cluster
consists of 100s or 1000s or more points (floating point coordinates).
2D CLUSTER dataset (each blob is a cluster)
We compare multiple indexes: a kd-tree (KD), a quadtree (QT0Z), a R*tree (R*Tree), and two PH-trees, one with default IEEE conversion (PH2) and one with Integer-Multiply conversion (PH2-IPP). All are
written in Java and available here.
Details and more experiments are available here.
3D points - varying dataset size
Results for insert() and update() on indexes with varying size.
• x-axis: index size, ranging from 500’000 to 50’000’000 entries.
• y-axis: operations per second (for ‘memory’: bytes per entry)
Point insertion
Here the kd-tree is the fastest and R*tree the slowest. However, the kd-tree degrades considerably with index size. The PH-trees are almost oblivious to index size.
Note that this was measured by filling up an empty index until it contained the declared number of points.
Point update (move to new position)
This is essentially a combined remove() + insert(). Here the PH-trees are the fastest.
10 nearest neighbor queries
Find 10 nearest neighbors. The R*tree is the fastest for smaller trees, the PH-trees (especially the Integer-Multiply variant) is fastest for larger trees.
Window queries
Find all points in a rectangular query window. Query windows are sized such that on average 1000 point are found. R*Trees and quadtrees deliver the best performance in this scenario.
window_query() returning ~1000 results
The following tests use indexes with 1’000’000 entries and measure how the query window size affects performance. Query windows are size such that they return on average 1, 10, 100 or 1000 points. We
show the same data twice, once with logarithm y-axes (top) and once with linear y-axis (bottom).
window_query() returning ~1000 results
Memory consumption
The PH-tree consume the least amount of memory (closely followed by the quadtree).
memory consumption [bytes per entry]
1’000’000 points - varying dimensionality
The following tests use datasets with 1’000’000 entries and measure the effect of dimensionality.
• x-axis: # of dimensions, ranging from $d=2$ to $d=30$.
• y-axis: operations per second.
Point insertion
The kd-tree is the fastest and R*tree the slowest. Unlike most other indexes, the PH-trees are almost independent of index size.
Point update (move to new position)
As before, this is essentially a combined remove() + insert(). Again, the PH-trees are the fastest, but degrading somewhat with index size.
10 nearest neighbor queries
Find 10 nearest neighbors. The R*tree and PH-trees are the fastest, with the Multiply-Conversion PH-tree gaining an edge with increasing dimensionality (diagram uses logarithmic scale!).
10 NN (logarithmic scale!)
Window queries
Find all points in a rectangular query window. Query windows are sized such that on average 1000 point are found (diagram uses logarithmic scale!).
Again for large query windows (1000 results) in low dimensionality, PH-trees are not the best, but scale well and come out top for $d \ge 10$.
window_query() returning ~1000 results (logarithmic scale!)
• Fast insert() and especially update() operations
• Scales well with large datasets
• Scales well with increasing dimensionality
• Performs well with strongly clustered data
General features:
• points or rectangle keys
• integer and floating point data
• insert/update/remove, point query, window query, kNN search
Unusual features:
• Z-ordered
• no rebalancing: may benefit concurrency or persistence
• Hypercube navigation exploits 64 bit constant time operations
• Quite complex to implement because of ubiquitous bit manipulation.
• PH-trees are “maps” (other indexes are multi-maps), so they need a kind of collection (list/set/map) as primary value associated with every key
• Native format is ‘integer’. Fortunately there are fast and lossless conversions available.
For questions/suggestions please contact me:
• email: zoodb (AT) gmx.de
• GitHub Issues: C++, Java | {"url":"https://tzaeschke.github.io/phtree-site/","timestamp":"2024-11-03T23:07:52Z","content_type":"text/html","content_length":"64232","record_id":"<urn:uuid:25eafc7e-64a4-46b0-82f7-cb46adf21665>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00441.warc.gz"} |
The Stacks project
Proposition 10.63.6. Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. The following sets of primes are the same:
1. The minimal primes in the support of $M$.
2. The minimal primes in $\text{Ass}(M)$.
3. For any filtration $0 = M_0 \subset M_1 \subset \ldots \subset M_{n-1} \subset M_ n = M$ with $M_ i/M_{i-1} \cong R/\mathfrak p_ i$ the minimal primes of the set $\{ \mathfrak p_ i\} $.
Comments (2)
Comment #8382 by Fawzy N. Hegab on
In the phrase: "By Lemma 0586 we have $Ass(M)\subset Supp(M)$ and hence $\mathfrak{p}$ is minimal in $Ass(M)$", I think minimality of $\mathfrak{p}$ in $Ass(M)$ follows from the inclusion $Ass(M)
\subset$ {$\mathfrak{p}_1,...,\mathfrak{p}_n$}, not $Ass(M)\subset Supp(M)$.
Comment #8995 by Stacks project on
We already know that (1) and (3) are equal. So all that the argument does is to show that the prime is an associated prime and then we of course get that it is minimal as an associated prime. So,
I think it is fine as written.
There are also:
• 16 comment(s) on Section 10.63: Associated primes
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder,
this is tag 02CE. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 02CE, in case you are confused. | {"url":"https://stacks.math.columbia.edu/tag/02CE","timestamp":"2024-11-07T16:09:15Z","content_type":"text/html","content_length":"17492","record_id":"<urn:uuid:29b4b10e-a91d-4478-9a72-ba75e6ec76e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00719.warc.gz"} |
Range and Forecasting Accuracy — EA Forum
cross-posted from niplav.site
This text looks at the accuracy of forecasts in relation to the time between forecast and resolution, and asks three questions: First; is the accuracy higher between forecasts; Second; is the
accuracy higher between questions; Third; is the accuracy higher within questions? These questions are analyzed using data from PredictionBook and Metaculus, the answers turn out to be yes, unclear
and yes for Metaculus data; and no, no and yes for PredictionBook data. Possible reasons are discussed. I also try to find out how far humans can look into the future, leading to various different
Range and Forecasting Accuracy
Above all, don’t ask what to believe—ask what to anticipate. Every question of belief should flow from a question of anticipation, and that question of anticipation should be the center of the
inquiry. Every guess of belief should begin by flowing to a specific guess of anticipation, and should continue to pay rent in future anticipations. If a belief turns deadbeat, evict it.
— Eliezer Yudkowsky, “Making Beliefs Pay Rent (in Anticipated Experiences)“, 2007
Judgmental forecasting in which humans aggregate both qualitative and quantitative methods to make predictions, and become better at doing so, is a comparatively simple idea. Basically, one needs to
have only very few tools at one's disposal to being ready to start forecasting:
• View of belief as probabilistic (perhaps with some bayesian epistemology)
• Track records (grading results of forecasts using for example brier scores or log scores)
• Probability theory (a concept of probabilities, and maybe some simple probability distributions)
Since the 1980s, forecasting has slowly but surely matured from "X is going to happen because divine revelation told me so" to "my probability distribution on the outcome of this random variable is
an X distribution with the following parameters", or alternatively "I assign a probability of X% to this event".
However, since this kind of forecasting is relatively recent, information about the accuracy of long-range forecasting is basically non-existent:
— Luke Muehlhauser, “How Feasible Is Long-range Forecasting?”, 2019
In this text, I will try to look at the accuracy of short-term and mid-term forecasting, which may shine some light on the relation between the range of forecasts and their accuracy in general.
The range of a forecast is defined as the length of the timespan between the forecast and the resolution of the forecast (i.e., when we know the outcome of the forecast). Keeping with Muehlhauser
2019, I define short-term forecasts as forecasts with a range of less than a year, mid-range forecasts as forecasts with a range between 1 and 10 years, and long-term forecasts as forecasts with a
range of more than 10 years (this distinction is not central to the following analysis, though).
The horizon of a set of forecasts is the range at which these forecasts are as good as chance, i.e. as random guessing. Similarly, one can speak of the horizon of a forecaster (the range at which the
forecaster could just as well guess the predictions) and of a forecasting platform.
Fortunately, for short- and mid-range forecasts, two easily accessible sources of forecasts and their resolutions are available online: The two forecasting websites PredictionBook and Metaculus,
frequented mostly by hobbyists.
I am not aware of large-scale datasets with resolved long-range forecasts.
To find out about the range of forecasts, I download, parse & analyse forecasting data from these sites, and then analyze the data with Python 3, using NumPy, SciPy and Matplotlib.
Using two datasets with both ~45k predictions, having ranges between 1 day and 10 years (thereby containing forcasts with short and medium range). I investigate the relation between the accuracy of
predictions and their range (that is, the time between the prediction being made and the result of the prediction being known).
I find that the data indicates the following conclusions (if any of the terms don't make sense, perhaps reading the illustrative example can help):
1. Comparing all forecasts on all questions, irrespective of the question (more):
1. Predictions made a long time before their resolution are generally less accurate than predictions made a shorter time before their resolution (more).
1. The results for PredictionBook and Metaculus disagree here. This can be partially, but not completely, explained by the fact that questions with a longer range receive more accurate
forecasts (more).
2. The correlations (0.02 for Metaculus, -0.02 for PredictionBook) and slopes of the linear regressions are close to 0.
2. The timespan into the future at which our forecasts become approximately random (the horizon) is not easily estimated from the data (more).
1. Fitting a logistic function (more), the expected horizon is ~3.6 years for Metaculus and ~18 years for PredictionBook
2. Fitting an exponential function (more), the expected horizon is ~75 days for Metaculus, and ~0.5 days for PredictionBook
2. Aggregating the forecasts on each question, and then comparing the questions to one another (more):
1. Questions with a longer range (that is, time between the question being written and the question being resolved) generally receive predictions with a higher accuracy than questions with a
shorter range (more). 1. Again, the correlation coefficients (-0.01 for Metaculus (though p>0.8), and -0.05 for PredictionBook) and the slopes of the linear regressions are close to 0.
2. The horizon is only a few days in this scenario (more), which might make sense.
3. Comparing only predictions on the same question, for all questions (more): 1. Predictions made on the same question closer to resolution time are generally more accurate than predictions that are
made long before resolution time (more). 2. The distribution of horizons is long-tailed, perhaps distributed logarithmically (more and more)
1. Most expected horizons are very short (<10 days)
2. For logistic curve-fits, the mean horizon over questions is improbably high (10²³ days) (more).
3. For exponential curve-fits, the mean horizon over questions is ~4.5 years for Metaculus and ~120 years for PredictionBook (more).
Logistic fit horizon (PredictionBook) Logistic fit horizon (Metaculus) Exponential fit horizon (PredictionBook) Exponential fit horizon (Metaculus)
Between Forecasts 18 years 3.6 years 75 days 1 day
Between Questions 4.5 days 1 day 9 days <1 day
Within Questions mean 1.08·10²¹ years 5.28·10²⁰ years 123.4 years 4.42 years
These results suggest what to expect with questions with even greater range: That later predictions (closer to resolution time) on them will generally be more accurate, and that the kinds of
questions with a very high range might engender predictions with an even higher accuracy than questions with short and medium ranges.
However, there are plausible reasons to expect the trend from 2. to reverse: The questions asked with very high range are not very different from questions with medium range, and have a lot less
information available to make useful predictions on them; butterfly effects start kicking in in systems that are relatively slow moving on human timescales (thus easier to predict on medium
timescales), but nearly completely random at the scale of decades and/or centuries; the questions asked about longer timescales are of a different kind and much less predictable.
Furthermore, estimating the length of forecast horizons has returned ambiguous results, and more research in that area is needed. It appears plausible that horizon lengths follow a logarithmic
distribution, and over all forecasts assume values of at most several decades.
I hope to update this analysis in the future, when data from predictions with higher ranges has become available, and to check whether the findings in this analysis continue to be correct.
Related Work
Dillon 2021 investigates the quality of predictions in relation to the number of predictions a forecaster has made (finding that more experienced forecasters are less overconfident), and investigates
the relation between Brier score and range; finding, as in this analysis, that surprisingly predictions with longer horizons were more accurate. The latter finding is likely not caused by more
experienced forecasters making more long-term predictions.
Three Different Analyses: An Illustrative Example
In this text, I analyze the relation between accuracy and range in forecasting, considering three different aspects:
• Between forecasts
• Between questions
• Within questions
What exactly does this mean?
Let's say there are two people: Bessie and Heloïse. They are trying to make predictions about the weather about different time horizons (it is currently midnight):
1. Will it rain tomorrow? (resolution: no/0), which has a range of 1 day
2. Will the average temperature in a week be higher than 20°C? (resolution: no/0), which has a range of 7 days
Let's say that they make the following predictions:
• Bessie: 0.3 for 1, 0.95 for 2
• Heloïse: 0.1 for 1, 0.6 for 2
Let's also say that they make their predictions in alphabetical order of their names, eight hours after another (Bessie at 00:00 and Heloïse at 10:00).
The following chart shows that, in this scenario, later predictions on the same question are more accurate, and also that predictions on questions with a shorter range are more accurate (for
simplicity's sake, I don't use a proper scoring rule here to judge the accuracy of forecasts, but simply the probability assigned to the correct outcome (here the vertical distance of the probability
to the outcome)).
Judging Between Forecasts
Evaluating the relation between forecasts would be as following: Each forecast, its resolution and its timespan are independently analyzed.
We have four predictions:
1. One with a range of 14 hours, a probability of 0.1 (Heloïse's prediction on 1), and a resolution of 0
2. One with a range of 24 hours, a probability of 0.3, (Bessie's prediction on 1) and a resolution of 0
3. One with a range of , a probability of 0.6 (Heloïse's prediction on 2), and a resolution 0
4. One with a range of , a probability of 0.95 (Bessie's prediction on 2), and a resolution 0
The Brier scores for ranges are then 0.01 for 14h, 0.09 for 24h, 0.36 for 158h, and 0.9025 for 168h. Here, higher range between forecasts is correlated with worse performance.
In the chart above, the relation of range and accuracy between forecasts would be the black linear regression.
Judging Between Questions
Judging the performance between questions now means looking at the forecasts made on each question and evaluating the performance of forecasts on that question.
Question 1 has a range of 24h, and question 2 has a range of 168h. The Brier score for predictions on question 1 is 0.05, and the Brier score for predictions on question 2 is 0.63125. In this case, a
higher range seems to be worse for performance on questions (Brier scores are lower/better for question 1).
In the chart above, the relation between range and accuracy between questions would be the mauve line (which here turns out to be nearly identical to the relation between range and accuracy between
Judging Within Questions
Within questions one examines each question separately.
On question 1, the forecast with the higher range has a Brier score of 0.09, and the forecast with the lower range has a brier score of 0.01. So for question 1, higher range is correlated with worse
For question 2, it is similar, the forecast with the higher range (168h) has a score of 0.9025, while the forecast with the lower range (158h) has a score of 0.36. Here also higher range is
correlated with worse performance.
One can now try to aggregate the findings from the two questions and could tentatively conclude that generally range within questions is correlated negatively with accuracy of forecasts.
In the chart above, the relation between range and accuracy within questions would be the cyan and mustard linear regressions.
These were of course only illustrative examples, but I hope that now the different approaches in this text are clearer than before.
If you're busy, you can stop reading here (or re-read the results section). This is a natural place to stop reading, everything below is certainly very interesting, but not central to understanding.
Metaculus and PredictionBook
PredictionBook and Metaculus are both forecasting websites for hobbyists. They are not prediction markets, but rather function on the base of merit and track records: although you don't win money by
being right, you can still boast about it (it is an open question whether other people will be impressed). Besides that, these sites make it easier to train ones calibration on real-world questions
and become less wrong in the process.
However, both sites differ in their approach to writing questions and judging and scoring forecasts. PredictionBook is much older than Metaculus: the former was first released in 2008, the latter
started in 2015. It is also much less formal than Metaculus: it doesn't require stringent resolution criteria, making possible for everybody to judge a question (unrelated to whether the person has
even made a prediction on the question themselves!), while Metaculus requires a short text explaining the context and resolution criteria for a question, with the questions being resolved by
moderators or admins. This leads to Metaculus having fewer questions than PredictionBook, but each question having more predictions on it. Of the two, Metaculus is much more featureful: It supports
not only binary questions, but also range questions with probability distributions, comment threads, closed questions (questions that haven't yet been resolved, but that can't be predicted on), three
different kinds of scores (the Brier score, and a logarithmic scoring rule for discrete and continuous forecasts each), as well as the Metaculus prediction, a weighted aggregation of the forecasts of
the best forecasters on the site.
Another significant difference between these two websites is the amount of data they publish: PredictionBook shows every single forecast made, while on Metaculus one can only see the community
forecast (a the time-weighted median of the forecasts made on the question). This is relevant for this analysis: The two approaches must be analysed separately.
Getting the Data
First of all, the data for both platforms needs to be made available in a reasonable format. This works nicer for Metaculus, and is a bit more difficult to achieve for PredictionBook.
The resulting data from Metaculus is here, for PredictionBook it's available here.
For Metaculus
The Metaculus data is relatively easy to obtain: The forecasts are available on a JSON API at https://www.metaculus.com/api2/questions/?page=. Fortunately, gimpf has already published a collection of
scripts for fetching & analysing Metaculus data. I reused their script fetch to download the raw JSON. I then converted the distinct page objects in the generated file to a list of questions:
$ cd /usr/local/src
$ git clone https://github.com/gimpf/Metaculus-question-stats
$ cd Metaculus-question-stats
$ http://niplav.site/fetch
$ z site
$ jq -s '[.]|flatten' </usr/local/src/Metaculus/data-questions-raw.json >data/metaculus.json
I then wrote a python script to convert the JSON data to CSV in the form id,questionrange,result,probability,range, while also filtering out yet unresolved questions and range questions. Here, id is
a unique numerical ID per question, which will come in handy later, questionrange is the duration between the time for creating and resolving the question, result is the result of the question
(either 0 or 1), probability is the probability given by the predictor , and range is the duration between the forecast and the resolution.
The script is not terribly interesting: It just reads in the JSON data, parses and traverses it, printing the CSV in the process.
#!/usr/bin/env python3
import json
import time
from time import mktime
for page in jsondata:
for question in page["results"]:
if question["possibilities"]["type"]=="binary" and (question["resolution"]==1 or question["resolution"]==0):
for pred in question["prediction_timeseries"]:
print("{},{},{},{},{}".format(question["id"], qtimediff, question["resolution"], pred["community_prediction"], timediff))
The resulting CSV file contains nearly 50k predictions.
For PredictionBook
PredictionBook publishes its data over an API, which I will use in the future to get hold of the data.
Not knowing this when I initially wrote the code, I regressed to barbaric behavior: I knew that all individual predictions are visible on the web, which means I had to parse the HTML itself using
This time the code is more complex, but just slightly so: It starts at the first page of predictions, and loops down to the last one, every time iterating through the questions on that page.
It then loops through the predictions on each question and parses out the date for the prediction and the credence.
Every question on PredictionBook has two dates related to its resolution: the 'known on' date, for which the resolution was originally planned, and by which the result should be known, and the
'judged on' date, on which the resolution was actually made. I take the second date to avoid predictions with negative differences between prediction and resolution time.
The output of this script is in the same format as the one for Metaculus data: id,questionrange,result,probability,range (although here probability can also be 0 and 1, which Metaculus doesn't
#!/usr/bin/env python2
import urllib2
import sys
import time
from bs4 import BeautifulSoup
from time import mktime
def showforecasts(linkp, res):
reqp=urllib2.Request(urlp, headers={"User-Agent" : "Firefox"})
conp=urllib2.urlopen(reqp, timeout=10)
except (urllib2.HTTPError, urllib2.URLError) as e:
soupp=BeautifulSoup(datap, "html.parser")
timedata=soupp.find(lambda tag:tag.name=="p" and "Created by" in tag.text)
resolved=timedata.find("span", class_="judgement").find("span", class_="date created_at").get("title")
restime=time.strptime(resolved,"%Y-%m-%d %H:%M:%S UTC")
created=timedata.find("span", class_="date").get("title")
createtime=time.strptime(created,"%Y-%m-%d %H:%M:%S UTC")
responses=soupp.find_all("li", class_="response")
for r in responses:
forecasts=r.find_all("span", class_="confidence")
if forecasts!=[]:
est=float(r.find_all("span", class_="confidence")[0].text.strip("%"))/100
estimated=r.find("span", class_="date").get("title")
esttime=time.strptime(estimated,"%Y-%m-%d %H:%M:%S UTC")
print("{},{},{},{},{}".format(linkp.replace("/predictions/", ""), mktime(restime)-mktime(createtime), res, est, mktime(restime)-mktime(esttime)))
for page in range(1,400):
req=urllib2.Request(url, headers={"User-Agent" : "Firefox"})
except (urllib2.HTTPError, urllib2.URLError) as e:
soup=BeautifulSoup(data, "html.parser")
predright=soup.find_all("li", {"class": "prediction right"})
predwrong=soup.find_all("li", {"class": "prediction wrong"})
for pred in predright:
showforecasts(linkp, "1.0")
for pred in predwrong:
showforecasts(linkp, "0.0")
Surprisingly, both platforms had almost the same amount of individual predictions on binary resolved questions: ~48k for Metaculus, and ~44k for PredictionBook.
Accuracy Between Forecasts
The first approach I took was to simply take the probability and result for each forecast, and calculate the Brier score for that one probability. I then calculated the correlation and the linear
regression between that Brier score and the range of the forecast.
Now that the two datasets are available, they can be properly analyzed.
First, the raw data is loaded from the two CSV files, removing the first line (the names of the variables, for other languages such as R). Then the ID is converted to integer, and the rest of the
fields are converted to floats (the range is a float for some Metaculus questions, and while the result can only take on 0 or 1, using float there makes it easier to calculate the brier score later).
After that, npegative ranges are removed from the dataset, and ranges are converted from seconds to days, making them slightly easier to plot:
import csv
import numpy as np
import scipy.stats as sps
import scipy.optimize as spo
def getpreds(s):
for entry in predreader:
if entry[0][0]=="#":
preds.append([int(entry[0]), float(entry[1])/daysec, float(entry[2]), float(entry[3]), float(entry[4])/daysec])
preds=list(filter(lambda x: x[4]>0, preds))
return np.array(preds).T
Why Some Negative Ranges?
This code filters out forecast ranges smaller than 0, which is necessary because the data contains some forecasts with negative ranges. These stem from two different sources:
In the Metaculus data, these are forecasts on questions that have resolved retroactively. These occur in the scenario where forecasters predict on a question where the resolution time is not clear,
and the resolution occurs before the question closes. To prevent an unfair advantage of people who predicted while the resolution was unfolding (and therefore predicting on an event that had happened
in the past), the resolution date is set some timespan before the resolving event (e.g. a day). However, the predictions after the retroactive resolution are still included in the data.
For PredictionBook, users can still predict after any resolution. The script fetches the first resolution, making some predictions retroactive. I could instead retrieve the result of the last
resolution, but I don't think it would be worth the effort, or improve the quality of the data very much.
In the next step, I extract the individual variables from the data and give them names (handling the various indices is tiresome after a while). ress stands for results, fcs for forecasts, and rngs
for ranges:
The Brier Score is a scoring rule for binary forecasts. It takes into account both the calibration and resolution of forecasts by calculating the mean squared error of forecasts () and outcomes ():
The Brier score is quite easy to implement:
def brier(x, y):
return np.mean((x-y)**2)
The first thing we can now do is to compare the forecasts from the two websites, and it turns out that Metaculus forecasts are slightly less good than PredictionBook forecasts:
>>> brier(metfcs, metress)
>>> brier(pbfcs, pbress)
But this comparison is not telling us much, since the questions on the two websites and the strictness for resolving questions are radically different, as explained here.
Now, one can calculate the Brier score for each of the forecasts and outcomes, with the mean being unnecessary, because we want to examine the score of each forecast individually:
First, one can check how high the range of these two datasets really is. The PredictionBook forecasts with the highest range span 3730 days (more than 10 years), for Metaculus it's 1387 days (nearly
4 years):
>>> np.max(metrngs)
>>> np.max(pbrngs)
One can now look at the correlation between range and Brier score first for Metaculus, and then for PredictionBook:
>>> np.corrcoef(metbriers, metrngs)
array([[1. , 0.02165924],
[0.02165924, 1. ]])
>>> np.corrcoef(pbbriers, pbrngs)
array([[ 1. , -0.02030743],
[-0.02030743, 1. ]])
For Metaculus, the results are not very surprising: The positive correlation tells us that the higher the range of a forecast, the lower the accuracy (or, poetically, at Metaculus the fogs of time
grow thicker the farther you want to look into the future).
However, for PredictionBook, the opposite is true (on this dataset): Forecasts with higher ranges give more accurate predictions, at least on average.
However, these correlations are quite weak, 0.02 could just be random noise. I would have to use a significance test to discern whether they are statistically significant.
Now, one can also perform a linear regression to gauge what the relation of range and accuracy of a forecast is:
>>> sps.linregress(metrngs, metbriers)
LinregressResult(slope=1.4921976403559925e-05, intercept=0.16753867328019442, rvalue=0.021659238937630332, pvalue=1.89939817752528e-06, stderr=3.1319561138899387e-06)
>>> sps.linregress(pbrngs, pbbriers)
LinregressResult(slope=-8.921762030379796e-06, intercept=0.16351703198845793, rvalue=-0.020307433721919746, pvalue=1.913246393632673e-05, stderr=2.0868414512480246e-06)
We can see that the rvalue is just the correlation, and that the pvalue is pretty good (<0.00001 and <.0001 for Metaculus and PredictionBook, respectively).
These are not particularly surprising. The inferred brier score at range 0 (the forecast directly before resolution) is ~0.16, which seems a bit pessimistic, but other than that, growth with higher
ranges for Metaculus data and lower accuracy for higher ranges for PredictionBook data match the correlation. The steepness of the regression is quite low because the ranges are in days.
Visualizing the accuracies of the forecasts with a scatterplot and linear regressions shows a very similar picture (red dots are for Metaculus forecasts, blue dots are for PredictionBook forecasts):
plt.xlabel("Range (days)")
plt.ylabel("Accuracy (Brier score)")
plt.plot(metrngs, metbriers, '.', color='red', markersize=1)
plt.plot(pbrngs, pbbriers, '.', color='blue', markersize=1)
The high amounts of noise are probably due to the low number of predictions for single days (or, in the case of weeks and months, for years/months with a high range, as not enough questions with this
range have resolved yet).
Non-Linear Curve-Fitting
Using a linear regression on the Brier score here, however, carries with it a deep issue: Unless the slope is 0, the linear regression will be below 0 or above 1 for some positive range—so one can't
use it to predict forecaster performance on questions with very long ranges.
(There is also the additional issue that in non-0-slope regressions, the linear regression might tell us that forecasters would perform worse than chance at some point in the future, that is, give an
expected Brier score >0.25, which is not what I expect to happen, unless reality is actively preventing us from making accurate long-term predictions).
Instead, I want to use functions that for positive values of x don't produce out-of-bounds errors (they at least return valid Brier scores).
I furthermore make some additional assumptions/desiderata about the the function to fit to the data:
1. For , it returns values in
2. For , it returns (at the time of resolution, we can predict the outcome perfectly, because we already know it)
1. It'd be technically elegant if for for (since we can perfectly predict things from the past (although there's a philosophical discussion to be had here about how much we can actually know
things from the past, and maybe it's even symmetric with the future)), but it's not super important
3. is monotonic for , that is
1. This is the one I'd be most willing to drop, since there might be weird non-monotonic effects in ability to predict.
Fitting a Logistic Function
The logistic function seems like an optimal candidate here: it fulfills at least desideratum 1 (if shrunk) and 3, and with some fiddling may even satisfy 2.
Because this is very different from a logistic regression (scaled values above the maximum (with a maximum of 0.25, some of the observed Brier scores are greater), continuous values to predict), I
curve-fit explicitely using scipy.optimize.curve_fit (why did I only learn about this function from scrolling through the scipy documentation‽ This is awesome!) with two parameters.
(Why not do a linear regression on the log-transformed data? Because the corresponding transformation ends up with with 0 inside a logarithm for PredictionBook data: The inverse logistic is , and
some PredictionBook Brier scores are 1).
So we instead use the following formula, which is just a squashed logistic function that the maximum is at 0.25:
def shrunk_logistic(x, slope, intercept):
return 0.25*1/(1+np.exp(slope*x+intercept))
We can now fit a curve with those parameters to the data, limiting the slope to negative values and the intercept to positive values (we want the function to be monotonically rising, and we want the
y-intercept to be below 0.125, that is we want the "middle" of the logistic function to be to the right of 0, even if we can't guarantee that the function will be ~0 for x=0).
>>> pblogifit=spo.curve_fit(shrunk_logistic, pbrngs, pbbriers, bounds=([-np.inf, 0], [0, np.inf]))
(array([-4.78706654e-04, 1.40345975e-20]), array([[ 1.69607043e-09, -4.52668529e-07],
[-4.52668529e-07, 4.22860649e-04]]))
>>> metlogifit=spo.curve_fit(shrunk_logistic, metrngs, metbriers, bounds=([-np.inf, 0], [0, np.inf]))
(array([-2.37260045e-03, 3.97380474e-19]), array([[ 7.35951274e-09, -1.08226199e-06],
[-1.08226199e-06, 3.59766672e-04]]))
The result can be plotted:
plt.title("Scatterplot with logistic-ish regression for Metaculus & PredictionBook forecasts by range (in days)")
plt.xlabel("Range (days)")
plt.ylabel("Accuracy (Brier score)")
fullrng=np.array(range(0, round(max(pbrngs))+1))
plt.plot(metrngs, metbriers, '.', color='red', markersize=1)
plt.plot(fullrng, shrunk_logistic(fullrng, metlogifit[0][0], metlogifit[0][1]), 'red', label='Metaculus shrunk logistic-ish regression', linewidth=2)
plt.plot(pbrngs, pbbriers, '.', color='blue', markersize=1)
plt.plot(fullrng, shrunk_logistic(fullrng, pblogifit[0][0], pblogifit[0][1]), 'blue', label='PredictionBook shrunk logistic-ish regression', linewidth=2)
I wonder whether the reason the Metaculus fit reaches the Metaculus data so much faster is because the Metaculus data ends earlier? (Also, yes, that is the logistic function and not a linear function
for the PredictionBook data, it's a really moderate slope).
Also, both plots start out with : not restricting the intercept to positive values returns negative intercepts (meaning that at 0, the value is of the logistic function is even higher than (in this
case) 0.125):
>>> pblogifit=spo.curve_fit(shrunk_logistic, pbrngs, pbbriers, bounds=([-np.inf, -np.inf], [0, np.inf]))
(array([-1.12830197e-14, -5.87766698e-01]), array([[ 1.32206792e-09, -4.11999218e-07],
[-4.11999218e-07, 4.67829989e-04]]))
>>> shrunk_logistic(0, -1.12830197e-1, -5.87766698e-01)
>>> metlogifit=spo.curve_fit(shrunk_logistic, metrngs, metbriers, bounds=([-np.inf, -np.inf], [0, np.inf]))
(array([-3.05026968e-04, -7.03162493e-01]), array([[ 3.73762741e-09, -7.74711069e-07],
[-7.74711069e-07, 3.76596526e-04]]))
>>> shrunk_logistic(0, -3.05026968e-04, -7.03162493e-01)
Here, the slopes are much steeper than in the more restricted case above.
Fitting an Exponential Function
Another function we could fit to the data might be of the form , with some (the function is decaying exponentially, but flipped so that it approaches 0, and then we scale it so that it always
converges toward 0.25).
We can guarantee this function to fulfill all three desiderata:
and (for )
which is the case.
In python, this is simply
def shift_exp(x, b):
return ((b**x)-1)/(-4)
We can now fit that kind of curve to the data:
>>> pbexpfit=spo.curve_fit(shift_exp, pbrngs, pbbriers, bounds=([0], [1]))
(array([1.22550795e-22]), array([[3.83266961e-18]]))
>>> metexpfit=spo.curve_fit(shift_exp, metrngs, metbriers, bounds=([0], [1]))
(array([0.95788506]), array([[6.50321645e-07]]))
As one can see, fitting this kind of curve suggests that the predictions become equivalent to random guesses almost immediately for PredictionBook, and for ranges >100 days for Metaculus. Perhaps
there are some problems with floating-point arithmetic at play here: the best fit would be at something like , but curve_fit doesn't know how to count that low?
This Is Cool
I believe that these findings are pretty cool: They give some sense of how long the range of forecasts needs to be for them to be approximately random.
We can do this by finding out at what point our function first predicts Brier scores sufficiently close to 0.25, let's take 0.24 as an arbitrary cutoff (which would be, on average, assigning a
probability of to events that take place).
The Horizon
Let's call this number the horizon: beyond it, our forecasts become random, we can neither steer nor see, the fogs of time have grown too thick. From our perspective, only chaos reigns there, and
every decision-theoretic ship that sails it is never to return with valuable information.
It is sometimes invoked by people when they want to talk about the inherent unknowability of the future, always without evidence of any actual number.
Then, for the squashed logistic function, we have to find the so that
Then, the logistic-ish forecasting horizon gives
>>> (np.log((1/0.96)-1)-metlogifit[0][1])/metlogifit[0][0]
>>> (np.log((1/0.96)-1)-pblogifit[0][1])/pblogifit[0][0]
which is ~3.6 years for Metaculus, and ~18 years for PredictionBook.
With the exponential fit, we know that
That gives
>>> np.log(0.04)/np.log(metexpfit[0][0])
>>> np.log(0.04)/np.log(pbexpfit[0][0])
less than a day for the PredictionBook predictive horizon, and ~75 days for the Metaculus predictive horizon.
Of course, don't believe these numbers too much: The difference in dataset range is probably causing a lot of the difference in fit, the exponential fit is way more pessimistic, and I haven't
performed any statistical tests to determine how to much believe these particular numbers.
But I'm really excited about testing these conclusions with growing datasets as forecasting platforms exist longer.
Overall, I like the logistic fit much better than the exponential one: in practice, we know that forecasters don't give quasi-random predictions for questions that are further out than 100 days (or,
as the PredictionBook fit suggests, that forecasting is impossible!).
But one can also take a look at the quality of the fit to the data: What is the mean squared error of the predicted and the actual Brier score for the observed data?
>>> np.mean((shrunk_logistic(pbrngs, pblogifit[0][0], pblogifit[0][1])-pbbriers)**2)
>>> np.mean((shrunk_logistic(metrngs, metlogifit[0][0], metlogifit[0][1])-metbriers)**2)
>>> np.mean((shift_exp(pbrngs, pbexpfit[0][0])-pbbriers)**2)
>>> np.mean((shift_exp(metrngs, metexpfit[0][0])-metbriers)**2)
The fits agree (very slightly) with me here: in both cases the logistic fit has a marginally smaller mean squared error in predicting the Brier score.
Why Assume Accuracy will Increase?
I believe that the finding for the PredictionBook data is quite surprising.
A priori, one would believe that beliefs about the near future are generally more accurate than beliefs about the far future: We can predict the weather in 2 minutes far better than the weather in 6
months, we can say much more about the position of a rock in an hour than in 100 years, more about the popularity of a political party in 2 months as opposed to 10 years. Even in reasonably chaotic
systems, one should expect to become more and more accurate the closer one comes to the expected time.
One example for this is a roulette wheel (the resolution being the number of the slot the ball eventually rolls into): I am able to give a much narrower probability distribution on values 100ms
before the ball falls into the slot than 1s before, and 5s before resolution my prediction is going to be very uniform. Information, like nearly everything else, has diminishing value, and posteriors
eventually converge towards the truth.
However, there is an interesting effect that takes place with systems that eventually reach equilibrium. Take, for example, a double pendulum in an environment with gravity: If I am at the start of
the swinging of the double pendulum, I can predict the state in 100ms better than in 1s (because it becomes more chaotic over time), but I am also better able to predict the state in 1h (or how long
it takes to reach equilibrium) than in 1s (because it reaches equilibrium in hanging straight down).
(I got this observation from “The World as Holocaust” by Stanisław Lem, though it is obvious in hindsight).
Possible Explanations
So, what is the reason for this rather weird finding? Several possible reasons come to mind.
Range and Biased Questions
The most obvious solution is that the analysis above is absolute bogus and completely meaningless: It compares questions about global catastrophic risks to questions about the extinction of banana
brands, very different kinds of questions with very different kinds of forecasts.
Here, one would assume that the longer-term questions asked are generally easier to predict, and that the effect goes away when one compares predictions among very similary questions (or, better,
within questions).
Generally, the long-term questions we prefer asking seem to be more menable to forecasting than short-term questions: development of population sizes, the climate, especially the movement of
interstellar bodies is much more thoroughly modelled than the development of markets, elections and the weather. This is of course only a weak trend, but one that could influence the questions (as
will be investigated in this section).
Simpson's Paradox
Simpson's Paradox is the phenomenon that while two features are correlated in a dataset, it can be true that the features are negatively correlated for some distinct subsets of the dataset.
It is best explained visually:
It might be the case that this analysis for PredictionBook data has come up against an instance of Simpson's paradox: The accuracy of forecasts is negatively correlated with range within the same
question, but the accuracy of forecasts is positively correlated with range across questions (because the kinds of questions with longer time horizons generally allow more accurate forecasts).
Unfortunately, whether Simpson's paradox applies or not can not always be easily judged from the scatterplot of datapoints.
However, below I check the correlation of range and forecast accuracy between questions, and find that they are negatively correlated, and furthermore find that they are positively related within
questions, which strongly indicates that the effect probably comes from questions with a long range receiving more accurate predictions (in the PredictionBook dataset).
Low Sample Sizes With High Ranges
Another question one might ask is: How big are the sample sizes at the tails when the range is high?
This is important: low sample sizes increase noise dramatically, and make findings much less reliable.
To get a rough overview over the sample sizes, on can look at the number of samples for each bucket. The sample sizes were calculated such that at position i in the array {pb,met}ss was the sample
size for week i:
I generated charts for the sample sizes in days:
fig=plt.figure(figsize=(8,8), clear=True)
plt.xlabel("Range (months)")
plt.ylabel("Number of datapoints)")
plt.plot(metss, '-', color='red')
plt.plot(pbss, '-', color='blue')
The red graphs stands for Metaculus sample sizes, the blue graph stands for PredictionBook sample sizes.
As one can see, the sample sizes have a drastical skew towards recent predictions, not surprising for relatively young platforms (although 10 years for PredictionBook is sizable by internet
standards, it's not that much compared to the expected range of some predictions on the platform, which might go into the thousands of years).
This can be seen in the data as well: The median range of Metaculus and PredictionBook predictions is only a couple of months, and less than 25% of questions have a range of more than one year:
>>> np.quantile(metrngs/365, 0.25)
>>> np.quantile(metrngs/365, 0.5)
>>> np.quantile(metrngs/365, 0.75)
>>> np.quantile(pbrngs/365, 0.25)
>>> np.quantile(pbrngs/365, 0.5)
>>> np.quantile(pbrngs/365, 0.75)
Statistical Significance of Truncated Datasets
Moved to Appendix B.
I hope that the dataset becomes richer the older these platforms become.
Because in the linear regression all datapoints are weighted equally, it could very well be that a tiny bit of noise at the tails dominates the entire regression.
Accuracy Between Questions
Another way to determine at the relation between forecasting accuracy and range is to look at the range of questions and not of individual forecasts.
In this case, this means taking the forecasts on all questions with a given range and calculating the brier score on these forecasts.
Determining the Range of a Question
The range of a question is determined by taking the time difference between the opening time (the time when the first prediction on the question could have been made) and the resolution time. One
could imagine other metrics to determine the range of a question: the mean range for forecasts of that question, the median range for forecasts on that question, time differences between writing/
opening and closing/resolution times of the question, and probably many more.
Here, the range of a question was set to the time difference between opening time and resolution time. The reasons for this were threefold:
First, I had no clear idea about the time when people were making forecasts on questions. Are most of the forecasts made just after opening, or just before closing? Or is the distribution uniform on
the time between opening and closing? And are these distributions different on long-range as opposed to short-range questions? Also, I was unsure whether taking the mean time for forecasts would just
be the same as comparing forecasts directly. So taking the median or the mean of the forecasts made was less preferable.
Second, what I cared about here was the uncertainty of questions at time of writing, not at time of prediction. This is much better tracked by opening time than by proxy on the forecasts.
Third, there was the question of data availability. Both Metaculus and PredictionBook publish opening/resolution times, but PredictionBook has no clear distinction between closing and resolution time
(there is, however, a distinction between effective resolution time and planned resolution time ("When was the question resolved?" vs. "When should the question have been resolved?")).
First, the dataset grouped by forecasts had to be grouped by the question ID, in both cases a positive integer. The resulting datastructure should have the structure
[[id, open-resolve-timediff, [outcomes], [forecasts], [forecast-resolve-timediffs]]*]
where the splat just indicates the inner list can be repeated. This is achieved by first finding the grouping of forecasts by question ID, then concatenating the ID, the question range, the list of
outcomes, the list of forecasts and the list of forecast ranges:
def group(d):
for e in np.unique(d[0]):
a.append([e, d[1][indices[0][0]], d[2][indices], d[3][indices], d[4][indices]])
return a
Strictly speaking, the outcomes could be a single element, since for every question there is only one well-defined outcome, but this makes it easier to later compute the brier score.
>>> metquestions[10]
[13.0, 119.99914351851852, array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]), array([0.2 , 0.4 , 0.2 , 0.3 , 0.15, 0.3 , 0.4 , 0.4 , 0.4 , 0.4 , 0.45,
0.4 , 0.44, 0.4 , 0.44, 0.4 , 0.38]), array([119.94258413, 118.91094545, 118.71650504, 115.99830505,
113.03583134, 89.66780818, 87.76008922, 87.12385685,
85.12093715, 85.08304395, 83.7246415 , 83.19617587,
82.69982874, 73.11635207, 71.99461814, 71.21645502,
>>> brier(metquestions[10][3],metquestions[10][2])
One can now also see how many questions there are in the two datasets (with the relatively unsurprising result that PredictionBook has much more resolved questions):
>>> len(metquestions)
>>> len(pbquestions)
The next step involves computing the Brier score for the forecasts on each question:
>>> metqbrier=np.array([[i[1], brier(i[3], i[2])] for i in metquestions])
>>> pbqbrier=np.array([[i[1], brier(i[3], i[2])] for i in pbquestions])
metqbrier is a list that contains sublists, one for each question, the sublist containing the range for the question and the brier score for all predictions on the question (pbqbrier has the same
Again I use linear regressions, correlation coefficients and scatter plots to inadequately analyze the data.
For accuracy between questions, the results were pretty surprising:
>>> np.corrcoef(metqbrier.T)
array([[ 1. , -0.0099402],
[-0.0099402, 1. ]])
>>> np.corrcoef(pbqbrier.T)
array([[ 1. , -0.05180824],
[-0.05180824, 1. ]])
>>> sps.linregress(metqbrier.T[0], metqbrier.T[1])
LinregressResult(slope=-5.199153608270726e-06, intercept=0.1751301126619239, rvalue=-0.009940204896962441, pvalue=0.8149259597777081, stderr=2.2200837795403376e-05)
>>> sps.linregress(pbqbrier.T[0], pbqbrier.T[1])
LinregressResult(slope=-2.1538793571528e-05, intercept=0.1952547647088438, rvalue=-0.05180823990580795, pvalue=2.0875245782500886e-09, stderr=3.5928014948058268e-06)
For Metaculus, the slope off the linear regression is approximately , compared that with for the slope for the linear regression between forecasts—the slope is less steep, but also negative. For
PredictionBook, the slope of the linear regression is , compared with for the data between forecasts, which is slightly steeper.
However, look at the p-value for the Metaculus correlation/linear regression! 0.8! So that number is basically worthless.
In both cases, there is a small negative correlation between the brier score and the range (to be precise, the larger the range, the lower the brier score/the higher the accuracy). For the Metaculus
data, this effect is not as pronounced as for the PredictionBook data, though both correlations are quite weak. The two linear regressions also show the same effect (lower accuracy at shorter ranges/
higher accuracy at higher ranges), but again the slope of the linear regression is not very steep.
And now: linear regressions and scatterplots!
The following are scatterplots with range on the X-axis and accuracy (calculated using the Brier score) on the Y-axis. Again, red dots/lines are for Metaculus data (twice as big as PredictionBook
data points, to make them visible in the sea of blue), and blue dots/lines are for PredictionBook data.
plt.xlabel("Range (days)")
plt.ylabel("Accuracy (Brier score)")
plt.plot(pbqbrier.T[0], pbqbrier.T[1], '.', color='blue', markersize=1)
plt.plot(pbqbrier.T[0], pbqintercept+pbqslope*pbqbrier.T[0], 'blue', label='PredictionBook linear regression', linewidth=1)
plt.plot(metqbrier.T[0], metqbrier.T[1], '.', color='red', markersize=2)
plt.plot(pbqbrier.T[0], mqintercept+mqslope*pbqbrier.T[0], 'red', label='Metaculus linear regression', linewidth=1)
The general trend seems to be: questions with a higher range tend to receive forecasts that have a higher accuracy than questions with a lower range. In itself, this is already a fascinating finding,
and might explain some of the effect seen with accuracy between forecasts in the previous section). On the other hand, the data is still very noisy, the correlations found are quite weak, and the
slopes of the linear regressions are are very near 0.
All in all, it's plausible that the relation of range and accuracy between questions explains a large part of the the weird relation for accuracy and range between forecasts, but I don't know enough
statistics to tease these out exactly. My intuition tells me that the effect on accuracy between questions is too small to explain the whole anomaly between forecasts.
Non-Linear Curve-Fitting
Again, one can fit the nonlinear exponential/logistic function defined above to the data between questions.
>>> pblogifit_betweenq=spo.curve_fit(shrunk_logistic, pbqbrier.T[0], pbqbrier.T[1], bounds=([-np.inf, 0], [0, np.inf]))
(array([-2.70329933e+00, 5.32716622e-52]), array([[ 0.16764075, -0.01981014],
[-0.01981014, 0.00898443]]))
>>> metlogifit_betweenq=spo.curve_fit(shrunk_logistic, metqbrier.T[0], metqbrier.T[1], bounds=([-np.inf, 0], [0, np.inf]))
(array([-7.92206883, 33.48197 ]), array([[ 199420.41507448, -811407.37948018],
[-811407.37948018, 3301492.9741521 ]]))
>>> pbexpfit_betweenq=spo.curve_fit(shift_exp, pbqbrier.T[0], pbqbrier.T[1], bounds=([-np.inf, 0], [0, 1]))
(array([4.77613047e-20]), array([[5.82829061e-18]]))
>>> metexpfit_betweenq=spo.curve_fit(shift_exp, metqbrier.T[0], metqbrier.T[1], bounds=([-np.inf, 0], [0, 1]))
(array([0.70814538]), array([[0.01386776]]))
But these numbers don't tell us much by themselves, do they become clearer when plotted?
plt.title("Scatterplot with logistic-ish regression for Metaculus & PredictionBook forecasts by range")
plt.xlabel("Range (days)")
plt.ylabel("Accuracy (Brier score)")
fullrng=np.array(range(0, round(max(pbrngs))+1))
plt.plot(pbqbrier.T[0], pbqbrier.T[1], '.', color='blue', markersize=1)
plt.plot(metqbrier.T[0], metqbrier.T[1], '.', color='red', markersize=2)
plt.plot(fullrng, shrunk_logistic(fullrng, metlogifit_betweenq[0][0], metlogifit_betweenq[0][1]), 'red', label='Metaculus shrunk logistic-ish regression', linewidth=2)
plt.plot(fullrng, shrunk_logistic(fullrng, pblogifit_betweenq[0][0], pblogifit_betweenq[0][1]), 'blue', label='PredictionBook shrunk logistic-ish regression', linewidth=2)
(The Metaculus regression is not visibile because it lies just under the PredictionBook regression, the short red line in the logistic plot is the Metaculus regression that starts at 0, while the
PredictionBook regression starts at 0.125).
Basically, the regressions here conclude that the best is that predictions on questions with any time horizons longer than a couple of days are indistinguishable from randomness, given the
assumptions made here.
This actually makes sense: We observe that the correlation between range and accuracy is positive, so the best curve that fits the data under the assumption of falling accuracy with higher range will
conclude that we're immediately in a domain with uniform randomness.
The predictive horizons here are
>>> (np.log(1/(0.96)-1)-metlogifit_betweenq[0][1])/metlogifit_betweenq[0][0]
>>> (np.log(1/(0.96)-1)-pblogifit_betweenq[0][1])/pblogifit_betweenq[0][0]
~4.5 days for Metaculus, and around a day for PredictionBook with logistic functions, and
>>> np.log(0.04)/np.log(metexpfit_betweenq[0][0])
>>> np.log(0.04)/np.log(pbexpfit_betweenq[0][0])
similarly short timespans for the exponential fit.
And, comparing the quality (mean squared error) of the nonlinear fits to one another reveals that the two methods are remarkably similar at fitting the data (which is not surprising, since they look
nearly identical):
>>> np.mean((shrunk_logistic(pbqbrier.T[0], pblogifit_betweenq[0][0], pblogifit_betweenq[0][1])-pbqbrier.T[1])**2)
>>> np.mean((shrunk_logistic(metqbrier.T[0], metlogifit_betweenq[0][0], metlogifit_betweenq[0][1])-metqbrier.T[1])**2)
>>> np.mean((shift_exp(pbqbrier.T[0], pbexpfit_betweenq[0][0])-pbqbrier.T[1])**2)
>>> np.mean((shift_exp(metqbrier.T[0], metexpfit_betweenq[0][0])-metqbrier.T[1])**2)
Why Longer Range Questions More Accurate?
The big question now is: Why do forecasts on predictions on questions with a higher range generally receive better Brier scores?
First, it's important to consider the p-value for the correlation with the Metaculus data. It's 80% likely we would have had the same result, given that the correlation was 0: not solid foundation to
base further understanding on. But we got the same result with the PredictionBook data, with a very decent p-value, so what is going on?
The explanation falls back to the considerations in the section on range and biased questions: the long-range questions we might be asking could be "easier" to predict, at least in the medium term,
than the short-range questions.
How could one test this? Metaculus contains categories for questions, and one might want to examine whether the negative trend between question range and accuracy of predictions on that question
still hold when questions in the same category are examined (although one might run into problems with the dataset size here—even the 557 questions in the dataset aren't enough to provide a decent
p-value). Unfortunately, no such categorization system exists for PredictionBook, one might try to analyze the titles of the questions, but it doesn't seem worth the effort.
This Partially Explains the Result Between Forecasts
For PredictionBook data, this explains why range and forecast accuracy were negatively correlated between forecasts: the negative correlation between range and accuracy between questions confounds
the relationship. We can test whether this is true by looking at the relation of range and accuracy within questions, where two forecasts at the same time are in some sense "equally difficult".
Accuracy Within Questions
If there exists any bias in regard to what kinds of questions get asked in relation to their range, how can we correct for this bias?
One approach could be to compare very similar questions, such as only questions about artificial intelligence, the cost & speed of gene sequencing or autonomous cars, and examine the relation of
range and accuracy within these categories. This might eliminate bias resulting from questions in different kinds of domains being easier or harder to forecast.
Here, I take a simpler approach. I examine the relation of range and accuracy within questions; are forecasts made on the same question later generally more accurate than forecasts made on a question
In order to do this, one can compute the Brier score for each prediction, and then perform one linear regression/compute the correlation per question to discern whether the relation is positive or
With metquestions and pbquestions, we already have the necessary data available to perform the analysis.
We can create a list of the form [[[brier_scores],[ranges]]*]:
wmetqbrier=[[i[4], (i[3]-i[2])**2] for i in metquestions]
wpbqbrier=[[i[4], (i[3]-i[2])**2] for i in pbquestions]
Since lreg can't deal with datasets of size 1, we have to filter those out of the dataset (the Metaculus dataset doesn't contain these, but I want to prepare for a possible future dataset where it
does), and they don't make much sense in our analysis anyway:
wmetqbrier=list(filter(lambda x: len(x[0])>1, wmetqbrier))
wpbqbrier=list(filter(lambda x: len(x[0])>1, wpbqbrier))
One can play around and calculate the correlation between range and accuracy for some questions:
>>> list(map(np.corrcoef, wmetqbrier[:4]))
[array([[1. , 0.53853205],
[0.53853205, 1. ]]),
array[[1. , 0.6569835],
[0.6569835, 1. ]]),
array([[1. , 0.05048498],
[0.05048498, 1. ]]),
array([[1. , 0.28412936],
[0.28412936, 1. ]])]
>>> list(map(np.corrcoef, wpbqbrier[:4]))
[array([[1. , 0.52609801],
[0.52609801, 1. ]]),
array([[1. , 0.89254317],
[0.89254317, 1. ]]),
array([[ 1. , -0.39887059],
[-0.39887059, 1. ]]),
array([[ 1., -1.],
[-1., 1.]])]
The perfect negative correlation come from the fact that some of the questions in the dataset have only two predictions, which all by chance anti-correlate with the range. This is not the case for
all questions, as one can see.
For the linear regression, one can simply map sps.linregress over the lists:
>>> wmetqregs=list(map(lambda x: sps.linregress(x[0], x[1]), wmetqbrier))
>>> wpbqregs=list(map(lambda x: sps.linregress(x[0], x[1]), wpbqbrier))
/usr/local/lib/python3.8/dist-packages/scipy/stats/_stats_mstats_common.py:130: RuntimeWarning: invalid value encountered in double_scalars
slope = r_num / ssxm
The result for wpbqbrier is unexpected. The culprits turn out to be a set of questions on which the same prediction has been made, twice, at the exact same second, which confuses the linear
regression algorithm:
>>> list(filter(lambda x: x[0][0]==x[0][1] and len(x[0]==2) and x[1][0]==x[1][1] and len(x[1])==2, wpbqbrier))
[[array([367.09616898, 367.09616898]), array([0.2025, 0.2025])], [array([367.09637731, 367.09637731]), array([0.2025, 0.2025])], [array([367.09899306, 367.09899306]), array([0.0225, 0.0225])], [array([367.09908565, 367.09908565]), array([0.25, 0.25])], [array([367.09936343, 367.09936343]), array([0.16, 0.16])], [array([367.10018519, 367.10018519]), array([0.0225, 0.0225])], [array([0.25236111, 0.25236111]), array([0.0025, 0.0025])], [array([0.36797454, 0.36797454]), array([0.25, 0.25])], [array([0.25259259, 0.25259259]), array([0.0625, 0.0625])], [array([0.36671296, 0.36671296]), array([0.04, 0.04])], [array([0.40542824, 0.40542824]), array([0.09, 0.09])]]
However, they can be filtered out pretty easily, and we recompute wpbqregs:
>>> wpbqbrier=list(filter(lambda x: not (x[0][0]==x[0][1] and len(x[0]==2) and x[1][0]==x[1][1] and len(x[1])==2), wpbqbrier))
>>> len(wpbqbrier)
>>> wpbqregs=list(map(lambda x: sps.linregress(x[0], x[1]), wpbqbrier))
Again, the results are split in three parts: linear regression, logistic curve-fit and exponential curve-fit.
Linear Regression
We can now visualise the linear regression for each question by setting plotting all linear regressions with random colors (the horizontal length of the linear regression indicates the time between
the first prediction and the last prediction on the question: a question that was opened three years ago, closed two years ago, and resolves now appears on the X-axis between 730 and 1095):
plt.xlabel("Range (days)")
plt.ylabel("Linear regression")
for i in range(0, len(wmetqregs)):
slope, intercept, _, _, _=r
cl=hex(random.sample(range(0, 256*256*256), 1)[0]) #random rgb code
#left padding with zeros, can't be bothered to read the formatting docs right now
plt.plot(rngs, intercept+slope*rngs, color=cl, linewidth=1)
Basically the same code for image generation is used also for the PredictionBook data:
Although the plots are kind of cool to look at, I'm not really sure what they can tell us. My guess would be that it somewhat shows a trend with higher ranges responding to higher Brier scores (and
therefore lower accuracy).
Aggregating Linear Regressions
We can test whether this suspicion is acually correct by calculating the average offset and the average ascension—if the ascension is positive, our suspicion is confirmed. We have to weight questions
by how many predictions they have received, otherwise the result is skewed by questions with very few predictions (if you're trying to find out whether, in basketball, making more free throws makes
you better at it, you'd also want to more strongly weight data from players with a larger number of shots).
This is done by computing the linear regression for range/accuracy for each question (we did that with w{met,pb}qregs), multiplying it by the number of predictions on that question, adding up the
linear regressions, and then dividing the result by the total number of predictions in the dataset (clean_{met,pb}forecasts):
>>> clean_metforecasts=np.sum([len(wmetqbrier[i][0]) for i in range(0, len(wmetqbrier))])
>>> awmetqslope=np.sum([len(wmetqbrier[i][0])*wmetqregs[i][0] for i in range(0, len(wmetqregs))])/clean_metforecasts
>>> awmetqintercept=np.sum([len(wmetqbrier[i][0])*wmetqregs[i][1] for i in range(0, len(wmetqregs))])/clean_metforecasts
>>> clean_pbforecasts=np.sum([len(wpbqbrier[i][0]) for i in range(0, len(wpbqbrier))])
>>> awpbqslope=np.sum([len(wpbqbrier[i][0])*wpbqregs[i][0] for i in range(0, len(wpbqregs))])/clean_pbforecasts
>>> awpbqintercept=np.sum([len(wpbqbrier[i][0])*wpbqregs[i][1] for i in range(0, len(wpbqregs))])/clean_pbforecasts
The PredictionBook data—how do I put this—simply makes no sense. I am pretty confident that this code is correct, but I think that the questions with very few predictions are producing incorrect
results, especially when the predictions are very close to each other. So let's arbitrarily exclude questions with less than ten predictions (actually an arbitrary choice I did not iterate over to
get a "desired" result):
>>> fwpbqbrier=list(filter(lambda x: len(x[0])>=10, wpbqbrier))
>>> len(fwpbqbrier)
>>> # Recomputing linear regressions
>>> clean_fpbforecasts=np.sum([len(fwpbqbrier[i][0]) for i in range(0, len(fwpbqbrier))])
>>> fwpbqregs=list(map(lambda x: sps.linregress(x[0], x[1]), fwpbqbrier))
>>> fawpbqslope=np.sum([len(fwpbqbrier[i][0])*fwpbqregs[i][0] for i in range(0, len(fwpbqregs))])/clean_fpbforecasts
>>> fawpbqintercept=np.sum([len(fwpbqbrier[i][0])*fwpbqregs[i][1] for i in range(0, len(fwpbqregs))])/clean_fpbforecasts
This looks much better.
So it is true that accuracy within question generally is higher with lower range for Metaculus data, and similar for PredictionBook data. Everything else would have been surprising.
plt.title("Mean of linear regressions on accuracy within questions (red is Metaculus data, blue is PredictionBook data)")
plt.xlabel("Range (days)")
plt.ylabel("Accuracy (Brier score)")
plt.plot(pbrngs, awmetqintercept+awmetqslope*pbrngs, 'red', label='Metaculus aggregate linear regression', linewidth=1)
plt.plot(pbrngs, fawpbqintercept+fawpbqslope*pbrngs, 'blue', label='PredictionBook aggregate linear regression', linewidth=1)
This chart, however, shows that the result is not as clean as one might hope: both linear regressions are very steep, predicting Brier scores of >1 for ranges of more than a year, which is clearly
This probably results from the probabilities being treated linearly, while handling them in logspace would be much more appropriate.
Logistic Curve-Fit
One can now similarly fit the logistic curve to the data within every question, yielding a list of parameters for the logistic function.
Doing this naively via a list comprehension fails:
>>> within_logi_fits=list(map(lambda x: spo.curve_fit(shrunk_logistic, x[0], x[1], bounds=([-np.inf, 0], [0, np.inf])), wmetqbrier))
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<stdin>", line 1, in <lambda>
File "/usr/local/lib/python3.8/dist-packages/scipy/optimize/minpack.py", line 799, in curve_fit
raise RuntimeError("Optimal parameters not found: " + res.message)
RuntimeError: Optimal parameters not found: The maximum number of function evaluations is exceeded.
To both find the culprit and then ignore it, we have to write the code iteratively:
for e in wmetqbrier:
within_logi_fits_met.append(spo.curve_fit(shrunk_logistic, e[0], e[1], bounds=([-np.inf, 0], [0, np.inf])))
except RuntimeError:
The resonsible data for the question looks completely innocuous:
[array([20.11263452, 19.95414332, 19.86404009, 19.80523882, 19.68123836,
19.30289307, 19.08148786, 18.67971381, 17.57324535, 16.17246518,
14.64708341]), array([0.49 , 0.5184, 0.49 , 0.4225, 0.3481, 0.4225, 0.3481, 0.3481,
0.16 , 0.3481, 0.16 ])]
I decide to just ignore any instances that give errors, and calculate within_logi_fits_pb the same way as above, just with fwpbqbrier. This removes data from 1 question from the Metaculus dataset,
and from 10 questions from the PredictionBook dataset:
>>> len(list(filter(lambda x: len(x)==0, within_logi_fits_met)))
>>> len(list(filter(lambda x: len(x)==0, within_logi_fits_pb)))
These can now be plotted, as the linear regressions were above:
plt.title("Logistic curve-fits for the accuracy of questions by range (only Metaculus data)")
plt.xlabel("Range (days)")
plt.ylabel("Logistic curve-fit")
for i in range(0, len(within_logi_fits_met)):
if len(r)==0:
slope, intercept=r[0][0], r[0][1]
cl=hex(random.sample(range(0, 256*256*256), 1)[0]) #random rgb code
#left padding with zeros, can't be bothered to read the formatting docs right now
plt.plot(fullrng_met, shrunk_logistic(fullrng_met, slope, intercept))
Again, basically the same code, but for PredictionBook data, gives us these plots:
These charts look like what I would have expected:
1. On many questions, the predictions might be more accurate with longer ranges, which results in sigmoids which go to 0.25 almost immediately.
2. On some questions, the slope is very slight, resulting in the lines parallel to the x-axis. This happens when there is basically no relation between range and accuracy.
3. A few questions were posed long ago, but also have more accurate predictions at higher ranges, and therefore their plot is approximately a step function around the time when they were posed.
4. And, finally, some questions have more accurate predictions at lower ranges, resulting in the functions that actually look like sigmoids.
I'm unsure about the best way to aggregate these different sigmoids into one, as I did with the linear regressions above.
Logistic Forecast Horizons for Questions
for there is in a god's face more of marvel than prediction can tell, and when that face is vaster than a great temple and seen looking down at sunset in the cryptic silences of that upper world
from whose dark lava it was divinely hewn of old, the marvel is so strong that none may escape it.
— Howard Phillips Lovecraft, “The Dream-Quest of Unknown Kadath”, 1943
We now can calculate the forecast horizon defined in this section for every question, and then calculate some summary statistics of the forecast horizons on all questions in the two datasets.
First, we have to compute the horizons discovered by the fit (ignoring the invalid fits represented by []):
>>> within_logi_fits_met_filt=list(filter(lambda x: len(x)>0, within_logi_fits_met))
>>> within_logi_fits_pb_filt=list(filter(lambda x: len(x)>0, within_logi_fits_pb))
>>> met_logi_horizons=[(np.log((1/0.96)-1)-f[0][1])/f[0][0] for f in within_logi_fits_met_filt]
>>> pb_logi_horizons=[(np.log((1/0.96)-1)-f[0][1])/f[0][0] for f in within_logi_fits_pb_filt]
So, what horizons do we have?
Mean Median Mode Variance Maximum Minimum
Metaculus 1.93e+23 4.18 4.18 1.42e+49 8.79e+25 0.0043
PredictionBook 3.93e+23 4.18 4.18 1.11e+50 3.04e+26 0.000137
This shouldn't be that surprising: some of the logistic fits were basically flat, and some were basically vertical, so it is to be expected that we have lots of outliers present. What is interesting
is that the median & mode in both datasets are the same number, ~4.17. This is probably due to many questions having a slope of -1 and an intercept 1, leading to the same logistic fit.
We can plot a histogram of the horizons for questions (with the horizon length being logarithmic, base 10):
plt.title("Horizons for logistic curve-fits within questions")
plt.xlabel("Horizon length")
plt.ylabel("Number of questions")
plt.hist([np.log10(met_logi_horizons), np.log10(pb_logi_horizons)], bins=20, color=('red', 'blue'))
From this we can glean that:
1. Most questions supposedly have really short horizon lengths (becoming impossible to predict basically immediately): len(list(filter(lambda x: x>1 and x<10, met_logi_horizons)))/len
2. Most others have decent horizon lengths (a few hundred to a few thousand days): len(list(filter(lambda x: x>=10 and x<10000, met_logi_horizons)))/len(met_logi_horizons)≈22.8%
3. A few are strong outliers (mostly in the direction of very long horizons): len(list(filter(lambda x: x<=1 or x>=10000, met_logi_horizons)))/len(met_logi_horizons)≈8.27%
Exponential Curve-Fit
We can now perform a very similar analysis, just using exponential fits. Here, for some reason, we don't run into problems with inscrutable errors:
within_exp_fits_met=[spo.curve_fit(shift_exp, e[0], e[1], bounds=([0], [1])) for e in wmetqbrier]
within_exp_fits_pb=[spo.curve_fit(shift_exp, e[0], e[1], bounds=([0], [1])) for e in fwpbqbrier]
This family of curves can now be plotted (first the exponential functions fitted to questions from the Metaculus data, and then to questions from the PredictionBook data):
plt.title("Exponential curve-fits for the accuracy of questions by range (only Metaculus data)")
plt.xlabel("Range (days)")
plt.ylabel("Exponential curve-fit")
for i in range(0, len(within_exp_fits_met)):
if len(r)==0:
cl=hex(random.sample(range(0, 256*256*256), 1)[0]) #random rgb code
#left padding with zeros, can't be bothered to read the formatting docs right now
plt.plot(fullrng_met, shift_exp(fullrng_met, coeff))
I personally believe that these plots are kind of gorgeous. Interesting are the two "rivers" in the Metaculus plots: they indicate that there are some horizons for which there are ~0 questions with
that horizon. But this is possibly just due to a small sample-size & randomness, as they don't really occur in the PredictionBook data.
Otherwise, these plots look as expected: Most exponential fits go to randomness very quickly (i.e. from an expected Brier score of 0 to a Brier score of 0.25 in a short range), and some show longer
Again, I don't quite know how to aggregate these, but I can calculate the expected horizons for the questions.
Exponential Forecast Horizons for Questions
Calculating the horizons is as easy as before:
met_exp_horizons=[np.log(0.04)/np.log(f[0][0]) for f in within_exp_fits_met]
pb_exp_horizons=[np.log(0.04)/np.log(f[0][0]) for f in within_exp_fits_pb]
And the summary statistics (in days):
Mean Median Mode Variance Maximum Minimum
Metaculus 1613.62 4.64 4.64 478494147 497370.21 0.064
PredictionBook 45041.92 4.64 4.64 8.29e+11 5.31e+7 0.064
The mean here is ~4.42 years for Metaculus data, and 123.4 years for PredictionBook data.
We can see that the expected horizons aren't as large (at least ~15 orders of magnitude smaller), but again we observe that both the median and the mode take on the same value of 4.64.
So we decide to plot a histogram of the expected horizons for the questions, scaled logarithmically:
plt.title("Horizons for expstic curve-fits within questions")
plt.xlabel("Horizon length (days, log₁₀)")
plt.ylabel("Number of questions")
plt.hist([np.log10(met_exp_horizons), np.log10(pb_exp_horizons)], bins=20, color=('red', 'blue'))
I wonder if the regularity I perceive (the nice declining "hill"-like patterns for horizons >10) is a spurious artifact, a result of the specific method of analysis, or actually inherent in the data.
If not, it indicates that PredictionBook contains more resolved questions with longer expected horizons (checks out: questions with longer ranges can be expected to have longer horizons, and
PredictionBook has more & older resolved questions than Metaculus).
Also, again, most questions have ranges ~10 (probably the 4.64 value from above), which is likely due to them becoming more accurate at higher ranges (discussed here).
Sample Sizes
One might, again, be interested in the sample sizes. How many predictions to questions receive?
As we know, the Metaculus dataset contains predictions on 557 questions, the PredictionBook dataset 13356, but there are way fewer questions with more than 1 unique prediction in the PredictionBook
>>> len(metquestions)
>>> len(pbquestions)
>>> len(wmetqbrier)
>>> len(wpbqbrier)
Let's first create sorted lists containing the numbers of forecasts on each question:
pblens=np.sort([len(x[0]) for x in wpbqbrier])
metlens=np.sort([len(x[0]) for x in wmetqbrier])
One can now look at some central values for those datasets: the maximum, mimimum, mean, median, and mode:
>>> import statistics
>>> np.min(metlens)
>>> np.max(metlens)
>>> np.mean(metlens)
>>> np.median(metlens)
>>> statistics.mode(metlens)
>>> np.min(pblens)
>>> np.max(pblens)
>>> np.mean(pblens)
>>> np.median(pblens)
>>> statistics.mode(pblens)
This is—surprising, to say the least. Metaculus makes creating new questions much harder, and more strongly encourages users to predict on existing questions, with an elaborate tagging system for
questions. PredictionBook on the other hand simplifies the questions creation process, leaving out moderation, complex resolution criteria etc. Still, I'm surprised—there must be at least one
PredictionBook question popular enough for 100 forecasts! But apparently not.
Interlude: It's Under 102
One result here is, to say the least, confusing—where did all those 101s come from in the Metaculus data‽ Surely, there are questions with more than 101 forecasts (which I know, this question about
2016 being the warmest year on record has 765 forecasts)!
I initially suspected a bug in my code, but to my surprise, after further investigation, it turns out that the Metaculus API returned timeseries with elements removed so that the length was always
I can think of two reasons to do this:
• Metaculus wants to prevent other entities from using the predictions to create stronger forecasting algorithms that could rival the Metaculus algorithm
• It was programmed in as a hard limit when Metaculus wasn't as big as it is now, and never changed
I mailed the support address on the site, asking for a full timeseries on resolved binary questions.
After the support address had not responded to my inquiry, I contacted one of the admins of the site on the Discord, but was informed that updating the API would be too difficult to do (which is
understandable, the Metaculus developers do not exist to cater to my whims, and are doing a phenomenal job).
More than a year later, I got hold off the private Metaculus data via my job, luckily the Metaculus admins also allowed me to use it for my private investigations.
I subsequently tried to replicate my previous findings with the private data, finding that the findings mostly replicate, or are only slightly different.
This analysis is still quite lacking in several aspects and could be significantly improved.
Metaculus Dataset is Only Community Timeseries
The Metaculus dataset and the PredictionBook dataset are quite different: For PredictionBook, the full dataset of all predictions is available, while the Metaculus API only offers data of the
weighted average of the community as a timeseries (with ≤101 datapoints). Due to this limitation, the PredictionBook results and the Metaculus results can't be easily compared.
This is the reason why I reported the results for the Metaculus dataset and the PredictionBook dataset separately, so that future work can work either with aggregated timeseries data or with full
datasets of individual forecasts.
PredictionBook Forecasts can be Resolved by Anyone
PredictionBook, unlike Metaculus, makes no attempts to generate a shared notion of ground truth: Any user can resolve any question as they like, with the question-writer having the final verdict.
This would make it quite easy to manipulate the dataset.
In contrast, Metaculus has a set of admins and moderators that share a notion of how the questions relate to events in the world, which keeps questions and resolutions consistent with each other.
I am grateful to Nuño Sempere for pointing out a fatal flaw in my previous version of this analysis, which caused me to rewrite it nearly completely.
Im am incredibly indebted to the Long-Term Future Fund, who gave me enough money for this project that I could justify to my parents that I wasn't wasting my time, and to pay my cryonics membership
for the year on top of that.
The code for image generation can be found here, the complete code for analyzing the data can be found here.
The code for previous versions was written in Klong, but abandoned for reasons concerning performance & replicability. The previous code for analysis can be found here, the previous code for image
generation can be found here (in some ways the previous code was much nicer, especially when calculating metquestions).
Appendix A: Replicating Metaculus Findings With Full Data
After receiving the private data from the Metaculus admins, I decided to check whether my previous findings would still bear out with the full dataset, or whether equally shrinking the data on
questions with many forecasts and fewer forecasts had introduced significant amounts of bias.
Some Predictions About The Results
This also gave me an excellent opportunity to test my own calibration on the findings: Would I be correct about which findings would and wouldn't hold up to scrutiny with nicer datasets? I used
PredictionBook to record my predictions about the results before running the code on the data (relying on the reader's trust that I hadn't just run the code beforehand and predicted with the benefit
of hindsight). The private dataset was much bigger and more up-to-date than the compressed one from the API I had used.
(Yo bro, I heard you like forecasts, so I made forecasts about my forecasting research…)
So, how well did I predict the outcome of my own research? I'll use the log score for a change:
probs=[0.1, 0.45, 0.4, 0.45, 0.4, 0.95, 0.2, 0.35, 0.4, 0.2, 0.55, 0.8, 0.6, 0.55, 0.55, 0.45, 0.48, 0.45, 0.98, 0.4, 0.1, 0.38, 0.5, 0.55, 0.99, 0.55, 0.45, 0.6, 0.85, 0.83, 0.52, 0.5, 0.52, 0.45, 0.55, 0.55, 0.48, 0.6, 0.55, 0.55]
outcs=[1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0]
>>> np.mean(list(map(lambda x: math.log(x[0]) if x[1]==1 else math.log(1-x[0]), zip(probs, outcs))))
Would a uniformly random guesser have beaten me?
>>> np.mean(list(map(lambda x: math.log(x[0]) if x[1]==1 else math.log(1-x[0]), zip([0.5]*40, outcs))))
Apparently yes. So either making predictions about the replication of your own research is hard, or I'm just bad at it. Yours to decide.
Analysis & Results
I use iqisa and the private Metaculus data to reproduce the analysis.
$ python3
>>> import metaculus
>>> import numpy as np
>>> import pandas as pd
>>> import scipy.stats as sps
>>> import scipy.optimize as spo
>>> daysec=24*60*60
>>> m=metaculus.load_private_binary('.http://niplav.site/prediction_data/metaculus/private.json')
>>> pmetbriers=(m['probability']-pd.to_numeric(m['outcome']))**2
>>> pmetrngs=m['resolve_time']-(pd.to_datetime(m['timestamp'], utc=True))
>>> pmetrngs=pmetrngs.values.astype(np.int64) / (10 ** 9 * daysec)
Again we filter out forecasts with negative ranges:
>>> m=m.loc[pmetrngs>0]
>>> pmetbriers=pmetbriers[pmetrngs>0]
>>> pmetrngs=pmetrngs[pmetrngs>0]
And can see that the Brier score isn't much different from the squashed data (even though there are many more forecasts:
>>> np.mean(pmetbriers)
>>> len(pmetbriers)
Analysis Between Forecasts
>>> np.corrcoef(pmetbriers, pmetrngs)
array([[1. , 0.08314111],
[0.08314111, 1. ]])
>>> sps.linregress(pmetrngs, pmetbriers)
LinregressResult(slope=6.814357486306815e-05, intercept=0.15699128148663716, rvalue=0.08314111171996655, pvalue=0.0, stderr=1.7072944201034568e-06, intercept_stderr=0.000589651206310534)
And now onto the horizons between forecasts:
>>> def shrunk_logistic(x, slope, intercept):
... return 0.25*1/(1+np.exp(slope*x+intercept))
>>> pmetlogifit=spo.curve_fit(shrunk_logistic, pmetrngs, pmetbriers, bounds=([-np.inf, 0], [0, np.inf]))
(array([-4.98313302e-03, 5.50014913e-18]), array([[ 9.12453976e-09, -8.09152901e-07],
[-8.09152901e-07, 1.53512973e-04]]))
The exponential fit is less stocky:
>>> def shift_exp(x, b):
... return ((b**x)-1)/(-4)
>>> pmetexpfit=spo.curve_fit(shift_exp, pmetrngs, pmetbriers, bounds=([0], [1]))
(array([0.97151306]), array([[7.70042249e-08]]))
And now onto the horizons:
>>> (np.log((1/0.96)-1)-pmetlogifit[0][1])/pmetlogifit[0][0]
>>> np.log(0.04)/np.log(pmetexpfit[0][0])
The horizon for the logistic fit is ~2 years, the one for the exponential fit is less than half a year.
The mean squared errors of the two methods are
>>> np.mean((shrunk_logistic(pmetrngs, pmetlogifit[0][0], pmetlogifit[0][1])-pmetbriers)**2)
>>> np.mean((shift_exp(pmetrngs, pmetexpfit[0][0])-pmetbriers)**2)
As one can see, the logistic fit barely beats the exponential fit.
Analysis Between Questions
First we have to rewrite the code that groups the forecasts by question.
for e in np.unique(m['question_id']):
pmetquestions.append([e, questionrange, outcomes, probabilities, franges])
We now have slightly more questions (nearly twice as many!):
>>> len(pmetquestions)
We define the Brier score again, and then already calculate the Brier scores for the questions:
>>> def brier(x, y):
... return np.mean((x-y)**2)
>>> pmetqbrier=np.array([[i[1], brier(i[3], i[2])] for i in pmetquestions])
And now we're ready to calcuate the results:
>>> np.corrcoef(pmetqbrier.T)
array([[1. , 0.0173564],
[0.0173564, 1. ]])
>>> sps.linregress(pmetqbrier.T[0], pmetqbrier.T[1])
LinregressResult(slope=1.1760476061621512e-10, intercept=0.17809615527756018, rvalue=0.01735639826497777, pvalue=0.5998458799770909, stderr=2.240922169316534e-10, intercept_stderr=0.0057047415146507935)
And now we can again look at the non-linear curve-fits:
>>> pmetlogifit_betweenq=spo.curve_fit(shrunk_logistic, pmetqbrier.T[0], pmetqbrier.T[1], bounds=([-np.inf, 0], [0, np.inf]))
(array([-1., 1.]), array([[0., 0.],
[0., 0.]]))
>>> pmetexpfit_betweenq=spo.curve_fit(shift_exp, pmetqbrier.T[0], pmetqbrier.T[1], bounds=([0], [1]))
(array([0.5]), array([[0.]]))
The exponential fit gives quite odd results—maybe a bug (another one‽)?
But in the meantime, here's the (resulting) predictive horizons:
>>> (np.log(1/(0.96)-1)-pmetlogifit_betweenq[0][1])/pmetlogifit_betweenq[0][0]
>>> np.log(0.04)/np.log(pmetexpfit_betweenq[0][0])
And the MSE for both methods (where something really odd happens): The two methods have the same MSE for fitting the data.
>>> np.mean((shrunk_logistic(pmetqbrier.T[0], pmetlogifit_betweenq[0][0], pmetlogifit_betweenq[0][1])-pmetqbrier.T[1])**2)
>>> np.mean((shift_exp(pmetqbrier.T[0], pmetexpfit_betweenq[0][0])-pmetqbrier.T[1])**2)
I have no idea why this should be the case. Nice thing that I wrote the resolution criterion for the question comparing MSE unambigiously (though I still loose Brier points on this one :-/).
Analysis Within Questions
First, one can again calculate the Brier scores for each forecast on each question.
>>> pwmetqbrier=[[i[4], (i[3]-i[2])**2] for i in pmetquestions]
>>> pwmetqbrier=list(filter(lambda x: len(x[0])>1, pwmetqbrier))
And now one can compute the correlations and linear regressions:
>>> pmetqcorrs=np.array(list(map(lambda x: np.corrcoef(x)[0][1], pwmetqbrier)))
>>> np.mean(pmetqcorrs)
>>> pwmetqregs=list(map(lambda x: sps.linregress(x[0], x[1]), pwmetqbrier))
>>> clean_pmetforecasts=np.sum([len(pwmetqbrier[i][0]) for i in range(0, len(pwmetqbrier))])
>>> pawmetqslope=np.sum([len(pwmetqbrier[i][0])*pwmetqregs[i][0] for i in range(0, len(pwmetqregs))])/clean_pmetforecasts
>>> pawmetqintercept=np.sum([len(pwmetqbrier[i][0])*pwmetqregs[i][1] for i in range(0, len(pwmetqregs))])/clean_pmetforecasts
The number of forecasts on each question shouldn't be a problem here:
>>> np.sort(list(map(lambda x: len(x[0]), pwmetqbrier)))
array([ 2, 7, 11, 12, 12, 13, 16, 17, 17, 20, 24,
25, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32,
32, 33, 34, 34, 36, 36, 37, 38, 39, 39, 39,
40, 40, 41, 42, 42, 43, 44, 44, 44, 44, 45,
900, 901, 912, 924, 946, 1001, 1007, 1032, 1035, 1075, 1083,
1178, 1191, 1201, 1271, 1319, 1346, 1405, 1472, 1706, 1978, 2290,
3144, 6483, 6543])
Now one can calculate the logistic-ish fits on the private metaculus data:
for e in pwmetqbrier:
within_logi_fits_pmet.append(spo.curve_fit(shrunk_logistic, e[0], e[1], bounds=([-np.inf, 0], [0, np.inf])))
except RuntimeError:
Again there are datapoints for which the logistic fit fails (two of them: len(list(filter(lambda x: len(x)==0, within_logi_fits_pmet)))==2).
Now, onto the logistic horizons:
>>> within_logi_fits_pmet_filt=list(filter(lambda x: len(x)>0, within_logi_fits_pmet))
>>> pmet_logi_horizons=[(np.log((1/0.96)-1)-f[0][1])/f[0][0] for f in within_logi_fits_pmet_filt]
>>> np.mean(pmet_logi_horizons)
>>> np.median(pmet_logi_horizons)
>>> import statistics
>>> statistics.mode(pmet_logi_horizons)
>>> np.var(pmet_logi_horizons)
>>> np.max(pmet_logi_horizons)
>>> np.min(pmet_logi_horizons)
And onto the exponential-ish horizons:
>>> within_exp_fits_pmet=[spo.curve_fit(shift_exp, e[0], e[1], bounds=([0], [1])) for e in pwmetqbrier]
>>> pmet_exp_horizons=[np.log(0.04)/np.log(f[0][0]) for f in within_exp_fits_pmet]
>>> np.mean(pmet_exp_horizons)
>>> np.median(pmet_exp_horizons)
>>> statistics.mode(pmet_exp_horizons)
>>> np.var(pmet_exp_horizons)
>>> np.max(pmet_exp_horizons)
>>> np.min(pmet_exp_horizons)
Replication Inbound?
And that concludes the replication attempt. Does it replicate the original findings?
• Between forecasts
□ The positive correlation between range and accuracy stays the same ✓
□ The linear regression is similar too (very small positive slope, intercept at ~0.16). ✓
□ The logistic-ish fit gives values within ~2 orders of magnitude for the slope & intercept, and both have the right sign. ✓
□ The exponential-ish fit, though, gives a much different value close to 1, while the original was one close to zero. ✗
□ The logistic-ish horizon is ~half as long as the one with the restricted data. ~
□ The exponential-ish horizon is ~60 times shorter than in the original finding. ✗
□ The logistic-ish fit is better than the exponential-ish fit, again. ✓
• Between questions
□ The correlation between range and accuracy flips sign, but stays very small. ✗
□ The slope of the linear regression also flips sign, but the intercept is remarkably similar. ~
□ The p-value is still abysmal. ✓
□ The sign of the logistic-ish fit is the same for both parameters, though the original parameters were more extreme. ~
□ The exponential-ish fit has a slightly smaller parameter, which discounts even more aggressively. ~
□ The logistic-ish predictive horizon is sightly smaller, but in the same ballpark. ✓
□ The exponential-ish horizon is ~half as long. ~
□ The mean squared errors of both methods are again remarkably close. ✓
• Within questions
□ Slope for linear regressions is surprising close, intercept off by a factor of ~2. Still ok. ✓
□ Mean logistic-ish horizon is ~3 orders of magnitude away from original estimate, median & mode are the same (?), variance is smaller (~5 orders of magnitude), maximum & minimum are kind of
close. Estimating the slope of a sigmoid is difficult. ~
□ Mean exponential-ish horizon is factor ~7 smaller than original estimate, median & mode again are the same, variance smaller, maximum smaller and minimum slightly larger. ~
Appendix B: Statistical Significance of Truncated Datasets
One could also be interested in how the statistical significance of the linear regression and correlation develops when we remove the forecasts with short ranges. This can be implemented quite easily
by creating a function val_shrinking_dataset which first sorts the pre-computed Brier scores by range, and then calculates p-values and correlation coefficients, afterwards removing the earliest
prediction from the dataset (I have no idea whether this is statistically acceptable, but it seemed like a reasonable thing to do, lest there's some problem here with p-hacking). The values are
concatenated into arrays, which are then returned.
def val_shrinking_dataset(briers, ranges):
for i in range(0, len(ranges)-2):
_, _, rval, pval, _=sps.linregress(chronranges, chronbriers)
return np.vstack([pvalues, rvalues, dropranges])
metpvals=val_shrinking_dataset(metbriers, metrngs)
pbpvals=val_shrinking_dataset(pbbriers, pbrngs)
The resulting data can be plotted (correlation cofficients on the left y-axis, p-values on the (logarithmic) right y-axis). Here, the datapoint at range x would be the correlation coefficient and its
p-value for all Brier scores after x (sorry for the legend in the upper right corner, I couldn't figure out how to move it to the right middle-bottom).
fig=plt.figure(figsize=(10,10), clear=True)
_, ax1 = plt.subplots()
ax1.set_xlabel("Range (months)")
ax1.set_ylabel("Correlation value")
ax1.plot(metpvals[2], metpvals[1], '-', linewidth=3, color='#ff4500', label="Metaculus truncated correlations")
ax1.plot(pbpvals[2], pbpvals[1], '-', linewidth=3, color='#00bfff', label="PredictionBook truncated correlations")
ax1.legend(loc='lower right')
ax2.set_ylabel("p value")
ax2.semilogy(metpvals[2], metpvals[0], '-', color='#ffa500', linewidth=1, label="Metaculus truncated p-values")
ax2.semilogy(pbpvals[2], pbpvals[0], '-', color='cyan', linewidth=1, label="PredictionBook truncated p-values")
ax2.legend(loc='upper right')
And the same chart, just without the Metaculus data to make it easier to read (allegedly):
These graphs are quite interesting in several regards. First, we can see that the correlation coefficients don't have a clear development as one removes forecasts with low ranges from the dataset:
for Metaculus the correlation first goes down, then up again. (This might be an artifact of having very little data for long ranges in the Metaculus dataset, though). PredictionBook data is a bit
more consistent: the correlation between range and accuracy rises the more early datapoints we remove (again with the coefficient flailing around in the end because it just doesn't have enough data).
But truly odd is the p-value here: The dip in the correlation coefficient for Metaculus data is judged to be extremely unlikely to have occurred randomly, down to , so that it makes the chart nearly
unreadable even on a log-scale.
I am…not quite sure what to make of this. Intuitively, I would expect the correlation between range and accuracy to start out strong with the whole dataset, and then become weaker, noisier and more
likely to be random the more values I remove from the start of the dataset. Perhaps the short-range Metaculus forecasts just introduce a bunch of noise, because people scramble to get a last
prediction in? But that doesn't explain why the correlation is then negative with an extremely low p-value with predictions with a range of less than 18 months removed.
The changes in Metaculus correlations might coincide with new years, perhaps?
For PredictionBook, there is a more straight-forward story to tell, namely that short-range predictions seem to just introduce noise, and the longer the range, the stronger the correlation (although
the relation to p-values doesn't look strong in any way, I wonder whether there is a way to test that).
Appendix C: Quotes About the Horizon of Forecasts
Demographers disagree about many things, but not that the further into the future we try to look, the more likely our forecasts are to be wrong.
— Richard J. Hernstein & Charles Murray, “The Bell Curve” ch. 15, 1994
The future is hard to predict. We may feel confident that eventually space will be colonized, or that eventually we'll make stuff by putting each atom just where we want it. But so many other
changes may happen before and during those changes that it is hard to say with much confidence how space travel or nanotechnology may affect the ordinary person. Our vision seems to fade into a
fog of possibilities.
The scenario I am about to describe excites me because it seems an exception to this general rule --[sic] more like a crack of dawn than a fog, like a sharp transition with sharp implications
regardless of the night that went before. Or like a sight on the horizon much clearer than the terrain inbetween. And, as scenarios go, this one seems rather likely. Here it is.
— Robin Hanson, “If Uploads Come First”, 1994
As time passes, the outcomes of most events become more predictable. It is therefore important to update probability estimates. We did this in the aggregation method by using an exponential decay
(a time constant of a couple days was optimal in most of our tests), so that out-of-date predictions counted less.
— Lyle Ungar et al., “The Good Judgment Project: A large scale test of different methods of combining expert predictions” p. 3, 2012
It was easiest to beat chance on the shortest-range questions that only required looking one year out, and accuracy fell off the further out experts tried to forecast—approaching the
dart-throwing-chimpanzee level three to five years out.
— Philip E. Tetlock & Dan Gardner, “Superforecasting” p. 12, 2015
Take something as seemingly simple as the relationship between time and predictability: it is generally true that the further we try to look into the future, the harder it is to see. But there
can be prolonged exceptions to the rule. Predicting the continuation of a long bull market in stocks can prove profitable for many years—until it suddenly proves to be your undoing. And
predicting that dinosaurs would continue to preside at the top of the food chain was a safe bet for tens of millions of years—until an asteroid set off a cataclysm that opened up ecological
niches for a tiny mammal that eventually evolved into a species that tries to predict the future.
The further out the forecaster tries to look, the more opportunity there is for chaos to flap its butterfly wings and blow away expectations. Big leaps in computing power and continued refinement
of forecasting models may nudge the limits a little further into the future but those advances gradually get harder and the payoffs shrink toward zero. How good can it get? No one knows. But
knowing the current limits is itself a success.
— Philip E. Tetlock & Dan Gardner, “Superforecasting” p. 20/21, 2015
Across all four years of the tournament, superforecasters looking out three hundred days were more accurate than regular forecasters looking out one hundred days. In other words, regular
forecasters needed to triple their foresight to see as far as superforecasters.
— Philip E. Tetlock & Dan Gardner, “Superforecasting” p. 94, 2015
Taleb, Kahneman, and I agree there is no evidence that geopolitical or economic forecasters can predict anything ten years out beyond the excruciatingly obvious—“there will be conflicts”—and the
odd lucky hits that are inevitable whenever lots of forecasters make lots of forecasts. These limits on predictability are the predictable results of the butterfly dynamics of nonlinear systems.
In my EPJ research, the accuracy of expert predictions declined toward chance five years out.
— Philip E. Tetlock & Dan Gardner, “Superforecasting” p. 234, 2015
The future is usually "absurd" - it is unstable in its surface rules over fifty-year intervals.
— Eliezer Yudkowsky, “Absurdity Heuristic, Absurdity Bias”, 2007
Successfully predicting the unimaginably far future - that is, more than 2 or 3 years out, or sometimes less - is something that human beings seem to be quite bad at, by and large.
— Eliezer Yudkowsky, “Biology-Inspired AGI Timelines: The Trick That Never Works”, 2021
Near-term is more predictable and hence conclusions are more likely to be true, but far futures may be more entertaining since more extreme things are possible.
—Anders Sandberg, “Grand Futures” p. 32, 2023
Two main indicators of forecasting ease are the forecast's size and time horizon. More extreme probability estimates reflect greater certainty, which may correlate with easier questions.
Nearer-term events may also be easier for analysts to predict.
—Barbara A. Mellers et al., “The Value of Precision in Probability Assessment: Evidence from a Large-Scale Geopolitical Forecasting Tournament” p. 7, 2018
niplav 1
I submit this here because this work was supported by the Long-Term Future fund. It's not fully done, but in (I hope) a presentable state (most notably, it's lacking a literature review and a look at
continuous predictions, as well as more & newer datasets).
niplav 5
Just read the results section? That's why I put it on top! (If it isn't clear enough, the illustrative example might be helpful as well).
Kids these days…
Mentioned in
More posts like this
Sorted by Click to highlight new comments since:
One-time pad -2
This report, by its very length, defends itself against the risk of being read.
Don't get me wrong, it sounds like a really interesting and important question with big implications for EA. Perhaps an executive summary would be helpful? | {"url":"https://forum.effectivealtruism.org/posts/nfEWwLH8qSqNATxmr/range-and-forecasting-accuracy","timestamp":"2024-11-06T08:04:11Z","content_type":"text/html","content_length":"1049255","record_id":"<urn:uuid:6fc31681-a44c-4d4c-80c4-439214c04d94>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00312.warc.gz"} |