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reduced suspension
Homotopy theory
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Paths and cylinders
Homotopy groups
Basic facts
Reduced suspension
For a pointed topological space $(X, x)$, its reduced suspension $\Sigma (X,x)$ is obtained from the plain suspension
$\mathrm{S}X \;\coloneqq\; \frac{ X \,\times\, [-1,\, +1] }{ \left( \begin{array}{c} X \times \{-1\}\mathrlap{,\,} \\ X \times \{+1\} \end{array} \right) }$
of the underlying topological space $X$ by collapsing the meridian through the basepoint $x$ itself to a point — this making $\Sigma(X,x)$ itself a pointed topological space with basepoint the
equivalence class of that meridian $mer(x)$:
$\Sigma (X,x) \;\coloneqq\; \frac{ \mathrm{S}X }{ \{x\} \times [-1,1] } \;\;\; \in \;\; Top^{\ast/} \,.$
(Notice that this identifies in particular also the two antipodal “poles” of the plain suspension.)
If $X$ admits the structure of a CW-complex then, under passage to the classical homotopy category of pointed topological spaces (cf. here) this construction models the homotopy pushout of the
terminal map $(X,x) \to (\ast,pt)$ along itself, which explains its prevalence in homotopy theory (especially in stable homotopy theory, see also at suspension spectrum).
Moreover, in this case of CW-complexes the underlying space of $\Sigma (X,x)$ (i.e. forgetting its basepoint) is weakly homotopy equivalent to the plain suspension $\mathrm{S} X$ of the underlying
space $X$ of $(X,x)$. In this sense, reduced suspension in the context of homotopy theory may be understood as just being plain suspension but with basepoints taken into account.
For $(X,x)$ a pointed topological space, then its reduced suspension $\Sigma X$ is equivalently the following:
• obtained from the standard cylinder $I\times X$ (product topological space with the closed interval $I = [0,1]$) by identifying the subspace $(\{0,1\}\times X) \cup (I\times \{x\})$ to a point,
i.e. the quotient space (this example)
$(X \times [0,1])/ ( ( X \times \{0,1\} ) \cup ([0,1] \times \{x\}) )$
(Think of crushing the two ends of the cylinder and the line through the base point to a point.)
• obtained from the plain suspension
$S X = (X \times [0,1])/( X \times \{0\}, X \times \{1\})$
of $X$ by passing to the quotient space which collapses $\{x\} \times I$ to a point (this example)
$\Sigma X \simeq S X / ( \{x\} \times I )$
For the purposes of generalized (Eilenberg-Steenrod) cohomology theory typically it does not matter whether one evaluates on the standard suspension or the reduced suspension. For example for
topological K-theory since $\{x\} \times I$ is a contractible closed subspace, then this prop. says that topological vector bundles do not see a difference as long as $X$ is a compact Hausdorff
• obtained from the reduced cylinder by collapsing the two ends, i.e. the cofiber
$\Sigma X \simeq cofib(X \vee X \to X \wedge (I_+))$
• the mapping cone in pointed topological spaces formed with respect to the reduced cylinder $X \wedge (I_+)$ of the map $X \to \ast$;
• the smash product $S^1\wedge X$, of $X$ with the circle (based at some point) with $X$.
$\Sigma X \simeq S^1 \wedge X \,.$
Relation to suspension
For CW-complexes $X$ that are also pointed, with the point identified with a 0-cell, then their reduced suspension is weakly homotopy equivalent to the ordinary suspension: $\Sigma X \simeq S X$.
Cogroup structure
suspensions are H-cogroup objects
Up to homeomorphism, the reduced suspension of the $n$-sphere is the $(n+1)$-sphere
$\Sigma S^n \simeq S^{n+1} \,.$
See at one-point compactification – Examples – Spheres for details.
Discussion of (reduced) suspension may be found in most introductions to homotopy theory (for discussion of unreduced suspension see also there).
For instance:
• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, §2.10 in: Algebraic topology from a homotopical viewpoint, Springer (2008) [doi:10.1007/b97586]
• Jeffrey Strom, §3.8 and §17 in: Modern classical homotopy theory, Graduate Studies in Mathematics 127, American Mathematical Society (2011) [doi:10.1090/gsm/127]
• Martin Arkowitz, Loop spaces and suspensions, §2.3 in: Introduction to Homotopy Theory, Springer (2011) [doi:10.1007/978-1-4419-7329-0]
Review in the context of stable homotopy theory: | {"url":"https://ncatlab.org/nlab/show/reduced%20suspension","timestamp":"2024-11-11T00:58:15Z","content_type":"application/xhtml+xml","content_length":"65856","record_id":"<urn:uuid:e95adbea-53c1-46be-b498-64e305c58dc1>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00876.warc.gz"} |
Quadratic Equation - Formula, Examples | Quadratic Formula - [[company name]] [[target location]], [[stateabr]]
Quadratic Equation Formula, Examples
If you going to try to figure out quadratic equations, we are enthusiastic about your journey in mathematics! This is indeed where the most interesting things starts!
The details can appear enormous at start. Despite that, offer yourself a bit of grace and space so there’s no pressure or stress while solving these questions. To be competent at quadratic equations
like an expert, you will require a good sense of humor, patience, and good understanding.
Now, let’s start learning!
What Is the Quadratic Equation?
At its heart, a quadratic equation is a math equation that portrays distinct scenarios in which the rate of change is quadratic or proportional to the square of few variable.
Though it seems like an abstract idea, it is simply an algebraic equation stated like a linear equation. It usually has two answers and utilizes intricate roots to work out them, one positive root
and one negative, using the quadratic equation. Unraveling both the roots the answer to which will be zero.
Meaning of a Quadratic Equation
Primarily, bear in mind that a quadratic expression is a polynomial equation that comprises of a quadratic function. It is a second-degree equation, and its standard form is:
ax2 + bx + c
Where “a,” “b,” and “c” are variables. We can employ this formula to solve for x if we plug these terms into the quadratic formula! (We’ll look at it next.)
All quadratic equations can be scripted like this, that makes working them out simply, comparatively speaking.
Example of a quadratic equation
Let’s contrast the ensuing equation to the previous formula:
x2 + 5x + 6 = 0
As we can see, there are 2 variables and an independent term, and one of the variables is squared. Therefore, linked to the quadratic equation, we can surely tell this is a quadratic equation.
Usually, you can observe these types of formulas when scaling a parabola, which is a U-shaped curve that can be graphed on an XY axis with the data that a quadratic equation gives us.
Now that we understand what quadratic equations are and what they appear like, let’s move forward to solving them.
How to Work on a Quadratic Equation Employing the Quadratic Formula
Although quadratic equations may look very complex when starting, they can be broken down into multiple simple steps utilizing a straightforward formula. The formula for working out quadratic
equations includes setting the equal terms and applying rudimental algebraic functions like multiplication and division to get two solutions.
Once all functions have been carried out, we can figure out the values of the variable. The results take us single step closer to discover result to our first problem.
Steps to Solving a Quadratic Equation Utilizing the Quadratic Formula
Let’s quickly put in the common quadratic equation once more so we don’t forget what it seems like
ax2 + bx + c=0
Ahead of working on anything, keep in mind to separate the variables on one side of the equation. Here are the three steps to work on a quadratic equation.
Step 1: Write the equation in conventional mode.
If there are terms on either side of the equation, add all alike terms on one side, so the left-hand side of the equation equals zero, just like the conventional mode of a quadratic equation.
Step 2: Factor the equation if workable
The standard equation you will conclude with should be factored, generally through the perfect square process. If it isn’t possible, put the terms in the quadratic formula, that will be your best
friend for solving quadratic equations. The quadratic formula seems similar to this:
Every terms correspond to the equivalent terms in a conventional form of a quadratic equation. You’ll be employing this significantly, so it is wise to remember it.
Step 3: Implement the zero product rule and work out the linear equation to eliminate possibilities.
Now once you possess 2 terms resulting in zero, work on them to obtain 2 answers for x. We possess two results due to the fact that the solution for a square root can either be negative or positive.
Example 1
2x2 + 4x - x2 = 5
Now, let’s fragment down this equation. First, streamline and put it in the conventional form.
x2 + 4x - 5 = 0
Now, let's recognize the terms. If we compare these to a standard quadratic equation, we will get the coefficients of x as follows:
To figure out quadratic equations, let's plug this into the quadratic formula and solve for “+/-” to involve each square root.
We figure out the second-degree equation to get:
Now, let’s clarify the square root to attain two linear equations and work out:
x=-4+62 x=-4-62
x = 1 x = -5
After that, you have your answers! You can review your work by checking these terms with the original equation.
12 + (4*1) - 5 = 0
1 + 4 - 5 = 0
-52 + (4*-5) - 5 = 0
25 - 20 - 5 = 0
This is it! You've figured out your first quadratic equation utilizing the quadratic formula! Congrats!
Example 2
Let's try one more example.
3x2 + 13x = 10
Initially, put it in the standard form so it equals 0.
3x2 + 13x - 10 = 0
To figure out this, we will substitute in the numbers like this:
a = 3
b = 13
c = -10
Work out x utilizing the quadratic formula!
Let’s clarify this as far as workable by solving it exactly like we executed in the prior example. Figure out all simple equations step by step.
You can work out x by taking the positive and negative square roots.
x=-13+176 x=-13-176
x=46 x=-306
x=23 x=-5
Now, you have your solution! You can check your workings using substitution.
3*(2/3)2 + (13*2/3) - 10 = 0
4/3 + 26/3 - 10 = 0
30/3 - 10 = 0
10 - 10 = 0
3*-52 + (13*-5) - 10 = 0
75 - 65 - 10 =0
And that's it! You will solve quadratic equations like a professional with a bit of patience and practice!
Given this summary of quadratic equations and their rudimental formula, children can now take on this challenging topic with assurance. By starting with this easy explanation, children gain a strong
understanding before taking on further complicated concepts ahead in their academics.
Grade Potential Can Help You with the Quadratic Equation
If you are fighting to understand these ideas, you may need a mathematics instructor to guide you. It is better to ask for guidance before you fall behind.
With Grade Potential, you can understand all the tips and tricks to ace your subsequent mathematics test. Become a confident quadratic equation solver so you are prepared for the ensuing intricate
ideas in your math studies. | {"url":"https://www.sanantonioinhometutors.com/blog/quadratic-equation-formula-examples-quadratic-formula","timestamp":"2024-11-15T02:36:24Z","content_type":"text/html","content_length":"79778","record_id":"<urn:uuid:4cf8931e-4011-4ae0-b8fc-a29a7458ef14>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00043.warc.gz"} |
Ensuring Randomness with Linux's Random Number Generator
Flickr/mark van de wouw
When building secure systems, having a source of random numbers is essential. Without them, most cryptographic systems break down and the privacy and authenticity of communications between two
parties can be subverted. For example, if you’re reading this using a link to https://blog.cloudflare.com then the SSL connection you are using will have required random numbers to ensure its
security (they were used as part of the establishment of the secure connection).
We’ve covered why secure systems require random numbers in a previous blog post, but getting random numbers from a computer is very hard. This blog post looks at Linux’s internal random number
generator and how it overcomes the problem of generating random numbers on a machine that’s anything but random.
CloudFlare’s servers require a good source of random numbers for authentication and to assure perfect forward secrecy in SSL. But, internally, the computers we all use are deterministic machines that
follow instructions and are required to do so in a predictable manner. Uncertainty and unpredictability are not built in: there is no easy way to tell a computer to go flip a coin or roll some dice.
To get randomness in a computer it has to be looked for in the outside world.
Consumer computers and mobile devices have a number of sensors that provide unpredictable input. The timing of keystrokes and mouse movements of a user will have some degree of randomness if measured
closely enough. Noise from microphones and cameras can also provide a lot of randomness. Mobile devices have even more sources including fluctuating wifi signals, motion sensors and GPS information.
Most of these sensors are not available on servers where random numbers are needed most. This is especially true for servers that run in virtualized environments that might not have access to a
precise system clock. For CloudFlare’s servers, we currently rely on the random number generator built into the Linux operating system.
Linux is one of the most popular operating systems in the world. It serves as the operating system for everything from the web servers and data centers of many the largest sites in the world (Google,
Facebook, Amazon, Apple, etc.), to desktop computers (Ubuntu, Chrome OS, etc.) to embedded devices (smart TVs, Android, etc.). CloudFlare’s software is built on the solid foundation of the Linux
operating system kernel.
Linux itself provides a random number service so that any program has access to random numbers at any time. Luckily for us, Linux is open source software and we can learn how it works by reading the
code. And verify that it provides a suitable source of random numbers for our cryptographic purposes.
Entropy and Randomness
Not all randomness is created equally. There are two sorts of randomness to think about: uniformity and unpredictability. A random number generator provides ‘uniform’ output if all numbers will come
up equally often if run long enough. That’s useful for modeling random processes, but not good enough for security.
For computer security, random numbers need to be hard to guess: they need to be unpredictable. The predictability of numbers is quantified in a measure called entropy.
If a fair coin is tossed it provides one bit of entropy: the coin lands with equal probability on heads or tails (which can be thought of as 0 and 1). Because the probability is equal there’s no
predictability in the coin’s ‘output’. We say it provides one bit of entropy.
An unfair coin toss provides less than one bit, since it’s much easier to guess when you know the bias. Flipping a coin with heads on both sides provides no entropy, since the result of a coin toss
can be guessed with absolute certainty.
Entropy is distinct from statistical randomness. Looking at the statistical properties of a stream of numbers does not guarantee that the stream contains any entropy. For example, the digits of pi
look random by almost any statistical measure, but contain no entropy since there is a well known formula to calculate them and perfectly predict the next value. (As an aside, pi is an example of a
normal number: one where all the digits will appear in equal quantities).
Flickr/foxtongue License: CC Attribution 2.0 Generic
Also, large numbers do not always have high entropy. You can take a small random number and turn it into a large random number and the entropy remains the same. For example, take a random number from
1 to 16 and compute its cryptographic hash with an algorithm like SHA-1. The resulting 160 bit number looks very random, but it is only one of only 16 possible such numbers. Guessing the number is
just as easy as guessing a random number from 1 to 16. It’s the size of the pool from which random numbers are drawn that matters.
For cryptographic keys, the amount of entropy used to create them is tied to how hard they are to guess. A 128 bit key created from a source with 20 bits of entropy is no more secure than a 20 bit
key. A good source of entropy is necessary to create secure keys.
Take a dip in the pool
On Linux, the root of all randomness is something called the kernel entropy pool. This is a large (4,096 bit) number kept privately in the kernel’s memory. There are 2^4096 possibilities for this
number so it can contain up to 4,096 bits of entropy. There is one caveat - the kernel needs to be able to fill that memory from a source with 4,096 bits of entropy. And that’s the hard part: finding
that much randomness.
The entropy pool is used in two ways: random numbers are generated from it and it is replenished with entropy by the kernel. When random numbers are generated from the pool the entropy of the pool is
diminished (because the person receiving the random number has some information about the pool itself). So as the pool’s entropy diminishes as random numbers are handed out, the pool must be
Replenishing the pool is called stirring: new sources of entropy are stirred into the mix of bits in the pool.
This is the key to how random number generation works on Linux. If randomness is needed, it’s derived from the entropy pool. When available, other sources of randomness are used to stir the entropy
pool and make it less predictable. The details are a little mathematical, but it’s interesting to understand how the Linux random number generator works as the principles and techniques apply to
random number generation in other software and systems.
The kernel keeps a rough estimate of the number of bits of entropy in the pool. You can check the value of this estimate through the following command:
cat /proc/sys/kernel/random/entropy_avail
A healthy Linux system with a lot of entropy available will have return close to the full 4,096 bits of entropy. If the value returned is less than 200, the system is running low on entropy.
The kernel is watching you
I mentioned that the system takes other sources of randomness and uses this to stir the entropy pool. This is achieved using something called a timestamp.
Most systems have precise internal clocks. Every time that a user interacts with a system, the value of the clock at that time is recorded as a timestamp. Even though the year, month, day and hour
are generally guessable, the millisecond and microsecond are not and therefore the timestamp contains some entropy. Timestamps obtained from the user’s mouse and keyboard along with timing
information from the network and disk each have different amount of entropy.
How does the entropy found in a timestamp get transferred to the entropy pool? Simple, use math to mix it in. Well, simple if you like math.
Just mix it in
A fundamental property of entropy is that it mixes well. If you take two unrelated random streams and combine them, the new stream cannot have less entropy. Taking a number of low entropy sources and
combining them results in a high entropy source.
All that’s needed is the right combination function: a function that can be used to combine two sources of entropy. One of the simplest such functions is the logical exclusive or (XOR). This truth
table shows how bits x and y coming from different random streams are combined by the XOR function.
Even if one source of bits does not have much entropy, there is no harm in XORing it into another source. Entropy always increases. In the Linux kernel, a combination of XORs is used to mix
timestamps into the main entropy pool.
Generating random numbers
Cryptographic applications require very high entropy. If a 128 bit key is generated with only 64 bits of entropy then it can be guessed in 2^64 attempts instead of 2^128 attempts. That is the
difference between needing a thousand computers running for a few years to brute force the key versus needing all the computers ever created running for longer than the history of the universe to do
Cryptographic applications require close to one bit of entropy per bit. If the system’s pool has fewer than 4,096 bits of entropy, how does the system return a fully random number? One way to do this
is to use a cryptographic hash function.
A cryptographic hash function takes an input of any size and outputs a fixed size number. Changing one bit of the input will change the output completely. Hash functions are good at mixing things
together. This mixing property spreads the entropy from the input evenly through the output. If the input has more bits of entropy than the size of the output, the output will be highly random. This
is how highly entropic random numbers are derived from the entropy pool.
The hash function used by the Linux kernel is the standard SHA-1 cryptographic hash. By hashing the entire pool and and some additional arithmetic, 160 random bits are created for use by the system.
When this happens, the system lowers its estimate of the entropy in the pool accordingly.
Above I said that applying a hash like SHA-1 could be dangerous if there wasn’t enough entropy in the pool. That’s why it’s critical to keep an eye on the available system entropy: if it drops too
low the output of the random number generator could have less entropy that it appears to have.
Running out of entropy
One of the dangers of a system is running out of entropy. When the system’s entropy estimate drops to around the 160 bit level, the length of a SHA-1 hash, things get tricky, and how they effect
programs and performance depends on which of two Linux random number generators are used.
Linux exposes two interfaces for random data that behave differently when the entropy level is low. They are /dev/random and /dev/urandom. When the entropy pool becomes predictable, both interfaces
for requesting random numbers become problematic.
When the entropy level is too low, /dev/random blocks and does not return until the level of entropy in the system is high enough. This guarantees high entropy random numbers. If /dev/random is used
in a time-critical service and the system has not incorporated a minimum amount of entropy, the delays could be detrimental to the quality of service.
On the other hand, /dev/urandom does not block. It continues to return the hashed value of its entropy pool even though there is little to no entropy in it. This data is not necessarily low-entropy.
As long as the entropy pool has incorporated enough entropy, it's likely not to be catastrophic to use. However, if the entropy pool has not been seeded, this data is not suited for cryptographic
The solution to the problem is to simply add more entropy into the system.
Hardware random number generation to the rescue?
Intel’s Ivy Bridge family of processors have an interesting feature called “secure key." These processors contain a special piece of hardware inside that generates random numbers. The single assembly
instruction RDRAND returns allegedly high entropy random data derived on the chip.
It has been suggested that Intel’s hardware number generator may not be fully random. Since it is baked into the silicon, that assertion is hard to audit and verify. As it turns out, even if the
numbers generated have some bias, it can still help as long as this is not the only source of randomness in the system. Even if the random number generator itself had a back door, the mixing property
of randomness means that it cannot lower the amount of entropy in the pool.
On Linux, if a hardware random number generator is present, the Linux kernel will use the XOR function to mix the output of RDRAND into the hash of the entropy pool. This happens here in the Linux
source code (the XOR operator is ^ in C).
Third party entropy generators
Hardware number generation is not available everywhere, and the sources of randomness polled by the Linux kernel itself are somewhat limited. For this situation, a number of third party random number
generation tools exist. Examples of these are haveged, which relies on processor cache timing, audio-entropyd and video-entropyd which work by sampling the noise from an external audio or video input
device. By mixing these additional sources of locally collected entropy into the Linux entropy pool, the entropy can only go up.
A diversity of sources
The main thing to understand is that better randomness comes through diversity. Taking a variety of sources of random data and mixing them together results in better random numbers. For servers, this
should include data local to the machine (hardware random number generator, network timing) along with sources derived externally in a safe location.
Looking ahead
In addition to the sources described above, there are many sources of random numbers to be harvested. These include lava lamps, space noise and the quantum properties of light. CloudFlare is working
on a system to ensure high quality random numbers to all of our servers by adding new sources into the system Linux currently provides. As these systems come online over the coming months, we will
share the details with the community.
Cloudflare's connectivity cloud protects entire corporate networks, helps customers build Internet-scale applications efficiently, accelerates any website or Internet application, wards off DDoS
attacks, keeps hackers at bay, and can help you on your journey to Zero Trust.
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To learn more about our mission to help build a better Internet, start here. If you're looking for a new career direction, check out our open positions. | {"url":"https://blog.cloudflare.com/ensuring-randomness-with-linuxs-random-number-generator/","timestamp":"2024-11-06T09:00:35Z","content_type":"text/html","content_length":"272853","record_id":"<urn:uuid:9780db43-02e7-40ab-8c22-92d627fc0e45>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00205.warc.gz"} |
Next-to-Leading-Order QCD Corrections to Higgs Boson Production Plus Three Jets in Gluon Fusion
Next-to-Leading-Order QCD Corrections to Higgs Boson Production Plus Three Jets
in Gluon Fusion
G. Cullen,1H. van Deurzen,2N. Greiner,2G. Luisoni,2P. Mastrolia,2,3E. Mirabella,2G. Ossola,4,5 T. Peraro,2and F. Tramontano6,7
1[Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany] 2
Max-Planck-Institut fu¨r Physik, Fo¨hringer Ring 6, 80805 Mu¨nchen, Germany
3[Dipartimento di Fisica e Astronomia, Universita` di Padova, and INFN Sezione di Padova, via Marzolo 8, 35131 Padova, Italy] 4[Physics Department, New York City College of Technology, The City
University of New York,]
300 Jay Street, Brooklyn, New York 11201, USA
5[The Graduate School and University Center, The City University of New York, 365 Fifth Avenue, New York, New York 10016, USA] 6[Dipartimento di Fisica, Universita` degli studi di Napoli ‘‘Federico
II’’, I-80125 Napoli, Italy]
7[INFN, Sezione di Napoli, I-80125 Napoli, Italy] (Received 24 July 2013; published 23 September 2013)
We report on the calculation of the cross section for Higgs boson production in association with three jets via gluon fusion, at next-to-leading-order (NLO) accuracy in QCD, in the infinite top-mass
approximation. After including the complete NLO QCD corrections, we observe a strong reduction in the scale dependence of the result, and an increased steepness in the transverse momentum
distributions of both the Higgs boson and the leading jets. The results are obtained with the combined use ofGOSAM,
SHERPA, and theMADDIPOLE-MADEVENTframework.
DOI:10.1103/PhysRevLett.111.131801 PACS numbers: 14.80.Bn, 12.38.Bx
The latest results reported by the ATLAS and CMS collaborations have confirmed with a higher confidence level the existence of a new neutral boson with mass of about 125–126 GeV and spin different
from one [1,2], and suggest that the new particle has indeed the features of a Higgs boson, thus confirming the validity of the electro-weak symmetry breaking mechanism. Although the evi-dence
accumulated so far is compatible with the hypothesis that the new resonance is the Higgs particle predicted by the Standard Model (SM) with the JP[¼ 0]þ[[][3][,][4][], in order] to confirm its
nature, further high-precision studies on spin, parity, coupling strengths, and branching ratios are mandatory.
In pp collisions, the dominant Higgs production mecha-nism proceeds via gluon fusion (GF), gg ! H, where the coupling of the Higgs boson to the gluons is mediated by a heavy quark loop.
Another important production channel for the Higgs boson is vector boson fusion (VBF), since it allows a direct measurement of the coupling of the Higgs boson to the massive electroweak bosons [5].
The cross section in the VBF channel is about an order of magnitude smaller than in GF, and even after applying specific cuts, the latter remains the main source of background for Higgs produc-tion
in VBF.
The experimental signature of this channel is character-ized by the presence of two jets separated by a large rapidity gap. Extra jet radiation is suppressed in VBF [6] and it is vetoed in the
experimental analysis to enhance the signal-to-background ratio. Furthermore, the NLO compu-tation for Higgs plus two jets in GF describes the distribu-tions for extra jets only at LO, and the effect
of a jet veto
challenges in general the validity of the numerical results, and especially of its uncertainty. The NLO computation of the cross section for the Higgs boson plus three jets in GF considered here can
be used to asses the effect of such extra jet-radiation veto, and reduce the theoretical error related to it.
The calculation of higher order corrections for the GF production of a Higgs boson in association with jets has received a lot of attention in the theory community over the past decades [7–9].
The leading order (LO) contribution to the production of a Higgs boson in association with two jets (Hjj), and three jets (Hjjj) have been computed respectively in Refs. [10,11], and in the recent
Ref. [12]. These calcula-tions have been performed retaining the full top-mass (mt) dependence, and showed the validity of the large top-mass approximation (mt! 1) whenever the mass of the Higgs
particle and the pT of the jets are not sensibly larger than the mass of the top quark. In this approximation, the Higgs boson coupling to two gluons, which at LO is mediated by a top-quark loop,
becomes independent of m[t], and it can be described by an effective operator [13], as
Leff ¼ geff
4 Htr ðGGÞ: (1) In theMS scheme, the coefficient geff reads [14,15]
g[eff] ¼ s 3v 1 þ[4]11s þ Oð3 sÞ; (2) in terms of the Higgs vacuum expectation value v, set to v ¼246 GeV. The operator (1) leads to new Feynman
rules, with vertices involving the Higgs field and up to four gluons.
The leading order contributions to Hjjj, both for VBF and GF (in the m[t]! 1 limit), have been calculated in [16]. However, while the VBF calculation is available also at NLO [6,17], the computation
of the Higgs plus three jets in GF is still missing.
Elaborating on the techniques employed in the recent calculation of the NLO contributions to Hjj production at the LHC [18], in this Letter we report on the calculation of the cross section for pp !
Hjjj in GF at NLO accuracy in QCD, within the infinite mtapproximation.
This calculation is challenging due to the complexity of both the real-emission contributions and of the virtual corrections, which involve more than 10 000 one-loop Feynman diagrams with up to
rank-seven hexagons.
A complete next-to-leading order calculation requires the evaluation of virtual and real emission contributions.
For the computation of the virtual corrections we use a code generated by the program package GOSAM [19],
which combines automated diagram generation and alge-braic manipulation [20–23] with integrand-level reduction techniques [24–30].
In order to deal with the complexity level of the consid-ered calculation, theGOSAMcode has been enhanced. On the one side, the generation algorithm has been improved by a more efficient diagrammatic
layout: Feynman dia-grams are grouped according to their topologies, namely, global numerators are constructed by combining diagrams that have a common set, or subset, of denominators,
irre-spectively of the specific particle content. On the other side, additional improvements in the performances of
GOSAM have been achieved by exploiting the optimized manipulation of polynomial expressions available in
FORM 4.0[31]. The new developments ofGOSAM, regarding
the improved generation and reduction algorithms, will be properly discussed in a dedicated communication.
Within the GOSAM framework the virtual corrections are evaluated using the d-dimensional integrand-level decomposition implemented in the SAMURAI library [32,33], which allows for the combined
determination of both cut-constructible and rational terms at once. Alternatively, a tensorial decomposition [34,35] via
GOLEM95 is used as a rescue system. After the reduction, all relevant master integrals are computed by means of
QCDLOOP[36,37],ONELOOP[38], orGOLEM95C[39]. The basic partonic processes contributing to Hjjj pro-duction are listed in TableI, together with the correspond-ing number of Feynman diagrams and the
approximate computing time per phase-space point after summing over color and helicities. Representative one-loop dia-grams are depicted in Fig.1.
The ultraviolet, the infrared, and the collinear singular-ities are regularized using dimensional reduction.
TABLE I. Number of Feynman diagrams and computing time per phase-space point for each subprocess, on a Intel i7 960 (3.20 GHz) CPU. The code is compiled with the Intel fortran compiler ifort (with
Subprocess Diagrams Time/PS-point [sec]
qq ! Hq0q0g 467 0.29
qq ! Hq qg 868 0.60
gg ! Hqqg 2519 3.9
gg ! Hggg 9325 20
FIG. 1. Sample hexagon diagrams which enter in the six-parton one-loop amplitudes for qq ! Hq qg and gg ! Hggg. The dot represents the effective ggH vertex.
0 µ / µ [pb]σ 0.5 1 1.5 2 2.5 3 LO NLO 0.25 0.5 1 2 4 2 T H = 0 µ
FIG. 2 (color online). Scale dependence of the total cross section at LO and NLO.
FIG. 3 (color online). Transverse momentum (pT) distribu-tions for the first, second, and third leading jet.
Ultraviolet divergences have been renormalized in theMS scheme. In the case of LO (NLO) contributions we describe the running of the strong coupling constant with one-loop (two-loop) accuracy.
The effective Hgg coupling leads to integrands that may exhibit numerators with rank larger than the number of the denominators. In general, for these cases, the parametriza-tion of the residues at
the multiple cut has to be extended and, as a consequence, the decomposition of any one-loop amplitude acquires new master integrals [29]. The extended integrand decomposition has been implemented in
Remarkably, for the processes at hand, it has been proven that the higher-rank terms are proportional to the loop momentum squared, which simplifies against a denominator, hence generating
lower-point integrands where the rank is again equal to the number of denomi-nators [18]. Consequently, the coefficients of the new master integrals have to vanish identically, as explicitly
verified. The available options inGOSAMfor the algebraic manipulation of the integrands allow for the automatic computation of the virtual corrections in two different ways. In the first
approach,GOSAMdecomposes the four-dimensional part of the numerators using the extended-rank decomposition, and adds the analytic results of the
rational terms (generated from the extra-dimensional part). In the second approach, the regular decomposition of
SAMURAI, without the higher rank extension, is employed
on the whole d-dimensional integrands. We checked that both approaches provide identical answers. In the follow-ing, we adopt the second strategy, which proved to be numerically more efficient.
The double and the single poles conform to the universal singular behavior of dimensionally regulated one-loop amplitudes [40]. We also checked that our results fulfill gauge invariance: when
substituting the polarization vec-tors of one or more gluons with the corresponding momenta, the result for the amplitudes, after summing over all diagrams, are indeed vanishing. Additional
infor-mation about the virtual contributions can be found in the Appendix.
Results for the cross section are obtained with a hybrid setup which combines the features of two different Monte Carlo (MC) tools. For the generation and integration of the Born and of the virtual
contributions, we used an automated framework for fixed order NLO QCD calcula-tions, based on the interplay of GOSAM andSHERPA[41], where the tree-level matrix elements are obtained with the
AMEGIC [42] library. The integration is carried out by generating Oð106Þ events, sampled on a MC grid trained on the Born matrix element, and weighted with the sum of the Born and the virtual
For the integration of the real-radiation terms, the dipole-subtraction terms, and the integrated dipoles, we employ a combination of MADGRAPH [43,44] (matrix elements), MADDIPOLE [45,46]
(subtraction terms), and
MADEVENT [47] (numerical integration). We verified the independence of our result under the variation of the so-called parameter that fixes the amount of subtractions around the divergences of the
real corrections.
We first proved the consistency of our hybrid MC inte-gration on pp ! Hjj, verifying that the full cross section at NLO agrees with the corresponding result for the inte-gration of both the virtual
and the real corrections obtained by the interplay ofSHERPAandGOSAMalone. Moreover, for the process under consideration, namely, pp ! Hjjj, we found excellent agreement between MADGRAPH and
SHERPAfor the LO cross section.
In the following, we present results for the integrated cross section of Higgs boson plus three jets production at
FIG. 4 (color online). Transverse momentum (pT) distribu-tions for the Higgs boson.
TABLE II. Benchmark phase space point for Higgs plus three jets production. Particles are ordered as in TableI.
Particle E px py pz p[1] 250.000 000 000 000 00 0.000 000 000 000 000 0 0.000 000 000 000 000 0 250.000 000 000 000 00 p[2] 250.000 000 000 000 00 0.000 000 000 000 000 0 0.000 000 000 000 000 0
250:000 000 000 000 00 p[3] 131.068 966 558 232 09 27.707 264 814 722 667 13:235 482 900 394 146 24.722 529 472 591 685 p[4] 164.744 201 405 974 25 129:375 840 986 751 83 79:219 260 486 951 597
64:240 582 451 932 028 p[5] 117.029 536 327 738 03 54.480 516 624 273 569 97.990 504 664 150 677 33:550 658 370 629 378 p[6] 87.157 295 708 055 642 47.188 059 547 755 266 5:535 761 276 804 790 6
73.068 711 349 969 661
the LHC, for a center-of-mass energy of 8 TeV. The mass of the Higgs boson is set to m[H] ¼ 125 GeV.
Jets are clustered using the anti-k[t] algorithm imple-mented in FASTJET [48–50] with radius R ¼0:5 and a
minimum transverse momentum of p[T;jet]>20 GeV and pseudorapidity jj <4:0. The LO cross section is com-puted with the LO parton-distribution functions cteq6L1, whereas at NLO we use cteq6mE [51].
Everywhere, but in the effective coupling of the Higgs to the gluons, the renormalization and factorization scales are set to F¼ R¼ ^H[T] 2 ¼ 12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2[H]þ p2[T;H] q þX i jpT;ij ;
where the sum runs over the final state jets. The strong coupling is therefore evaluated at different scales accord-ing to 5s ! 2sðmHÞ3sð ^HT=2Þ. The theoretical uncertain-ties are estimated by
varying the scales by factors of 0.5 and 2.0, respectively. In the effective coupling the scale is kept at m[H]. Within this setup we obtain the following total cross section at LO and NLO:
LO½pb ¼ 0:962þ0:51[0:31]; NLO½pb ¼ 1:18þ0:01[0:22]: The scale dependence of the total cross section, depicted in Fig. 2, is strongly reduced by the inclusion of the NLO contributions.
In Figs. 3 and 4, we show the pT distributions of the three jets and of the Higgs boson, respectively. The NLO corrections enhance all distributions for pT values lower than 150–200 GeV, whereas
their contribution is negative at higher pT. This behavior is explicitly shown in the lower part of Fig.4for the case of the Higgs boson.
This study also shows that the virtual contributions for pp ! Hjjj generated by GOSAM can be successfully
paired with available Monte Carlo programs to aim at further phenomenological analyses.
We thank Thomas Hahn and Gudrun Heinrich for dis-cussions and comments on the manuscript, and Marek Scho¨nherr for assistance with the usage of SHERPA. The work of G. C. was supported by DFG SFB-TR-9
and the EU TMR Network LHCPHENOnet. The work of H. v. D., G. L., P. M., and T. P. was supported by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovaleskaja Award, endowed by
the German Federal Ministry of Education and Research. G. O. was supported
in part by the National Science Foundation under Grant No. PHY-1068550. F. T. acknowledges partial support by MIUR under Project No. 2010YJ2NYW. G. C. and G. O. wish to acknowledge the kind
hospitality of the Max-Planck-Institut fu¨r Physik in Munich at several stages during the completion of this project. This research used computing resources from the Rechenzentrum Garching and the
New York City College of Technology.
Results for the virtual contributions
The numerical values of the one-loop subamplitudes, defined as
2RefMtreelevel[M]oneloop[g] ð[s]=2ÞjMtreelevel[j]2
a[2] 2 þ
þ a0; (A1) and evaluated at the nonexceptional phase space point given in TableII, are collected in TableIII. The values of the double and the single poles conform to the universal singular behavior
of dimensionally regulated one-loop amplitudes [40]. The precision of the finite parts is esti-mated by reevaluating the amplitudes for a set of momenta rotated by an arbitrary angle about the axis
of collision.
In Fig.5, we present the results for the finite part a[0] of the virtual matrix elements for the various subprocesses calculated along a certain one-dimensional curve in the space of final state
momenta. Starting from the phase space point in TableII, in which the initial partons lie along the z axis, we generate new configurations by rotating the final state momenta by an angle 2 ½0; 2
about the y axis.
TABLE III. Numerical results for the four subprocesses listed in TableIevaluated at the phase space point of TableII. The accuracy of the result is indicated by the underlined digits.
gg ! Hggg gg ! Hqqg qq ! Hq qg qq ! Hq0q0g
a[0] 41:228 787 667 416 85 48:684 241 349 894 78 69:323 511 404 746 95 15:792 627 671 779 15 a[1] 47:167 154 191 326 59 36:082 777 280 772 28 29:988 629 329 636 59 32:353 205 870 739 68 a[2] 14:999
999 999 999 91 11:666 666 666 666 83 8:333 333 333 333 339 8:333 333 333 333 398
FIG. 5 (color online). Finite-term a[0] of the virtual matrix elements for qq ! Hq0q0g (green), qq ! Hq qg (blue), gg ! Hqqg (orange), gg ! Hggg (red).
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[51] J. Pumplin, D. R. Stump, J. Huston, H.-L. Lai, P. Nadolsky, and W.-K. Tung, J. High Energy Phys. 07 (2002) 012. | {"url":"https://123dok.org/document/7qvgkxdq-leading-order-corrections-higgs-boson-production-gluon-fusion.html","timestamp":"2024-11-02T21:17:51Z","content_type":"text/html","content_length":"161894","record_id":"<urn:uuid:225f01d4-6639-4ef3-b467-b1dcd120fa45>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00259.warc.gz"} |
ESP Biography
ALEXANDRE GAUTHIER, I'm a graduate student in Applied Physics
Major: Applied Physics
College/Employer: Stanford
Year of Graduation: 2021
Brief Biographical Sketch:
Not Available.
Past Classes
(Clicking a class title will bring you to the course's section of the corresponding course catalog)
C6430: Introduction to Special Relativity in Splash Spring 2018 (May. 05 - 06, 2018)
Special relativity describes the behavior of systems moving at speeds close to the speed of light. Once you start moving so fast, a lot of weird stuff starts to happen. We will discuss Einstein's
postulates, the two statements which form the foundation of special relativity. Then we will introduce time dilation - the bizarre idea that "moving clocks tick slower."
C5963: Introduction to Special Relativity in Splash Fall 2017 (Nov. 11 - 12, 2017)
Special relativity describes the behavior of systems moving at speeds close to the speed of light. Once you start moving so fast, a lot of weird stuff starts to happen. We will discuss Einstein's
postulates, the two statements which form the foundation of special relativity. Then we will introduce time dilation - the bizarre idea that "moving clocks tick slower."
C5198: The Exciting World of Lasers in Splash Fall 2016 (Dec. 03 - 04, 2016)
Lasers have innumerable applications, from laser tag to laser eye surgery. We will discuss the unique properties that make laser light different from ordinary light, and learn about some interesting
applications of lasers.
P4811: Introduction to Special Relativity in Splash Spring 2016 (Apr. 09 - 10, 2016)
Special relativity describes the behavior of systems moving at speeds close to the speed of light. Once you start moving so fast, a lot of weird stuff starts to happen. We will discuss Einstein's
postulates, the two statements which form the foundation of special relativity. Then we will introduce time dilation - the bizarre idea that "moving clocks tick slower."
P4527: Introduction to Special Relativity in Splash Fall 2015 (Nov. 07 - 08, 2015)
Special relativity describes the behavior of systems moving at speeds close to the speed of light. For some reason, this subject has a reputation for being very difficult to understand. But really,
you don't need any fancy math or physics to learn the basic principles of relativity - all you need is algebra. This course will introduce some of the bizarre behaviors of relativistic systems,
including time dilation and length contraction. These phenomena will be algebraically derived, and apparent paradoxes will be discussed. We will also discuss the historical events leading to the
discovery of special relativity. | {"url":"https://stanfordesp.learningu.org/teach/teachers/agauth/bio.html","timestamp":"2024-11-11T07:11:24Z","content_type":"application/xhtml+xml","content_length":"19581","record_id":"<urn:uuid:cc44ef28-9efd-4257-9b70-c1ec62f15329>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00805.warc.gz"} |
Matrix algebra – The Dan MacKinlay stable of variably-well-consider’d enterprises
Matrix algebra
Maybe also some operator algebra
July 9, 2018 — March 11, 2024
functional analysis
linear algebra
Algebra over matrices, which are the things that define the linear operators that we care about when we define operators over vectors.
If we admit infinitesimal matrices, then this gets us matrix calculus, and various nice matrix representations, matrix inverses and such.
This page mostly exists to bookmark tools that I have found useful to do abstract matrix algebra, e.g. without pre-committing to a size or whatever.
This is not wildly esoteric or difficult, but there is a lot of fiddly book-keeping, so a symbolic mathematics package helps. Unfortunately, those packages have wildly esoteric and difficult
documentation to understand. This page is mostly to bookmark those packages and relevant manual pages.
I may yet put some other results in though.
1 Ore algebras
Look interesting. TBC
3 Tooling
The keyword that we are looking for is non-commutative algebra, since matrix algebras are non-commutative. There are many non-commutative algebras and most of them are more complicated than the
matrix ones.
3.1 Mathematica
Lots of handy extensions for non-commutative algebras in general. See NCAlgebra et al. If you have a license, start here.
3.2 SymPy
SymPy has a non-commutative algebra package, see Quantum Mechanics, which provides operators over complex fields. Restricting ourselves to the reals probably gets us what we want.
3.3 Sage
I think that this is probably powerful, but I have gotten rather lost in the documentation.
We can go bareback and simply define Algebras with the commutative=False option, I think. Then what do we do?
Noncommutative Algebras in Sage introduces several ways of solving problems, and notes that they possibly all depend upon PLURAL.
Specific algebra that might be of interest:
mkauers/ore_algebra supports Ore Algebra operations in Sage. Maybe that does what I want?
4 References
Bhatia. 1997.
Matrix Analysis
. Graduate Texts in Mathematics.
Golub, and van Loan. 1983. Matrix Computations.
Petersen, and Pedersen. 2012.
“The Matrix Cookbook.”
Searle. 2014.
“Matrix Algebra.”
Wiley StatsRef: Statistics Reference Online
Searle, and Khuri. 2017. Matrix Algebra Useful for Statistics.
Seber. 2007. A Matrix Handbook for Statisticians.
Tropp. 2019. Matrix Concentration & Computational Linear Algebra / ENS Short Course.
Yurtsever, Tropp, Fercoq, et al. 2021.
“Scalable Semidefinite Programming.” SIAM Journal on Mathematics of Data Science | {"url":"https://danmackinlay.name/notebook/matrix_algebra.html","timestamp":"2024-11-08T20:39:27Z","content_type":"application/xhtml+xml","content_length":"35161","record_id":"<urn:uuid:de9bd595-306b-43d7-805e-2d8a152a706f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00451.warc.gz"} |
Derivative of e^x^2 by First Principle and Chain Rule - iMath
Derivative of e^x^2 by First Principle and Chain Rule
The function e to the power x^2 is written as $e^{x^2}$ and its derivative is $2xe^{x^2}$. In this post, we will find the derivative of e to the power x square by the first principle and chain rule
of derivatives.
Recall the first principle of derivatives: The derivative of a function f(x) by first principle is defined by the limit:
$\dfrac{d}{dx}(f(x))$ = lim[h→0] $\dfrac{f(x+h)-f(x)}{h}$ $\cdots$ (I)
We will use this to find the differentiation of e to the power x^2.
Derivative of $e^{x^2}$ by First Principle
Let us put $f(x)=e^{x^2}$ in the above formula (I).
So we obtain that
$\dfrac{d}{dx}(e^{x^2})$ = lim[h→0] $\dfrac{e^{(x+h)^2}-e^{x^2}}{h}$
= lim[h→0] $\dfrac{e^{x^2+2xh+h^2}-e^{x^2}}{h}$. Here we have used the formula (a+b)^2 = a^2+2ab+b^2.
= lim[h→0] $\dfrac{e^{x^2}e^{2xh+h^2}-e^{x^2}}{h}$
= lim[h→0] $\dfrac{e^{x^2}(e^{2xh+h^2}-1)}{h}$
= lim[h→0] $\dfrac{e^{x^2}(e^{h(2x+h)}-1)}{h}$
= lim[h→0] $[\dfrac{e^{x^2}(e^{h(2x+h)}-1)}{h(2x+h)}$ $\times (2x+h) ]$
$=e^{x^2}\lim\limits_{h \to 0}$ $\dfrac{e^{h(2x+h)}-1}{h(2x+h)}$ $\times \lim\limits_{h \to 0} (2x+h)$
Let u=h(2x+h). Then u tends to zero when h tends to 0.
$=e^{x^2}\lim\limits_{u \to 0}$ $\dfrac{e^{u}-1}{u}$ $\times \lim\limits_{h \to 0} (2x+h)$
$=e^{x^2} \times 1$ $\times (2x+0)$ as the limit of (e^x-1)/x is 1 when x tends to 0.
Thus, the derivative of e^x^2 by first principle is 2xe^x^2.
Derivative of $e^{x^2}$ by Chain Rule
Let u=x^2.
Then $\dfrac{du}{dx}=2x$.
Now, the derivative of $e^{x^2}$ by the chain rule is equal to
$\dfrac{d}{dx}(e^{x^2})$ $=\dfrac{d}{dx}(e^u)$
$= \dfrac{d}{du}(e^u) \cdot \dfrac{du}{dx}$
$=e^u \cdot 2x$ as du/dx=2x.
$=2xe^{x^2}$ as u=x^2.
Therefore, the differentiation of e to the power x^2 is equal to the product of 2x and e^{x^2}.
Also Read:
Derivative of root x + 1 by root x
Q1: What is the derivative of e^x^2?
Answer: The derivative of $e^{x^2}$ is equal to $2x e^{x^2}$. | {"url":"https://www.imathist.com/derivative-of-ex2-by-first-principle-and-chain-rule/","timestamp":"2024-11-09T16:48:47Z","content_type":"text/html","content_length":"180381","record_id":"<urn:uuid:a116d43c-5f58-40e5-8217-b83418e266dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00867.warc.gz"} |
Scientific Visualization Color
Color coding
A D V E R T I S E M E N T
Colors and light are essential to visualization. Most visualization techniques contain a step in which data values are mapped to colors to make the range of the data visible (see the next sections).
The interpretation of results produced by these visualization techniques depends crucially on the mapping of data to colors because the human eye is more sensitive to some parts of the visible
spectrum of light than to
other parts and the brain may interpret different color patterns differently. Since the mapping of data values to colors involves color coding, we will describe this in some detail. There exist quite
a few color coding systems. We discuss here the RGB, CMY, and HSV systems.
The RGB code system is mainly used in television sets and computer screens. Color is determined by the three independent components Red (R), Green (G), and Blue (B). These components correspond with
the red, green, and blue cathode ray tubes. Each component has values in the interval [0,1]. A value of 0 corresponds to black and a value of 1 corresponds with full color. For example {R=0, G=0, B=
0}is black, {R=1, G=0, B=0}is full red, {R=0, G=1, B=1}is full cyan, and {R=1, G=1, B=1}is white. Every color can be constructed by taking the right combination of the three components. However, the
number of values that can be represented by cathode ray tubes for each component in the interval [0, 1] is finite. Most graphical display systems currently use 256 different values, which means that
256**3 = 16,777,216 different colors are supported: more than enough for most applications.
Another color coding system is the CMY system. Color is determined by the three independent components Cyan (C=1-R), Magenta (M=1-G), and Yellow (Y=1-B). It is widely used in the publishing world.
The CMY system is related to the RGB system as follows: C=1-R, M=1-B, and Y=1-G. It is clear that these systems are complementary to each other.
The color coding which is used most frequently in visual environments is the HSV system. The three components Hue (H), Saturation (S), and Value (V) have values between 0 and 1. Whereas one can view
the volume spanned up by the three independent components of the RGB and CMY color systems as a unit cube, the HSV volume is an inverted cone. The Hue is the angular coordinate in a plane
perpendicular to the symmetry axis, where H=0 corresponds to red and an angle of 0 degrees, H=0.33 corresponds to green and an angle of 120 degrees, H=0.66 corresponds to blue and an angle of 240
degrees, and H=1.0 corresponds again to red. The Saturation determines the saturation of the color. Saturation is the radial coordinate in the plane perpendicular to the symmetry axis, so that S=1
corresponds to full saturation and S=0 corresponds to no color (i.e. white), S=0 is situated on the symmetry axis and S=1 corresponds to the maximum radius in the plane, i.e. the boundary of the
cone. The Value determines the intensity of the color. Value is the coordinate along the symmetry axis of the cone, where V=0 corresponds to no intensity (i.e. black), and V=1 corresponds to maximum
intensity. V=0 is at the top of the cone and V=1 is at the base of the cone. We will not discuss the relation between the RGB and HSV system as it is a complicated one. Interested people are referred
to refs. [2] and [3]. HSV is frequently used in visual environments, because the effects of manipulating the components are much more predictable than in RGB and CMY. HSV corresponds to a more
natural experience of colors than the other systems provide.
In most visualization techniques, colors from red to blue are used to reveal transitions in some quantity. If one wants to use the complete set of available colors one needs to map the range of
values to this complete set. Storing this mapping consumes too much memory.
Therefore, one normally chooses N colors of the available set (typically N=256) and maps the range of values to these N colors. This mapping of the values to colors will be denoted as the colormap in
the rest of this report.
It is important to notice that using colors to reveal transitions in quantities can be misleading if there are no transitions at all, e.g. for a monotonically increasing quantity. For the kind of
phenomena in which the quantity changes smoothly it is better to use continuously changing schemes like gray scales, saturation scales, and intensity scales. This immediately shows another advantage
of the HSV code
system: saturation and intensity are natural variables in the HSV system.
Surface rendering techniques
This section briefly describes a general set of 3D scalar and vector surface rendering techniques. The first four descriptions deal with scalar field techniques and the other two with vector field
Scalar glyphs
Scalar glyphs is a technique which puts a sphere or a diamond on every data point. The scale of the sphere or diamond is determined by the data value. The scalar glyphs may be colored according to
the same scalar field or according to another scalar field. In this way correlations can be found. As no interpolations are needed for this technique it consumes few CPU seconds.
This technique produces surfaces in the domain of the scalar quantity on which the scalar quantity has the same value, the so-called isosurface value. The surfaces can be colored according to the
isosurface value or they can be colored according to another scalar field using the texture technique. The latter case allows for the search for correlation between different scalar quantities.
There are different methods to generate the surfaces from a discrete set of data points. All methods use interpolation to construct a continuous function. The correctness of the generated surfaces
depends on how well the constructed continuous function matches the underlying continuous function representing the discrete data set. The method which is implemented in the software packages
described in chapter 3, is the Marching Cube Algorithm.
Cutting planes
This technique makes it possible to view scalar data on a cross-section of the data volume with a cutting plane. One defines a regular, Cartesian grid on the plane and the data values on this grid
are found by interpolation of the original data. A convenient colormap is used to make the data visible.
Orthogonal slicers
It often occurs that one wants to focus on the influence of only two independent variables (i.e. coordinates). Thus, the other independent variables are kept constant. This is what the orthogonal
slicer method does. For example, if the data is defined in spherical coordinates and one wants to focus on the angular dependences for a specific radius, the orthogonal slicer method constructs the
corresponding sphere. No interpolation is used since the original grid with the corresponding data is inherited. A convenient colormap is used to make the data visible.
Vector glyphs
This technique uses needle or arrow glyphs to represent vectors at each data point. The direction of the glyph corresponds to the direction of the vector and its magnitude corresponds to the
magnitude of the vector. The glyphs can be colored according to a scalar field.
Streamlines, streaklines, and particle advection
This is a set of methods for outlining the topology, i.e. the field lines, of a vector field. Generally, one takes a set of starting points, finds the vectors at these points by interpolation, if
necessary, and integrates the points along the direction of the vector. At the new positions the vector values are found by interpolation and one integrates again. This process stops if a
predetermined number of integration steps has been reached or if the points end up outside the data volume. The calculated points are connected by lines.
The difference between streamlines and streaklines is that the streamlines technique considers the vector field to be static whereas the streaklines technique considers the vector field to be time
dependent. Hence, the streakline technique interpolates not only in the spatial direction, but also in the time direction. The particle advection method places little spheres at the starting points
representing massless particles. The particles are also integrated along the field lines. After every integration step each particle is drawn together with a line or ribbon tail indicating the
direction in which the particle is moving.
This is a technique to color arbitrary surfaces, e.g. those generated by the isosurface techniques, according to a 3D scalar field. An interpolation scheme is used to determine the values of the
scalar field on the surface. A colormap is used to assign the color.
Volume Visualization
Volume rendering is used to view 3d data without the usual intermediate step of deriving a geometric representation which is then rendered. The volume representation uses voxels, or volume elements
to determine visual properties, such as opacity, color, shading at each point in the computational domain. Several images are created by slicing the volume perpendicular to the viewing axis at a
regular interval and compositing together the contributing images from back to front, thus summing voxel opacities and colors at each pixel. By rapidly changing the color and opacity transfer
functions, various structures are interactively revealed in the spatial domain.
Volumetric rendering
Volumetric rendering allows the entire data set to be viewed at once, and lets the user "see inside" the data. For each pixel in an image created using volumetric rendering, a ray is cast through the
semi-transparent volume. The resulting color at the pixel is a composite of all the voxels the ray has intersected. As a consquence, such images tend to be blurry. Another characteristic volumetric
rendering is that it is typically slower than surface rendering techniques. Therefore, volumetric rendering of adata set is often not well suited for realtime visualization. However, it does provide
features that are obscured by surface rendering techniques.
Volume rendering techniques
Volume rendering techniques have been developed to overcome problems of the accurate representation of surfaces in the isosurface techniques. In short, these problems are related to making a decision
for every volume element whether or not the surface passes through it and this can produce false positives (spurious surfaces) or false negatives (erroneous holes in surfaces), particularly in the
presence of small or poorly defined features. Volume rendering does not use intermediate geometrical representations, in contrast to surface rendering techniques. It offers the possibility for
displaying weak or fuzzy surfaces. This frees one from the requirement to make a decision whether a surface is present or not.
Volume rendering involves the following steps: the forming of an RGBA volume from the data, reconstruction of a continuous function from this discrete data set, and projecting it onto the 2D viewing
plane (the output image) from the desired point of view. An RGBA volume is a 3D four-vector data set, where the first three components are the familiar R, G, and B color components and the last
component, A, represents opacity. An opacity value of 0 means totally transparent and a value of 1 means totally opaque. Behind the RGBA volume an opaque background is placed. The mapping of the data
to opacity values acts as a classification of the data one is interested in. Isosurfaces can be shown by mapping the corresponding data values to almost opaque values and the rest to transparent
values. The appearance of surfaces can be improved by using shading techniques to form the RGB mapping. However, opacity can be used to see the interior of the data volume too. These interiors appear
as clouds with varying density and color. A big advantage of volume rendering is that this interior information is not thrown away, so that it enables one to look at the 3D data set as a whole.
Disadvantages are the difficult interpretation of the cloudy interiors and the long time, compared to surface rendering, needed to perform volume rendering.
We will describe two implementations of volume rendering: ray casting and splatting. These implementations are used in the four visualization packages we have compared (see chapter 3). The two
methods differ in the way the RGBA volume is projected onto the 2D viewing plane.
Ray casting
Several implementations exist for ray casting. We describe the implementation used in Visualization Data Explorer. For every pixel in the output image a ray is shot into the data volume. At a
predetermined number of evenly spaced locations along the ray the color and opacity values are obtained by interpolation. The interpolated colors and opacities are merged with each other and with the
background by compositing in back-to-front order to yield the color of the pixel. These compositing calculations are simply linear transformations. Specifically, the color of the ray Cout as it
leaves each sample location, is related to the color Cin of the ray, as it enters, and to the color c(xi) and the opacity a(x) at that sample location by the transparency formula :
Performing this formula in a back-to-front order, i.e. starting at the background and moving towards the image plane, will produce the pixel color. It is clear from the above formula that the opacity
acts as a data selector. For example, sample points with opacity values close to 1 hide almost all the information along the ray between the background and the sample point and opacity values close
to zero transfer the information almost unaltered. This way of compositing is equal to the dense-emitter model, where the color indicates the instantaneous emission rate and the opacity indicates the
instantaneous absorption rate.
This technique was developed to improve the speed of calculation of volume rendering techniques like ray casting, at the price of less accurate rendering. We will not go into detail here as this
technique is rather complicated. It differs from ray casting in the projection method. Splatting projects voxels, i.e. volume elements, on the 2D viewing plane. It approximates this projection by a
so-called Gaussian splat, which depends on the opacity and on the color of the voxel (other splat types, like linear splats can be used also). A projection is made for every voxel and the resulting
splats are composited on top of each other in back-to-front order to produce the final image.
Data Types
• Hierarchical Data Formats (HDF)
• Network Common Data Format (netCDF)
• Databases- The currently accepted storage method for most scientific data is the Relational Database Management System. This is the format used by many commercial databases, such as Oracle. Data
can be extracted using the Standard Query Language (SQL) commands.
Animation techniques
These techniques simulate continuous motion by rapidly displaying images. The viewer is given the impression that he is watching a continuous motion. To achieve this impression the graphical hardware
needs image display rates of at least 25 images per second, since otherwise motion will look shaky. As most graphical hardware can not reach that display rate for moderate sized images (i.e. 256x256
pixels), one uses video hardware. One either sends every image to a framebuffer to write one videoframe at a time to videotape or one stores the images on a fast accessible device, CMY a laserdisk,
and, after all images have been stored, displays them on a television screen from where they can be put on a videotape. There are two kinds of animation which we will describe below.
Flipbook animation
This is a well known technique. The generated images are displayed one after the other. Its name is attached to the thumbing or flipping through a series of images.
Keyframe animation
For this technique one only has to generate so-called keyframes. Keyframes mark changes in the characteristics of the motion, for example the sudden change in the direction of motion of an electron
due to a collision with an ion. Interpolation techniques are used to generate a set of images between two keyframes. The larger the interpolated set of images the smoother the conversion from one
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Joseph Fourier the Man Behind the Discovery of Greenhouse Effect » Famous Mathematicians Vedic Math School
Home » Blog » Famous Mathematicians » Joseph Fourier the Man Behind the Discovery of Greenhouse Effect
Joseph Fourier the Man Behind the Discovery of Greenhouse Effect
Famous Mathematicians, Famous Scientists / By Prince Jha / 5 minutes of reading
Baron Jean-Baptiste-Joseph Fourier (1768-1830) was also known as a Fourier transformer, Egyptologist, and Administrator. Fourier was a French mathematician, and also a physicist. His major influence
was on The Analytical Theory of Heat in 1822.
Time and Place of birth
Joseph Fourier was born on March 21, 1768, in Auxerre, France.
Joseph Fourier Early life
Joseph Fourier was the son of a tailor, in his childhood, He attended the local military school and was taught by Benedictine monks at The Convent of St. Mark. As his capability in mathematics was
outstanding, he became a teacher in the same school.
At the age of 9, he was an orphan. In the year 1780, he went to the Ecole Royale Militaire and mathematics became his real interest. Also, received first prize for his study on Bossut’s in the year
He decided to get trained for the priesthood in the year 1787 but continued doing mathematics.
Fourier Joseph Adulthood
In the year 1793 Fourier joined politics in the local Revolutionary Committee. The results for the politics weren’t good and he tried to quit politics but that wasn’t possible and he was stuck in the
revolution. In 1795 Fourier was chosen to go to Paris and study in the Ecole Normale.
And he joined the first batch and soon after the completion of the course, he joined as a teacher in Ecole Normale but soon switched to Ecole Polytechnique, and in between, he got arrested because of
his political involvement but after 1795 he again got back into teaching Polytechnique.
At the Institute of Cairo, he was elected as the secretary till he was in Egypt, side by side kept working on mathematics. Fourier joined Napoleon’s army in 1798. And later, deserted his army and
returned to Paris. From 1804 to 1807 he started his important mathematical works.
Must Read These Also.
Fourier Joseph Death
Jean-Baptiste Fourier Joseph died on 16 May 1830 he had a severe heart condition for quite a while and resulted in his death at the age of 62. He was buried in the Pere Lachaise Cemetery, in Paris.
In Auxerre, a bronze statue was built.
Joseph Fourier Education and Career
In his childhood, he went to a military school, and then became a mathematics teacher in the same school and for higher studies went to Ecole Normale Superieure.
Fourier Joseph was a mathematician with a great distinction. In the Scientific corps, he was denied the position but continued his studies and research in mathematics. Fourier is popular for the
Fourier transforms as it includes heat transfer and vibrations.
In one of his papers he explained the study of the conduction of heat, his reasoning was on Newton’s law of cooling which was two-dimensional objects, this work of Fourier Joseph was in his
well-known work “Fourier series”.
Must Read These Also.
The Greenhouse Effect
The first person to discover the greenhouse effect was Fourier; he studied Earth’s temperature from a mathematical point of view. He calculated the temperature day and night, and in summer and
winter, and concluded that it would be much warmer.
He also calculated that it would be much colder than it is from the radiation of the sun.
When the energy from the sun peeks out and when the atmosphere is transparent to this radiation, which passes by and gives warmth to land and ocean, this energy, in turn, radiates as infra-red
radiation. He also suggested that to sustain life on earth The Greenhouse Effect is necessary.
Unravel Sophie Germain’s pioneering math and explore more trailblazing women reshaping the world of STEM! Delve into related inspiring stories now
Awards & Achievements
• At the Academie Francaise and at the Academe de Medecin Fourier Joseph was elected in the year 1826.
• At the Royal Swedish Academy of Sciences, He was elected as a foreign member in the year 1830 shortly before his death.
• Fourier was made a Baron by Napoleon in 1809.
Joseph Fourier was undeniably the most influential mathematician. All of his works, his significant discoveries have made an influence in modern times, his researches have made a huge impact
especially the Greenhouse Effect and the Fourier transforms.
Books published
In the year 1822, The Analytical Theory of Heat was published by Jean-Baptiste Joseph Fourier
The profound study of nature is the most beautiful source of mathematical discoveries.
Mathematical analysis is as extensive as nature itself.
Must Read These Also.
What is Fourier Joseph known for?
Fourier Joseph was well-known for one of his major discoveries The Greenhouse Effect and also was known as a French mathematician, administrator, and Egyptologist.
Where did Jean-Baptiste Fourier Die?
Fourier died in Paris, France on 16 May 1830.
Was Jean-Baptiste Joseph Fourier Married?
Joseph Fourier never got married.
What are the two types of Fourier series?
The two types of Fourier series are the Trigonometric series and the Exponential series, by Fourier Joseph
Must Read These Also.
Joseph Fourier for his whole life worked on the theories and one of them we know now is the greenhouse effect, he worked in politics but his interest was always more on mathematics, researches on
theories, etc., The great mathematician who has given us so much, on the Eiffel Tower his name has been inscribed.
I hope my article was helpful to all you people. It was well-searched and written. Please leave your comments. And if you want to know more about anyone else do let me know in the comment box I’ll
come up with that particular article.
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Design Optimisation Approach of an Outer Rotor Multiphase PM Actuator for Multirotor Aerial Vehicle Applications
ESTACA, ESTACA’Lab—Paris-Saclay, F-78180 Montigny-le-Bretonneux, France
GeePs Group of Electrical Engineering-Paris, UMR CNRS 8507, CentraleSupélec, Université Paris-Saclay, 91192 Gif Sur Yvette, France
Author to whom correspondence should be addressed.
Submission received: 18 December 2023 / Revised: 9 February 2024 / Accepted: 9 February 2024 / Published: 13 February 2024
The electric urban air mobility sector has gained significant attraction in public debates, particularly with the proliferation of announcements demonstrating new aerial vehicles and the
infrastructure that goes with them. In this context, the development of new methodologies for the design and sizing of actuation systems, ensuring high performances of these aerial vehicles, remains
an important task in this process. This will allow for better integration within this transport sector. In this paper, a robust design optimisation approach of multiphase fault-tolerant (FT) outer
rotor (OR) permanent magnets (PM) for multirotor aerial vehicle applications is proposed. In order to show the effectiveness and the robustness of the proposed design methodology, the number of
stator winding phases, with a fractional slot concentrated winding (FSCW) configuration, as well as the PM configuration are considered as variables. Thus, four cases for the number of phases are
considered, namely 3, 5, 6 and 7 phases, where for each number of phases case, the PM takes 3 configurations, namely surface PM, interior V-shape PM and interior spoke PM. First, a pre-sizing step is
carried out, consisting of selecting the optimal combinations slot/pole, designing the multiphase FSCW layout, and estimating the electric motor (EM) geometry using analytical computations to obtain
a preliminary validation of the design specifications. Second, constrained multiobjective optimisation is considered in order to optimise the EM performances, such as motor efficiency and weight,
under constraints where the FEMM/Matlab based Finite Element Analysis (FEA) tool is used to perform this optimisation. Finally, results analysis and performance comparisons of different EM
configurations are carried out in order to assess the design parameters, such as phases number, PM position, and harmonic currents in the EM design and consequently to select the best configuration
for the considered application.
1. Introduction
Through the use of fully electric or hybrid propulsion systems, vertical take-off and landing (eVTOL) aerial vehicle technologies represent a good solution to transport cargo or a small number of
passengers from point to point in highly congested cities and areas. This will help avoid traffic and provide a more ecological means of transportation. Their capabilities to hover and to perform
VTOL, as well as their great maneuverability, make multirotor eVTOL technologies an excellent choice for the urban air mobility (UAM) market [
]. However, with the rapid evolution of transport market requirements, especially in terms of efficiency, performance, safety, and large endurance, the development of new, precise, and rapid
approaches to the design and optimisation of these aerial vehicles remains an important task in this process of integration. Electric motors (EMs) play a fundamental role in enhancing the
performances and safety of the electric propulsion systems, and thus upgrading the performances and safety of all aerial vehicle [
As reported in [
], EMs for electric, more-electric or hybrid aircraft and airspace applications, are defined by their major requirements (MR), namely weight, which is a very important indicator that directly affects
the overall performance of the vehicle; volume, which is linked to the limited volume reserved for the motorization; safety, which directly affects the mission success, performance, and
dispatchability; efficiency, which also becomes an MR for energy saving and performance and finally the cost, which indicates the accessibility of a new solution. According to this same reference and
to [
] PM synchronous motors remain the best choice to achieve these MR, in comparison with induction motors, switched reluctance motors, synchronous reluctance motors, and wound field synchronous motors.
This is principally explained by the high torque and power density, low rotor and stator losses and the high-speed capability that the PM motor could offer.
Multiphase motors with a number of phases higher then 3, in comparison with conventional three-phase ones, can improve power density and fault tolerance due to the redundant design, reducing the cost
of circuit isolation by reducing the DC bus voltage, offering more degree of freedom for design and control [
]. These intrinsic characteristics make multiphase motors a suitable solution for eVTOL aerial vehicle applications, where reliability and safety as well as performance are highly required.
Multiphase PM fault tolerant motors for electric and more-electric aircraft and airspace applications are researched in [
], where the different phase numbers 5-phase, 6-phase, 9-phase, and 15-phase are commonly considered. In [
], authors propose multi-3-phase PM motors in order to enhance the weight as well as the fault tolerance of an electromechanical actuator (EMA) for helicopter primary flight control. A 5-phase
concentrated winding PM motor was proposed for the electric steering of a commercial aircraft nose landing gear in [
]. A differential evolution (DE) optimisation technique was used in order to minimize the weight with respect to torque and efficiency targets. Authors in [
] have proposed a 6-phase PM motor for helicopter tail rotor applications, where an outer rotor with two interior PM configurations, spoke configurations and a V-shape configuration are designed, and
their respective performances are compared in order to select the suitable configuration.
State of art regarding EM for eVTOLs applications are reported in [
], where the power and torque densities constraints are considered. NASA in [
] aims to present a design optimisation approach of 3-phase PM motors with an efficiency constraint higher than
$96 %$
, and a power density of 13 kw/kg for eVTOLs applications. The motor design is based on the genetic optimisation of the motor fitness function, which is defined using specific power, mission
efficiency, peak winding temperature, and the thermochemical aging of the winding insulation over 10,000 missions. The performances are presented and discussed. Authors in [
] propose a 9-phase PM motor with a fractional slot concentrated winding (FSCW) configuration. The static performances of the motor, namely the torque and the torque ripple, the efficiency, and
losses are presented and discussed.
However, these multiphase EM motors cited in these papers are unique cases, where the influence of increasing the phase number is not evaluated in combination with the PM configurations that they
could take. Additionally, the multiphase winding configuration favors the appearance of higher odd harmonics in the EMF waveform, which leads to improving the static performance as well as the FT
capabilities of the EM. Nevertheless, a methodology to extract the main harmonics of the current, with their corresponding optimal ratio, is necessary to reach these goals and to avoid the torque
ripple increasing.
In this paper, a robust design optimisation approach for direct drive multiphase PM OR motors is presented and formulated. The motor, through direct coupling with the propeller, ensures the
propulsion of a fixed-pitch multirotor aerial vehicle. Through the matching of the advantages of multiphase motors, FSCW configuration, and the consideration of harmonics higher than the fundamental
in the injected control currents, the performances and reliability of the motor are improved, where a multiobjective optimisation is carried out using the motor efficiency and the motor active
components mass as the objective function. This article takes four different configurations of the number of phases and slot/pole ($s / p$) combinations, namely 3-phase 44/48 $s / p$, 5-phase 40/44
$s / p$, 6-phase 40/48 $s / p$, and 7-phase 48/56 $s / p$. The consideration of high pole number with FSCW allows to improve the motor performances and to reduce the motor mass. For each
configuration, 3 PM positions are chosen, which are surface PM, Spoke PM, and V-shape PM. Through these different configurations, a performance analysis and comparative study are carried out to
assess the effect of each parameter on the performances of the EM.
This article is organised as follows,
Section 2
provides details about the design requirements and characteristics of the multiphase actuator. The design methodology as well as the basic structural and analytical pre-sizing step is presented in
Section 3
Section 4
is devoted to the formulation of the optimisation problem;
Section 5
analyses and compares the performances of each configuration in order to select the suitable configuration. Conclusions and perspectives about the work are presented in
Section 6
2. Design Requirements
The application context of this motor is the propulsion of a multirotor aerial vehicle, with a gross take-off weight of 450 kg (
$G T O W$
). This aerial vehicle is composed of 6 propulsion chains (
$N p$
), including a propeller, EM, electronic speed controller (ESC), and energy storage system (ESS) that could feed several propulsion chains. Moreover, this vehicle could carry a payload of
$23 %$
of the
$G T O W$
(roughly 100 kg), with a maximum cruising speed
$V c$
of 54 km/h at altitude
of 500 m. For a multirotor aerial vehicle, the power flight mission, as illustrated in
Figure 1
, is divided into 3 phases of take-off/hovering, cruise, and landing/hovering. Among these phases, it is remarkable that the power required during the take-off and landing phases is much higher than
during the cruise phase.
During the take-off and landing, the required force for the lift
$F l i f t$
(N) is given by:
$F l i f t = G T O W · ( a + g ) + F d r a g ,$
) and
$F d r a g$
(N) are, respectively, the vehicle acceleration, the gravitational acceleration and the drag force which is a function dependent on the vehicle speed and geometry.
The acceleration of the aerial vehicle is fixed as
$a = ( V c 2 2 · h$
). In order to take into account the drag force and motor efficiency, which is supposed to be a small drag profile, an efficiency
$η = 0.75$
is considered. Thus, the motor power
$P m$
(kW), during the take-off and the landing segment flight operation, must satisfy:
$P m ≥ G T O W · ( a + g ) · V c N p · η ≈ 15 kW$
During the cruise flight mission, the thrust is equal to the drag force. In this case, the lift force is given by $F l i f t = G T O W · g$, and motor power must satisfy $P m ≥$ 13 kW, with a thrust
to weight ratio of $T / W = 1.2$.
Regarding the propeller sizing in order to satisfy the mission power flight, a two blade fixed pitch carbon fiber-based material is chosen. This category of propellers is characterized by its
stiffness and lightweight. The propeller parameters are the diameter
$D p$
, the blade number
$B p$
and the pitch angle
$φ p$
or the pitch which is related to the pitch angle by
$H p = D p · tan ( φ p )$
and its model, which describes the propeller thrust
$T p$
(N) and torque
$M p$
(N · m) in terms of the other parameters given by [
$T = C T · ρ · N 60 2 · D p 4 M = C M · ρ · N 60 2 · D p 5 ,$
$C T$
$C M$
, and
(rpm) are, respectively, air density, thrust coefficient, torque coefficient, and propeller velocity. The air density,
, is determined by both the local temperature
$T t$
(°C) and the air pressure
, which is further determined by altitude
(m). The thrust and torque coefficients are dependent on the propeller blade airfoil shape. Their model and their approximative values are presented in [
]. Based on the sizing methodology presented in [
], it is suggested that with a carbon fiber propeller of
$D p = 1.2$
m and
$H p = 0.456$
m, the developed thrust is 74 kg at a velocity of 3300 rpm. The propeller parameters and its thrust and torque in terms of the velocity are, respectively, given by
Table 1
Figure 2
. It is noticeable, as shown in
Figure 2
, that the working motor torque is 36 N · m.
3. Design Methodology and Elements for Optimal Design
3.1. Design Methodology
The design optimisation method presented in this paper is based on a combination of analytical and numerical optimisations design methods for OR multiphase PM electric actuators. As shown in
Figure 3
, this methodology offers the possibility of considering several topologies in terms of motor number of phases
$n p h$
, the
$s / p$
combinations, winding configuration, and positions of PM. This approach takes as input the rated range speed
$ω m$
with the maximum torque
$T m$
, which is defined through the AV flight mission, the rated bus dc voltage
$U d c$
, and constraints in terms of embedded mass and maximum volume reserved for the actuator (
$W m , V m$
). It should be noted that in aircraft and airspace applications such as the one considered in this paper, the constraints in terms of mass and volume are critical and directly influence the AV
In summary, this methodology is resumed in three main steps. The first step, which includes the number of phases selection and FSCW design, consists of performing choices for the electric motor
optimal design. The number of phases selection is subjective to the application constraints, especially in terms of the power level of the application as well as the number of open-circuit (OC)
phases that the EM could tolerate. In this paper, 4 number EM phase cases are considered, which are the 3, 5, 6, and 7 phases. This will allow us to assess its effect on the EM electric and magnetic
performances. The
$F S C W$
configuration design is based on the
$s / p$
combination selection. The
Section 3.2
Section 3.3
give respective details for an optimal selection of the
$s / p$
combination and design of the corresponding winding layout.
The second step of this approach, which is presented in
Section 3.4
Section 3.5
, is devoted to the EM pre-sizing, including PM position selection, and analytical sizing. The analytical sizing is based on empirical parameters, namely the quantity of torque per rotor volume (
$T R V$
) and the shape ratio (
$S R$
). This step allows the validation of the initial specifications input of the approach, especially the motor output torque, using finite element analysis (
$F E A$
) validation, in addition, it makes it possible to limit the search space of the optimisation algorithm, which leads to reducing the calculation time.
The final step consists of carrying out a global multi-objective optimisation of the EM based on the Direct algorithm as explained in
Section 4.2
. This optimisation consists of both maximizing and minimizing, respectively, the EM efficiency
$( η m o t ( % ) )$
and the EM weight
$( W m o t$
(kg)) in a single function with the use of weighting coefficients. Constraints on the electromagnetic torque
$( T m o t )$
, on the flux density
$( B m )$
, and on the electromotive force
$( E r m s )$
are considered in the optimisation problem as shown in
Section 4.1
. In the case of multiphase motor configuration, it is possible to consider the injection of currents containing higher harmonics than the fundamental, which results in a compact EM structure with a
high torque density. Thus, in this paper, the formulation of the optimisation problem considers two scenarios, a scenario of optimisation with the injection of purely sinus currents and a scenario of
optimisation with non-sinus of currents. A new methodology for selecting the optimal current harmonics ratio that allows the creation of torque without the pulsation effect is developed and presented
Section 4.3
. The design methodology is enclosed by the analysis and comparison of the EM magnetic and electrical performances.
3.2. Selection fo the $s / p$ Combination
The optimal selection of s/p combinations depends on several parameters such as winding factor, torque density, torque cogging, motor efficiency, rotor losses, and noise level [
]. In the multiphase concentrated winding, the
$s / p$
combinations that allow the realisation of balanced winding should fulfil the following condition [
$Q s G C D ( Q s , 2 p ) = n p h · k$
$Q s$
is the number of slots,
is the number of pair poles,
$n p h$
number of EM phases,
is an integer number, and
$G C D$
is the great common divisor. The torque cogging and noise level are directly conditioned by the
$L C M$
(least common divisor) and
$G C D$
of the
$s / p$
combination. The larger the value of
$G C D$
, the smaller the value of the torque cogging, and the larger the value of the
$L C M$
, the smaller the level of noise and the vibration. Furthermore, regarding the integer
, which is also called the key winding factor in [
], the higher it is, the more the magneto-motive force (MMF) distribution would contain lower amplitude of higher harmonics which leads to lower torque ripper. The winding factor
$K w$
is also used as an indicator for the
$s / p$
combination selection, where the larger it is, the torque density and the efficiency of the motor are improved [
]. In this paper, the following
$s / p$
combination is selected for each phase number: 3-phase
$48 / 40$
, 5-phase
$40 / 44$
, 6-phase
$48 / 44$
and 7-phase
$56 / 44$
Table 2
gives the
$L C M$
$G C D$
, and
$K w$
, including the third(
$K w 3$
) harmonic for the 5-phase and 6-phase machines and the third and fifth(
$K w 5$
) harmonic for the 7-phase machine for each
$s / p$
3.3. Winding Configuration Design
Multiphase fractional slot concentrated winding (FSCW) configuration has gained much attention in the field of electric motors, particularly for aeronautical applications. This interest could be
explained by the multiple advantages that present this type of winding configuration, in particular, in terms of a high motor torque density, improvement of the motor efficiency by reducing copper
losses, improving the fault tolerance by reducing the coupling between phases, and ease of manufacturing and installation [
]. However, this kind of winding layout leads to a significant appearance of subharmonics and higher harmonics in the MMF distribution, which could be limited, as explained in the
Section 3.1
, by an appropriate selection of the
$s / p$
combination. There are two main types of FSCW, namely single-layer (SL) winding and double-layer (DL) winding. SL winding configurations have stator slots that are each occupied by the coil sides of
a single stator phase, while each slot in a DL winding configuration is split equally between coil sides from two phases. In this paper, the DL configuration is considered for the FSCW configuration
Regarding the optimal winding layout, it is based on the
$C r o s$
$V i a r o u g e$
methodology presented in [
], which was applied to a 3-phase PM Machine with an FSCW. In this paper, this methodology is extended to a number of phases greater than 3. The first step is to reduce the number of slots per pole
per phase (
$N S P P$
) to a fraction of two prime integers
, where:
A repeatable sequence of 0 and 1 specific to the winding can be derived from this relation. It is a list of y numbers which characterizes the winding distribution under $y / n p h$ poles. The
structure of the whole winding can be derived from a periodic distribution of the structure under y poles described by $n p h$ consecutive repeatable sequences if y is an even number. If y is an odd
number, this distribution is antiperiodic. The number of 1 in the sequence is equal to y, and the number of 0 sequences is equal to $y − x$.
As an example, in the case of 5-phase with the
$s / p$
combination of
$40 / 44$
the following steps explain how the whole winding layout is obtained:
• Step 1: The initial repeatable sequence is: 11000000000;
• Step 2: The optimal repeatable sequence is: 10000010000;
• Step 3: The usual phase sequence is associated with the hole sequence. In the case of 5-phase, given by: $A C ′ E B ′ D A ′ C E ′ B D ′$ ($A ′$ characterizes the return conductor corresponding to
coil of phase A);
• Step 4: The conductors associated with the numbers 1 of the sequence are selected to make the first layer of winding. This allows to obtain the first layer winding. The second layer winding is
obtained by reproducing and shifting the initial layer by a tooth or a slot width.
Figure 4
gives the winding layout for a quarter of the machines (
$Q s = 10$
). The whole winding configuration is completed by an antiperiodic symmetry, as
$y = 11$
is an odd number in this case.
The same process is applied to other configurations, namely 3-phase
$48 / 40$
, 6-phase
$48 / 44$
, and 7-phase
$56 / 44$
Figure 5
a–d gives their whole winding configuration, respectively.
3.4. PM Configurations
There are many configurations for installing magnets on or in the rotor. These topologies can be divided into two main categories: surface magnet (SPM) rotor and interior magnet rotor (IPM). In
comparison with other topologies, the SPM rotor configuration is widely adopted as an efficient solution, due to its simple structure, and ease of manufacture; the absence of magnetic salience and
reduced tooth effect minimizes torque oscillation and makes control much easier [
]. In the case of the IPM rotor, the magnets are inserted in the rotor, which makes them intrinsically well protected mechanically and magnetically [
Figure 6
provides the 3 PM configurations which will be considered in this paper. It is noticeable that the consideration of several PM configurations will allow us to assess their influence on EM
performances, especially the motor efficiency, torque density and torque rippling.
3.5. Analytical Pre-Sizing
The analytical pre-sizing stage allows an initial validation of the specifications, particularly in relation to the maximum torque that the EM must provide. This step also makes it possible to limit
the search space that the optimization algorithm must consider to locate the
$P M$
actuator optimal geometry. The analytical sizing of the EM is determined using two variables, namely the torque per rotor volume
$( T R V )$
and the shape ratio
$( S R )$
. Thus, the outer rotor diameter
$D o r$
and the stack length
$L s t k$
are determined. These variables are, respectively, given by:
$T R V = T m a x π 4 · D o r 2 · L s t k , S R = L s t k D o r ,$
$T m a x$
is the
$E M$
maximum torque which is fixed to be 36 N·m. The approximative value of the
$T R V$
is dependent on
$P M$
materials. In the case of
$N d F e B$
OR PM motors with air cooling, the
$T R V$
value is about 28 kN/m
]. Regarding the
$S R$
value, in the case of the outer runner motor, it must avoid the motor shearing. In this paper, a
$S R$
initial value of
$1 / 5$
is considered. It is remarkable that in the case of the outer rotor configuration, the torque density (
$T D = T m a x π 4 · D o r 2 · L s t k$
$T R V$
coincide with each other. Then, the outer rotor diameter
$D o r$
and the stack length
$L s t k$
are, respectively, given by:
$D o r = T m a x π 4 · T R V · S R 3 L s t k = T m a x · S R 2 π 4 · T R V 3$
Thus, the rotor yoke, stator yoke and tooth thickness are, respectively, given by:
$h r y = B g · π · D o r 4 · p · B r y = B g · π 4 · p · B r y · T m a x π 4 · T R V · S R 3 h r y = B g · π · D o r 4 · p · B s y = B g · π 4 · p · B s y · T m a x π 4 · T R V · S R 3 h r y = B g ·
π · D o r Q s · B t = B g · π Q s · B t · T m a x π 4 · T R V · S R 3$
$B g$
$B r y$
$B s y$
$B t$
are, respectively, the airgap flux density, the rotor yoke flux density, the stator flux density, and the teeth width flux density. The inner rotor and the outer stator diameters are, respectively,
$D i r$
, given by:
$D i r = D o r − 2 · h r y = 1 − 2 · B g · π 4 · p · B r y · T m a x π 4 · T R V · S R 3 D o s = D i r − 2 · l g = 1 − 2 · B g · π 4 · p · B r y · T m a x π 4 · T R V · S R 3 − 2 · l g$
$l g$
is the airgap length where an initial value of 0.8 is taken. It is remarkable that in the case of
$S P M$
, the thickness of PM must be considered in the calculation of the outer stator diameter. The PM width
$h m$
for the three configurations, is given by:
$h m ( S P M ) = π · D o r 2 · p · α m = α m · π 2 · p · T m a x π 4 · T R V · S R 3 h m ( I P M ) = sin ( α m ) cos ( α m ) · D o r = sin ( α m ) cos ( α m ) · T m a x π 4 · T R V · S R 3 h m ( S p
o k e ) = B g · π · D o r Q s · B t = B g · π Q s · B t · T m a x π 4 · T R V · S R 3$
$α m$
as illustrated in
Figure 6
, the polar arc or the pole pitch is fixed at
in the case of IPM and SPM configurations. The
$P M$
thickness, in the spoke configuration case, is equal to the rotor yoke thickness.
The number of spires per coil
$N s$
is given by:
$N s = π · p · E m a x n c · K w · ω m · B r · R o s · L s t k · sin ( α m · π 2 )$
$E m a x ( V )$
$n c$
$ω m$
(rad/s), and
$B r ( T )$
are, respectively, the maximum value of the electromotive force (EFM), the number of coils per phase, the mechanical rotor velocity, and the remanent flux density of the PM.
A threshold of $3 2 · U d c$ is taken as a reference value for the FEM maximum value, where $U d c$ is the DC bus voltage. For the $N d F e B$ PM material, the remanent flux density value is about
$1.2 T$. The number of coils per phase depends on the winding configuration.
The slot area
$A s$
is defined as follows:
$A s = 2 N s · A c K f i l l$
, where
$A c$
$K f i l l$
are, respectively, the conductor section and the filling factor, which is fixed at
. The conductor section is given by
$A c = I m a x J$
. With
$J ( A / m 2 )$
, and
$I m a x$
, respectively, the current density, which is chosen based on the cooling technology, and the maximum current. In the pre-sizing
is fixed at
$10 A / m 2$
. The maximum current can be estimated as:
$I m a x = T m a x · ω m η · n p h · U d c$
is the global efficiency of the propulsion chain.
The slot height
is defined using the following condition:
$A s ≤ A S P − A t o o t h$
$A S P$
$A t o o t h$
, as illustrated in
Figure 7
are, respectively, the pole slot area, and the tooth area. The term
$A S P − A t o o t h$
presents the geometrical slot area. These area expressions are given by:
$A P S ≈ π Q s · R o s − d 1 2 − R o s − d 1 − d 2 − h 2 A t o o t h ≈ h t · h + W t · d 2 − π 2 · d 2 2$
The tooth head width
$W t$
is calculated using:
$τ s = D o s Q s$
$b s$
are, respectively, the stator pole and slot opening, which is initially estimated to
$b s = 2 · l g$
. The resolution of the Equation (
) is performed numerically, using a Matlab code. The inner stator diameter is given by:
$D i s = D o s − 2 · ( h s y + d 1 + d 2 + h ) ,$
Table A1
Table A2
Table A3
of the
Appendix A
summarize, respectively, the geometric data for the rotor parameters, PM parameters for the 3 configuration, and the stator parameters. It is notable that the phase number influences the stator
electric parameters and geometrical parameters, especially the maximum injected current, the spires number and the slot surface.
4. Multiobjective Optimisation Problem
4.1. Problem Formulation
The EM optimisation is an important step in the proposed sizing methodology, as it allows to obtain the best motor performances with respects to the imposed constraints. The objective function was
constructed by combining the motor efficiency
$η m o t$
and weight
$W m o t$
, as given in Equation (
), which are the representatives of the most important motor variables in the application context.
$F o b j = α · η m o t + β · 1 W m o t$
The coefficients
are used to prioritize both variables between them, with a complementarity condition
$( α + β = 1 )$
and thus, the optimisation algorithm will maximise the prioritized variable. In this paper, the motor efficiency and weight are chosen to have the same priority in the optimisation problem, which
means that
$α = β = 0.5$
. The motor efficiency
$η m o t$
is given by:
$η m o t = P u P u + P c o p p e r + P c o r e + P m e c h$
$P u ( W )$
$P c o p p e r ( W )$
$P c o r e ( W )$
, and
$P m e c h ( W )$
are, respectively, the useful power, copper losses, core losses, and mechanical losses. Their respective expressions are given by:
$P u = T m · ω m P c o p p e r = n p h · R p h · I m a x 2 P c o r e = ( C h · f + C e · f 2 ) · B m 2 · V P m e c h = 0.3 · P u · V t 2 · 10 − 5$
$T m$
$ω m$
$R p h ( Ω )$
$C h$
· Hz· T
$C e$
· Hz
· T
$B m$
$V t$
(m/s) are, respectively, the motor torque, speed, a phase resistance, hysteresis loss coefficient, eddy current loss coefficient, frequency, flux density peak amplitude, core volume, and tangential
speed. The phase resistance is directly estimated in terms of winding and stator parameters as:
$R p h = ( p · q · N p h · L s p · N s ) R c u 20 · ( 1 + α ( T s + T 0 ) ) S c$
as the number of slots per pole per phase given by
$q = Q s 2 · p · N p h$
$L s p$
as the spire length given by:
$L s p = 2 · ( L s t k + L C o H )$
, where
$L C o H$
is the coil head length, which is estimated by
$L C o H = π · R o s Q s$
. The
$R c u 20 = 1.75 × 10 − 8$
(Ω · m) and
$S c = I m a x J$
are, respectively, the copper resistivity and the conductor section.
$T s ($
$T 0 = 20 ($
°C), and
$α = 3.93 × 10 − 3 ( 1 /$
°C) are, respectively, the stator winding temperature, the ambient temperature, and the copper temperature coefficient. The core losses are supposed to be decomposed into losses due to hysteresis,
which are proportional to the frequency, and losses due to eddy current, which are proportional to the square of the frequency [
]. In this case, the stack material is supposed to be homogeneous and isotropic, with a uniform field. The coefficients
$C h$
$C e$
depend on stack material, which could be given by the manufacturer. In this paper, the ferromagnetic
$M 19 − 29$
gauge steel is considered as reference material. The corresponding hysteresis and eddy current coefficient are given by:
$C h = 156 ( W / ( m 3 · H z · T 2 ) ) C e = 0.144 ( W / ( m 3 · H z 2 · T 2 ) )$
The mechanical losses are due to the ventilation and aerodynamics that occur during the revolution between the airbag and the rotor, and friction in the bearings. Their analytical calculation is
complicated, because of the vortex flows during rotation. An empirical expression [
], given in Equation (
), is used to estimate these losses, which are in this case proportional to the useful motor power
$P u$
and to the square of the rotor tangential speed
$V t = R o r · ω m$
. The EM optimisation problem considers three constraints functions, which are: a constraint on the output motor torque
$T m$
that must correspond to the torque imposed by the propeller at a base speed of
$ω b a s e = 4000$
rpm; a constraint on the electromotive force at the base speed, in order to avoid the flux weakening and to maintain the
$E M$
performances and a constraint the maximal core flux density that must stay below of a reference value of
$2 T$
in order to avoid over saturation. Thus, the formulation problem of the EM optimisation is given as follows:
$m a x ( F o b j ( x ) ) T m o t ≥ 36 N . m E r m s = ( U d ) 2 + ( U q ) 2 ≤ 3 2 · U d c B m = ( B x ) 2 + ( B y ) 2 ≤ 2 T$
is the vector of the optimisation parameters, and is composed of a common parameters vector
$x m$
for the four winding configuration cases, as explained in
Section 3.2
, and it is given by:
$85 ≤ R o r ≤ 105 , 35 ≤ L s t k ≤ 45 , 2 ≤ b s ≤ 5 , 3 ≤ h s y ≤ 6 , 0.3 ≤ l g ≤ 0.7 , 10 ≤ N s ≤ 20 , 1 ≤ d 1 ≤ 3 , 1 ≤ d 2 ≤ 4 , a n d J ≤ 12$
. A second vector optimisation, specific to each PM topology
$x P M$
, is given by:
• SPM configuration: $8 ≤ h m ≤ 12 , 3 ≤ t m ≤ 5$, and $3 ≤ h r y ≤ 6$.
• V-shape configuration: $0.7 ≤ l P M ≤ 0.9 , 1 ≤ t m ≤ 2.5$, and $3 ≤ h r y ≤ 6$.
• Spoke configuration: $4 ≤ t m ≤ 6 ,$ and $8 ≤ h r y = h m ≤ 12$.
It is noticeable that in the of V-shape configuration, the variable $l P M$ represents the ratio of the $P M$ length $h m$ to the rotor yoke thickness.
4.2. Optimisation Algorithm
The optimisation is based on the Direct algorithm, belonging to the global non-linear constrained optimisation family. Its working principle consists of minimising a black-box objective function with
a bounded normalised search space. The normalisation of search space has no influence on the algorithm convergence as the optimisation variables are bounded. Thus, the search space becomes the unit
hyper-cube. DIRECT works by partitioning the unit hypercube into subrectangles with the property that the objective function has been evaluated at each rectangle’s center point. In each iteration,
certain “potentially optimal” rectangles are selected for further search; these rectangles are then subdivided, and the function is evaluated at the center points of the newly-formed subrectangles [
Figure 8
describes the way the algorithm moves towards the optimum of the EM optimisation problem. The algorithm, as well as the EM formulation problem, are implemented using Matlab functions, where the
Direct algorithm is explored from the optimisation toolbox of Matlab. The electromagnetic model of the motor is implemented using FEMM, where the inputs are the EM geometry data with the required
4.3. Optimal Harmonic Current Injection Ratio
In the case of multiphase winding configuration with
$n p h > 3$
, it is possible to consider the current injection containing harmonics. This makes it possible to improve the torque density as well as the reliability of the EM. This characteristic comes from the
fact that a multiphase motor voltage vector and output torque in the decoupled plan are given by [
$V d q = R s · I d q + [ L d q ] · d I d q d t + E d q T = 1 ω m ( e → · i → ) = 1 ω m · ∑ g = 1 m ( e g → · i g → )$
represents the number of the fictious machine, with two phases each, depending on the
$n p h$
as follows:
$m = n p h − 1 2 i f n p h i s o d d m = n p h − 2 2 i f n p h i s e v e n$
Thus, this formulation makes it possible to transform virtually the multiphase motor to
fictitious machines coupled in series, where each one is associated with a family of odd harmonics.
Figure 9
gives the equivalent representation of a multiphase PM motor in the general case of
$n p h$
. From this representation, the number of main fictitious and homopolar machines are deduced for the 4 number of phases cases considered in this paper.
The term
$2 k + 1$
is corresponding to the EMF harmonics, which must be equal to the injected current harmonic in order to create an effective torque. The following expression of the electromagnetic torque explains the
relation between the current and the EMF or airgap flux density harmonics:
$T e m = n p h · K ω m · B 1 · I 1 + B 3 · I 3 + B 5 · I 5 + ⋯ + B 2 k + 1 · I h = n p h · K ′ ω m · E 1 · I 1 + E 3 · I 3 + E 5 · I 5 + ⋯ + E 2 k + 1 · I h$
$K ′$
$B 2 k + 1$
$E 2 k + 1$
, and
$I h$
are, respectively, a parameter depending on the geometrical and electrical motor parameters, the
$( 2 k + 1 ) t h$
harmonic of the airgap flux density or the EMF, and
$h t h$
harmonic of the injected current. However, the maximisation of the motor output torque, in the case of
$n p h > 3$
, requires an optimal harmonic current injection. The sinusoidal
$( S i n )$
current and sinusoidal current with the consideration of higher harmonics
$( S i n + h )$
are expressed, respectively, as follows:
$i j ( θ m ) = I S · sin p · θ m − 2 π n p h j i j ( θ m ) = I N S · ∑ h = 0 ∞ A 2 h + 1 · sin ( 2 h + 1 ) · p · θ m − 2 π n p h j$
$I S$
$I N S$
$A 2 h + 1$
$θ m$
$j = 0 , 1 , ⋯ , n p h − 1$
are, respectively, current peak amplitude in the
$S i n$
case, current peak amplitude in the
$S i n + h$
case, ratio of the (
$2 h + 1 ) t h$
harmonic current to the fundamental one, rotor mechanical position, and the number of the phase starting form 0. The determination of the optimal harmonic current injection with the corresponding
$I N S$
is performed in two manners, either to maintain the same
$R M S$
value as case
$S i n$
or to maintain the same peak amplitude of the injected current as in case
$S i n$
. In [
], the authors present an analytical approach to derive the optimal third harmonic current for a 5-phase SPM machine under constraints of peak and RMS value. However, in the case of a higher phase
number, especially when
$n p h ≥ 7$
, the analytical development is not an efficient solution. In this paper, a new numerical method is developed in the general case. This makes it possible to determine the optimal harmonic current
that allows to maximise the motor output torque with consideration peak/RMS constraints. The method is described as follows:
• In the case of peak value, this constraint is expressed as:
$I S = I N S · m a x ∑ h = 0 ∞ A 2 h + 1 · sin ( 2 h + 1 ) · p · θ m − 2 π n p h j$
Thus, the term
$m a x ∑ h = 0 ∞ A 2 h + 1 · sin ( 2 h + 1 ) · p · θ m − 2 π n p h j$
must be minimised, in order to maximise the motor output torque. This constraint, in this case, consists of finding the ratios
$A 2 h + 1$
that satisfy the following problem:
$A = min m a x ∑ h = 0 ∞ A 2 h + 1 · sin ( 2 h + 1 ) · p · θ m − 2 π n p h j I N S = I S A w i t h 0 ≤ A 2 h + 1 < 1 a n d A 1 = 1$
Based on this formulation, the optimal harmonic ratio $A 2 h + 1$, in the case of 5 and 6-phase, the optimal third harmonic current injection is $A 3 = 0.167$. In the case of 7-phase, the optimal
third and fifth harmonics current injection are $A 3 = 0.24 , A 5 = 0.07$.
• In the case of RMS value, this constraint is expressed as:
$I S 2 = I N S 2 · ∑ h = 0 ∞ ( A 2 h + 1 ) 2$
As in the previous case, in order to maximise the motor torque, the term
$∑ h = 0 ∞ ( A 2 h + 1 ) 2$
must be minimised. This constraint, in this case, consists of finding the ratios
$A 2 h + 1$
that satisfy the following problem:
$A = min ∑ h = 0 ∞ ( A 2 h + 1 ) 2 I N S = I S A w i t h 0 ≤ A 2 h + 1 < 1 a n d A 1 = 1$
It is remarkable that the minimum of the considered function is attainable for
$A 2 h + 1 = 0 f o r h > 1$
; however, it is possible to inject higher harmonics in a limit of 20% as shown in [
]. Thus, in the optimisation part, they will be the consideration of the current injection in both cases
$S i n$
$S i n + h$
with peak constraint.
Figure 10
gives the wave form of the current injection in the cases
$S i n$
$S i n$
$h 3$
$S i n$
$h 3$
$h 5$
with the peak constraint. In order to assess the influence of these harmonics, the case
$5 − p h a s e$
optimisation is carried out using current injection with
$S i n$
$S i n$
$h 3$
5. Results Analysis and Performances Comparison
The optimisation problem, formulated in the previous section, is applied for each configuration of the EM while respecting each scenario of phase number and PM configuration. The optimisation for the
$n p h = 5$
, 6, and 7 was carried out by the injection of current harmonics, where the optimal ratios of the harmonic injection of currents are defined following the method presented in
Section 4.3
. The cases
$n p h = 3 ,$
and 5 were optimised by the injection of sine currents. It should be noted that the injection of the currents considered in the optimization occurs directly in the original plan, i.e., (
$a , b , c , d ,$
) for the case
$n p h = 5$
. However, it is possible to consider the injection in the decoupled plane
$( d k , q k )$
. A validation of the designed and optimised EM is performed using FEA, where the results and their analysis, as well as a comparative study of the number of phases, PM configuration, and harmonic
current effects on the EM performance, are presented in the following sections. It is notable that every three subfigures of the same column, given below in
Section 5.1
Section 5.2
Section 5.3
Section 5.4
show, respectively, the field lines and flux density in no-load conditions with the corresponding EMFs and its harmonic composition. The optimised geometrical parameters for the whole cases are
reported in the
Table A4
Table A5
Table A6
of the
Appendix A
5.1. Design Optimisation Results of the Case $n p h = 3$
The optimisation of the 3-phase with
$s / p : 48 / 40$
is carried out using a peak current of
$I s = 28$
A, a current density
$J = 11$
, and DC bus voltage
$U d c = 400$
Figure 11
a–i gives, respectively, the optimisation results for the 3 PM configuration. It is remarkable that in this case, the maximum core flux density amplitude stays below 2 T for the 3 PM positions, which
allows avoiding the core saturation. Moreover, the EMF harmonic spectrum shows a weak presence of higher harmonics in comparison to the fundamental frequency
$F s = p · ω m 60$
5.2. Design Optimisation Results of the Case $n p h = 5$
Regarding the case of the 5-phase with
$s / p : 40 / 44$
, the optimistion is performed using a peak current of
$I s = 17 A$
, a current density
$J = 11$
, and DC bus voltage
$U d c = 400$
Figure 12
a–i gives, respectively, the optimisation results for the 3 PM positions, where every three subfigures of the same column show the field lines and flux density in no-load conditions with the
corresponding EMFs and its harmonic composition. The obtained EMFs, as excepted, present a non-sinusoidal waveform, where the fifth harmonic presents higher amplitude for the 3 PM positions. The
amplitude of the third harmonic increases by passing from the SPM configuration to the IPM (Spoke and V-shape) configuration, which is explained by the higher harmonic component of the IPM
configuration. The core saturation is avoided, as the maximum core flux density amplitude remains below 2 T for the 3 PM positions
5.3. Design Optimisation Results of the Case $n p h = 6$
Regarding the case of the 6-phase with
$s / p : 48 / 44$
, the optimistion is performed using a peak current of
$I s = 14$
A, a current density
$J = 11$
, and DC bus voltage
$U d c = 400$
V. The optimisation results for the 3 PM positions are reported in
Figure 13
a–i, where every three subfigures of the same column show the field lines and flux density in no-load conditions with the corresponding FEM and its harmonic composition. The obtained EMFs present a
weak amplitude for the fifth, explaining the low effect on its waveform, especially in the SPM case. However, the amplitude of the third and fifth harmonics increase by passing from the SPM
configuration to the IPM (Spoke and V-shape) configuration, which is explained by the higher harmonic component of the IPM configuration.
5.4. Design Optimisation Results of the Case $n p h = 7$
The optimistion, in the case of the 7-phase with
$s / p : 56 / 44$
, is performed using a peak current of
$I s = 12$
A, a current density
$J = 11$
, and DC bus voltage
$U d c = 400$
V. The optimisation results for the 3 PM positions are reported in
Figure 14
a–i, where the same conditions are considered as the previous cases. For this motor topology, the obtained EMFs present non-sinusoidal waveform, where the third harmonic amplitude is more present
than the fifth and seventh for the 3 PM positions.
5.5. Assessment of Number of Phases and PM Configuration Effects on the EM Performances
In order to assess the effect of the number of phases and PM configuration, an FEA magnetostatic simulation is carried out for different configurations.
Table 7
Table 8
Table 9
compare these configurations in terms of average torque
$T a v g$
(N·m), torque ripple (
$T r i p$
(%)), torque mass density (
$T m d$
(N/kg)), torque volume density (
$T v d$
(N·m/L)), motor active mass (
$M m o t$
(kg)), PM mass (
$M P M$
(kg)), motor efficiency
$η m o t$
(%) at
$ω m = 4000$
rpm, maximum core flux density (
$B m$
(T)), core losses
$P c o r e$
(W), copper losses
$P c o p p e r$
(W), and mechanical losses
$P m v$
(W). Each table compares the EM performances, for a given PM configuration, for each case of the number of phases.
The increase in the number of phases, passing from the 3-phase to the higher phase number case, makes it possible to improve the motor performance, such as torque mass and volume motor densities,
motor efficiency, weight, volume and losses. However, it is remarkable that the increase in the number of phases by more than five phases causes a drop in the performances as well as an increase in
the motor weight, which is explained by the decreased maximum amplitude of the injected current. Regarding the PM configuration effect, it is noticeable that their effects are not conserved by the
increasing phase number. For instance, in the case of the 3 and 5 phases, the spoke configuration allows obtaining the best motor in terms of performance and compactness; however, in the cases of the
6 and 7 phases, it is the SPM configuration that allows to obtain the best EM performances.
5.6. Assessment of Current Harmonics Effects on the EM Performances
The case of the 5-phase motor with a spoke configuration is taken as an example to evaluate the effect of harmonic current ratio, where the optimisation, in this case was carried out considering the
pure sinusoidal current and non-sinusoidal current. Moreover, the designed PM spoke machine, considering the sinusoidal waveform current in the optimisation part, is used to validate the design
approach under a load of
$T m = 36$
N·m at
$ω m = 4000$
rpm. This load represents the operating point imposed by specifications.
Figure 15
gives the instantaneous output torque using
$S i n$
$S i n + h$
As expected, the consideration of higher harmonic in the current waveform allows for an improvement in the mean torque density of the EM of more than 18 %, in comparison with the classical case of
$S i n$
current. This result is confirmed by
Figure 15
, which shows, respectively, the output instantaneous torque in terms of the current density and the rotor mechanical position for each optimisation case, as well as a comparison of their respective
performances, is shown in
Table 10
, with a current density of
$J = 11$
and peak amplitude of the injected current of
$I m a x = 18$
Through the plots of
Figure 16
, it is remarkable that, for both cases, the instantaneous torque increases with the increasing current density, where the torque peak values in the case of optimisation with non-sinusoidal current
(b) are much higher than the case with sinusoidal currents (a).
Table 10
gives more details about their respective performance, where it is shown that torque, torque mass and volume densities, in the case (b), are improved by 3.28 %, 20%, and 20 %, respectively, with a
weight gain of more than 600 g, as well as an increase in the torque ripple.
6. Conclusions
This paper proposed a general and robust design optimisation approach of multiphase PM OR actuators for multirotor aerial vehicle applications. The number of phases and the PM positions were taken as
input for design, where 12 motor topologies were designed and analysed. This made it possible to assess their influence on the EM performances, as shown in the comparative study.
The motor design requirements are formulated, where a two-blade fixed pitch propeller was sized in order to assume the considered flight mission. The pre-sizing step, including $s / p$ combination
selection, winding design, and analytical sizing, made it possible to validate the motor topology as well as the specifications, especially in terms of the motor output torque. These results are
refined using a multiobjective constrained optimisation of the motor efficiency and weight, where constraints on the output torque, EMF, and core maximum flux density amplitude are considered. It was
shown that, in comparison with the 3-phase motor, the higher number of phases allows for improved EM performance, torque, torque densities and compactness, as well as reliability with the redundant
winding configuration. Moreover, the injection of current, including higher harmonics in the case of multiphase configuration, allowed us to confirm these results.
Some perspectives will be considered in future work, e.g., the integration of FT capabilities and thermal constraints in the multiphase EM design step, considering scenarios of single OC fault,
multiple OC fault of adjacent or non-adjacent winding phases and inter-turn short circuit fault (ITSC).
Author Contributions
Conceptualisation, S.C.; methodology, S.C.; software, S.C. and G.K.; formal analysis, S.C.; resources, S.C.; discussion: S.C., G.K., C.M., R.S. and A.A.; writing—original draft preparation, S.C.;
writing—review and editing; S.C., G.K., C.M. and A.A. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Data Availability Statement
The data are available upon request to the corresponding author, Saad Chahba:
[email protected]
Conflicts of Interest
The authors declare no conflicts of interest.
Appendix A
$n ph$ $n ph = 3$ $n ph = 5$ $n ph = 6$ $n ph = 7$
$s / p$ combination $48 / 40$ $40 / 44$ $48 / 44$ $56 / 44$
Stator outer diameter $D o s$ (mm) $190.4$ $190.4$ $198.4$ $198.4$
Stator inner diameter $D i s$ (mm) $120.08$ $155.0$ $153.59$ $164.87$
Stator yokeless thickness $h s y$ (mm) 3 3 3 3
Stator pole pitch $τ s$ (mm) $12.46$ 15 $12.46$ $10.68$
$P M$ width $h m$ in the $S P M$ case 11.78 10.71 10.71 10.71
PM materiel NdFeB
teeth width $h t$ (mm) 3 3 3 2
Tooth hight h (mm) 27 8 6.5 5
Airgap length $l g$ (mm) 0.8 0.8 0.8 0.8
Slot opening $b s$ (mm) 3 3 3 3
Tooth head width $W t$ (mm) 9.64 12.0 9.46 7.68
Number of turns per coil $N s$ 20 17 14 12
Maximum currents $I m a x$ (A) 28 17 14 12
Currents density $J m a x$ (A/mm^2) 12
DC bus voltage $U d c$ (V) 400
Stator materiel Pure Iron (M19 Jauge 1.9 mm)
Stator flux density $B s y$ 0.9 T
Teeth width flux density $B t$ 1.8 T
$n ph$ $n ph = 3$ $n ph = 5$ $n ph = 6$ $n ph = 7$
$s / p$ combination $48 / 40$ $40 / 44$ $48 / 44$ $56 / 44$
Outer rotor diameter $D o r$ (mm) 206 206 206 206
Stack length $L s t k$ (mm) $41.2$ $41.2$ $41.2$ $41.2$
Inner rotor diameter $D i r$ (mm) 200 196 200 200
$s / p$ combination 3 3 3 3
Rotor materiel Pure Iron (M19 Jauge 1.9 mm)
Rotor flux density $B r y$ 0.9 T
$n ph$ $n ph = 3$ $n ph = 5$ $n ph = 6$ $n p h = 7$
$s / p$ combination $48 / 40$ $40 / 44$ $48 / 44$ $56 / 44$
Stator outer diameter $D o s$ (mm) $198.4$ $198.4$ $198.4$ $198.4$
Stator inner diameter $D i s$ (mm) $142.72$ $164.11$ 170 $173.7$
Stator yokeless thickness $h s y$ (mm) 3 3 3 3
Stator pole pitch $τ s$ (mm) 13 15 13 $11.13$
$P M$ width $h m$ in the $I P M$ case 12.55 11.41 11.41 11.41
$P M$ width $h m$ in the $s p o k e$ case 3 3 3 3
PM materiel NdFeB
Tooth width $h t$ (mm) 3 3 3 2
Tooth hight h (mm) 24.27 5.3 6 4.2
Airgap length $l g$ (mm) 0.8 0.8 0.8 0.8
Slot opening $b s$ (mm) 3 3 3 3
Tooth head width $W t$ (mm) 10 12.58 10 8.13
Number of turns per coil $N s$ 20 17 14 12
Maximum currents $I m a x$ (A) 28 17 14 12
Currents density $J m a x$ (A/mm^2) 12
DC bus voltage $U d c$ (V) 400
Stator materiel Pure Iron (M19 Jauge 1.9 mm)
$n ph$ SPM Configuration
$R or$ $h m$ $L stk$ $b s$ $h ry$ $h sy$ $t m$ $l g$ $N S$ $d 2$ $d 3$
3 102.00 11.33 40.78 1.78 3.42 3.42 3.17 0.30 6 1.50 1.50
5 91.00 8.67 37.34 2.00 4.78 4.78 4.17 0.25 15 1.11 2.70
6 92.89 10.00 37.11 2.70 3.42 3.42 3.17 0.38 14 2.00 1.50
7 93.00 11.00 40.83 2.33 3.42 3.42 4 0.30 14 1.67 1.12
$n ph$ Spoke Configuration
$R or$ $h m$ $L stk$ $b s$ $h ry$ $h sy$ $l g$ $N S$ $d 2$ $d 3$
3 86.67 7.05 42.00 1.4 12.12 5.33 0.38 4 1.40 1.33
5 80.33 10.11 37.50 1.80 6.02 3.93 0.38 15 2.50 1.33
6 93.00 8.33 34.50 2.7 8.00 3.93 0.35 13 1.89 2.00
7 80.33 8.33 39.89 1.5 8.00 3.93 0.38 12 1.50 1.33
$n ph$ V-Shape Configuration
$R or$ $h m$ $L stk$ $b s$ $h ry$ $h sy$ $t m$ $l g$ $N S$ $d 2$ $d 3$ $α m$
3 103.78 10.00 45.89 1.20 13.00 4.00 2.00 0.37 5 2.00 1.33 24.50
5 80.33 10.1 37.5 1.78 6.03 3.93 2.00 0.38 15 2.00 1.33 24.50
6 91.00 9.50 38.56 1.80 9.11 3.75 2.33 0.30 12 1.50 1.33 26.67
7 92.00 9.50 41.33 2.00 9.00 3.75 2 0.30 14 1.80 1.11 26.00
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Figure 11. FEA analysis for the case $n p h = 3$. (a) Field lines and flux density of the SPM case, (b) Field lines and flux density of the Spoke case, (c) Field lines and flux density of the V-shape
case, (d) No-load EMF of the SPM case, (e) No-load EMF of the Spoke case, (f) No-load EMF of the V-shape case, (g) EMF harmonic composition, (h) EMF harmonic composition, (i) EMF harmonic
Figure 12. FEA analysis for the case $n p h = 5$. (a) Field lines and flux density of the SPM case, (b) Field lines and flux density of the Spoke case, (c) Field lines and flux density of the V-shape
case, (d) No-load EMF of the SPM case (e) No-load EMF of the Spoke case, (f) No-load EMF of the V-shape case, (g) EMF harmonic composition, (h) EMF harmonic composition, (i) EMF harmonic composition.
Figure 13. FEA analysis for the case $n p h = 6$. (a) Field lines and flux density of the SPM case, (b) Field lines and flux density of the Spoke case, (c) Field lines and flux density of the V-shape
case, (d) No-load EMF of the SPM case, (e) No-load EMF of the Spoke case, (f) No-load EMF of the V-shape case, (g) EMF harmonic composition, (h) EMF harmonic composition, (i) EMF harmonic
Figure 14. FEA analysis for the case $n p h = 7$. (a) Feild lines and flux density of the SPM case, (b) Feild lines and flux density of the Spoke case, (c) Feild lines and flux density of the V-shape
case, (d) No-load EMF the SPM case, (e) No-load EMF the Spoke case, (f) No-load EMF the V-shape case, (g) EMF harmonic composition, (h) EMF harmonic composition, (i) EMF harmonic composition.
Specifications Value
Propulsion chain number $N p$ 6
Propeller diameter $D p$ (m) 1.2
Propeller pitch $H p$ (m) 0.465
Thrust coefficient $C t$ 0.09787
Torque coefficient $C m$ 0.00402
Gross take-off weight $G T O W$ (kg) 445
Thrust T (kg) 74
Propeller speed $N p$ (rpm) 3300
Propeller torque $M p$ N · m 36
$n ph$ $s / p$ Combination $LCM$ $GCD$ k $K w$
$K w 1$ $K w 3$ $K w 5$
3 48/40 240 8 2 0.933 - -
5 40/44 440 4 2 0.976 0.794 -
6 48/44 528 4 2 0.950 0.604 -
7 56/44 616 4 2 0.891 0.717 0.988
Injected Harmonic Current Corresponding Odd Harmonics
$h = 1$ (Main fictious machine) $1 , 5 , 7 , 9 , 17 , 19 , ⋯ ( 2 · k = 3 c ± 1 )$
Injected Harmonic Current Corresponding Odd Harmonics
$h = 1$ (1st fictious machine) $1 , 9 , 11 , 19 , 21 , 29 , ⋯ ( 2 k + 1 = 5 c ± 1 )$
$h = 3$ (2nd fictious machine) $3 , 7 , 13 , 17 , 23 , 27 , ⋯ ( 2 k + 1 = 5 c ± 2 )$
Injected Harmonic Current Corresponding Odd Harmonics
$h = 1$ (1st fictious machine) $1 , 5 , 7 , 11 , 13 , 17 , ⋯ ( 2 k + 1 = 6 c ± 1 )$
$h = 3$ (2nd fictious machine) $3 , 9 , 15 , 21 , 27 , 33 , ⋯ ( 2 k + 1 = 6 c ± 3 )$
Injected Harmonic Current Corresponding Odd Harmonics
$h = 1$ (1st fictious machine) $1 , 5 , 7 , 11 , 13 , 17 , ⋯ ( 2 k + 1 = 7 c ± 1 )$
$h = 5$ (2nd fictious machine) $5 , 9 , 19 , 23 , 33 , 37 , ⋯ ( 2 k + 1 = 7 c ± 2 )$
$h = 3$ (third fictious machine) $3 , 11 , 17 , 25 , 31 , ⋯ ( 2 k + 1 = 7 c ± 3 )$
$n ph$ SPM Configuration
$T avg$ $T rip$ $T md$ $T vd$ $M mot$ $M PM$ $η mot$ $B m$ $P core$ $P copper$ $P mv$
3 36.26 12.77 8.79 66.38 4.13 0.51 93.78 2.028 662.05 262.35 83.18
5 $36.96$ $1.92$ $12.03$ $90.6$ $3.08$ $0.46$ $93.97$ $1.95$ 426.01 499.51 67.49
6 $36.3182$ 5 $13.05$ $98.5$ $2.78$ $0.41$ $93.77$ $1.96$ 387.44 554.10 69.09
7 $37.83$ $0.64$ $11.53$ $87.23$ $3.29$ $0.62$ $91.76$ $2.05$ 476.71 873.47 72.15
$n ph$ Spoke Configuration
$T avg$ $T rip$ $T md$ $T vd$ $M mot$ $M PM$ $η mot$ $B m$ $P core$ $P copper$ $P mv$
3 35.708 16.01 9.13 68.98 3.91 0.68 91.64 2.08 1130.5 174.32 59.09
5 $37.11$ $6.30$ $12.82$ $95.50$ $2.89$ $0.42$ $93.60$ $2.02$ 453.66 559.04 52.81
6 $37.82$ $4.25$ $10.41$ $77.49$ $3.64$ $0.47$ $93.35$ $2.02$ 578.88 481.89 68.04
7 $36.91$ $4.23$ $9.36$ $69.79$ $3.94$ $0.49$ $91.03$ $2.036$ 669.94 786.28 66.40
$n ph$ V-Shape Configuration
$T avg$ $T rip$ $T md$ $T vd$ $M mot$ $M PM$ $η mot$ $B m$ $P core$ $P copper$ $P mv$
3 36.00 15.27 6.90 52.01 5.21 0.83 91.13 2.50 1133.4 249.28 85.49
5 $36.82$ $5.19$ $9.91$ $73.98$ $3.72$ $0.49$ $92.25$ $2.06$ 707.05 522.93 65.76
6 $36.78$ $9.81$ $8.18$ $60.78$ $4.50$ $0.52$ $90.51$ $2.59$ 924.74 624.36 67.16
7 $37.21$ $3.86$ $10.59$ $79.43$ $3.51$ $0.56$ $88.69$ $2.50$ 949.54 968.31 69.44
Performanaces of the 5-Phase Spoke EM
$T avg$ $T rip$ $T md$ $T vd$ $M mot$ $M PM$ $η mot$ $B m$ $P core$ $P copper$ $P mv$
(a) 36.58 1.42 10.25 76.41 3.57 0.60 93.01 2.03 600.41 488.88 61.53
(b) $37.11$ $6.30$ $12.82$ $95.50$ $2.89$ $0.42$ $93.60$ $2.02$ 453.66 559.04 52.81
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Chahba, S.; Krebs, G.; Morel, C.; Sehab, R.; Akrad, A. Design Optimisation Approach of an Outer Rotor Multiphase PM Actuator for Multirotor Aerial Vehicle Applications. Aerospace 2024, 11, 150.
AMA Style
Chahba S, Krebs G, Morel C, Sehab R, Akrad A. Design Optimisation Approach of an Outer Rotor Multiphase PM Actuator for Multirotor Aerial Vehicle Applications. Aerospace. 2024; 11(2):150. https://
Chicago/Turabian Style
Chahba, Saad, Guillaume Krebs, Cristina Morel, Rabia Sehab, and Ahmad Akrad. 2024. "Design Optimisation Approach of an Outer Rotor Multiphase PM Actuator for Multirotor Aerial Vehicle Applications"
Aerospace 11, no. 2: 150. https://doi.org/10.3390/aerospace11020150
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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fractions Archives - Simply Science
Have students play Go Fish, Memory Games, or online games to practice converting fractions to decimals to percents.
A fun way to review an important math skill.
Let’s Play Memory Games!
Do your students need to practice a basic math skill? Do you want them to be more confident using that skill Would you like to send a game home so students can practice math with family members?
While playing a Memory Game, students match up numbers with names, computation problems with their answers, fractions with their equivalents and much, much more! Print a set of cards and an answer
key, zip both in a bag, and you have an easy to store game that’s an excellent way to spend a few minutes in class, review an important skill, or reward carefully completed work. When you create
games …
Let’s Play a Bingo Game!
Students in one of my classes are learning fractions so we’re playing Fraction Bingo. Bingo makes it fun for students practice a math skill. Visit Simply Math to find a variety of bingo games
that your students will look forward to playing as they practice their math skills. Print the cards, cut squares of scrap paper, and then put both in a zip bag. Students keep their bingo bags in
their desks so they can play when there’s just a few minutes to play one game. It’s the “If we finish our work and we’ve done it carefully we can play a game of bingo” approach. It can be a …
Need More Time for Science?
Is your day just too short for science? Busy teaching reading, writing, and math and feeling a bit guilty because there are other subjects you want to teach but that dismissal bell keeps ringing too
soon? Consider ways to integrate math and science. Students practice their math skills and you have a few more minutes each week for science. Students see the connection between math and science.
Seem simple? It can be with some planning. Here are suggestions, with resources, that can get you started. Graphing – Oh Deer! Line graphs - temperature change Bar graphs - precipitation SymmetryÂ
-Â light reflecting using mirrors Angles and measurement … | {"url":"https://simplyscience.com/tag/fractions/","timestamp":"2024-11-09T03:19:12Z","content_type":"text/html","content_length":"48648","record_id":"<urn:uuid:03053517-360e-4bc0-9c16-e08403207304>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00429.warc.gz"} |
What Is Statistical Power Analysis & Why Does It Matter?
- A bigger sample size. More samples tested means more data collected and therefore more power.
- A larger effect sample size. If your effect is easier to detect, the statistical power of your study should increase.
- Higher level of significance. This will effectively make your test more sensitive, and thereby increase its power.
- Measure more accurately. By reducing your margins of measurement error, you'll also minimize variability in your study and make it more effective and reliable. | {"url":"https://www.visualizedata.app/articles/statistical-power-analysis/","timestamp":"2024-11-12T09:39:19Z","content_type":"text/html","content_length":"94709","record_id":"<urn:uuid:fec7d36b-e9c1-47d3-864c-254a974d4711>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00788.warc.gz"} |
3rd grade math puzzle worksheet
3rd Grade Math Puzzles Worksheets
3rd Grade Math Puzzles Worksheets
3rd Grade Math Puzzles Worksheets
3rd Grade Math Puzzles Worksheets
Third Grade Math Worksheets - Free Printable Math PDFs | edHelper.com
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Math Puzzle Worksheets 5th Grade Kids Will Love! Get It Free Now!
Free Math Puzzles Worksheets pdf printable | Math Champions
Math Crossword Puzzle +3 Worksheet for 1st - 2nd Grade | Lesson Planet
50+ Math Puzzles worksheets for 3rd Grade on Quizizz | Free ...
3rd Grade Math Puzzles Worksheets | {"url":"https://worksheets.clipart-library.com/3rd-grade-math-puzzle-worksheet.html","timestamp":"2024-11-07T16:44:03Z","content_type":"text/html","content_length":"24343","record_id":"<urn:uuid:7c12fb18-fbcb-48e4-898c-bb392fc60fe4>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00139.warc.gz"} |
1 Minute Multiplication Worksheet
Math, particularly multiplication, forms the cornerstone of many scholastic techniques and real-world applications. Yet, for several students, grasping multiplication can position an obstacle. To
resolve this obstacle, educators and parents have accepted an effective device: 1 Minute Multiplication Worksheet.
Intro to 1 Minute Multiplication Worksheet
1 Minute Multiplication Worksheet
1 Minute Multiplication Worksheet -
Our multiplication worksheets are free to download easy to use and very flexible These multiplication worksheets are a great resource for children in Kindergarten 1st Grade 2nd Grade 3rd Grade 4th
Grade and 5th Grade Click here for a Detailed Description of all the Multiplication Worksheets Quick Link for All Multiplication Worksheets
These basic math fact multiplication worksheets are similar to the RocketMath Mad Math Minutes or Mastering Math Facts multiplication programs used at many schools These are typically one minute
timed tests Try this countdown timer Spaceship Math Multiplication A Any Number Times One Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4
Relevance of Multiplication Technique Understanding multiplication is essential, laying a solid structure for advanced mathematical principles. 1 Minute Multiplication Worksheet use structured and
targeted technique, cultivating a deeper understanding of this fundamental math operation.
Advancement of 1 Minute Multiplication Worksheet
38 Printable Math Multiplication Worksheets Photos Worksheet For Kids
38 Printable Math Multiplication Worksheets Photos Worksheet For Kids
1 Minute Multiplication Looking for a stimulating challenge to help your child commit their multiplication facts to memory See how many of these one digit multiplication problems your young
mathematician can solve in one minute
On this page you will find Multiplication worksheets for practicing multiplication facts at various levels and in a variety of formats This is our most popular page due to the wide variety of
worksheets for multiplication available
From typical pen-and-paper exercises to digitized interactive styles, 1 Minute Multiplication Worksheet have actually progressed, dealing with diverse knowing designs and preferences.
Kinds Of 1 Minute Multiplication Worksheet
Fundamental Multiplication Sheets Simple workouts focusing on multiplication tables, assisting students develop a strong arithmetic base.
Word Problem Worksheets
Real-life situations integrated right into troubles, boosting critical thinking and application abilities.
Timed Multiplication Drills Tests made to improve rate and accuracy, helping in rapid psychological math.
Benefits of Using 1 Minute Multiplication Worksheet
Multiplication Worksheets
Multiplication Worksheets
Free 1 Minute Multiplication printable Math worksheets for 3rd Grade students Click on the image to view or download the PDF version Related posts
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Boosted Mathematical Skills
Regular practice develops multiplication effectiveness, boosting general math capacities.
Enhanced Problem-Solving Talents
Word problems in worksheets develop logical reasoning and strategy application.
Self-Paced Knowing Advantages
Worksheets accommodate individual knowing speeds, cultivating a comfortable and adaptable learning setting.
Just How to Produce Engaging 1 Minute Multiplication Worksheet
Incorporating Visuals and Colors Dynamic visuals and shades catch focus, making worksheets aesthetically appealing and involving.
Consisting Of Real-Life Circumstances
Connecting multiplication to daily situations adds significance and functionality to workouts.
Tailoring Worksheets to Various Skill Levels Personalizing worksheets based upon varying effectiveness degrees makes certain inclusive understanding. Interactive and Online Multiplication Resources
Digital Multiplication Tools and Gamings Technology-based resources use interactive discovering experiences, making multiplication interesting and satisfying. Interactive Internet Sites and
Applications On the internet systems provide varied and obtainable multiplication practice, supplementing traditional worksheets. Personalizing Worksheets for Various Learning Styles Aesthetic
Learners Aesthetic help and representations help understanding for learners inclined toward visual discovering. Auditory Learners Verbal multiplication issues or mnemonics satisfy learners who grasp
ideas through auditory means. Kinesthetic Students Hands-on activities and manipulatives sustain kinesthetic students in comprehending multiplication. Tips for Effective Implementation in Learning
Consistency in Practice Routine practice enhances multiplication abilities, promoting retention and fluency. Stabilizing Repeating and Variety A mix of repetitive exercises and diverse issue formats
maintains interest and understanding. Providing Useful Responses Comments help in determining areas of enhancement, motivating ongoing progress. Obstacles in Multiplication Method and Solutions
Inspiration and Engagement Hurdles Monotonous drills can result in uninterest; cutting-edge techniques can reignite inspiration. Getting Over Anxiety of Mathematics Negative assumptions around
mathematics can prevent progression; developing a favorable learning setting is important. Effect of 1 Minute Multiplication Worksheet on Academic Efficiency Studies and Research Study Searchings For
Research indicates a favorable relationship in between regular worksheet usage and improved math efficiency.
1 Minute Multiplication Worksheet become versatile devices, promoting mathematical efficiency in students while accommodating diverse discovering designs. From standard drills to interactive on-line
resources, these worksheets not just improve multiplication skills but additionally advertise important reasoning and problem-solving capacities.
Pin On Matematyka
Multiplication Worksheets Mad Minute PrintableMultiplication
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Multiplication Math Facts Worksheets One Minute
These basic math fact multiplication worksheets are similar to the RocketMath Mad Math Minutes or Mastering Math Facts multiplication programs used at many schools These are typically one minute
timed tests Try this countdown timer Spaceship Math Multiplication A Any Number Times One Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4
Multiplication Drills Superstar Worksheets
Over 42 One minute multiplication drill worksheets for numbers 0 13 Our simple single page multiplication worksheets can be used in a variety of ways Use them to teach drill memorize repeat test and
so much more Print and go with our easy downloadable PDF math worksheets One Minute Multiplication Drills Worksheet Pack Subscriber Freebie
These basic math fact multiplication worksheets are similar to the RocketMath Mad Math Minutes or Mastering Math Facts multiplication programs used at many schools These are typically one minute
timed tests Try this countdown timer Spaceship Math Multiplication A Any Number Times One Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4
Over 42 One minute multiplication drill worksheets for numbers 0 13 Our simple single page multiplication worksheets can be used in a variety of ways Use them to teach drill memorize repeat test and
so much more Print and go with our easy downloadable PDF math worksheets One Minute Multiplication Drills Worksheet Pack Subscriber Freebie
Multiplication Drills 1 12 Free Printable
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FAQs (Frequently Asked Questions).
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Just how often should pupils practice utilizing 1 Minute Multiplication Worksheet?
Constant technique is crucial. Regular sessions, ideally a few times a week, can yield considerable enhancement.
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Logic Theorist - (Cognitive Computing in Business) - Vocab, Definition, Explanations | Fiveable
Logic Theorist
from class:
Cognitive Computing in Business
The Logic Theorist is a pioneering computer program developed in 1955 that was designed to mimic the problem-solving skills of a human mathematician by proving mathematical theorems. It represents a
significant milestone in artificial intelligence, showcasing early efforts to automate reasoning and demonstrate how computers could engage in logical deduction similar to human thought processes.
congrats on reading the definition of Logic Theorist. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The Logic Theorist was created by Allen Newell and Herbert A. Simon and is often considered one of the first artificial intelligence programs.
2. It successfully proved 38 out of the first 52 theorems in 'Principia Mathematica', a foundational work in mathematical logic by Alfred North Whitehead and Bertrand Russell.
3. The program utilized a form of symbolic logic to represent problems, making it easier for the computer to process and deduce conclusions.
4. Logic Theorist's approach laid the groundwork for future developments in AI, including the development of more advanced theorem proving systems.
5. Its success demonstrated that computers could not only perform calculations but also engage in complex reasoning tasks, fundamentally changing perceptions of what machines could achieve.
Review Questions
• How did the Logic Theorist influence the development of artificial intelligence as a field?
□ The Logic Theorist marked a critical advancement in artificial intelligence by demonstrating that computers could mimic human reasoning processes. Its ability to prove mathematical theorems
highlighted the potential for machines to engage in complex problem-solving tasks beyond mere calculations. This groundbreaking work paved the way for future AI research and applications,
inspiring further exploration into automating reasoning and decision-making.
• In what ways did the Logic Theorist utilize mathematical logic to achieve its theorem-proving capabilities?
□ The Logic Theorist utilized mathematical logic by representing problems in symbolic form, allowing it to apply formal rules of deduction to derive conclusions. By leveraging these logical
structures, the program could systematically explore potential proofs for theorems, mimicking how human mathematicians approach problem-solving. This representation was crucial for enabling
the program to handle complex logical relationships effectively.
• Evaluate the long-term implications of the Logic Theorist's approach on modern cognitive technologies and automated reasoning systems.
□ The Logic Theorist's innovative approach set important precedents for modern cognitive technologies by demonstrating that logical reasoning could be automated through computer programming.
Its methods laid the foundation for advanced theorem provers and other AI systems that rely on formal logic to solve complex problems today. As cognitive technologies have evolved, they
continue to build on these principles, integrating heuristic methods and machine learning techniques to enhance reasoning capabilities in diverse applications, from natural language
processing to automated decision-making.
"Logic Theorist" also found in:
Subjects (1)
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The Monty Hall Problem
by , May 16th, 2011 at 07:29 PM (25769 Views)
The Monty Hall problem is a statistical problem which originates from the television Game show Lets Make a Deal, hosted by Monty Hall. The game is simple: a contestant is presented 3 closed
doors, behind one of which is a valuable prize (oftentimes described as a car, whereas the other doors have goats behind them). A contestant chooses a door. The host then opens one of the doors
you did not choose, which does NOT contain the prize. Then the host asks - do you want to change your decision? What is the probability you will win if you choose to change doors? What is the
probability you win if you choose to remain with your decision.
Simplistically, one might guess your probability of winning went from 1/3, to 1/2. This however, is incorrect. If you stay, your probability of winning is still 1/3. If you change, your chances
of winning is 2/3...How does this counter-intuitive result play out? One can tabulate all the possibilities, and the contestants decision:
Door1 Door 2 Door 3 Switch Stay
Prize Nothing Nothing Prize Nothing
Nothing Prize Nothing Nothing Prize
Nothing Nothing Prize Nothing Prize
Looking at the chart carefully*: if you switch your chances go from 1/3 to 2/3 of winning. Still not convinced? Below is a simple simulation of this problem. This iterates through a game if n
number of doors a defined number of times, tallying up whether you win or loose based upon staying
or changing...and the simulation backs everything up - average for 3 doors: stay: 33%, change 66%.
import java.util.Random;
* Simulates the MontyHall problem. 3 doors, 2 with goats and 1 with car. You choose a door,
* Monty hall opens one of the other two to reveal a goat. How often will you be correct if you
* stay? How often if you switch?
* @author copeg
public class MontyHallSimulation implements Runnable{
/*Random number generator */
private static final Random RANDOM = new Random();
/*Number of rounds to simulate*/
private int rounds;
/*Number of doors total*/
private int doors;
/*Rate for staying*/
private double stayRate = 0;
/*Rate for changing*/
private double changeRate = 0;
* Constructs a MontyHallSimulation with 3 doors, to iterate 1000 times.
public MontyHallSimulation(){
* Constructs a MontyHallSimulation with the number of rounds to simulate, and
* 3 doors.
* @param rounds The number of rounds to simulate.
public MontyHallSimulation(int rounds){
this(rounds, 3);
* Constructs a MontyHallSimulation with the number of rounds and doors to use in
* the simulation.
* @param rounds The number of rounds to simulate
* @param doors The number of doors to use in the simulation.
* @throws IllegalArgumentException if doors is less than 3.
public MontyHallSimulation(int rounds, int doors){
if ( doors < 3 ){
throw new IllegalArgumentException("Cannot simulate the problem with less than 3 doors.");
this.rounds = rounds;
this.doors = doors;
* Implementation of the Runnable interface. Simulates the Monty Hall Problem.
* This loops doors number of times, determining whether staying or changing
* results in a correct answer.
public void run(){
int stayCount = 0;
int changeCount = 0;
for ( int i = 0; i < rounds; i++ ){
int choose = RANDOM.nextInt(doors);//choose a door at random
int solution = RANDOM.nextInt(doors);//find a random place where the car will be.
if ( solution != choose ){//Car is in the other door - if you change you win
}else{//If you stay you win.
stayRate = stayCount/(double)rounds;
changeRate = changeCount/(double)rounds;
* Retrieves the rate one will be correct if one stays. This method returns
* zero unless run has been called.
* @return
public double getStayRate(){
return stayRate;
* Retrieves the rate one will be correct if one changes. This method returns
* zero unless run has been called.
* @return
public double getChangeRate(){
return changeRate;
* Application entry point.
* @param args
public static void main(String[] args){
MontyHallSimulation sim = new MontyHallSimulation(1000, 1000);
System.out.println("Choose to stay: percent correct - " + sim.getStayRate());
System.out.println("Choose to change: percent corect - " + sim.getChangeRate());
How about the same problem with 1000 doors? You are virtually guaranteed to win if you change your decision (what are the chances you chose the right door in the first place?).
A fun statistical problem to investigate and simulate...happy coding!
*The chart does not translate well in this format, however the link below contains a better, more readable version to inspect.
Wikipedia: The Monty Hall Problem
Description of the Monty Hall Problem | {"url":"https://www.javaprogrammingforums.com/blogs/copeg/13-monty-hall-problem.html","timestamp":"2024-11-13T08:46:08Z","content_type":"application/xhtml+xml","content_length":"62353","record_id":"<urn:uuid:852e310f-6217-48d1-8bfa-9f8c9c8dd3ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00004.warc.gz"} |
Ordinal Numbers - Fun2Do LabsOrdinal Numbers
Ordinal Numbers
Ordinal numbers are used every day in our daily lives. We don’t even realise this. It is therefore essential that students learn these concepts in a way that they can remember them. Ordinal numbers
are also referred to as ranking numbers. Ordinal numbers always represent rank or position. Ordinal numbers are written using ‘st’, ‘nd’, ‘rd’, and ‘th’ as superscripts with the numbers. For example,
1st tells us that the object at number 1 is actually first in the position.
Ordinal Number Names
Ordinal number names are always written using the suffix st (first), nd (second), third (rd) for numbers ending in 1, 2 and 3. For the rest of the numbers ‘th’ is used as a suffix. For example :
fourth, fifth, tenth, eleventh, etc. For example, we express 51 as fifty-first, 52 as fifty-second, and 53 as fifty-third.
Cardinal Numbers
Cardinal numbers are the counting numbers that represent the number of objects or people. Cardinal numbers show quantity. Example : There are eight apples in the basket, Cage contains two beautiful
Teaching ordinal numbers with kid friendly, clear and easy to understand posters from Uncle Math School by Fun2Do Labs :
Ignite kids’ curiosity with engaging stories for role play and skits, making the learning of this concept an exciting and effective experience. Teaching ordinal numbers through stories from Uncle
Math School by Fun2Do Labs :
Text of Stories
Learning ordinal numbers can be made enjoyable by incorporating interactive games and activities.
Arrange and see :
One of the easiest activities for early learners is lining objects up in a row. The best thing about this activity is that you can use anything of which you have an abundance. Example : Lego, blocks,
toys, cars, etc.
Integrating with other topics :
Teaching days of the week and months of the year using ordinal numbers : Kids gain knowledge of the ordinal numbers as well as the sequence of the days in the week or months of the year.
Help your kids to practise ordinal numbers with interesting and fun worksheets and solutions from Uncle Math School by Fun2Do Labs.
Explore related guides : | {"url":"https://fun2dolabs.com/ordinal-numbers/","timestamp":"2024-11-07T14:07:34Z","content_type":"text/html","content_length":"51844","record_id":"<urn:uuid:b74c109d-5060-430a-ae89-262be0a380bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00705.warc.gz"} |
Data Types for Circuits | Axiom V2 Developer Docs
Data Types for Circuits
In an Axiom client circuit, it is important to distinguish between values that should be constant in the circuit regardless of what inputs it is given, and values that are variable and depend on the
inputs to the circuit. We use different data types to distinguish these notions. In particular we have the primitive types ConstantValue, CircuitValue, and CircuitValue256. All circuit inputs must be
built from CircuitValue and CircuitValue256 types to delineate that they are variable and may change with each new proof.
Constant Data Types
export type ConstantValue = string | number | bigint;
Circuit functions can take in the ConstantValue type. These values are inferred to be fixed and immutable in your circuit. You should use ConstantValue types when the value should not change based on
the inputs to your circuit.
In some editors, IntelliSense will show RawCircuitInput in function signatures. This is an internal type alias that is the same as ConstantValue.
Circuit Data Types
There are two protected classes to represent variable values inside a circuit: CircuitValue and CircuitValue256.
Due to some specifics of our ZK proving system, a CircuitValue represents variable values that can be at most 253 bits. Since EVM slots are 256 bits, we've created a CircuitValue256 class that
represents 256 bit values in hi-lo form (hi is the most significant 128 bits, lo is the 128 least significant bits).
Here are the methods available on a CircuitValue object:
class CircuitValue {
// converts the CircuitValue to a CircuitValue256 inside the circuit
toCircuitValue256(): CircuitValue256;
// returns the value as a bigint
value(): bigint;
// returns the value as a number
number(): number;
// returns the value as an address string
address(): string;
The toCircuitValue256() function can be used to convert the object from CircuitValue type to CircuitValue256 type:
• toCircuitValue256() -- constructs a new CircuitValue256 object where the lo field is the original CircuitValue and the hi field is loaded as a constant 0.
If necessary, you can cast from ConstantValue to CircuitValue by using the constant() function.
The functions value(), number(), address() should only be used for logging/debugging purposes and should not be used in any ZK primitives or Axiom subqueries.
• value() -- returns the internal value as a bigint
• number() -- tries to cast the internal value to Number and returns it. Will throw an error if the value is too large to fit in a Number.
• address() -- converts the internal value to a hex string starting with 0x representing 20 bytes exactly. Will left pad with zeros if the value is less than 20 bytes. Will truncate if the value is
greater than 20 bytes.
Since EVM slots are 256 bits, we've created a CircuitValue256 class that represents 256 bit values in hi-lo form (hi is the most significant 128 bits, lo is the 128 least significant bits).
Here are the methods available on a CircuitValue256 object.
class CircuitValue256 {
// returns the `hi` CircuitValue
hi(): CircuitValue;
// returns the `lo` CircuitValue
lo(): CircuitValue;
// constrains that the CircuitValue256 can fit in 253 bits
// and constrains out = 2**128 * hi + lo
toCircuitValue(): CircuitValue;
// returns the value as a bigint
value(): bigint;
// returns the value as a hex string
hex(): string;
We have the following functions that return CircuitValue:
• hi() -- returns the hi field of the CircuitValue256 object, where hi refers to the most significant 128 bits of the value.
• lo() -- returns the lo field of the CircuitValue256 object, where lo refers to the least significant 128 bits of the value.
• toCircuitValue() -- uses ZK primitives to constrain that the CircuitValue256 has at most 253 bits and then computes and returns 2**128 * hi + lo as a CircuitValue.
By default all Axiom Subqueries return a CircuitValue256. To use these results in other ZK primitives that take CircuitValues, you must call .toCircuitValue(). If you know that the data you are
querying is less than 253 bits (ie. block number, address, uint128, etc.), this is totally safe. For values which may overflow 253 bits (such as storage slots, bytes32), your proof will fail to
verify if the value actually overflows (in particular, if the hi CircuitValue exceeds 125 bits).
The functions value(), hex() should only be used for logging/debugging purposes and should not be used in any ZK primitives or Axiom subqueries.
• value() -- returns the internal value as a bigint
• hex() -- returns the internal value as a hex string starting with 0x representing 32 bytes exactly. Will left pad with zeros if the value is less than 32 bytes. | {"url":"https://docs.axiom.xyz/sdk/typescript-sdk/axiom-circuit/circuit-types","timestamp":"2024-11-11T19:46:50Z","content_type":"text/html","content_length":"39850","record_id":"<urn:uuid:43aad2f0-1a56-44ec-b91e-9b48810ec101>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00521.warc.gz"} |
The Task abstraction
Neil Mitchell, Simon Peyton Jones and I have just finished a paper describing a systematic and executable framework for developing and comparing build systems. The paper and associated code are
available here: https://github.com/snowleopard/build. The code is not yet well documented and polished, but I’ll bring it in a good shape in April. You can learn more about the motivation behind the
project here.
(Update: the paper got accepted to ICFP! Read the PDF, watch the talk.)
In this blog post I would like to share one interesting abstraction that we came up with to describe build tasks:
type Task c k v = forall f. c f => (k -> f v) -> k -> Maybe (f v)
A Task is completely isolated from the world of compilers, file systems, dependency graphs, caches, and all other complexities of real build systems. It just computes the value of a key k, in a
side-effect-free way, using a callback of type k → f v to find the values of its dependencies. One simple example of a callback is Haskell’s readFile function: as one can see from its type FilePath →
IO String, given a key (a file path k = FilePath) it can find its value (the file contents of type v = String) by performing arbitrary IO effects (hence, f = IO). We require task descriptions to be
polymorhic in f, so that we can reuse them in different computational contexts f without rewriting from scratch.
This highly-abstracted type is best introduced by an example. Consider the following Excel spreadsheet (yes, Excel is a build system in disguise):
A1: 10 B1: A1 + A2
A2: 20 B2: B1 * 2
Here cell A1 contains the value 10, cell B1 contains the formula A1 + A2, etc. We can represent the formulae (i.e. build tasks) of this spreadsheet with the following task description:
sprsh1 :: Task Applicative String Integer
sprsh1 fetch "B1" = Just ((+) <$> fetch "A1" <*> fetch "A2")
sprsh1 fetch "B2" = Just ((*2) <$> fetch "B1")
sprsh1 _ _ = Nothing
We instantiate the type of keys k with String (cell names), and the type of values v with Integer (real spreadsheets contain a wider range of values, of course). The task description sprsh1 embodies
all the formulae of the spreadsheet, but not the input values. Like every Task, sprsh1 is given a callback fetch and a key. It pattern-matches on the key to see if it has a task description (a
formula) for it. If not, it returns Nothing, indicating that the key is an input. If there is a formula in the cell, it computes the value of the formula, using fetch to find the value of any keys on
which it depends.
The definition of Task and the above example look a bit mysterious. Why do we require Task to be polymorphic in the type constructor f? Why do we choose the c = Applicative constraint? The answer is
that given one task description, we would like to explore many different build systems that can build it and it turns out that each of them will use a different f. Furthermore, we found that
constraints c classify build tasks in a very interesting way:
• Task Applicative: In sprsh1 we needed only Applicative operations, expressing the fact that the dependencies between cells can be determined statically; that is, by looking at the formulae,
without “computing” them — we’ll demonstrate this later.
• Task Monad: some tasks cannot be expressed using only Applicative operations, since they inspect actual values and can take different computation paths with different dependencies. Dependencies
of such tasks are dynamic, i.e. they cannot be determined statically.
• Task Functor is somewhat degenerate: the task description cannot even use the application operator <*>, which limits dependencies to a single linear chain. Functorial tasks are easy to build, and
are somewhat reminiscent of tail recursion.
• Task Alternative, Task MonadPlus and their variants can be used for describing tasks with non-determinism.
Now let’s look at some examples of what we can do with tasks.
Given a task, we can compute the value corresponding to a given key by providing a pure store function that associates keys to values:
compute :: Task Monad k v -> (k -> v) -> k -> Maybe v
compute task store = fmap runIdentity . task (Identity . store)
Here we do not need any effects in the fetch callback to task, so we can use the standard Haskell Identity monad (I first learned about this trivial monad from this blog post). The use of Identity
just fixes the ‘impedance mismatch’ between the function store, which returns a pure value v, and the fetch argument of the task, which must return an f v for some f. To fix the mismatch, we wrap the
result of store in the Identity monad: the function Identity . store has the type k → Identity v, and can now be passed to a task. The result comes as Maybe (Identity v), hence we now need to get rid
of the Identity wrapper by applying runIdentity to the contents of Maybe.
In the GHCi session below we define a pure key/value store with A1 set to 10 and all other keys set to 20 and compute the values corresponding to keys A1 and B1 in the sprsh1 example:
λ> store key = if key == "A1" then 10 else 20
λ> compute sprsh1 store "A1"
λ> compute sprsh1 store "B1"
Just 30
As expected, we get Nothing for an input key A1 and Just 30 for B1.
Notice that, even though compute takes a Task Monad as its argument, its application to a Task Applicative typechecks just fine. It feels a bit like sub-typing, but is actually just ordinary
higher-rank polymorphism.
Now let’s look at a function that can only be applied to applicative tasks.
Static dependencies
The formula A1 + A2 in the sprsh1 example statically depends on two keys: A1 and A2. Usually we would extract such static dependencies by looking at the syntax tree of the formula. But our Task
abstraction has no such syntax tree. Yet, remarkably, we can use the polymorphism of a Task Applicative to find its dependencies. Here is the code:
dependencies :: Task Applicative k v -> k -> [k]
dependencies task key = case task (\k -> Const [k]) key of
Nothing -> []
Just (Const ks) -> ks
Here Const is the standard Haskell Const functor. We instantiate f to Const [k]. So a value of type f v, or in this case Const [k] v, contains no value v, but does contain a list of keys of type [k]
which we use to record dependencies. The fetch callback that we pass to task records a single dependency, and the standard Applicative instance for Const combines the dependencies from different
parts of the task. Running the task with f = Const [k] will thus accumulate a list of the task’s dependencies – and that is just what dependencies does:
λ> dependencies sprsh1 "A1"
λ> dependencies sprsh1 "B1"
["A1", "A2"]
Notice that these calls to dependencies do no actual computation. They cannot: we are not supplying any input values. So, through the wonders of polymorphism, we are able to extract the dependencies
of the spreadsheet formula, and to do so efficiently, simply by running its code in a different Applicative! This is not new, for example see this paper, but it is cool.
Dynamic dependencies
Some build tasks have dynamic dependencies, which are determined by values of intermediate computations. Such tasks correspond to the type Task Monad k v. Consider this spreadsheet example:
A1: 10 B1: IF(C1=1,B2,A2) C1: 1
A2: 20 B2: IF(C1=1,A1,B1)
Note that B1 and B2 statically form a dependency cycle, but Excel (which uses dynamic dependencies) is perfectly happy. The diagram below illustrates how cyclic dependencies are resolved when
projecting them on conditions C1=1 and C1=2 (rectangles and rounded rectangles denote inputs and outputs, respectively). Incidentally, my PhD thesis was about a mathematical model for such
conditional dependency graphs, which was later developed into an algebra of graphs.
We can express this spreadsheet using our task abstraction as:
sprsh2 :: Task Monad String Integer
sprsh2 fetch "B1" = Just $ do c1 <- fetch "C1"
if c1 == 1 then fetch "B2"
else fetch "A2"
sprsh2 fetch "B2" = Just $ do c1 <- fetch "C1"
if c1 == 1 then fetch "A1"
else fetch "B1"
sprsh2 _ _ = Nothing
The big difference compared to sprsh1 is that the computation now takes place in a Monad, which allows us to extract the value of C1 and fetch different keys depending on whether or not C1 = 1.
We cannot find dependencies of monadic tasks statically; notice that the application of the function dependencies to sprsh2 will not typecheck. We need to run a monadic task with concrete values that
will determine the discovered dependencies. Thus, we introduce the function track: a combination of compute and dependencies that computes both the resulting value and the list of its dependencies in
an arbitrary monadic context m:
track :: Monad m =>
Task Monad k v -> (k -> m v) -> k -> Maybe (m (v, [k]))
track task fetch = fmap runWriterT . task trackingFetch
trackingFetch :: k -> WriterT [k] m v
trackingFetch k = tell [k] >> lift (fetch k)
We use the standard^(*) Haskell WriterT monad transformer to record additional information — a list of keys [k] — when computing a task in an arbitrary monad m. We substitute the given fetch with a
trackingFetch that, in addition to fetching a value, tracks the corresponding key. The task returns the value of type Maybe (WriterT [k] m v), which we unwrap by applying runWriterT to the contents
of Maybe. Below we give an example of tracking monadic tasks when m = IO:
λ> fetchIO k = do putStr (k ++ ": "); read <$> getLine
λ> fromJust $ track sprsh2 fetchIO "B1"
C1: 1
B2: 10
λ> fromJust $ track sprsh2 fetchIO "B1"
C1: 2
A2: 20
As expected, the dependencies of cell B1 from sprsh2 are determined by the value of C1, which in this case is obtained by reading from the standard input via the fetchIO callback.
A simple build system
Given a task description, a target key, and a store, a build system returns a new store in which the values of the target key and all its dependencies are up to date. What does “up to date” mean? The
paper answers that in a formal way.
The three functions described above (compute, dependencies and track) are sufficient for defining the correctness of build systems as well as for implementing a few existing build systems at a
conceptual level. Below is an example of a very simple (but inefficient) build system:
busy :: Eq k => Task Monad k v -> k -> Store k v -> Store k v
busy task key store = execState (fetch key) store
fetch :: k -> State (Store k v) v
fetch k = case task fetch k of
Nothing -> gets (getValue k)
Just act -> do v <- act; modify (putValue k v); return v
Here Store k v is an abstract store datatype equipped with getValue and setValue functions. The busy build system defines the callback fetch so that, when given a target key, it brings the key up to
date in the store, and returns its value. The function fetch runs in the standard Haskell State monad, initialised with the incoming store by execState. To bring a key k up to date, fetch asks the
task description task how to compute k. If task returns Nothing the key is an input, so fetch simply reads the result from the store. Otherwise fetch runs the action act returned by the task to
produce a resulting value v, records the new key/value mapping in the store, and returns v. Notice that fetch passes itself to task as an argument, so that the latter can use fetch to recursively
find the values of k‘s dependencies.
Given an acyclic task description, the busy build system terminates with a correct result, but it is not a minimal build system: it doesn’t keep track of keys it has already built, and will therefore
busily recompute the same keys again and again if they have multiple dependants. See the paper for implementations of much more efficient build systems.
For a few monads more
We have already used a few cool Haskell types — Identity, Const, WriterT and State — to manipulate our Task abstraction. Let’s meet a few other members of the cool-types family: Proxy, ReaderT,
MaybeT and EitherT.
The Proxy data type allows us to check whether a key is an input without providing a fetch callback:
isInput :: Task Monad k v -> k -> Bool
isInput task = isNothing . task (const Proxy)
This works similarly to the dependencies function, but in this case we do not even need to record any additional information, thus we can replace Const with Proxy.
One might wonder: if we do not need the fetch callback in case of input, can we rewrite our Task abstraction as follows?
type Task2 c k v = forall f. c f => k -> Maybe ((k -> f v) -> f v)
Yes, we can! This definition is isomorphic to Task. This isn’t immediately obvious, so below is a proof. I confess: it took me a while to find it.
toTask :: Task2 Monad k v -> Task Monad k v
toTask task2 fetch key = ($fetch) <$> task2 key
fromTask :: Task Monad k v -> Task2 Monad k v
fromTask task key = runReaderT <$> task (\k -> ReaderT ($k)) key
The toTask conversion is relatively straightforward, but fromTask is not: it uses a ReaderT monad transformer to supply the fetch callback as the computation environment, extracting the final value
with runReaderT.
Our task abstraction operates on pure values and has no mechanism for exception handling. It turns out that it is easy to turn any Task into a task that can handle arbitrary exceptions occurring in
the fetch callback:
exceptional :: Task Monad k v -> Task Monad k (Either e v)
exceptional task fetch = fmap runExceptT . task (ExceptT . fetch)
The exceptional task transformer simply hides exceptions of the given fetch of type k → f (Either e v) by using the standard ExceptT monad transformer, passes the resulting fetch callback of type k →
ExceptT e f v to the original task, and propagates the exceptions by runExceptT. Using MaybeT, one can also implement a similar task transformer that turns a Task Monad k v into the its partial
version Task Monad k (Maybe v).
Our final exercise is to extract all possible computation results of a non-deterministic task, e.g. B1 = A1 + RANDBETWEEN(1,2) that can be described as a Task Alternative:
sprsh3 :: Task Alternative String Integer
sprsh3 fetch "B1" = Just $ (+) <$> fetch "A1" <*> (pure 1 <|> pure 2)
sprsh3 _ _ = Nothing
We therefore introduce the function computeND that returns the list of all possible results of the task instead of just one value (‘ND’ stands for ‘non-deterministic’):
computeND :: Task MonadPlus k v -> (k -> v) -> k -> Maybe [v]
computeND task store = task (return . store)
The implementation is almost straightforward: we choose f = [] reusing the standard MonadPlus instance for lists. Let’s give it a try:
λ> store key = if key == "A1" then 10 else 20
λ> computeND sprsh3 store "A1"
λ> computeND sprsh3 store "B1"
Just [11,12]
λ> computeND sprsh1 store "B1"
Just [30]
Notice that we can apply computeND to both non-deterministic (sprsh3) as well as deterministic (sprsh1) task descriptions.
Non-deterministic tasks are interesting because they allow one to try different algorithms to compute a value in parallel and grab the first available result — a good example is portfolio-based
parallel SAT solvers. This shouldn’t be confused with a deterministic composition of tasks, which is also a useful operation, but does not involve any non-determinism:
compose :: Task Monad k v -> Task Monad k v -> Task Monad k v
compose t1 t2 fetch key = t1 fetch key <|> t2 fetch key
Here we simply compose two task descriptions, picking the first one that knows how to compute a given key. Together with the trivial task that returns Nothing for all keys, this gives rise to the
Task monoid.
Final remarks
We introduced the task abstraction to study build systems, but it seems to be linked to a few other topics, such as memoization, self-adjusting computation, lenses and profunctor optics, propagators
and what not.
Have we just reinvented the wheel? It might seem so, especially if you look at these type signatures from the lens library:
type Lens s t a b
= forall f. Functor f => (a -> f b) -> s -> f t
type Traversal s t a b
= forall f. Applicative f => (a -> f b) -> s -> f t
Our implementations of functions like dependencies are heavily inspired by — or to be more accurate — stolen from the lens library. Alas, we have been unable to remove the Maybe used to encode
whether a key is an input, without complicating other aspects of our definition.
The task abstraction can be used to express pure functions in a way that is convenient for their memoization. Here is an example of encoding one of the most favourite functions of functional
fibonacci :: Task Applicative Integer Integer
fibonacci fetch n
| n >= 2 = Just $ (+) <$> fetch (n-1) <*> fetch (n-2)
| otherwise = Nothing
Here the keys n < 2 are input parameters, and one can obtain the usual Fibonacci sequence by picking 0 and 1 for n = 0 and n = 1, respectively. Any minimal build system will compute the sequence with
memoization, i.e. without recomputing the same value twice.
Interestingly, the Ackermann function — a famous example of a function that is not primitive recursive — can’t be expressed as a Task Applicative, since it needs to perform an intermediate recursive
call to determine one of its dependencies:
ackermann :: Task Monad (Integer, Integer) Integer
ackermann fetch (n, m)
| m < 0 || n < 0 = Nothing
| m == 0 = Just $ pure (n + 1)
| n == 0 = Just $ fetch (m - 1, 1)
| otherwise = Just $ do index <- fetch (m, n - 1)
fetch (m - 1, index)
Now that we’ve seen examples of applicative and monadic tasks, let us finish with an example of a functorial task — the Collatz sequence:
collatz :: Task Functor Integer Integer
collatz fetch n | n <= 0 = Nothing
| otherwise = Just $ f <$> fetch (n - 1)
f k | even k = k `div` 2
| otherwise = 3 * k + 1
So here is a claim: given a Task, we can memoize, self-adjust, propagate and probably do any other conceivable computation on it simply by picking the right build system!
Update: how to handle failures
Sjoerd Visscher’s comment (below) pointed out that the fetch callback is defined to be total: it has type k → f v and returns a value for every key. It may be useful to allow it to fail for some
keys. I know of three ways of modelling failure using the Task abstraction:
(1) Include failures into the type of values v, for example:
data Value = FileNotFound | FileContents ByteString
This is convenient if tasks are aware of failures. For example, a task may be able to cope with missing files, e.g. if fetch “username.txt” returns FileNotFound, the task could use the literal string
“User” as a default value. In this case it will depend on the fact that the file username.txt is missing, and will need to be rebuilt if the user later creates this file.
In many cases this approach is isomorphic to choosing v = Either e v’.
(2) Include failures into the computation context f, for example:
cells :: Map String Integer
cells = Map.fromList [("A1", 10), ("A2", 20)]
fetch :: String -> Maybe Integer
fetch k = Map.lookup k cells
We are choosing f = Maybe and thanks to the polymorphism of Task, any task can be executed in this context without any changes. For example, sprsh1 fetch “B1” now returns Just (Just 30), but sprsh1
fetch “B2” fails with Just Nothing.
This is convenient if tasks are not aware of failures, e.g. we can model Excel formulas as pure arithmetic functions, and introduce failures “for free” if/when needed by instantiating Task with an
appropriate f. Also see the function exceptional defined above, which allows us to add arbitrary exceptions to a failure-free context f.
(3) Finally, the task itself might not want to encode failures into the type of values v, but instead demand that f has a built-in notion of failures. This can be done by choosing a suitable
constraint c, such as Alternative, MonadPlus or even better something specific to failures e.g. MonadZero or MonadFail. Then both the callback and the task can reuse the same failure mechanism as
shown below:
class Monad m => MonadFail m where
fail :: String -> m a
sprsh4 :: Task MonadFail String Integer
sprsh4 fetch "B1" = Just $ do
a1 <- fetch "A1"
a2 <- fetch "A2"
if a2 == 0 then fail "division by 0" else return (a1 `div` a2)
sprsh4 _ _ = Nothing
Are there any other types of failure that are not covered above?
^(*) Beware: as of writing, standard WriterT transformers have a space leak which may be an issue if a task has many dependencies. You might want to consider using a more efficient CPS-based WriterT
5 thoughts on “The Task abstraction”
1. I wonder why you require from the callback to always return a value for any key? Wouldn’t it make more sense to allow it to fail too? Then you get:
type Task c k v = forall f. c f => (k -> Maybe (f v)) -> k -> Maybe (f v)
But you could also require that f has failure built in. Then Task Applicative becomes Task Alternative (with empty instead of Nothing) and Task Monad becomes Task MonadPlus (with mzero instead of
Nothing) and Task becomes a proper optic:
type Task c k v = forall f. c f => (k -> f v) -> k -> f v
If your original f doesn’t support failure you can replace it with Compose Maybe f.
1. Thanks for your comment, Sjoerd! We’ve been thinking along similar lines, and perhaps the Task abstraction can still be improved by finding a better way to express various failures.
There are currently three ways of modelling callback failure using our Task abstraction:
(1) Include failures into the type of values v, for example:
data FileContents = FileNotFound | Contents ByteString
This is convenient when the task has a built-in notion of failures. For example, a build rule may be designed to cope with some missing files, e.g. if fetch “username.txt” returns
FileNotFound, a build rule could use the literal string “User” as a default value. In this case it will depend on the fact that the file “username.txt” is missing, and will be rebuilt if the
user later creates this file.
In many cases the result is isomorphic to choosing v = Maybe v’.
(2) Include failures into the context f, for example:
values :: Map String Integer
fetch :: String -> Maybe Integer
fetch k = Map.lookup k values
Now we are choosing f = Maybe.
This is convenient when the task itself has no built-in notion of failures, e.g. we can model Excel formulas as pure arithmetic functions, and introduce failures “for free” if/when needed by
instantiating Task with an appropriate f.
(3) Finally, the task itself might not want to encode failures into the type of values v, but instead demand that f has a built-in mechanism for expressing failures. This can be done by
choosing an appropriate c, e.g ApplicativeZero, MonadZero, Alternative, MonadPlus, etc.
Now, let’s look at the type you propose:
type Task c k v = forall f. c f => (k -> Maybe (f v)) -> k -> Maybe (f v)
Here the first Maybe is outside f, so a failure is known ‘statically’ by simply looking at the given key (similar to the above fetch example with Map.lookup). This will not work in cases when
you need to perform some effects in order to determine if the lookup has failed, e.g. to check if the file is actually missing on disk. Here you want something like IO (Maybe v) instead, and
I see no way of fitting this into the above type signature.
Your second type:
type Task c k v = forall f. c f => (k -> f v) -> k -> f v
is essentially the same as ours with respect to failures, but it doesn’t have a mechanism to (statically) indicate that a given key is an input. Are you suggesting to indicate this using the
failure mechanism? While this works for some examples, I don’t think it’s sufficient in general: some of our build systems exploit the fact that it is possible to determine whether a key is
an input statically, without performing an actual computation, i.e. they require the function isInput implemented above. After removing Maybe from the result, it becomes harder or impossible
to implement isInput.
I think I should add the above explanation of three ways of modelling failure into the blog post.
2. I’ve added a section on failures to the blog post.
2. > This definition is isomorphic to Task. This isn’t immediately obvious, so below is a proof. I confess: it took me a while to find it.
When I read this I wondered if you could derive one from the other using the algebraic structure of ADTs (“algebra of algebraic data types”), to go from one to the other but of course that
doesn’t work:
(1 + (fv ^ (fv ^ k))) ^ k
((1 + fv) ^ k) ^ (fv ^ k)
And I’m sure the existentially quantified term needs to be treated differently, but no idea how. In any case I would be really interested if there was some way to reason algebraically either to
determine that there must exist some isomorphism to Task2, or to derive the implementation.
In poking at this I did notice a couple things that might not really be that interesting…
First, as you probably knew `toTask` doesn’t require any constraint, and this definition is a little more terse:
toTask :: Task2 Unconstrained k v -> Task Unconstrained k v
toTask = flip . fmap sequenceA
For a second I was excited thinking that we could have:
fromTask :: Task Unconstrained k v -> Task2 Unconstrained k v
fromTask = fmap sequenceA . flip
But that of course is impossible, since we can’t have Traversable ((->) a)
On a whim though a hayoo search turned up http://hackage.haskell.org/package/countable-1.0/docs/Data-Searchable.html#v:assemble
So I guess an alternative would have `(f v)` and `k` both be constrained to `Finite` but that’s probably not useful at all.
1. Hello Brandon! Many thanks for sharing your observations — they are very interesting and useful. As soon as my versions of toTask and fromTask successfully compiled I happily switched to
other things and never looked back. But it’s interesting to see if we can find a more principled approach to finding such transformations — mine was: think really really hard and try out
monad transformers until it works 🙂 Surely we can do better.
Also thanks a lot for the reference to `assemble` and `Finite`. This is something that might come in handy in my work on selective functors (https://blogs.ncl.ac.uk/andreymokhov/selective). | {"url":"https://blogs.ncl.ac.uk/andreymokhov/the-task-abstraction/","timestamp":"2024-11-11T06:47:51Z","content_type":"text/html","content_length":"108373","record_id":"<urn:uuid:6eb81944-8424-4c54-b56c-ece5490133ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00624.warc.gz"} |
Holy Bible Aionian Edition
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pronoun who? what?
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in the others:—Dialectal forms: [
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κι in διεκί, ποκκί (which see);
neuter plural Doric dialect
Boeotian dialect
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τίω, τίοισι (see. infr. B). (I.-[
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genitive česo
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τις, τι,
genitive Ionic dialect
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], but found in [
Refs 3rd c.BC+
], revived by the Atticists, [
Refs 1st c.BC+
]; τινος [
Refs 5th c.BC+
dative Ionic dialect
τεῳ [
Refs 8th c.BC+
]; Trag. and
Attic dialect
τῳ (also in [
Refs 8th c.BC+
]; τινι [
Refs 8th c.BC+
] in the form οὔ τινι [
Refs 8th c.BC+
τινα [
Refs 8th c.BC+
τι [
Refs 8th c.BC+
τινες [
Refs 8th c.BC+
Doric dialect
τινεν [
Refs 3rd c.BC+
accusative neuter
τινα (ὅτινα [
Refs 8th c.BC+
], never in Trag, [
Refs 5th c.BC+
]; ἄσσα (which see) [
Refs 8th c.BC+
], never in Trag. or [
Refs 5th c.BC+
Attic dialect
ἄττα first in [
LXX+5th c.BC+
], revived by the Atticists, [
Refs 1st c.BC+
genitive Ionic dialect
τεων [
Refs 5th c.BC+
]; τινων not in [
Refs 5th c.BC+
τισι, τισιν, first in [
Refs 5th c.BC+
]; N.-W.
Doric dialect
τινοις [
Refs 3rd c.BC+
Ionic dialect
τεοισι [
Refs 5th c.BC+
] (for τεοις and τεον see at {τεός});
τινας [
Refs 8th c.BC+
τινα (see. above):—
any one, any thing, some one, some thing
; and as
adjective any, some
, and serving as the Indef.
Article a, an
; θεός νύ τίς ἐστι κοτήεις [
Refs 8th c.BC+
) special usages:
some one
(of many), i.e.
many a one
, ὧδε δέ τις εἴπεσκεν [
Refs 8th c.BC+
]: sometimes with meiosis, implying
Refs 8th c.BC+
]; so in Prose, [
Refs 5th c.BC+
any one concerned, every one
, εὖ μέν τις δόρυ θηξάσθω [
Refs 8th c.BC+
]; ἀλλά τις αὐτὸς ἴτω let
every man
come himself,[
]; so in Trag. and
Attic dialect
, even with the
, τοῦτό τις. ἴστω S [
Refs 5th c.BC+
]; τοὺς ξυμμάχους αὐτόν τινα κολάζειν that
every man
should himself chastise his own allies, [
Refs 5th c.BC+
]; ἄμεινόν τινος better than
any others
, [
Refs 4th c.BC+
]:—this is more fully expressed by adding other pronominal words, τις ἕκαστος [
Refs 8th c.BC+
]. In these senses, τις is frequently combined with
words, οἱ κακοὶ. οὐκ ἴσασι, πρίν τις ἐκβάλῃ, for πρὶν ἐκβάλωσι, [
Refs 5th c.BC+
]; οἷς ἂν ἐπίω, ἧσσόν τις πρόσεισι, for ἧσσον προσίασι, [
Refs 5th c.BC+
]; especially after εἴ or ἤν τις, [
Refs 5th c.BC+
) in reference to a definite person, whom one wishes to avoid naming, οὐκ ἔφασαν ἰέναι, ἐὰν μή τις χρήματα διδῷ (i.e. Cyrus) [
Refs 5th c.BC+
]; so also euphemistic for something bad, ἤν τι ποιῶμεν [
Refs 5th c.BC+
]: hence for the
1st pers.
2nd pers. pronoun
, ἅ τιν᾽ οὐ πείσεσθαι ὀΐω [
Refs 8th c.BC+
]; ποῖ τις τρέψετα; for ποῖ τρέψομα; [
Refs 5th c.BC+
) indefinitely, where we say
, French
, sometimes with an ironical force, φοβεῖταί τις [
Refs 4th c.BC+
]; as
, τὸν Πλοῦτον ἔξω τις κάλει call P. out,
, [
Refs 5th c.BC+
) τις, τι may be opposed, expressly or by implication, to οὐδείς, οὐδέν, and mean
somebody, something
, by meiosis for
some great one, some great thing
, ηὔχεις τις εἶναι you boasted that you were
, [
Refs 5th c.BC+
]; κἠγών τις φαίνομαι ἦμεν after all I too am
, [
NT+3rd c.BC+
]; also in
, οἴονταί τι εἶναι ὄντες οὐδενὸς ἄξιοι [
Refs 5th c.BC+
) τις is sometimes opposed to to another word, ἀελλοπόδων μέν τιν᾽ εὐφραίνοισιν ἵππων τιμαί, τέρπεται δὲ καί τις. [
LXX+5th c.BC+
]; ἔστιν οὖν οὐ πᾶν τὸ ταχύ, ἀλλά τι (sic codices BT) αὐτοῦ ἀγαστόν [
Refs 5th c.BC+
]; τὸ μεῖζον τοῦθ᾽ ὅπερ ἐστὶν ἑτέρου λέγεται· τινὸς γὰρ λέγεται μεῖζον greater than
, [
]; πότερον τῷ τυχόντι ἢ τισί; [
) with (Proper name)s τις commonly signifies
one named
so-and-so, ἦν δέ τις ἐν Τρώεσσι Δάρης [
Refs 8th c.BC+
]; with a sense of contempt, Θερσίτης τις ἦν there was
Thersites, [
Refs 5th c.BC+
one of the same sort
, converting the (Proper name) into an appellative, ἤ τις Ἀπόλλων ἢ Πάν
Apollo or
Pan, [
Refs 4th c.BC+
]; [πόλιες] ταὶ μέλονται πρός τινος ἢ Διὸς ἢ γλαυκᾶς Ἀθάνας Lyric poetry in [
Refs 5th c.BC+
]; ἰσθμόν τιν᾽ [
Refs 5th c.BC+
) with
τις combines to express the idea of a
used as predicate, ὥς τις θαρσαλέος καὶ ἀναιδής ἐσσι προΐκτης
bold and impudent beggar, [
Refs 8th c.BC+
]; ἐγώ τις, ὡς ἔοικε, δυσμαθής
, [
Refs 5th c.BC+
]; φόβου πλέα τις εἶ
, [
Refs 4th c.BC+
]; ὡς ταχεῖά τις. χάρις διαρρεῖ in what swift
(={ταχέως πως}), [
Refs 5th c.BC+
]; δεινόν τι ποιεύμενος thinking it
, [
) with numerals and
expressing number, size, or the like, εἷς δέ τις ἀρχὸς ἀνὴρ. ἔστω
one man, [
NT+8th c.BC+
]; sometimes the τις softens the definiteness of the numeral, ἑπτά τινες
seven, seven
or so
, [
Refs 5th c.BC+
]; so without an actual numeral, ἡμέρας τινάς
days, i.e.
, [
]; στρατῷ τινι
of a certain amount, considerable
, [
]; ἐνιαυτόν τινα a year
or so
, [
]; so οὐ πολλοί τινες, τινὲς οὐ πολλοί, [
Refs 5th c.BC+
]; ὀλίγοι τινές or τινὲς ὀλίγοι [
]; οὔ τινα πολλὸν χρόνον no
long time, [
Refs 5th c.BC+
]; so also ὅσσος τις χρυσός what
store of gold, [
Refs 8th c.BC+
) with Pronominal words, ἀλλά τί μοι τόδε θυμὸς. μερμηρίζει
, namely this, [
Refs 8th c.BC+
]; οἷός τις what sort of
man, [
Refs 8th c.BC+
) with the Article,
) when a noun with the
is in
with τις, as ὅταν δ᾽ ὁ κύριος παρῇ τις when the person in authority,
whoever he be
, is here, [
Refs 5th c.BC+
]; τοὺς αὐτοέντας. τιμωρεῖν τινας (variant τινα) [
) in Philosophic writers, τις is added to the
to show that the
is used to denote a particular individual who is not specified in the general formula, although he would be in the particular case, ὁ τὶς ἄνθρωπος
the individual
man (
whoever he may be), this or that
man, opposed to ἄνθρωπος (man in general), ὁ τὶς ἵππος, ἡ τὶς γραμματική, [
Refs 4th c.BC+
]; τὸ τὶ μέγεθος, opposed to ὅλως τὸ μέγεθος, [
Refs 5th c.BC+
], the
is used as in [
Refs 8th c.BC+
] cc. (which see) ὁ, ἡ, τό [
Refs 5th c.BC+
], δεῦρο ὅ τις θεός, ὄφθητί μοι in a general formula of invocation, [
) frequently in opposed clauses, ὁ μέν τις, ὁ δὲ. [
Refs 5th c.BC+
], etc: also combined with other alternative words, ὁ μέν τις, ὁ δέ τις, ἕτερος δέ τις. [
]; ὁ μὲν, ἕτερος δέ τις, ὁ δὲ, etc, [
Refs 5th c.BC+
]: also in
, τὸ μέν τι, τὸ δέ τι. [
Refs 5th c.BC+
]; in
sense, τὸ μὲν, τὸ δέ τι.
partly, partly.
, [
Refs 2nd c.BC+
]; and τι remains unaltered even when the
, τὰ μέν τι μαχόμενοι, τὰ δὲ καὶ ἀναπαυόμενοι [
Refs 5th c.BC+
]; also τὸ δέ τι. but
in some
measure, without τὸ μέν preceding, [
Refs 5th c.BC+
) later τις is used as in b above but without the
, γράψον. ὅτι τι καί τι εἴληφας that you have received
things, [
Refs 2nd c.AD+
]; τίς τινι χαίρειν [
Refs 2nd c.AD+
) the
τι is used,
) collectively, ἦν τι καὶ ἐν ταῖς Συρακούσαις there was
a party.
, [
Refs 5th c.BC+
]; so perhaps τῶν ἄλλων οὔ πέρ τι πεφυγμένον ἐστ᾽ Ἀφροδίτην, οὔτε θεῶν, οὔτ᾽ ἀνθρώπων no
] (but
τις in [
) euphemistic for something bad, see above [
) joined with Verbs,
somewhat, in any degree, at all
, ἦ ῥά τί μοι κεχολώσεαι [
Refs 8th c.BC+
]; οὐ πάνυ τι, πολύ τι, σχεδόν τι, see at {πάνυ} [
]; also in conjunction with οὐδέν, μηδέν, οὐδέν τι πάντως [
Refs 5th c.BC+
]; οὐδέν, μηδέν τι μᾶλλον, [
Refs 5th c.BC+
] —also καί τι καὶ. ὑποψίᾳ
in part
also from suspicion, [
Refs 5th c.BC+
) τίς τε frequently in [
Refs 8th c.BC+
) ἤ τις ἢ οὐδείς
or none,
next to none
, [
Refs 5th c.BC+
]; ἤ τι ἢ οὐδέν
or nothing, [
Refs 5th c.BC+
) τις is
in such phrases as οὐδέν τι or μηδέν τι, see above [
) repeated in successive clauses, ὅσα λέγει τις ἢ πράσσειτις ἢψέγειν ἔχει [
Refs 5th c.BC+
] (whereas τις is sometimes omitted in the first clause, οὔτε φωνὴν οὔτε του μορφὴν βροτῶν [
Refs 5th c.BC+
], the repetition is pleonastic, as also in [
Refs 4th c.BC+
) τις is sometimes omitted, οὐδέ κεν ἔνθα τεόν γε μένος καὶ χεῖρας ὄνοιτο (i.e. τις) [
Refs 8th c.BC+
]; ὡς δ᾽ ἐν ὀνείρῳ οὐ δύναται (i.e. τις) φεύγοντα διώκειν [
Refs 5th c.BC+
]: τις must often be supplied from what goes before,[
) sometimes also τις is omitted before a
case which must depend upon it, as ἢ [τις] τᾶς ἀσώτου Σισυφιδᾶν γενεᾶς [
Refs 5th c.BC+
]; ἢν γαμῇ ποτ᾽ αὐτὸς ἢ [τις] τῶν ξυγγενῶν [
Refs 5th c.BC+
) Accentuation and position of τις:
) accentuation: τις is normally enclitic, but in certain uses is orthotone, i.e. theoretically oxytone (τίς, τινά, τινές, τινῶν, etc,[
Refs 4th c.AD+
] or τις, τινὰ, τινὲς, τινῶν, etc.). According to [
) at the beginning of a sentence, τίς ἔνδον; is
any one
within? [
Refs 4th c.BC+
]; τί φημ; ={λέγω τι}; am I saying
Refs 5th c.BC+
]; <τίς ἦλθ;> ἦλθέ τις has
come? [
Refs 2nd c.BC+
]; τὶς κάθηται, τὶς περιπατεῖ,
so and so
is sitting (walking), [
Refs 2nd c.AD+
]; τὶς αἰπόλος καλούμενος Κομάτας [
Refs 5th c.BC+
]; τι οὖν (τὶς ἂν εἴποι) ταῦτα λέγει; [
Refs 4th c.BC+
) when τις is opposed to to another τις or to some other word, τισὶ μὲν συμφέρει, τισὶ δ᾽ οὐ συμφέρει [
Refs 5th c.BC+
] for
a certain person
, [
Refs 5th c.BC+
]. Codices are not consistent; in
] they make it enclitic; in
]; sometimes enclitic and orthotone in the same sentence, πάντα δὲ τὰ γιγνόμενα ὑπό τέ τινος γίγνεται καὶ ἔκ τινος καὶ τί [
Refs 5th c.BC+
) position:
) τις is rarely first word in the sentence, and rarely follows a pause (see. above [
]; it may stand second word, ἔσκε τις ἐνθάδε μάντις ἀνήρ [
Refs 8th c.BC+
]; but in general its position is not far before or after the word to which it belongs in sense, ἀλλ᾽ ἄγε δή τινα μάντιν ἐρείομεν [
) in
Ionic dialect
Prose it sometimes stands between its genitive and the Article of that genitive, τῶν τις Περσέων [
Refs 5th c.BC+
]; so also in late Prose, [
Refs 2nd c.AD+
) it stands between the
Refs 5th c.BC+
) τίς τι is the correct order, not τί τις, “IG” 12.110.46, 5th c.BC: Thucydides Historicus 7.10, 5th-6th c.BC: Xenophon Historicus “Anabasis” 4.1.14 (codices dett.), 4th c.BC: Demosthenes Orator
22.22, etc.
) whereas in _Attic dialect_ the order ἐάν τις is compulsory, in _Doric dialect_ the usual order is αἴ τίς κα, [
Refs 5th c.BC+
]: later
Doric dialect
εἴ τί κα [
]; καἴ τι ἂν (={καὶ εἴ τι ἂν}) [
Refs 1st c.BC+
], see below [
Refs 5th c.BC+
Doric dialect
order influenced the Koine, as in the rare εἴ τις ἂν [
Refs 1st c.AD+ | {"url":"https://www.aionianbible.org/Strongs/Aramaic---Syriac-Peshitta/strongs-g5100","timestamp":"2024-11-02T14:30:11Z","content_type":"text/html","content_length":"56926","record_id":"<urn:uuid:ae4290e1-e905-4871-b1b7-25ee886b1ed3>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00836.warc.gz"} |
USA (United States) Archives - Quran Mualim
Category USA (United States)
PDF Books, All Grade 1 to Grade 12 etc.
The American Mathematics Competitions (AMC) are the primary of a sequence of competitions in center college and excessive school mathematics that result in the United States group for the
International Mathematical Olympiad (IMO). AMC has three tiers: The AMCs cause International Math… | {"url":"https://www.quranmualim.com/category/usa/","timestamp":"2024-11-06T17:26:47Z","content_type":"text/html","content_length":"149163","record_id":"<urn:uuid:ea5ff9b7-249e-4f03-bf35-587e700e19f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00672.warc.gz"} |
NCERT Solutions Class 6 Maths Chapter 3 Number Play Exercise 3.9 PDF
Class 6 Maths NCERT Solutions for Chapter 3 Exercise 3.9 Maths FREE PDF Download
Chapter 3 of NCERT Class 6 Maths Number Play introduces students to the fascinating world of number patterns, prime numbers, composite numbers, and number operations. Exercise 3.9 Playing with Number
Patterns focuses on helping students recognize, predict, and apply various number patterns in problem-solving. By engaging in the exercises, students sharpen their arithmetic skills, learn the
significance of prime and composite numbers, and develop the ability to understand and apply divisibility rules.
1. Class 6 Maths NCERT Solutions for Chapter 3 Exercise 3.9 Maths FREE PDF Download
2. Glance on Class 6 Maths Chapter 3 Number Play Exercise 3.9 Number Play
3. Access NCERT Solutions for Class 6 Maths Chapter 3 Number Play
4. Benefits of NCERT Solutions for Class 6 Maths Ex 3.9
5. Class 6 Maths Chapter 3: Exercises Breakdown
6. Important Study Material Links for Maths Chapter 3 Class 6
8. Chapter-wise NCERT Solutions Class 6 Maths
9. Related Important Links for Class 6 Maths
Our Class 6 Maths NCERT Solutions PDF breaks the lesson into easy-to-understand explanations, making learning fun and interactive. Students will develop essential language skills with engaging
activities and exercises. Check out the revised CBSE Class 6 Maths Syllabus and start practising the Maths Class 6 Chapter 3.
Glance on Class 6 Maths Chapter 3 Number Play Exercise 3.9 Number Play
• Introduction to number patterns
• Identifying and working with odd and even numbers
• Understanding prime and composite numbers
• Exploring factors and multiples
• Divisibility rules for numbers
• Introduction to sequences and recognizing patterns in numbers
FAQs on NCERT Solutions for Class 6 Maths Chapter 3 Exercise 3.9 Number Play
1. What is the focus of NCERT Class 6 Maths, Chapter 3: Number Play?
The chapter focuses on understanding number patterns, prime and composite numbers, factors, multiples, and divisibility rules.
2. What is a prime number as discussed in Chapter 3: Number Play?
A prime number is a number that has only two factors – 1 and the number itself, like 2, 3, 5, etc.
3. What is the difference between a prime and a composite number as per Chapter 3: Number Play?
Prime numbers have only two factors (1 and the number itself), while composite numbers have more than two factors.
4. What are factors and multiples in the context of Chapter 3: Number Play?
A factor divides a number exactly, while a multiple is the product of a number with another number.
5. How do divisibility rules help in solving problems in Chapter 3: Number Play?
Divisibility rules help determine if one number can be divided by another without performing actual division, making calculations quicker and easier.
6. What is the greatest common divisor (GCD) according to Chapter 3: Number Play?
GCD is the largest number that can exactly divide two or more numbers.
7. How does Exercise 3.9 help in identifying number patterns in Chapter 3: Number Play?
Exercise 3.9 provides problems where students recognise, predict, and apply different number patterns.
8. What is a sequence in terms of number patterns in Chapter 3: Number Play?
A sequence is an ordered list of numbers that follow a specific pattern or rule.
9. Why are multiples important in Chapter 3: Number Play?
Multiples help in understanding concepts such as least common multiple (LCM), which is essential in problem-solving.
10. How do Vedantu’s NCERT Solutions help with Chapter 3: Number Play?
Vedantu’s solutions simplify complex concepts, provide clear explanations of number patterns, and offer practice questions to ensure thorough understanding. | {"url":"https://www.vedantu.com/ncert-solutions/ncert-solutions-class-6-maths-chapter-3-exercise-3-9","timestamp":"2024-11-04T08:15:59Z","content_type":"text/html","content_length":"320999","record_id":"<urn:uuid:556d2e62-9b91-40c2-92f6-f5da1362610d>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00816.warc.gz"} |
ball mill critical speed working principle
WEBBall Mill Critical Speed Working Principle; Have a Question? Ask our expert. Speak your question. Please enter your question. Watch Video. Stainless Steel Laboratory Ball Mill, For
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WEBJul 2, 2020 · In recent research done by AmanNejad and Barani [93] using DEM to investigate the effect of ball size distribution on ball milling, charging the mill speed with 40% small balls
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WEBFactors Affecting Tumbling Mill Efficiency Mill Speed. The speed of the mill is a critical factor affecting its efficiency. The optimal speed of the mill depends on the size of the grinding
media and the materials to be ground. A higher speed can generate more impact and attrition, but it can also cause excessive wear and tear on the mill ...
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WEBThis article aims to describe the working principle of a jet mill. A jet mill, also called fluid energy mill, is used for solid material micronization. ... Feed particle size is critical,
restricted by the size of the feed injector. For mills of 200300 mm, the feed size can be a maximum of mm. ...
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WEBMay 24, 2019 · The document summarizes the ball mill, which is a grinder used to grind and blend materials. It discusses the basic parts of a ball mill including the hollow cylinder and balls.
It then explains the principle of operation through impact and attrition. The document also covers the theory behind maintaining critical speed for optimum efficiency.
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WEBMay 11, 2021 · Working of Ball Mill. The material to be ground is kept in a hollow cylinder. The material is placed up to 60% of the volume. A fixed number of balls is placed in the cylinder
and then the cylinder is closed. The mill is allowed to rotate. Speed of rotation is an important point of consideration.
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WEBThe formula to calculate critical speed is given below. N c = /sqt(Dd) N c = critical speed of the mill. D = mill diameter specified in meters. d = diameter of the ball. In practice Ball Mills
are driven at a speed of 5090% of the critical speed, the factor being influenced by economic consideration.
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WEBJun 6, 2016 · Such mills are common in South African operations; mills are sometimes referred to as tube mills or ROM ball mills and are also operated both autogenously and semiautogenously.
Many of these mills operate at higher mill speeds (nominally 90% of critical speed) and often use "grid" liners to form an autogenous liner surface.
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WEBApr 22, 2017 · Consistency of Pulp in Ball Mill. The ratio of moisture to solids is important in ball mill work. From actual operation it has been observed that fine grinding is best done when
water constitutes 33 to 40 per cent, of the pulp, or the waterto .
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WEBThere exists a critical speed at which grinding occurs. This speed is described as the minimum speed required to rotate the balls. The working of a ball mill can be understood as follows: A
powder mix is positioned in ball mill and is subjected to high energy collision by the balls (Figure 1). The ball milling process can be summed up as: a.
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WEBJul 4, 2023 · Therefore, the critical speed of the mill can be defined as the speed at which the powder blender just starts centrifuging. At the critical speed, the gravitational force acting
on the ball due to the weight of the ball is balanced by the centrifugal force (Fig. ). If the mass of the ball is m, then, at critical speed
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WEBApr 1, 2002 · The large mill runs at a much lower absolute speed than the small mill, even though the percent critical mill speed value is the same. For example, at 60% critical speed, the
charge is very clearly divided into cascading and aracting zones in the case of a large mill, while for the same speed only a aracting zone exists in a small mill.
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WEBMar 23, 2022 · where g is m/s 2, R radius of the cylinder (m), r radius of the ball (m), and n c critical speed (rps). The operating speed of the ball mill is kept at 65–80% of the critical
speed. The lower values are kept for the wet grinding in viscous solution, while a higher value is kept for dry grinding. Burr Mill or Plate Mill
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WEBThe working principle of the hammer mill is simple to understand. The principle is illustrated in Fig. 1 (a). It only requires choosing an appropriate motor, crushing hammers/knives and
material to be crushed. It operates on the principle of impact between rapidly moving hammers mounted on the rotor and the stationary powder bed.
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WEBBall Mill Critical Speed Working Principle. 1263. Added 1 year ago anonymously in action GIFs Source: ...
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WEBWorking Principle of a Ball Mill. The working principle of a ball mill is based on the rotation of the drum, which causes the grinding media to fall onto the material to be ground. ... The
speed of rotation is crucial for the milling process, as it affects the impact and friction forces acting on the material. Appliions of Ball Mills. 1 ...
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WEBBall Mill Principle, Appliion, Uses, Critical Speed, Diagram Important Info 2024 Filtration Pharmacy Notes PPT PDF 2024 Important Principle, Construction, and Working of Hammer Mill and Ball
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WEBThat does not mean that the preference is speed control over feed rate control. Each ore is different and each plant is different. As equipment designers, we have to provide each of the
features that the operators could use to optimize their process. In my experience, SAG Mills are normally variable speed, and Ball Mills are normally fixed speed.
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WEBThe ball mill consists of a metal cylinder and a ball. The working principle is that when the cylinder is rotated, the grinding body (ball) and the object to + (86) . ... This is the critical
speed of the 180 litre wet mill .
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WEBSep 17, 2023 · The ball mill for the installation of horizontal rotating cylinder, outer edge of which lies Gear, two warehouses, latticetype ball mill. Feed materials from the device into the
compound empty ...
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WEBAug 3, 1999 · Rose and Sullivan showed critical rotation speed Nc, to reach the final stage,, centrifugal motion: N c = 1 2π 2g D−2r where D is the inner diameter of a jar and r is the radius
of balls. All throughout, they used the critical rotation speed as the constant value for given conditions of ballmill [5]. After this work, the critical ...
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WEBOct 5, 2021 · Ball Mill. Works on the principle of Impact and Attrition. The mill consists of a hollow cylinder fixed on a metallic frame in such a way that it can rotate around its horizontal
axis. The cylinder contains metallic balls occupying around 30 to 50% of total capacity. The balls are usually made up of rubber, porcelain or metal.
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WEBSep 27, 2022 · Advantages of Ball Mill. Produces a very fine powder – particle size less than or equal to 10 microns. Suitable for milling toxic materials as it can be used in an enclosed
form. A wide range of ...
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WEBNov 8, 2023 · The working of a pharmaceutical hammer mill is based on the principles of highspeed impact and controlled particle size reduction. By adjusting parameters such as hammer size,
rotor speed, and screen perforation size, operators can precisely control the final particle size of pharmaceutical materials, ensuring consistency and quality in ...
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WEBAug 14, 2019 · Working principle of ball mill back to top. The ball mill is usually composed of a horizontal cylinder, a hollow shaft and a grinding head. The barrel is a long cylinder, which
is equipped with ball grinding medium. ... D—the inner diameter of the ball mill, M; N0 critical speed of revolutions, r /min. When the nonsmooth ball mill liner is ...
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WEBIn most cases, the ideal mill speed will have the media tumbling from the top of the pile (the shoulder) to the bottom (the toe) with many impacts along the way. The ideal mill speed is
usually somewhere between 55% to 75% of critical speed. Critical Mill Speed. Critical Speed (left) is the speed at which the outer layer of media centrifuges ...
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WEBJan 1, 2022 · The filling levels M* was taken as 30%, 40% and 50% of full mill and the mill speed N* was selected as, and of the critical speed. The critical speed is the speed at which a mill
drum rotates such that balls are stick to the drum, which is given by 2 g / D − d where D and d are the mill diameter and particle diameter in meters ...
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WEBBall Mill Critical Speed Working Principle
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How to Write the Squared Symbol in Excel - EasyClick AcademyHow to Write the Squared Symbol in Excel
This video tutorial offers a step-by-step guide on how to write the squared symbol in Excel. We’ll have a look at two ways to do this – within text and with numbers, too.
Would you rather watch this tutorial? Click the play button below!
The first way to write the squared symbol relies on the use of formatting. Let’s say we want to type the sign for the square metre. We can simply type in the cell ‘m2’ and format the number 2 to
display correctly.
First, we need to select the number 2 written within the text, then we go to the Home tab and click here, on the bottom right-hand corner, to open ‘Font Settings’. You’ll see this window where you’ll
find the Effects section and tick ‘Superscript’.
Unfortunately, to write the squared symbol the way we just did doesn’t work with numbers. So, if the cell is formatted as ‘General’ or ‘Number’, you’ll simply have to use the way we’re about to have
a look at.
If you need to make a quick fix, you need to change the cell formatting to ‘Text’.
But this is not necessary, because there’s one more way to insert the squared symbol in a cell. And this one works perfectly well with both – texts and numbers.
Let’s delete the cell B4 and return the formatting to ‘General’. And here we go with how to write the squared symbol through the option of inserting symbols in Excel.
If we need to write three squared, type in the cell the number 3 and now we’re ready to insert the squared symbol.
Go to the ‘Insert’ tab and click on ‘Symbols’ at the very end. This option is handy if you need to use special symbols and signs in Excel.
We need the squared symbol, so we look it up, click on it and insert it through the ‘Insert’ button. It appears right where it’s supposed to be – next to the number 3.
You can follow these steps to insert any symbol or sign you need in Excel.
Once the symbol is in its place, close the window with the button ‘Close’ and that’s all it takes!
If you’d like to know more on how to square a number in Excel, have a look at more of our video tutorials – the link is in the list below!
If you found this tutorial helpful, give us a like and watch other tutorials by EasyClick Academy. Learn how to use Excel in a quick and easy way!
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Curved Line
A “Curved line” or simply a ” Curve” is a line that is not straight. We see curves everywhere around us. Be it art or decoration or a general thing, and curves can be seen around us. In this article,
we are going to learn the definition of a curved line, different types of curved lines with many examples.
What is a Curved Line?
A curved line is one that is not straight and is bent. Ideally, it is smooth and continuous. In other words, a curve is defined as a group of points that resemble a straight line that falls between
two neighbouring points. We know that the curvature of the straight line is zero. Hence, if the curvature of a line is not zero, then we can call it a curved line. The following figure shows the
different types of curved lines.
Difference Between Straight and Curved Line
Straight Line Curved Line
A straight line is the shortest line that joins any two points. A bent line that is not straight is called a Curved Line.
It always moves in one direction. It doesn’t move in one direction.
Examples of Curved Lines
There are many examples of curved lines like the alphabets – C and S. Whereas the letters A, M, N, L, etc are not examples of curves since they can be formed by joining the line segments (or straight
Also, Read:
Different Types of Curved Lines
The curved lines can be classified into different types. They are:
• Simple Curve
• Non-simple Curve
• Algebraic Curve
• Transcendental Curve
Simple Curve
A simple curve is defined as a curve that doesn’t cross itself. We know that the open curve has two endpoints whereas a closed curve has no endpoints. A closed curve creates a path that may begin
from any point and terminate at the same point. Thus, the simple curve may be open or closed.
Non-simple Curve
The non-simple curve is a type of curve that intersects with itself while changing its direction. Like simple curves, the non-simple curves can also be open or closed.
Algebraic Curve
A plane curve where a set of points are located on the Euclidean plane and are represented in terms of polynomials is called Algebraic Curve. The polynomial’s degree denotes the degree of the curve.
C = {(a, b) ∈ R^2: P(a, b) = 0}
Transcendental Curve
This curve is different from the algebraic curve. The curve that does not represent the algebraic form, then it is called a transcendental curve. This curve might have many intersecting points
together with the straight line. Hence, a transcendental curve is not a polynomial based on a and b.
Practice Question
Question: Identify the open and closed curves from the below figure.
To learn more Maths-related concepts, stay tuned with BYJU’S – The Learning App and download the app today to learn all Maths concepts easily by exploring more videos.
Frequently Asked Questions on Curved Line
What is a curved line?
A curved line is a type of straight line with bent. In other words, it is a geometrical object similar to the line having curvature.
What are the examples of curved lines?
There are many examples of curved lines in the English alphabet such as C, S and O.
Why we use curved lines?
Curved lines are used mainly in the graphical representation of different types of functions.
What are the types of curves?
There are two types of curves namely simple open curves and simple closed curves. For example, “C” is the simple open curve and “O” is the simple closed curve that we can see in alphabets.
What is a simple curve?
A curve that doesn’t cross itself is called a simple curve, otherwise, it is a complex (or non-simple) curve. For example, “U” is the simple curve and “8” is the non-simple curve. | {"url":"https://mathlake.com/Curved-Line","timestamp":"2024-11-06T02:55:32Z","content_type":"text/html","content_length":"13297","record_id":"<urn:uuid:4d408eb7-ca9d-4202-81e3-e0281d207403>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00343.warc.gz"} |
Poker Hands 5 Card Draw Rip’s Applied Arithmetic Blog
If you may be happy together with your holding and don’t wish to draw any cards, you «stand pat.» This strategy will work and be very simple to program, however it’s wasteful in house (size of this
system executable). The next step is 64 bits and will take the hit of the type rather than double the scale of the key. Pocket tens are one other premium pair with which you need to typically be
prepared to commit a lot of money.
Thus the following three examples point to the same poker hand. The solely distinction is the order in which the playing cards are dealt. There are 2,598,960 many possible 5-card Poker palms. Thus
the chance of acquiring anyone specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million).
High Card
So there are forty hands for straight flush in whole. A flush is a hand with 5 cards in the identical go nicely with however not in consecutive order (or not in sequence). Thus the requirement for
flush is considerably extra relaxed than a straight flush.
• This is definitely a really rare occasion (less than 0.05% chance of happening).
• That is, «pair» vs. «pair.» If you detect a special hand configuration («two pair»), that places the scores into separate groups.
• If aces usually are not low, simply rotate the hand descriptions in order that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the worst hand.
• The following table shows the variety of combos for two to 10 playing cards from a single 52-card deck, with no wild cards.
If aces usually are not low, simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand and ace-high replaces king-high because the worst hand. Given are five numbers a, b,
c, d and e, every from 0 to 51, representing the five cards within the hand. Is there a way to get the important thing to 32 bits or beneath and never need to type the 5 card tuple. Five cards the
entire similar suit, but not so as, such as Q-K of spades.
Poker Palms Rating
Now you must be acquainted with the different poker 5 card hands. So, it’s time to transfer forward and find out how the sport takes place and fundamental poker rules to guarantee you don’t feel
overwhelmed whereas enjoying poker online. Let us perceive gameplay and 5 playing cards poker rules. A Straight Flush is the second-best among all the 5 card poker arms and consists of 5 cards of the
same go nicely with in sequential order. For example – a 3, 4, 5, six and 7, all of ♣. If you and an opponent have the same five-card poker hand, then the pot is divided equally between you.
I wanted to include one thing a bit extra exciting in this article, so here is the highest 20 No Limit Hold’em beginning hands in phrases of raw all-in fairness (or percentages). Both gamers have a
pair of kings, but the winner of the pot is Player B as a result of he has Player A ‘out-kicked’. ○ Fold- If you aren’t happy with your playing cards, you may give again your playing cards face all
the means down to the dealer, which implies you’re quitting that match of 5-card poker.
Poker Palms & Poker Hand Rankings Chart
However suits are typically ranked in other video games like Bridge, the place the ranking order of fits is spades, hearts, diamonds, and golf equipment. The most valuable hand is a Royal Flush, a
Straight Flush made from the very best value cards within the recreation. Once you’ve mastered poker hands, try our different guides to help you turn out to be a better participant. For example, a
participant holding a pair of 9s with a King, eight and 5, would beat a participant with a pair of 9s alongside a Jack, 8 and 5. If kickers are additionally tied, the subsequent highest card decides
the winner, and if all playing cards are tied the pot is evenly break up between the winning gamers. The following tables show the variety of mixtures and chance for each poker hand utilizing one of
the best five cards from out of 5 to 10 playing cards.
Some of the higher ranked poker palms are in a single go nicely with but with further strict necessities. After that if there’s multiple participant remaining, a showdown occurs during which the
participant with one of the best five-card poker hand wins. The variety of distinct poker hands is even smaller. However, despite the fact that the hands aren’t similar from that perspective, they
still form equal poker arms because every hand is an A-Q excessive card hand.
Two Pair
The greatest non-pair is also identified as massive slick. Players typically complain about missing flops with ace-king, so we wrote How to Play Ace-King When You Miss the Flop. One pair consists of
two cards of the same value, and three extra playing cards.
Hand rankings in poker correspond to the chance of constructing such arms. Many contemplate poker much less of a playing sport than other casino video games. For that to be true, players need to
enhance their understanding of recreation play and the technique required to be a winning player. From a royal flush to high card, understand what beats what in Poker with our free Poker Hand
Rankings Guide. Use as a reference for yourself, or obtain and print to provide them out at your Home Games. Five of a Kind – This is the very best possible hand and might occur solely the place a
minimum of one card is wild, similar to a joker.
Two pair consists of two cards of equal value, one other two cards of equal value, and one further card. Also generally known as ‘trips’, three of a sort is three cards of the same value and 2
aspect cards of different values. As you may have found, when you add collectively the values of the playing cards in the greatest way you have proposed then you will get ambiguities. Without
normalization, it’s extremely troublesome to create a novel integer worth for each hand and assure that a pair of 4’s will at all times beat a pair of three’s. The result will be a singular value for
every hand, and you’re certain that a Flush will all the time beat a Straight and a king-high Full House will beat a 7-high Full House, and so on.
So eliminating similar arms that ignore relative suit values, there are solely 134,459 distinct palms. In poker, the chance of every sort of 5-card hand may be computed by calculating the proportion
of palms of that sort among all attainable palms. The following table shows the variety of combinations if every card was dealt from a separate deck, which would be mathematically equal to an
infinite number of decks.
• Should you make a remark that a player only opens with a pair of jacks or stronger on the button, you presumably can easily fold a pair of nines within the blinds instead of calling.
• Meanwhile, if you have just one pair but your opponent keeps checking to offer you a free play on the pot, you may well have the strongest hand and will bet your hand.
• The variety of distinct poker palms is even smaller.
• Since a sport of poker uses a 52-card deck of French playing cards, there are 2,598,960 totally different possible mixtures (aka. poker hands).
• The Royal Flush is the most effective hand in poker, so no one other palms beat this one.
• Then, you possibly can multiply the special prime values of every card together to produce a novel worth for every attainable hand.
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Coffeeshop Physics
A few weeks ago, I wrote an article for Fermilab Today about the spin of fundamental particles and the Higgs boson in particular. My cats were eager to demonstrate how photons emerge from a Higgs
decay in an anticorrelated state, so I included them as Figure 1. It was wildly popular. I was even asked to present it as a talk, which gave me a chance to expand on the topics that I had raced
through to stay within my 800 word budget.
Spin is interesting for a lot of reasons. At the moment, it is perhaps the most important unknown parameter of the new particle discovered last July. Spin is also at the heart of quantum weirdness,
because it’s an “amount of rotation” that is somehow quantized like a light switch: on or off, and never in between. It seems like we could just put a particle on a slow enough turntable to dial up a
non-quantized angular momentum, but nature has a way of enforcing its rules. Talking about spin also makes for a nice bridge between the world of subatomic particles and the world of everyday
experience, since it has some macroscopic consequences.
I don’t harbor the illusion that the popularity of my article was due to anything but Cats On The Internet, however.
Angular momentum
In the original Karate Kid movie, the kid who wants to learn karate is frustrated by his teacher’s insistence that he spend his time painting fences and waxing cars. Only later is it revealed that
the student had learned the fundamental moves without realizing it, blocking punches with “wax on, wax off.” In much the same way, the deepest mysteries of physics are taught in Physics 101, but
they’re hard to recognize in everyday objects like bicycle wheels and spinning tops.
One of these deep principles is the conservation of angular momentum. Angular momentum is the amount of rotation an object has, taking into account its mass and size, and it is curiously constant. A
spinning figure skater has a constant angular momentum as she contracts her arms and twirls faster because a fast-spinning small object as as much angular momentum as a slow-spinnging large object. A
cloud of interstellar dust, stately drifting in a slow inward spiral, can eventually collapse into a pulsar the size of a city block, feverishly revolving around its axis a thousand times per second.
The angular momentum stays the same.
Fundamental particles are, as far as we know, infinitesimal points of zero size. If a figure skater or a dust cloud could shrink down to a single point, would they rotate infinitely fast? Is it even
meaningful to say that an object rotates when it doesn’t have any extension in space? Whether the particles are actually moving or not, they have angular momentum and conserve it just as a
macroscopic object would. The fact that they have angular momentum and maybe not motion suggests that angular momentum is the more foundational concept.
Another odd thing about the angular momentum of individual particles is that it is quantized. An electron always has an angular momentum of ^-35 Joule-seconds, is also the minimum uncertainty in
Heisenberg’s Uncertainty Principle:
which means that the uncertainty in a particle’s position (
Phase space
Phase space is another workaday tool in classical physics that turns out to have deep implications in quantum mechanics. It is simply a plot of momentum (velocity times mass) versus position. If
we’re only interested in one spatial dimension, then phase space is two-dimensional. If we’re interested in three spatial dimensions, then phase space is six-dimensional, since there’s a component of
momentum for each component of position. Phase space is not another world, just a way of representing problems.
To give a sense of how phase space works, I’ve put together a demo that plots momentum versus position for a simulated pendulum that you can move with your mouse. It requires a plug-in from Unity (a
video game engine) that installs fairly easily on non-Linux laptops. (Unity is wonderful: you just set up physical objects and let it simulate all of their complex interactions. I’ve been having fun
As you can see in the demo, the pendulum occupies a single point in phase space at a given time, and that point is always moving. In the center of its swing, the position is near zero but its
momentum is large. At the turn-around points, the position is far from zero but the momentum is passing through zero. A nice, smooth swing traces an ellipse in phase space, but if you grab the ball
and bounce it, you get more complex patterns.
If a system has an unpredictable or even fractal phase space pattern, it is called “chaotic.” Although chaotic systems are classical, with no quantum uncertainty, their outcomes cannot be predicted
because they are hypersensitive to the initial position and momentum of the system. No position or momentum can be specified with infinite precision, just arbitrarily high precision (in classical
My pendulum demo is not chaotic: if you let go of the ball in a slightly different place, it might bounce a bit more at first, but it eventually settles into a regular orbit. A chaotic Lorenz
attractor (see figure), orbits one point for a while, then switches unpredictably to another, and back again. Some unstable systems start in a smooth orbit but eventually fly away with no fixed
This chaotic/non-chaotic dichotomy is a mathematical abstraction. At a very small scale, phase space is granular, which moots the question of sensitivity to arbitrarily small regions of phase space.
A real, quantum mechanical system never occupies a single point on the plane, but a region with area equal to
One way to think about this extension in phase space is to call it uncertainty. We imagine that the true position and momentum are somewhere in the blob, and we don’t know where. If the momentum is
very precise (middle figure), then the position must be uncertain; if the position is very precise (right figure), then the momentum must be uncertain. That is the meaning of Heisenberg’s Uncertainty
However, many experiments in quantum mechanics force us to say that the system is occupying all points within the blob simultaneously. I prefer to think of the system as actually spreading out over
that area.
Quantized angular momentum
The formal definition of angular momentum (
which means that angular momentum can be represented as an area in phase space. It must therefore be quantized in units of
Each type of particle has an intrinsic angular momentum, called spin. For brevity, spin
spin-0 spin-1/2 spin-1 spin-2
Higgs bosons Quarks, electrons, muons, taus, neutrinos Photons, W, Z bosons, gluons Gravitons
Particles bound in combinations, such as the three quarks in a proton, have additional angular momentum from the particles orbiting each other. This, too, is quantized. In fact, if any angular
momentum in the universe were not quantized, then all of it would not be quantized, since we would be able to transfer the fractional part from one system to another.
The spin of photons, particles of light, is known to photographers as circular polarization. A photon’s spin can point in the same direction as its motion (“spin +1”) or opposite to its motion (“spin
−1”), and these are the “right-handed” and “left-handed” polarizations.
My objection and Nature’s rebuttal
It seems ludicrous for something like rotation to be quantized. If a particle has spin-1/2, why couldn’t we just put it on a record player and rotate it the other way, at a rate less than
Physics is an experimental science, so we can just try it and see what happens. When you try to spin down a single particle or look at its spin from different angles, it occupies both the spin-up
state and the spin-down state with different probabilities. To make any conclusion, one must measure event rates with an ensemble of particles, and that doesn’t completely answer my appeal to common
Some quantum systems are big enough to see by eye, such as a bucket of liquid helium, and this makes quantum mechanics almost tangible. When cold enough, the whole body of liquid acts as one giant
quantum state with quantized angular momentum. On a turntable, it resists attempts to rotate it by forming vortices that spin the other way. The vortices are quantized, and they add up to the
appropriate angular momentum by arranging themselves in a triangular grid.
Nature finds a way.
Spin of the newly discovered particle
Only a few facts are known about the particle discovered this July: it decays to
• two photons,
• two Z bosons,
• probably two W bosons,
• maybe two b quarks,
• and maybe, just maybe, two taus.
When a particle decays, conservation of angular momentum requires the spin of all the decay products to add up to the spin of the original particle. The decay products can even emerge from the decay
orbiting each other, which adds to the total angular momentum. Nevertheless, some decays are impossible due to angular momentum mismatches, and we can use this fact to deduce the spin of the original
Of the decay modes listed above, the most restrictive is the decay to two photons. Photons are spin-1 and massless; the only ways to satisfy the accounting are:
(spin 0) = (spin +1) + (spin −1)
(spin +2) = (spin +1) + (spin +1) or
(spin −2) = (spin −1) + (spin −1) or
The photons are either spinning in opposite ways (original particle was spin-0) or the same way (original particle was spin-2).
Unfortunately, we can’t use something like a polarization filter to measure the spin of these photons: they’re 30 billion times more energetic than visible light and register in the detector as a
shower of high-energy particles, rather than a well-behaved wave.
Instead, we use the fact that spin affects the pattern of particle decays. When a heavy particle decays into two light particles, the angle of the decay products is random, but there is a trend in
the randomness. With enough events, we can see if the decays match one pattern better than the others. We do not yet have enough events to tell whether the newly discovered particle is a spin-0 Higgs
boson or a spin-2 graviton, but the datasets are growing.
This is the main reason that physicists are cautious about calling it a Higgs boson. If it’s not spin-0, then it can’t solve the problem that the Higgs mechanism was invented for. It would be some
new, unexpected particle— certainly welcome, but not the Higgs. | {"url":"http://coffeeshopphysics.com/articles/2012-11/17_spin/","timestamp":"2024-11-03T02:24:47Z","content_type":"text/html","content_length":"19337","record_id":"<urn:uuid:19f20a1e-2b7b-48b8-94c7-ed79d9c99636>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00815.warc.gz"} |
How to use Graph and BFS in C#?
In this article, you will learn what a Graph Data Structure is. Also, you will find representations of a Graph, the types, and applications of the Graphs. Moreover, breadth-first search (BFS) and how
it works will be explained thoroughly.
It’s recommended to read what is data structure first.
What is Graph in Data structure?
A Graph is a non-linear data structure like a Tree, and it is a collection of nodes that are connected to other nodes by edges.
The nodes are sometimes also referred to as vertices and the edges are lines that connect any two nodes or vertices in the Graph. Take a look at the following Graph.
In the above Graph, we have vertices that contain the data (number) and the edges are connecting between the vertices. We can say a Graph is a pair of sets (V, E) where V is vertices and E is an edge
connecting the pairs of vertices.
V = {1, 2, 3, 4, 5} //we have 5 Vertices.
E = { (1, 2), (1, 3), (2, 4), (4, 3), (4, 5) } //we have 6 Edges.
G = {V, E} // the Graph
Graph Representation
We can represent Graph in two different ways:
1) Adjacency Matrix
An adjacency matrix is a 2D array. Each row and column represent a vertex.
If the value of any element a[i][j] is 1, it represents that there is an edge connecting vertex i and vertex j.
The adjacency matrix for the Graph in the diagram above is.
Since it is an undirected Graph, for edge (1,3) we also need to mark edge (3,1) making the adjacency matrix symmetric about the diagonal.
2) Adjacency List
An adjacency list represents a Graph as an array of linked lists.
The index of the array represents a vertex and each element in its linked list represents the other vertices that form an edge with the vertex.
The adjacency list for the Graph in the diagram above is.
An adjacency list is efficient in terms of storage because we only need to store the values for the edges. And this way is useful when we have a lot of vertices like millions.
Types of Graphs in Data Structure
There is a lot of types but here we will mention the popular types.
1) Undirected
A Graph in which all the edges do not point in a specific direction.
2) Directed
A Graph in which all the edges are pointed in a single direction.
3) Weighted Graph
A Graph that has a value associated with every edge. The values corresponding to the edges are called weights. A value in a weighted Graph can represent things depending on what you will use such as
distance, and time.
Application of Graph in Data Structure
Graphs have a lot of applications. And below we will mention the most popular applications:
• Used in social networks such as Facebook and Linkedin. In Linkedin, it’s suggested to follow x, because x is your friend's follower. And Linkedin knew this relationship by the Graph.
• Used in Google maps for building transportation systems. In google maps, the intersection of two or more roads represents the vertex while the road connecting two vertices represents an edge.
• Used in the World Wide Web where the web pages represent the nodes.
Traversal means to visit each node of a Graph. For Graphs, there are two types of traversals:
1. Depth First traversal.
2. Breadth-First traversal.
In this section, we are going to learn Breadth-first traversal/Search or BFS in detail.
What is Breadth-First Search (BFS)?
BFS is an algorithm for searching used in Tree and Graph. Breadth-First Search is a traversal technique in which we traverse all the nodes of the Graph in a breadth-wise motion. In BFS, we traverse
one level at a time and then jump to the next level.
In a Graph, the traversal can start from any node and cover all the nodes level-wise. The BFS algorithm makes use of the queue data structure for implementation.
Graphs may contain cycles, so we may come to the same node again. To avoid processing a node more than once, we use a boolean visited array
How does BFS work?
To understand how BFS works, there are rules:
1. Visit the adjacent unvisited vertex. Mark it as visited. Display it. Insert it in a queue.
2. If no adjacent vertex is found, remove the first vertex from the queue.
3. Repeat Rule 1 and Rule 2 until the queue is empty.
We will dig deeper into these rules by taking an example and explaining the example step by step.
Step 1
Consider we have the following Graph having 5 vertices {A, B, C, D, E}. and we have Queue.
Step 2
We start by visiting A (starting node), marking it as visited, and also enqueue it.
Step 3
We then see an unvisited adjacent node from A. In this example, we have three nodes it does not matter choose anything, we choose B, mark it as visited, and enqueue it.
Step 4
Next, the unvisited adjacent node from A is C. We mark it as visited and enqueue it.
Step 5
Next, the unvisited adjacent node from A is D. We mark it as visited and enqueue it.
Step 6
Now, A is left with no unvisited adjacent nodes. So, we dequeue and find B.
Step 7
From B we have E as an unvisited adjacent node. We mark it as visited and enqueue it.
By doing that we have all the vertices are visited.
You can visualization the BFS at Data Structure Visualizations
Example: Implementing Code of the Breadth-First Search in the Graph using C#
In this example, you will learn how to create a Graph and add vertices with their edges, traversal each vertex, and print it using BFS.
class Graph
// No. of vertices
private int _V;
//Adjacency Lists
LinkedList<int>[] _adj;
public Graph(int V)
_adj = new LinkedList<int>[V];
for (int i = 0; i < _adj.Length; i++)
_adj[i] = new LinkedList<int>();
_V = V;
// Function to add an edge into the graph
public void AddEdge(int v, int w)
// Prints BFS traversal from a given source s
public void BFS(int s)
// Mark all the vertices as not visited
bool[] visited = new bool[_V];
for (int i = 0; i < _V; i++)
visited[i] = false;
// Create a queue for BFS
Queue<int> queue = new Queue<int>();
// Mark the current node as
// visited and enqueue it
visited[s] = true;
while (queue.Any())
// Dequeue a vertex from queue
// and print it
s = queue.First();
Console.Write(s + " ");
// Get all adjacent vertices of the
// dequeued vertex s. If a adjacent
// has not been visited, then mark it
// visited and enqueue it
LinkedList<int> list = _adj[s];
foreach (var val in list)
if (!visited[val])
visited[val] = true;
static void Main(string[] args)
Graph graph = new Graph(5);
graph.AddEdge(0, 1);
graph.AddEdge(0, 2);
graph.AddEdge(1, 2);
graph.AddEdge(2, 0);
graph.AddEdge(2, 3);
graph.AddEdge(3, 4);
Console.Write("Following is Breadth First Traversal(starting from vertex 2)\n");
Following is Breadth First Traversal(starting from vertex 2)
See Also | {"url":"https://debug.to/3320/how-to-use-graph-and-bfs-in-c","timestamp":"2024-11-13T11:32:18Z","content_type":"text/html","content_length":"60700","record_id":"<urn:uuid:eca240c6-bf07-41f7-8a45-82612329b7b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00569.warc.gz"} |
How to calculate interest rate on a loan amount
Use the loan calculator to determine your monthly payments for a simple loan. Input your loan amount, interest, and term in the loan calculator to see how much you'll pay each month. Simple Loan
Calculator. 1 Loan Amount. 2 Interest Rate . Home loan interest calculation Assuming you have an outstanding loan amount of $500,000 and an interest rate of 5% APR, your interest payment for one 15
Jul 2019 For example, if you were considering a mortgage for $200,000 with a 6% interest rate, your annual interest expense would amount to $12,000,
To calculate the periodic interest rate for a loan, given the loan amount, the number of payment periods, and the payment amount, you can use the RATE function. In the example shown, the formula in
C10 is: = RATE ( C7 , C6 , - C5 ) * 12 Loans have One use of the RATE function is to calculate the periodic interest rate when the amount, number of payment periods, and payment amount are known. For
this example, we want to calculate the interest rate for $5000 loan, and with 60 payments of $93.22 each. The NPER function is configured as follows: Figure the monthly interest by multiplying the
monthly rate by the loan balance at the start of the month ($100,000 multiplied by 0.5% equals $500 for the first month). Subtract the interest costs from the monthly payment. Keep a running tally in
an additional column if you want to track interest over time. Interest-Only Loan Payment Formula Calculating payments for an interest-only loan is easier. Multiply the amount you borrow (a) by the
annual interest rate (r), then divide by the number of payments per year (n). Or, multiply the amount you borrow (a) by the monthly interest rate, which is the annual interest rate (r) divided by 12:
Free calculator to find the interest rate as well as the total interest cost of an amortized loan with fixed monthly payback amount. Also learn more about interest cost, experiment with other
interest and loan calculators, or explore many more calculators on topics such as finance, math, fitness, and health. Then, multiply that number by the total number of time periods since the loan
began to find the simple interest. For example, if the principal is $55,000, the interest rate is 0.03 percent, and the number of time periods since the loan began is 10 years, first you’d multiply
55,000 by 0.03 to get 1,650. Interest Rate. Nearly all loan structures include interest, which is the profit that banks or lenders make on loans. Interest rate is the percentage of a loan paid by
borrowers to lenders. For most loans, interest is paid in addition to principal repayment. Loan interest is usually expressed in APR, or annual percentage rate, which include both interest and fees.
The rate usually published by banks for saving accounts, money market accounts, and CDs is the annual percentage yield, or APY.
29 Sep 2017 To get the most accurate rates using our Explore Interest Rates tool, you'll need to put in your state, and depending on your loan amount and
Student-Platform Student Loan CalculatorEMI CalculatorMaturity Value Calculator. Show More. Repayment Annual Rate of Interest. | 4%| 6% | 8% | 10 %| To calculate your own home equity, just subtract
the amount you owe from the market value of the property. Learn About. Interest Rate. When you have a mortgage Figure out how much your home loan repayments on a property will be. Simply enter the
details into the calculator and it will do the rest. Months. Interest Rate Use Westpac's Latest Rates. — % p.a. Total number of payments. Total interest You should be able to see your monthly
payments with different loan interest rates, amounts and terms. Then, you can decide on a monthly payment size that fits
You should be able to see your monthly payments with different loan interest rates, amounts and terms. Then, you can decide on a monthly payment size that fits
8 Aug 2014 The mathematical formula to calculate EMI is: EMI = P × r × (1 + r)n/((1 + r)n - 1) where P= Loan amount, r= interest rate, n=tenure in number of View disclaimer. What's a good interest
rate for a car loan? This depends on a number of factors,
Online calculator to calculate interest rate of a product using david cantrell's approximate solution method. Calculate the monthly payment to be paid with the given number of payments, interest
rate, and loan amount.
Calculator Rates Loan Breakdown Calculator. This calculator will help you to determine the principal and interest breakdown on any given payment number. Enter the loan's original terms (principal,
interest rate, number of payments, and monthly payment amount) and click on the "Calculate" button.
Interest Rate. Nearly all loan structures include interest, which is the profit that banks or lenders make on loans. Interest rate is the percentage of a loan paid by borrowers to lenders. For most
loans, interest is paid in addition to principal repayment. Loan interest is usually expressed in APR, or annual percentage rate, which include both interest and fees. The rate usually published by
banks for saving accounts, money market accounts, and CDs is the annual percentage yield, or APY.
Our 'Interest Rate Calculator' takes into account the size of your home loan and the loan term to calculate the interest rate you may be eligible for. It also provides Step 1. Multiply each loan
amount by its interest rate to obtain the per loan weight factor. $20,000 * 6.80% = $1,360. $10,000 * 7.90% Loan amount. Total amount of your loan. Payment. Payment for this loan. Interest rate.
Annual interest rate for Student-Platform Student Loan CalculatorEMI CalculatorMaturity Value Calculator. Show More. Repayment Annual Rate of Interest. | 4%| 6% | 8% | 10 %| To calculate your own
home equity, just subtract the amount you owe from the market value of the property. Learn About. Interest Rate. When you have a mortgage Figure out how much your home loan repayments on a property
will be. Simply enter the details into the calculator and it will do the rest. Months. Interest Rate Use Westpac's Latest Rates. — % p.a. Total number of payments. Total interest
29 Sep 2017 To get the most accurate rates using our Explore Interest Rates tool, you'll need to put in your state, and depending on your loan amount and 22 Aug 2019 The Annual Percentage Rate (APR)
is a calculation of the overall charges are added to the loan amount before interest is calculated. All lenders are required to quote the interest rate on a loan or credit card as an APR. | {"url":"https://binaryoptionstxiaq.netlify.app/sender77737wa/how-to-calculate-interest-rate-on-a-loan-amount-320","timestamp":"2024-11-11T23:17:04Z","content_type":"text/html","content_length":"35628","record_id":"<urn:uuid:7362768e-8603-4bb3-8946-c64b9d14e640>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00181.warc.gz"} |
Dynamic Programming in Java
Dynamic Programming is typically used to optimize recursive algorithms, as they tend to scale exponentially. The main idea is to break down complex problems (with many recursive calls) into smaller
subproblems and then save them into memory so that we don't have to recalculate them each time we use them.
What is Dynamic Programming?
Dynamic programming is a programming principle where a very complex problem can be solved by dividing it into smaller subproblems. This principle is very similar to recursion, but with a key
difference, every distinct subproblem has to be solved only once.
To understand what this means, we first have to understand the problem of solving recurrence relations. Every single complex problem can be divided into very similar subproblems, this means we can
construct a recurrence relation between them.
Let's take a look at an example we all are familiar with, the Fibonacci sequence! The Fibonacci sequence is defined with the following recurrence relation:
Note: A recurrence relation is an equation that recursively defines a sequence where the next term is a function of the previous terms. The Fibonacci sequence is a great example of this.
So, if we want to find the n-th number in the Fibonacci sequence, we have to know the two numbers preceding the n-th in the sequence.
However, every single time we want to calculate a different element of the Fibonacci sequence, we have have certain duplicate calls in our recursive calls, as can be seen in following image, where we
calculate Fibonacci(5):
For example, if we want to calculate F(5), we obviously need to calculate F(4) and F(3) as a prerequisite. However, to calculate F(4), we need to calculate F(3) and F(2), which in turn requires us to
calculate F(2) and F(1) in order to get F(3) – and so on.
This leads to many repeated calculations, which are essentially redundant and slow down the algorithm significantly. To solve this issue, we're introducing ourselves to Dynamic Programming.
In this approach, we model a solution as if we were to solve it recursively, but we solve it from the ground up, memoizing the solutions to the subproblems (steps) we take to reach the top.
Therefore, for the Fibonacci sequence, we first solve and memoize F(1) and F(2), then calculate F(3) using the two memoized steps, and so on. This means that the calculation of every individual
element of the sequence is O(1), because we already know the former two.
When solving a problem using dynamic programming, we have to follow three steps:
• Determine the recurrence relation that applies to said problem
• Initialize the memory/array/matrix's starting values
• Make sure that when we make a "recursive call" (access the memoized solution of a subproblem) it's always solved in advance
Following these rules, let's take a look at some examples of algorithms that use dynamic programming.
Rod Cutting Algorithm
Let's start with something simple:
Given a rod of length n and an array that contains prices of all pieces of size smaller than n. Determine the maximum value obtainable by cutting up the rod and selling the pieces.
Naive Solution
This problem is practically tailor-made for dynamic programming, but because this is our first real example, let's see how many fires we can start by letting this code run:
public class naiveSolution {
static int getValue(int[] values, int length) {
if (length <= 0)
return 0;
int tmpMax = -1;
for (int i = 0; i < length; i++) {
tmpMax = Math.max(tmpMax, values[i] + getValue(values, length - i - 1));
return tmpMax;
public static void main(String[] args) {
int[] values = new int[]{3, 7, 1, 3, 9};
int rodLength = values.length;
System.out.println("Max rod value: " + getValue(values, rodLength));
Max rod value: 17
This solution, while correct, is highly inefficient. Recursive calls aren't memoized so the poor code has to solve the same subproblem every time there's a single overlapping solution.
Dynamic Approach
Utilizing the same basic principle from above, but adding memoization and excluding recursive calls, we get the following implementation:
public class dpSolution {
static int getValue(int[] values, int rodLength) {
int[] subSolutions = new int[rodLength + 1];
for (int i = 1; i <= rodLength; i++) {
int tmpMax = -1;
for (int j = 0; j < i; j++)
tmpMax = Math.max(tmpMax, values[j] + subSolutions[i - j - 1]);
subSolutions[i] = tmpMax;
return subSolutions[rodLength];
public static void main(String[] args) {
int[] values = new int[]{3, 7, 1, 3, 9};
int rodLength = values.length;
System.out.println("Max rod value: " + getValue(values, rodLength));
Max rod value: 17
As we can see, the resulting outputs are the same, only with different time/space complexity.
We eliminate the need for recursive calls by solving the subproblems from the ground-up, utilizing the fact that all previous subproblems to a given problem are already solved.
Performance Boost
Just to give a perspective of how much more efficient the Dynamic approach is, let's try running the algorithm with 30 values.
The Naive solution took ~5.2s to execute whereas the Dynamic solution took ~0.000095s to execute.
Simplified Knapsack Problem
The Simplified Knapsack problem is a problem of optimization, for which there is no one solution. The question for this problem would be - "Does a solution even exist?":
Given a set of items, each with a weight w1, w2... determine the number of each item to put in a knapsack so that the total weight is less than or equal to a given limit K.
So let's take a step back and figure out how we will represent the solutions to this problem. First, let's store the weights of all the items in an array W.
Next, let's say that there are n items and we'll enumerate them with numbers from 1 to n, so the weight of the i-th item is W[i].
We'll form a matrix M of (n+1)x(K+1) dimensions. M[x][y] corresponds to the solution of the knapsack problem, but only including the first x items of the beginning array, and with a maximum capacity
of y.
Let's say we have 3 items, with the weights being w1=2kg, w2=3kg, and w3=4kg.
Utilizing the method above, we can say that M[1][2] is a valid solution. This means that we are trying to fill a knapsack with a capacity of 2kg with just the first item from the weight array (w1).
While in M[3][5] we are trying to fill up a knapsack with a capacity of 5kg using the first 3 items of the weight array (w1,w2,w3). This isn't a valid solution, since we're overfitting it.
Matrix Initialization
There are 2 things to note when filling up the matrix:
Does a solution exist for the given subproblem (M[x][y].exists) AND does the given solution include the latest item added to the array (M[x][y].includes).
Therefore, initialization of the matrix is quite easy, M[0][k].exists is always false, if k > 0, because we didn't put any items in a knapsack with k capacity.
On the other hand, M[0][0].exists = true, because the knapsack should be empty to begin with since k = 0, and therefore we can't put anything in and this is a valid solution.
Furthermore, we can say that M[k][0].exists = true but also M[k][0].includes = false for every k.
Note: Just because a solution exists for a given M[x][y], it doesn't necessarily mean that that particular combination is the solution. In the case of M[10][0], a solution exists - not including any
of the 10 elements. This is why M[10][0].exists = true but M[10][0].includes = false.
Algorithm Principle
Next, let's construct the recurrence relation for M[i][k] with the following pseudo-code:
if (M[i-1][k].exists == True):
M[i][k].exists = True
M[i][k].includes = False
elif (k-W[i]>=0):
if(M[i-1][k-W[i]].exists == true):
M[i][k].exists = True
M[i][k].includes = True
M[i][k].exists = False
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So the gist of the solution is dividing the subproblem into two cases:
1. When a solution exists for the first i-1 elements, for capacity k
2. When a solution exists for the first i-1 elements, but for capacity k-W[i]
The first case is self-explanatory, we already have a solution to the problem.
The second case refers to knowing the solution for the first i-1 elements, but the capacity is with exactly one i-th element short of being full, which means we can just add one i-th element, and we
have a new solution!
In this implementation, to make things easier, we'll make the class Element for storing elements:
public class Element {
private boolean exists;
private boolean includes;
public Element(boolean exists, boolean includes) {
this.exists = exists;
this.includes = includes;
public Element(boolean exists) {
this.exists = exists;
this.includes = false;
public boolean isExists() {
return exists;
public void setExists(boolean exists) {
this.exists = exists;
public boolean isIncludes() {
return includes;
public void setIncludes(boolean includes) {
this.includes = includes;
Now we can dive into the main class:
public class Knapsack {
public static void main(String[] args) {
Scanner scanner = new Scanner (System.in);
System.out.println("Insert knapsack capacity:");
int k = scanner.nextInt();
System.out.println("Insert number of items:");
int n = scanner.nextInt();
System.out.println("Insert weights: ");
int[] weights = new int[n + 1];
for (int i = 1; i <= n; i++) {
weights[i] = scanner.nextInt();
Element[][] elementMatrix = new Element[n + 1][k + 1];
elementMatrix[0][0] = new Element(true);
for (int i = 1; i <= k; i++) {
elementMatrix[0][i] = new Element(false);
for (int i = 1; i <= n; i++) {
for (int j = 0; j <= k; j++) {
elementMatrix[i][j] = new Element(false);
if (elementMatrix[i - 1][j].isExists()) {
} else if (j >= weights[i]) {
if (elementMatrix[i - 1][j - weights[i]].isExists()) {
The only thing that's left is reconstruction of the solution, in the class above, we know that a solution EXISTS. However, we don't know what it is.
For reconstruction we use the following code:
List<Integer> solution = new ArrayList<>(n);
if (elementMatrix[n][k].isExists()) {
int i = n;
int j = k;
while (j > 0 && i > 0) {
if (elementMatrix[i][j].isIncludes()) {
j = j - weights[i];
i = i - 1;
System.out.println("The elements with the following indexes are in the solution:\n" + (solution.toString()));
Insert knapsack capacity:
Insert number of items:
Insert weights:
The elements with the following indexes are in the solution:
[5, 1]
A simple variation of the knapsack problem is filling a knapsack without value optimization, but now with unlimited amounts of every individual item.
This variation can be solved by making a simple adjustment to our existing code:
// Old code for simplified knapsack problem
else if (j >= weights[i]) {
if (elementMatrix[i - 1][j - weights[i]].isExists()) {
// New code, note that we're searching for a solution in the same
// row (i-th row), which means we're looking for a solution that
// already has some number of i-th elements (including 0) in it's solution
else if (j >= weights[i]) {
if (elementMatrix[i][j - weights[i]].isExists()) {
The Traditional Knapsack Problem
Utilizing both previous variations, let's now take a look at the traditional knapsack problem and see how it differs from the simplified variation:
Given a set of items, each with a weight w1, w2... and a value v1, v2... determine the number of each item to include in a collection so that the total weight is less than or equal to a given
limit k and the total value is as large as possible.
In the simplified version, every single solution was equally as good. However, now we have a criteria for finding an optimal solution (aka the largest value possible). Keep in mind, this time we have
an infinite number of each item, so items can occur multiple times in a solution.
In the implementation we'll be using the old class Element, with an added private field value for storing the largest possible value for a given subproblem:
public class Element {
private boolean exists;
private boolean includes;
private int value;
// appropriate constructors, getters and setters
The implementation is very similar, with the only difference being that now we have to choose the optimal solution judging by the resulting value:
public static void main(String[] args) {
// Same code as before with the addition of the values[] array
System.out.println("Insert values: ");
int[] values = new int[n + 1];
for (int i=1; i <= n; i++) {
values[i] = scanner.nextInt();
Element[][] elementMatrix = new Element[n + 1][k + 1];
// A matrix that indicates how many newest objects are used
// in the optimal solution.
// Example: contains[5][10] indicates how many objects with
// the weight of W[5] are contained in the optimal solution
// for a knapsack of capacity K=10
int[][] contains = new int[n + 1][k + 1];
elementMatrix[0][0] = new Element(0);
for (int i = 1; i <= n; i++) {
elementMatrix[i][0] = new Element(0);
contains[i][0] = 0;
for (int i = 1; i <= k; i++) {
elementMatrix[0][i] = new Element(0);
contains[0][i] = 0;
for (int i = 1; i <= n; i++) {
for (int j = 0; j <= k; j++) {
elementMatrix[i][j] = new Element(elementMatrix[i - 1][j].getValue());
contains[i][j] = 0;
elementMatrix[i][j].setValue(M[i - 1][j].getValue());
if (j >= weights[i]) {
if ((elementMatrix[i][j - weights[i]].getValue() > 0 || j == weights[i])) {
if (elementMatrix[i][j - weights[i]].getValue() + values[i] > M[i][j].getValue()) {
elementMatrix[i][j].setValue(M[i][j - weights[i]].getValue() + values[i]);
contains[i][j] = contains[i][j - weights[i]] + 1;
System.out.print(elementMatrix[i][j].getValue() + "/" + contains[i][j] + " ");
System.out.println("Value: " + elementMatrix[n][k].getValue());
Insert knapsack capacity:
Insert number of items:
Insert weights:
Insert values:
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 1/1 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 2/1 0/0 1/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 3/1 0/0 0/0 2/0 6/2 1/0 0/0 5/1 9/3
0/0 0/0 0/0 0/0 3/0 0/0 0/0 2/0 6/0 1/0 4/1 5/0 9/0
0/0 0/0 0/0 5/1 3/0 0/0 10/2 8/1 6/0 15/3 13/2 11/1 20/4
Value: 20
Levenshtein Distance
Another very good example of using dynamic programming is Edit Distance or the Levenshtein Distance.
The Levenshtein distance for 2 strings A and B is the number of atomic operations we need to use to transform A into B which are:
1. Character deletion
2. Character insertion
3. Character substitution (technically it's more than one operation, but for the sake of simplicity let's call it an atomic operation)
This problem is handled by methodically solving the problem for substrings of the beginning strings, gradually increasing the size of the substrings until they're equal to the beginning strings.
The recurrence relation we use for this problem is as follows:
$$ lev_{a,b}(i,j)=min\begin{cases} lev_{a,b}(i-1,j)+1\\lev_{a,b}(i,j-1)+1\\lev_{a,b}(i-1,j-1)+c(a_i,b_j)\end{cases} $$
c(a,b) being 0 if a==b, and 1 if a!=b.
If you're interested in reading more about Levenshtein Distance, we've already got it covered in Python in another article: Levenshtein Distance and Text Similarity in Python
public class editDistance {
public static void main(String[] args) {
String s1, s2;
Scanner scanner = new Scanner(System.in);
System.out.println("Insert first string:");
s1 = scanner.next();
System.out.println("Insert second string:");
s2 = scanner.next();
int n, m;
n = s1.length();
m = s2.length();
// Matrix of substring edit distances
// example: distance[a][b] is the edit distance
// of the first a letters of s1 and b letters of s2
int[][] distance = new int[n + 1][m + 1];
// Matrix initialization:
// If we want to turn any string into an empty string
// the fastest way no doubt is to just delete
// every letter individually.
// The same principle applies if we have to turn an empty string
// into a non empty string, we just add appropriate letters
// until the strings are equal.
for (int i = 0; i <= n; i++) {
distance[i][0] = i;
for (int j = 0; j <= n; j++) {
distance[0][j] = j;
// Variables for storing potential values of current edit distance
int e1, e2, e3, min;
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
e1 = distance[i - 1][j] + 1;
e2 = distance[i][j - 1] + 1;
if (s1.charAt(i - 1) == s2.charAt(j - 1)) {
e3 = distance[i - 1][j - 1];
} else {
e3 = distance[i - 1][j - 1] + 1;
min = Math.min(e1, e2);
min = Math.min(min, e3);
distance[i][j] = min;
System.out.println("Edit distance of s1 and s2 is: " + distance[n][m]);
Insert first string:
Insert second string:
Edit distance of s1 and s2 is: 3
Longest Common Subsequence (LCS)
The problem goes as follows:
Given two sequences, find the length of the longest subsequence present in both of them. A subsequence is a sequence that appears in the same relative order, but not necessarily contiguous.
If we have two strings, s1 = "MICE" and s2 = "MINCE", the longest common substring would be "MI" or "CE", however, the longest common subsequence would be "MICE" because the elements of the resulting
subsequence don't have to be in consecutive order.
Recurrence Relation and General Logic
$$ lcs_{a,b}(i,j)=min\begin{cases} lcs_{a,b}(i-1,j)\\lcs_{a,b}(i,j-1)\\lcs_{a,b}(i-1,j-1)+c(a_i,b_j)\end{cases} $$
As we can see, there is only a slight difference between Levenshtein distance and LCS, specifically, in the cost of moves.
In LCS, we have no cost for character insertion and character deletion, which means that we only count the cost for character substitution (diagonal moves), which have a cost of 1 if the two current
string characters a[i] and b[j] are the same.
The final cost of LCS is the length of the longest subsequence for the 2 strings, which is exactly what we needed.
Using this logic, we can boil down a lot of string comparison algorithms to simple recurrence relations which utilize the base formula of the Levenshtein distance.
public class LCS {
public static void main(String[] args) {
String s1 = new String("Hillfinger");
String s2 = new String("Hilfiger");
int n = s1.length();
int m = s2.length();
int[][] solutionMatrix = new int[n+1][m+1];
for (int i = 0; i < n; i++) {
solutionMatrix[i][0] = 0;
for (int i = 0; i < m; i++) {
solutionMatrix[0][i] = 0;
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
int max1, max2, max3;
max1 = solutionMatrix[i - 1][j];
max2 = solutionMatrix[i][j - 1];
if (s1.charAt(i - 1) == s2.charAt(j - 1)) {
max3 = solutionMatrix[i - 1][j - 1] + 1;
} else {
max3 = solutionMatrix[i - 1][j - 1];
int tmp = Math.max(max1, max2);
solutionMatrix[i][j] = Math.max(tmp, max3);
System.out.println("Length of the longest continuous subsequence: " + solutionMatrix[n][m]);
Length of longest continuous subsequence: 8
Other Problems that Utilize Dynamic Programming
There are a lot more problems that can be solved with dynamic programming, these are just a few of them:
• Partition Problem (coming soon)
• Given a set of integers, find out if it can be divided into two subsets with equal sums
• Subset Sum Problem (coming soon)
• Given a set of positive integers, and a value sum, determine if there is a subset of the given set with sum equal to given sum.
• Coin Change Problem (Total number of ways to get the denomination of coins, coming soon)
• Given an unlimited supply of coins of given denominations, find the total number of distinct ways to get a desired change.
• Total possible solutions to linear equation of k variables (coming soon)
• Given a linear equation of k variables, count the total number of possible solutions of it.
• Find Probability that a Drunkard doesn't fall off a cliff (Kids, do not try this at home)
• Given a linear space representing the distance from a cliff, and providing you know the starting distance of the drunkard from the cliff, and his tendency to go towards the cliff p and away from
the cliff 1-p, calculate the probability of his survival.
• Many more...
Dynamic programming is a tool that can save us a lot of computational time in exchange for a bigger space complexity, granted some of them only go halfway (a matrix is needed for memoization, but an
ever-changing array is used).
This highly depends on the type of system you're working on, if CPU time is precious, you opt for a memory-consuming solution, on the other hand, if your memory is limited, you opt for a more
time-consuming solution for a better time/space complexity ratio. | {"url":"https://stackabuse.com/dynamic-programming-in-java/","timestamp":"2024-11-04T20:33:28Z","content_type":"text/html","content_length":"153712","record_id":"<urn:uuid:57816a64-54a0-494d-8fa1-e5c8f731d4e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00513.warc.gz"} |
Calculus With Analytic Geometry
Calculus With Analytic Geometry
Calculus With Analytic Geometry
• ISBN 13:
• ISBN 10:
• Edition: 5th
• Format: Hardcover
• Copyright: 01/01/1995
• Publisher: WILEY JOHN & SONS INC
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The aim of this major revision is to create a contemporary text which incorporates the best features of calculus reform yet preserves the main structure of an established and well-tested calculus
course. The multivariate calculus material is completely rewritten to include the concept of a vector field and focuses on major physics and engineering applications of vector analysis. Covers such
new topics as Jacobians, Kepler's laws, conics in polar coordinates and parametric representation of surfaces. Contains expanded use of calculator computations and numerous exercises. | {"url":"https://www.knetbooks.com/calculus-analytic-geometry-5th-anton/bk/9780471594956","timestamp":"2024-11-11T13:30:43Z","content_type":"application/xhtml+xml","content_length":"88403","record_id":"<urn:uuid:6f3ce9ec-e7fb-41c7-9f54-5f20d20336a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00416.warc.gz"} |
Optimization Research Based on Elastic Matrix in the Big Data Perspective
Proceedings of the 2nd International Conference on Mathematical Statistics and Economic Analysis, MSEA 2023, May 26–28, 2023, Nanjing, China
Research Article
Optimization Research Based on Elastic Matrix in the Big Data Perspective
• @INPROCEEDINGS{10.4108/eai.26-5-2023.2334375,
author={Zhiying Hu},
title={Optimization Research Based on Elastic Matrix in the Big Data Perspective},
proceedings={Proceedings of the 2nd International Conference on Mathematical Statistics and Economic Analysis, MSEA 2023, May 26--28, 2023, Nanjing, China},
keywords={big data; hesse matrix; elastic matrix; optimization},
• Zhiying Hu
Year: 2023
Optimization Research Based on Elastic Matrix in the Big Data Perspective
DOI: 10.4108/eai.26-5-2023.2334375
In today's era of big data, the basic idea of using mathematical methods to solve practical problems is to analyze and process the data, establish a mathematical model to solve the problem, find out
the functional relationship of the problem, and then use specific mathematical tools to find the corresponding solution. the basic idea of using mathematical methods to solve practical problems is to
analyze and process the data, establish a mathematical model to solve the problem, find out the functional relationship of the problem, and then use specific mathematical tools to find the
corresponding solution.Calculus has a wide range of applications in economics and can solve many economic problems quickly and accurately. Based on calculus, this paper uses Hesse matrix and elastic
matrix as tools respectively to solve the optimization of economic problems from different aspects in the form of examples, which has certain theoretical and practical value.
Copyright © 2023–2024 EAI | {"url":"https://eudl.eu/doi/10.4108/eai.26-5-2023.2334375","timestamp":"2024-11-05T22:14:32Z","content_type":"text/html","content_length":"9407","record_id":"<urn:uuid:6367e966-76a5-49b9-aa39-751ffb287453>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00319.warc.gz"} |
Statistics Questions with Solutions
Question 20
Views: 5,846
The following table gives the distribution of the life time of neon lamps: Find the median life time of a lamp.
Question 19
Views: 5,987
The lengths of leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table:Find the median length of the leaves.
Question 18
Views: 6,273
A life insurance agent found the following data for distribution of ages of policy holders. Calculate the median age, if policies are only given to person having age years onwards but less than
Question 17
Views: 5,536
If the median of the distribution given below is , find the values of and .
Question 16
Views: 5,964
The following frequency distribution gives the monthly consumption of electricity of consumers of a locality. Find the median, mean and mode of the data and compare them.
Question 15
Views: 6,269
A student noted the number of cars passing through a spot on a road for 100 periods each of 3 utes and summarised it in the table given below. Find the mode of the data:
Question 14
Views: 5,891
The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches.
Find the mode of the data.
Question 13
Views: 5,460
The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures.
Question 12
Views: 5,617
The following data gives the distribution of total monthly household expenditure of families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure:
Question 11
Views: 5,660
The following data gives the information on the observed lifetimes (in hours) of electrical components:
Detere the modal lifetimes of the components.
Ask my first question
Question 10
Views: 6,227
The following table shows the ages of the patients admitted in a hospital during a year.
Find the mode and the mean of the data given above. Compare and interpret the measures of central tendency.
Question 9
Views: 6,267
The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
Question 8
Views: 5,674
A class teacher has the following absentee record of students of a class for the whole term. Find the mean number of days a student was absent.
Question 7
Views: 5,523
To find out the concentration of in the air (in parts per million, i.e., ppm), the data was collected for localities in a certain city and is presented below:
Find the mean concentration of in the air.
Question 6
Views: 6,180
The table below shows the daily expenditure on food of households in a locality.
Find the mean daily expenditure on food by a suitable method.
Question 5
Views: 5,791
In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of
If mean number of mangoes kept in a pocket box. Which method of finding the mean did you choose?
Question 4
Views: 5,953
Thirty women were exaed in a hospital by a doctor and the number of heart beats per ute were recorded and summarized as follows. Find the mean heart beats per ute for these women, choosing a suitable
Question 3
Views: 6,225
The following distribution shows the daily pocket allowance of children of a locality, The mean pocket allowance is Rs.18. Find the missing frequency f.
Question 2
Views: 6,554
Consider the following distribution of daily wages of 50 workers of a factory.
Find the mean daily wages of the workers of the factory by using an appropriate method.
Question 1
Views: 6,324
A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in houses in a locality.
Find the mean number of plants per house.
Which method did you use for finding the mean, and why?
Get instant study help from an expert tutor 24/7
Download Filo | {"url":"https://askfilo.com/questions/class-10/math/statistics","timestamp":"2024-11-06T08:20:16Z","content_type":"text/html","content_length":"776092","record_id":"<urn:uuid:eab6b8c2-a719-4581-96bd-32a69ee7e640>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00422.warc.gz"} |
Solve equations, graphs, and handwritten sums with Math Notes | iOS 18 Guide
• Calculator can solve math problems with graphs and variables
• All your calculations are stored in the Notes app
• With Apple Pencil you can even solve handwritten equations
Apple always said it wouldn’t bring the Calculator app to the iPad until it could do something special with it, and this might just be it. After 15 years, there’s finally a Calculator app in iPadOS
18 – and a beefed up Calculator experience for iPhone too. Chief among the new features is the impressive Math Notes, which allows users to solve equations, create graphs, and more.
This feature is available as part of the Apple Intelligence Beta in iOS 18.1, and will require an AI-compatible device.
Solving math
To access the feature, open the Calculator app and press the calculator button in the lower left corner. Choose Math Notes.
Tap the New Note button in the corner to get started. Type out a sum and the app will automatically give you the answer, updating as you add more steps. Tap the return key to insert an equals sign
and commit the answer. Simple.
Use variables
Try declaring a variable, like “y = 12” or “rent = $1750”. You can then use these figures on subsequent lines, for example to calculate yearly expenses with “y x rent”. Change the variables and the
figures will magically update anywhere they’re referenced across the note. It’s really handy for quick expense calculations, working out tips, doing homework, or keeping track of changing data.
Instant graphs
You can even mock up graphs based on your equations. Type an equation that defines both axes, some variation of y = x. Then tap the equals button and look for Create Graph from the context menu. Once
created, you can tap the graph for a few settings. Use the nodes around the edges to resize it, and pinch to adjust the scale. There’s a pop-up menu that will allow you to customize colors and so on,
Take note
All the Math Notes you create will be automatically saved to the regular Notes app, and it’s worth noting you can actually kick off a calculation session from there as well. Create a new note and
start typing a sum – once you add an equals sign, the app will catch on and switch to a Math Note.
Handwritten sums
This is all well and good on iPhone. But Math Notes gets even more magical when combined with iPad and Apple Pencil. Here, users can scribble down mathematical expressions and see them solved in
their own handwriting. You can use all the mathematical expressions you’d expect, or draw a line under a list of numbers to get their total.
You’ll need to be relatively neat, else you’ll see a red border indicating the app doesn’t understand what you’ve written. If this happens, use the eraser or undo button and try again. But when it
works, it feels fantastic to jot down numbers as if on a notepad and see the answers appear as you write. | {"url":"https://www.tapsmart.com/tips-and-tricks/math-notes/","timestamp":"2024-11-10T05:07:07Z","content_type":"text/html","content_length":"57569","record_id":"<urn:uuid:a3312c5d-73b5-44c5-a0a1-efbde90f2cb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00568.warc.gz"} |
July 2021
Lets walk through an implementation of huffman encoding and decoding in Python. For the purposes of demonstrating key ideas, I’m going to just treat bits as plaintext strings just to keep things
simple. While this means that the output isn’t truly compressed as bytes, you will hopefully take away a deeper understanding of how it works under the hood.
Theoretical Overview
Huffman encoding works by exploiting the unequal distribution of character occurrences in text. Rather than encoding every single character with the same number of bites, it encodes characters that
occur more frequently with smaller number of bits and those that occur less frequently with greater number of bits.
For example, lets say we have the text abc.
Without compression, each character in abc will take up a fixed number of bits – lets say a byte (8 bits) per character using an ASCII character set. That’s 3 bytes for 3 characters or 24 bits total.
If we used a variable length encoding, we can instead use as many bits as we need to identify a character. To illustrate this concept, lets map each character to a variable number of bits like so:
a = 0
b = 10
c = 110
Then abc will be 010110 which is only 6 bits. That’s 18 bits (75%) less compared to the uncompressed version!
But here’s the catch: we need to make sure that these codes are prefix free, meaning that no code in our set is a prefix of another code. Why? This is best understood with an example. Lets add
another character, d, to our previous set.
a = 0
b = 01
c = 110
d = 10
Now consider if we wanted to encode acb, we would have 011001. But upon decoding it, it can be misinterpreted as bdb (01 – 10 – 01). That’s because the bits for b contains the prefix of a – so if you
read from left to right, you can either read 0 and stop (which gives you a) or read both 0 and 1 (which gives you b). When do you stop reading?
Unless we introduce a delimiter into our output bit stream, we can’t tell where the bits of one character ends and another starts. The only way to tell without a delimiter is to have codes that
introduce no ambiguity, and you can accomplish that by ensuring that the codes are prefix free.
This presents two additional challenges that the creator of the huffman encoding solved:
1. How do we generate these prefix free codes?
2. How do we generate optimal prefix free codes such that we are able to assign shorter codes to higher frequency characters?
The prefix free codes are created by constructing a binary trie. The edges in the tree represent 0’s and 1’s and the leaf nodes of this binary tree represent a unique character in a text. Therefore,
the paths represent the code for the character at the leaf. Since the characters are at the leaf nodes, all the paths to those nodes are unique and non-overlapping, making the codes prefix free. To
attain optimatily, the trie is constructed bottom up, starting with characters that occur the least often so that the eventual codes (made up of paths from the root of the trie to leaf nodes) are
shortest for those that occur the most often.
Implementation Overview
Here’s an overview of both compression and decompression steps:
1. read the text and figure out character frequency
2. use frequency to build trie – this generates the bit sequence in the form of trie paths for each character
3. generate a code table using trie – this lets us find a corresponding bit sequence code for a character
4. encode our text using table to produce a bit stream
5. encode our trie as bits. this will be used by decoder
6. write both to a file
1. read the trie bits portion (header) to re-construct the trie. we’ll need this to decode the encoded text
2. read the body / text bits portion (this is the encoded form of the actual value we’re trying to get)
The Trie Node
We’ll be using this to construct our trie. this will be referenced throughout the implementation.
class Node:
def __init__(self, char="", left=None, right=None, freq=0):
self.char = char
self.left = left
self.right = right
self.freq = freq
def to_binary(self):
return "{:08b}".format(ord(self.char))
def is_leaf(self):
return (self.left is None and self.right is None)
# necessary for heapq comparisons
# heapq falls back on object comparison when priority keys are equal
def __lt__(a, b):
return a.char < b.char
def __eq__(self, other):
if isinstance(other, Node):
return (
(self.char == other.char) and
(self.left == other.left) and
(self.right == other.right)
return False
def __repr__(self):
return "None" if self.char is None else self.char
Code language: HTML, XML (xml)
Compression Process
This is the main method for compression – we’re encoding the text and then we’re including some header metadata for the decoder. The essense of the header metadata is the serialized trie that we
constructed for the purposes of encoding.
def compress(text):
trie_tree = build_trie(text)
table = build_code_table(trie_tree)
trie_bits = serialize_trie_to_binary(trie_tree)
header = "{:016b}{}".format(len(trie_bits), trie_bits)
body = encode_text(table, text)
return header + body
Code language: JavaScript (javascript)
The following method uses a min heap to ensure that the most frequently occuring characters (via the freq attribute) are included in our trie structure last.
def build_trie(text):
from collections import Counter
from heapq import heappush, heappop
char_count = Counter(text)
queue = []
for char, freq in char_count.items():
node = Node(char=char, freq=freq)
heappush(queue, (node.freq, node))
while len(queue) > 1:
freq1, node1 = heappop(queue)
freq2, node2 = heappop(queue)
parent_node = Node(
freq=freq1 + freq2
heappush(queue, (parent_node.freq, parent_node))
freq, root_node = heappop(queue)
return root_node
Code language: JavaScript (javascript)
This method constructs our character to code hash table. Our trie lets using decode an encoded stream by allowing us to follow the binary node paths to the characters using bit values in a stream.
However, we need to create a character to code mapping in order for our constructed trie to be useful in the encoding process. Otherwise, we would need to scan our entire trie using either DFS or BFS
searching for a target character (for every character we want to encode).
def build_code_table(node):
table = {}
def map_char_to_code(node, value):
if node.is_leaf():
table[node.char] = value
map_char_to_code(node.left, value + "0")
map_char_to_code(node.right, value + "1")
map_char_to_code(node, "")
return table
Code language: JavaScript (javascript)
In order for a decoder to decode our encoded text, it needs to know the character-to-code mapping we used so this method serializes the trie used in the encoding into bits. It uses a pre-order
traversal to encode our trie. If it’s a non-leaf node, we prefix the output with a zero. Otherwise, we prefix it with a 1 followed by the actual bits representing the character.
def serialize_trie_to_binary(node):
if not node:
return ""
if node.is_leaf():
return "1" + node.to_binary()
return "0" + serialize_trie_to_binary(node.left) + serialize_trie_to_binary(node.right)
Code language: JavaScript (javascript)
This method makes use of our character-to-code table to convert characters into bits. This represents our compressed text!
def encode_text(table, text):
output = ""
for x in text:
output += table[x]
return output
Code language: JavaScript (javascript)
Decompression Process
Here’s the main method for decompression. It essentially re-constructs the trie in memory used the bits in the input representing the trie. Then it uses that in-memory trie to decode the bits of the
input that represent our encoded text.
def decompress(bit_string):
trie_size = int(f"{bit_string[0:16]}", 2)
trie_range_end = 16 + trie_size
trie = deserialize_binary_trie(bit_string[16:trie_range_end])
body = bit_string[trie_range_end:]
return decode_text(trie, body)
Code language: JavaScript (javascript)
This function does the reverse of serialize_trie_to_binary. The recursion here takes advantage of the fact that 1 bits are leafs of our trie, therefore it can be used as a base case to continue
de-serializing the next trie path. The curr_pos is used in this function to act as a pointer into our current read position so we know when to start and stop reading input.
def deserialize_binary_trie(bits):
def read_bits(curr_pos):
if curr_pos >= len(bits):
return None, curr_pos
bit = bits[curr_pos]
if bit == "1":
char_range_left = curr_pos+1
char_range_right = char_range_left + 8
char_bits = bits[char_range_left:char_range_right]
return Node(
char=chr(int(char_bits, 2))
), char_range_right
left_node, pos = read_bits(curr_pos + 1)
right_node, pos = read_bits(pos)
return Node(
left = left_node,
right = right_node
), pos
node, pos = read_bits(0)
return node
Code language: JavaScript (javascript)
Finally, with our trie object on hand, this function follows the bits of the encoded text using the trie to find the characters.
def decode_text(node, data):
out = ""
root = node
curr_node = root
for bit in data:
if bit == "0":
curr_node = curr_node.left
curr_node = curr_node.right
if curr_node.is_leaf():
out += curr_node.char
curr_node = root
return out
Code language: PHP (php)
That completes the overview of this basic python representation of the huffman algorithm. In practice, some implementations may used pre-existing code tables rather than generating them on the fly as
we did here. For example, if you need fast encoding and know about average frequencies of the text you’re encoding, you may not want to be constructing a new trie on every encode operation.
Here’s a couple of resources I used in writing this implementation – I highly encourage you to check them out to understand huffman in even greater depth. | {"url":"https://linisnil.com/posts/2021/07/","timestamp":"2024-11-14T05:31:07Z","content_type":"text/html","content_length":"49734","record_id":"<urn:uuid:94b79f22-17c7-4f2f-bb76-36ca28be5393>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00380.warc.gz"} |
ECSE4540 Homework#1 solution
1. (30 points.) Create the following 500×500 8-bit images in Matlab; include a picture and the Matlab commands you used in your writeup. Note that each of these only requires 1-3 lines of Matlab
code, no for loops, and no if statements! Use the image (not Cartesian) coordinate system; that is, x indexes the rows from top to bottom and y the columns from left to right. Use imshow(im,[0 255])
for grayscale image display. Use uint8(im) for commands that don’t produce an integer output. You will be penalized if your codeistoocomplicated.
(a) Agrayscaleimageofconstantintensity60 (b) Agrayscaleimagewithalternatingblackandwhiteverticalstripes,eachofwhichis2pixelswide (c)
Agrayscaleimagewherethelefthalfhasintensity32andtherighthalfhasintensity200 (d) Agrayscaleimagewitharampintensitydistribution,describedby I(x,y)=x/2 (e)
AgrayscaleimagewithaGaussianintensitydistributioncenteredat(64,64),describedby I(x,y)=255exp−³(x−64)2+(y−64)2 2002 ´(f) A color image where the upper left quadrant is white, the lower left quadrant
is magenta, the upper rightquadrantiscyan,andthelowerrightquadrantisblue
2. (20points.) (a) A 4K HDR Netflix video stream transmits frames of size 4096×2160 at 30 frames per second with 10 bits per color channel. Compute the number of bits it would take to stream the
season premiere of StrangerThings2,a48-minuteepisode. (b) Suppose your internet provider caps your download rate at 30 Mbps. Based on your answer in part (a), compute a lower bound on the compression
ratio that must be achieved by Netflix’s video compressionalgorithminordertodeliverthestream.
3. (20 points.) Consider the image in Problem 1e, which has 256 gray levels. Create quantized versions of this image with 128, 64, 32, and 16 gray levels (this involves the round command). At what
point can you visuallydetectfalsecontouring?
4. (15 points.) We model the camera in the iPhone 8 as a pinhole placed at (0 m, 0 m, 0 m) in world coordinates. Suppose the focal length is 28mm and a 6.29×5.21 mm CCD array that has 4608 pixels in
the x direction and 2592 pixels in the y direction is placed at the focal plane. What image pixel does the world coordinate (-0.05 m, 0.1 m, 2.5 m) project to, assuming (0,0) is at the upper left
corner (not the center) of thearray?
5. (15points.) ThecolorofRey’slightsaberisgivenbytheHTMLRGBcode #A8D0E8. Representthiscoloras:
(a) AnRGBtripletwhereeachvalueisintherange[0,1]. (b) ACMYtripletwhereeachvalueisintherange[0,1]. (c) An HSI (hue-saturation-intensity) triplet where each value is in the range [0,1]. See Gonzalez and
Woods (Section 6.2 3rd ed., Section 7.2 4th ed.) for how to do the conversion and show your work! Note that Gonzalez and Woods define the intensity component slightly differently than Photoshop
andotherpaintprograms,andthatHSIisnotthesameasHSV! (d) aninterpretationoftheHSItripletinwords(e.g.,similarto“adeep,darkred”.) | {"url":"https://jarviscodinghub.com/product/ecse4540-homework1-solution/","timestamp":"2024-11-02T02:26:15Z","content_type":"text/html","content_length":"102847","record_id":"<urn:uuid:c16a159a-34c0-4991-8931-78a1d233377a>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00576.warc.gz"} |
Zero absolute vorticity plane Couette flow as an hydrodynamic representation of quantum energy states under perpendicular magnetic field
Here we extend the Madelung transformation of the Schrödinger equation into a fluid-like form to include the influence of an external electromagnetic field on a charged particle. The vorticity of the
Madelung fluid is then in the opposite direction to the imposed magnetic field and equal in magnitude to the cyclotron angular frequency. When the particle motion is confined to a plane,
perpendicular to an imposed magnetic field, the equivalent flow dynamics is that of zero absolute vorticity obtained in a quasi-two-dimensional rotating frame, where the cyclotron frequency plays a
role equivalent to that of the Coriolis frequency in a rotating frame. We show how the Landau levels and the extended modes in the integer quantum Hall effect are all mapped into such zero absolute
vorticity-like plane Couette flows, where the latter exhibit a geostrophic-like balance between the magnetic force and the gradients of the quantum (Bohm) potential and the electric force.
Funders Funder number
Center for Ocean Research in Hong Kong and Macau
RGC 16304021
Hong Kong University of Science and Technology
Qingdao National Laboratory for Marine Science and Technology
Dive into the research topics of 'Zero absolute vorticity plane Couette flow as an hydrodynamic representation of quantum energy states under perpendicular magnetic field'. Together they form a
unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/zero-absolute-vorticity-plane-couette-flow-as-an-hydrodynamic-rep","timestamp":"2024-11-13T01:32:31Z","content_type":"text/html","content_length":"51139","record_id":"<urn:uuid:0e131881-39e9-432f-8b87-40a8a2e21792>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00324.warc.gz"} |
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Stright Lines, January 2001, Vol. 4-1
The Official Newsletter of the IUP Mathematics Department
January, 2001___________________Volume 4, Issue 1
Welcome to another issue of Stright Lines. For any of you receiving this as a first issue and thinking those IUP Mathematics Faculty can't spell "straight", I remind you that the Mathematics
Department is located in Stright Hall.
I am sad to report that we have received no letters from graduates to include in this newsletter. We still hope to hear from you.
In the last issue Joe Kirchner recalled several humorous stories from his days at IUP. We hoped to get some more stories from alumni or retired faculty. Joe thought "it might be a little tough
getting funny stories from a bunch of math majors" and he seems to have been right since we have received no contributions. We still would welcome humorous stories about your days at IUP. Jim Reber,
IUP Graduates are Involved!
In one of the first issues of Stright Lines I extolled the professional activities of IUP graduates in mathematics education. IUP graduates continue to make a difference in education. For example,
check out the web page of the Mathematics Council of Western Pennsylvania at www.mcwp.org. It is maintained by David Taylor, an IUP graduate. David taught for 3.5 years in Maryland and then returned
to Western Pennsylvania to teach mathematics at South Fayette Township Jr.-Sr. High School. One of his responsibilities at the school has been the Cooperative Satellite Learning Project, a
cooperative effort among the school, NASA, Goddard Space Flight Center, and Allied Signal Technical Services Corporation. This fall David became Director of Information Technology at South Fayette.
This year from March 15 - 17, 2001, the 50th Annual Meeting of the Pennsylvania Council of Teachers of Mathematics (PCTM) will be held in Pittsburgh at Greentree's Radisson and Holiday Inn hotels.
IUP graduates are certainly prominent among committee chairs, presenters and presiders. Dave Depner is Co-chair of Local Arrangements and Susan Stonebraker is Chair of Meals and Functions. Many
alumni who have earned their degrees at IUP are sharing their knowledge and expertise by presenting programs. These alumni include Linda Brecht, Elaine Carbone, Patty Flach, Rhonda Fedyk-Foust, Nina
Girard, Bill Hadley, Jennifer Landsman, Peggy Lunardini, Majory Maher, Rita McMinn, Mary Lou Metz, Mary Lynn Raith, Shannon Relihan-Rieger, Cathy Schloemer, Eli Shaheen, Anita Smith, Kirstie Trump,
John Uccellini, and Mark Zelinskas. Many of the presenters are also presiding over sessions, as are Adrienne Kapisak, Dorothy Mullin, and nine student teachers. (I wonder who twisted the arms of the
latter!) Anyway, these student teachers deserve mention; they are Tracy Birchall, Leah Drane, Jessica Feerst, Melissa Luckey, Shawn Moorhead, Doug Murdoch, John Nelson, Denise Shade, and Jane
Shumaker. (If I have forgotten anyone, please advise me, and I will give you proper coverage in the next Stright Lines.)
If you are attending the PCTM meeting, please look on the message board by the registration table for IUP announcements. We will try to plan some get-togethers.
You may be interested in knowing where some of our recent graduates have taken positions. We are now fortunate to have many of our graduates getting their first positions in Pennsylvania. Recent
graduates who are now teaching in Pennsylvania are as follows: Shelly Huston, Shady Side Academy, Pittsburgh; Ralph Santilli, Butler; Matt Rodkey, Homer Center in Homer City; Jeff Ziegler, Pittsburgh
Public Schools; Brad Baker, Beaver; LeeAndrea McCullough, Quaker Valley, Sewickly; Karin Rabenold, Marion Center; and Lisa Sargent, Manheim Twp., Lancaster. Among those who have gone to other states
are Kim White, Concord, North Carolina; Janel Hartzok, Westminster, MD; and Joyce George, Ocean City, MD.
Periodically we get e-mail messages from former students who are recruiting mathematics teachers for their schools. Two of these have come from Chris Clark at Manassas Park, Virginia and Chris
O'Rourke at McLean, Virginia. Although our mathematics department does not operate a placement bureau, we are always happy to share job postings. If you are searching for a job or trying to fill a
position, please forward information to us.
Ann Massey asmassey@grove.iup.edu
News about Graduates
Dr. Buriok received a note from John A. Miller (J.Miller@connect.xerox.com) who graduated from IUP in 1977 and is now Managing Principal, Document Management and Imaging, with Xerox Connect in
Pittsburgh. John noted that he gave one of the commencement speeches in the department. He also mentioned that Xerox Connect is growing quickly and hires many college seniors.
Mark Rayha (Class of 1993) resigned his position as a Business Systems Analyst with Citistreet (formally known as the Copeland Companies) and accepted a position as a Lead Systems Analyst at
Schering-Plough, a pharmaceutical company. His home email address is m.rayha@gte.net.
Tracie A. Moreland (Class of 1996) finished the Applied Statistics graduate program at Villanova University (while working full time). She is currently with Merck & Co. as a marketing analyst.
Dr. Rebecca Stoudt received a note from Aurele Houngbedji (amhst44+@pitt.edu) who graduated from IUP with an M.S. degree in August, 1996. Aurele graduated with his Ph.D. on April 30, 2000. He will be
working at Ohio Savings Bank in the Capital Markets Department as a Quantitative Analyst. The position is related directly to Aurele's research, which is stochastic modeling in Finance. He will be
doing quantitative research, financial data analysis, derivatives trading and risk management.
Cindy Venturino Biedrycki wrote Dr. Massey from Prince William County in Virginia where she and Stephanie Clifton are teaching. Both finished their masters degrees in Curriculum and Instruction at
Virginia Tech last August.
Alumni Bulletin Board
Available at our Web site
If you go to the IUP Mathematics Department web site, http://www.ma.iup.edu/, you can leave a message on the Alumni Bulletin Board. One recent posting is from Kirstie Trump (MDteach4u2@aol.com) on 09
/27/00 :
Hello everyone! I am currently teaching 8th grade math and algebra in Carroll County, Maryland. IUP prepared me well for teaching and I am grateful to all of my professors and classmates for always
supporting me. Carroll County is always looking for good math teachers and loves to recruit IUP graduates. Please email me if you would like more info!
Mullin Receives Award
Last year Dorothy Mullin received the Award for Outstanding Contributions to MCWP (Mathematics Council of Western Pennsylvania). Dorothy has served as a member of the MCWP board and chairperson of
many committees for PCTM and NCTM regional meetings as well as for MCWP meetings. Always she has been willing to give of herself to make professional events successful.
Dorothy received her bachelor's degree in mathematics education from IUP and both her masters in mathematics and her doctorate in mathematics education at the University of Pittsburgh. She taught at
Penn State, McKeesport for more than 20 years. We were fortunate to have Dorothy in our department when she returned to her Alma Mater as a temporary instructor for a year.
Dale Shafer died in Florida on March 21, 1999. His master's degree was from Columbia University and his doctorate of education degree was from the University of Oklahoma. He taught for two years in
the Oley Valley School District, for three years at Slippery Rock College, and for 30 years in the IUP Mathematics Department from 1964 - 1994. He was the executive secretary of the School, Science
and Math Association for 10 years. At IUP he often taught statistics courses.
Richard "Dick" Wolfe died in South Carolina on January 24, 2000 from injuries suffered in a traffic accident. His master's and doctorate degrees were from the University of Illinois in
Champaign-Urbana. He taught at Waynesboro High School and then here at IUP from 1967 until his retirement in 1991. He taught mathematics education courses and supervised numerous student teachers
over the years.
I. "Ike" Leonard Stright died on February 9, 2000. He received his Ph.D. degree from Case Western Reserve University. He taught mathematics in high school and at Baldwin Wallace College and Northern
Michigan University. He became Professor of Mathematics at IUP in 1947 and was Dean of the Graduate School from 1957 until 1971. The building which houses the Mathematics Department, and hence this
newsletter, is named for Dr. Stright.
Word from Daniel Griffith,
Class of 1970
Dr. Daniel A. Griffith, now Professor of Geography at Syracuse University, sent us two recent publications. One article appeared in the Journal of Statistical Planning and Inference. He noted that
his IUP mathematics education prepared him very well for earning an M.S. in statistics (1985). The other article appeared in Linear Algebra and Its Applications. This article draws upon his
undergraduate and graduate work in mathematics at IUP (B.S., 1970; graduate work 1970-72). Daniel observes that training by three of his IUP instructors - Mr. D. McBride (retired), Dr. J. Hoyt
(retired) and Mr. C. Maderer - helped make this second article possible. In closing he notes that he continues to appreciate the mathematics training he receive at IUP that has enabled him to both
publish in statistics journals and contribute to the linear algebra literature.
The SPIRAL Project
By Rebecca A. Stoudt and Roberta M. Eddy
SPIRAL (Science/Mathematics/ Technology Preparation Involving Real-world Active Learning) is a teacher professional development project funded by the Eisenhower Professional Development Program and
IUP matching funds. SPIRAL is a multi-disciplinary program that SPIRALs concepts from K through 12 and out across the disciplines. The disciplines involved are Mathematics, Biology, Chemistry,
Geoscience, and Physics. The use of a wide variety of technology is woven throughout the program.
The project is co-directed by Rebecca Stoudt (Mathematics) and Roberta Eddy (Chemistry). Other SPIRAL faculty are Janet Walker and Gary Stoudt (Mathematics), Terry Peard (Biology), John Wood
(Chemistry), Connie Sutton (Geoscience), and Norman Gaggini and Ken Hershman (Physics). Kent Jackson (Special Education), Mary Ann Rafoth (Educational and School Psychology), and Len Lehman
(Curriculum Consultant) complete the SPIRAL staff.
The central focus of this project is an 8-day, intensive, residential, summer institute (SI) where preservice teachers, inservice teachers, and administrators come together to learn instructional
strategies and to conduct field-tested activities consistent with state and national standards. The SI emphasizes two SPIRAL models, LIGHT and ECOSYSTEM. An awareness of special needs students and
diverse learning styles in science and mathematics is stressed throughout the SI. Furthermore the incorporation of SPIRAL activities into the school district's curricula is facilitated by two SI
synthesis and curriculum incorporation sessions. SPIRAL also includes ongoing professional development activities such as follow-up workshops (fall and spring), development of portfolios, and a joint
ARIN/SPIRAL Academic Alliance for educators of mathematics and science.
A 5-member SPIRAL school district team ideally consists of an administrator (can be a principal, assistant principal, curriculum director, or head of department), a special needs or learning support
instructor, and three K-12 teachers of mathematics and science (specifically an elementary teacher, a middle school teacher, and a high school teacher). When each team arrives at the SI, it is linked
with two IUP preservice teachers, one elementary and one secondary. The preservice teachers are majoring or concentrating in mathematics and/or science.
SPIRAL participants use standard-based models of teaching that emphasize the inquiry approach and cooperative learning. As a result, the participants' content knowledge in all SPIRAL disciplines has
increased significantly in every SI since the beginning of SPIRAL (1998). This significant increase was measured by the pre/post-test scores of 93 inservice and 51 preservice teachers. In fact, for
each SI, the post-test score mean was at least double the pre-test score mean.
The Eisenhower Professional Development Program has awarded SPIRAL approximately $597,000 since the projects beginning. These awards have been matched with approximately $349,000 from IUP (College of
Natural Sciences and Mathematics, College of Education, Graduate School and Research), Texas Instruments, and ARIN IU-28. Hence, SPIRAL is almost a $1 million project to date.
A large portion of the grant money is spent on supplies and materials for the teams to take back to their home schools so that they can easily implement SPIRAL activities in their curricula. Each
team receives over $4000 of equipment which includes but is not limited to: (1) TI-83 Plus calculator/viewscreen; (2) CBL2 kit with set of probes--biology gas pressure sensor, dissolved oxygen,
colorimeter, pH system; (3) CBR system; (4) digital camera; (5) various CD-ROMs; (6) aquatic kick net; (7) Silica Gel GF thin layer chromatography plates; (8) UV lamp; (9) HACH Color Cube kits (iron,
nitrogen-nitrate, phosphorous orthophosphate); (10) HACH Color Disc Kits (iron, nitrogen-nitrate, phosphorous orthophosphate); (11) light, image, shadow kits; (12) topographic and geologic maps; (13)
Guide Book to Rocks and Soil; (14) rock/mineral set; (15) fossil set;
(continued on page 4)
(continued from page 3)
(16) pocket gem field magnifier; (17) pH tester; (18) fluorescent experiment kit, (19) lightsticks; (20) cool blue light and goofy glowing gel kits; (21) color filters; (22) mirror set; (23) soil
percolation kit; (24) bar magnet set; (25) student clinometer; (26) refracting telescope kit; (27) Ecneics kit; (28) solar system floor puzzle; (29) star chart; (30) solar system/planet poster; (31)
spectrum analysis chart; (32) spectroscope and (33) numerous activity books.
IUP's Curriculum Through the Years, Part 3
by Gary Stoudt
In the last two issues we looked at the opening of the Indiana Seminary and Normal School and the State Normal School of the Ninth District, in Indiana, Pennsylvania. One of the texts used in the
curriculum of the Indiana Seminary and Normal School was Ray's Algebra. Thanks to Dr. Ed Donley who loaned me a copy of the book, I can tell you something about the book in order to help you get a
feel for what the mathematical studies at the Normal School where like. Unless otherwise stated, quotes in this article are from this book.
Dr. Donley's copy is of the 1875 edition, so it is most likely that this was the text used at the time of the school's founding in 1875. The full title of the book is Elements of Algebra for
Colleges, Schools, and Private Students, Second Book. The author is Joseph Ray, M.D., professor of mathematics at Woodward College. Woodward College was located in Cincinnati but no longer exists.
The publisher was Wilson, Hinkle and Co. in Cincinnati. There are very few diagrams in the text, although it is typeset using modern notation. According to Miami Valley Vignettes, by George C. Crout:
Ray wrote a series of texts which made arithmetic understandable to elementary pupils. Joseph Ray was a professor at Woodward College, later becoming its president. In addition to his work at the
Cincinnati college, he was a state leader in education. Ray compiled a set of three texts in mathematics, taking the student from simple processes to advanced ones. His third book was used in both
high school and colleges. The series was published in Cincinnati. Even after his death in 1865, the Ray textbook series dominated the textbook field in mathematics until the early 1900's.
The text has a wonderful Preface, part of which is reproduced here.
Algebra is justly regarded one of the most interesting and useful branches of education, and an acquaintance with it is now sought by all who advance beyond the more common elements. To those who
would know Mathematics, a knowledge not merely of its elementary principles, but also of its higher parts, is essential; while no one can lay claim to that discipline of mind which education confers,
who is not familiar with the logic of algebra.
It is both a demonstrative and a practical science - a system of truths and reasoning, from which is derived a collection of Rules that may be used in the solution of an endless variety of problems,
not only interesting to the student, but many of which are of the highest possible utility in the arts of life.
Those were the days! This sentiment is still alive today in the current debate concerning "algebra for all." Of course, we also still make the claim that algebra is "useful."
The text starts with definitions, notation, and the fundamental rules of arithmetic, including operations with polynomials, all in Chapter 1. The description of operations with monomials is much like
a modern text with the exception of the use of the vinculum (a horizontal bar) along with parentheses. For example, There is no mention of FOIL, but there is an interesting method of multiplying and
dividing polynomials called the "method of detached coefficients." Ray states "this method is applicable where the powers of the same letter increase or decrease regularly." For example, to multiply
by :
1 - 3 + 0 + 1
1 + 0 - 1
1 - 3 + 0 + 1
- 1 + 3 - 0 - 1
1 -3 -1 + 4 -0 -1
the answer is .
In the next two chapters we move into factoring (factoring of quadratic trinomials is done "by inspection") and working with algebraic fractions, which we would call rational expressions. This is all
done in the fairly standard "modern" way. The lone exception is the work done on converting fractions into infinite series. For example, ( 1 - x ) / (1 + x ) is written as an infinite series using
long division. (continued on page 5)
(continued from page 4)
In Chapters 4 and 5 Ray moves into solving equations, starting with the "simple equation" (linear equation). This is done in the usual way, but Ray includes some interesting word problems, as in this
example: "A smuggler had a quantity of brandy, which he expected would sell for 198 shillings; after he had sold 10 gallons, a revenue officer seized one third of the remainder, inconsequence of
which, what he sold brought him only 162 shillings. Required the number of gallons he had, and the price per gallon." There are also included many problems that we would recognize (Plus a change...).
Classic problems such as division of items ("a sum of money is to be divided among five persons so that ..."); work problems ("If A does a piece of work in 10 days..."); traveling problems ("There
are two places, 154 miles distant from each other, from which two persons, A and B, set out at the same instant..."); number problems ("There are three numbers whose sum is 187..."); and purchasing
problems ("If 10 apples cost a cent, and 25 pears cost 2 cents, ..."). Ray then discusses systems of two linear equations (no solution by graphing, though) and literal equations.
We now move on to powers and roots in Chapter 6. Interestingly, the binomial theorem is stated (as Newton's Theorem) but Pascal's triangle is nowhere to be found. Ray shows how to find square roots
and cube roots of numbers and polynomials. (For the younger folks out there, send me an email if you want to know the method!) The sections that follow deal with radicals, including fractional
exponents and "imaginary, or impossible quantities." The chapter ends with a section on simple inequalities.
The solution of quadratic equations begins in Chapter 7. First we solve the pure quadratic, which "contains only the second power of the unknown quantity, and known terms" and then the "affected
quadratic," which "contains the first and second power of the unknown quantity, and known terms." The affected quadratic is first solved by completing the square. Next the affected quadratic is
solved by the "Hindoo [sic] Method." This method was known to Brahmagupta (b. 598) and Ray describes it much as Brahmagupta did, except Ray uses modern notation.
1st. Reduce the equation to the form 2nd. Multiply both sides by four times the coefficient of .
3rd. Add the square of the coefficient of x to each side, extract the square root, and finish the solution.
As an example, consider .
Multiply both sides by 8: .
Add 25 to both sides: .
Extract the root: 4x - 5 = , etc.
Next in the text is a discussion of the theory of quadratic equations, a look at equations that are quadratic in form, theorems concerning the roots of quadratic equations, theorems concerning
imaginary roots, and so on. The chapter ends with a discussion of the solution of two simultaneous quadratic equations in two variables.
Chapter 8 is concerned with ratios, proportion and progressions. Included here is a discussion of the mean proportion of two numbers, alternation, inversion, and composition of proportions, harmonic
proportions, arithmetical, geometrical, and harmonic progressions, including the sums of arithmetic and geometric series. In Chapter 9 Ray discusses permutations, combinations, and the binomial
theorem. The notation for combinations is not used. Instead Ck is used, where it is assumed n is known.
Infinite series is the topic covered in Chapter 10, along with the general Binomial theorem and decomposition of fractions into partial fractions, which Ray calls "decomposition of rational
fractions." This topic is in this chapter because of its relationship to the technique of indeterminate coefficients for finding the terms of a series expansion. Work with series is done in the
spirit of Newton: treating infinite sums as finite sums with respect to performing algebraic operations on them. Here is an example.
Thus it is required to develop 1/ (3x - x2) and we assume the series to be , etc., we have after clearing of fractions [multiply both sides by 3x - x2] ,
, etc.
from which, by equation the coefficients of the same powers of x, 1 = 0, 3A = 0, etc.
The first equation, 1 = 0, being absurd, we infer that the expression cannot be developed under the assumed form. But, Putting ,etc., clearing of fractions, and equating the coefficients of the like
powers of x, we find
, , , , etc. Hence Or, since the division of 1 by the first term of the denominator gives , or 3 x -1 we ought to have assumed , etc.
(continued on page 6)
(continued from page 5)
Work done with series is also done in the spirit of Leibniz, using the so-called "differential method of series." This method is based on sequences of differences.
Let the series [Ray's term] be a, b, c, d, e,... ; then the respective orders of differences are,
first order b - a, c - b, d - c, e - d, ...
second order c - 2b + a, d - 2c + b, e - 2d + c, ...
third order d - 3c + 3b - a, e - 3d +3c - b, ...
fourth order e - 4d + 6c - 4b + a, ....
If we denote the first terms of the 1st, 2nd, 3rd, 4th, etc., orders of differences by D1, D2, D3, D4, etc., and invert the order of the letters we have D1 = - a + b; D2 = a - 2b + c;
D3 = - a + 3b - 3c + d; D4 = a - 4b + 6c - 4d + e, etc. Here, the coefficients of a, b, c, d, etc., in the nth order of differences, are evidently those of the terms of a binomial raised to the nth
power; and their signs are alternately positive and negative.
From this the author shows how to find the nth term of a series a, b, c, d, e,... using differences:
D1 = - a + b; whence b = a + D1
D2 = a - 2b + c; whence c = a + 2D1+D2
D3 = - a + 3b - 3c + d; whence
d = a + 3D1+3D2+D3
D4 = a - 4b + 6c - 4d; whence
e = a + 4D1+6D2+4D3+D4.
This technique is then applied to counting the number of balls in triangular and rectangular piles of cannon balls. This is the only place in the book where illustrations appear; there are
illustrations of piles of cannon balls! The chapter concludes with a look at "recurring series," what we would call recursive sequences.
In Chapter 11 Ray discusses continued fractions, logarithms, exponential equations, interest, and annuities. In the logarithm sections, time is spent on computing common logarithms using a table of
logarithms. Next there is a brief section on the rules of single and double position. These are techniques for solving linear equations of the form ax + b = m. These techniques were known to the
ancient Egyptians and were used in medieval Europe under the name "regla falsi," or "false position." The technique involves making two guesses x1 and x2 and finding the differences e1 and e2 between
ax1 + b and m and ax2 + b and m. This section is placed here in the text because this technique is used to solve exponential equations of the form xx = a. An example is given, to solve xx = 100.
Begin by rewriting as x log x = 2.
First supposition Second supposition
x = 3.5; log x = .544068 x = 3.6; logx = .556303
x log x = 1.904238 x log x = 2.002690
a = 2 a = 2
error = -.095762 error = .002690
Diff results : diff assumed nos. : :
Error 2nd result : Its cor.
.098452 : 0.1 : : .002690 : 0.00273
Hence x = 3.6 - .00273 nearly.
The sections on interest and annuities are very similar to what is in books now, with the exception that all the formulas are derived using properties of series, instead of just being given.
In Chapter 12 the general theory of equations is discussed, including the relationship between the coefficients and roots of equations, the factor theorem, the Fundamental Theorem of Algebra,
Descartes' rule of signs, the transformation of equations, and Sturm's theorem. Chapter 13 ends the book with a discussion of numerical solutions of polynomial equations, including Horner's and
Newton's methods. Also included is Cardano's rule for solving cubics!
This is quite a text. We do not know how much of the book was covered in the course that used it. It is important to keep in mind that this course was required of all students at Indiana Normal.
I hope you enjoyed this look into the past. It would be interesting to learn how many of these topics were covered in later years. You can help by going through your old textbooks (or just going
through your memories) and dropping us a note. As always, let me know what you think and please feel free to get involved. Send (via email, FAX or U.S. Mail) what mathematics/education courses you
took, the professors' names, what textbooks you used, and when to:
Gary Stoudt
Department of Mathematics
Stright Hall
Indiana University of PA
Indiana, PA 15705
FAX (724) 357-7908
We will get to your era soon enough!
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Isogeometrical Analysis based Shape Optimization
DI Rainer Schneckenleitner
Nov. 28, 2017, 3:15 p.m. S2 416-1
Shape optimization problems arise in many different scientific and engineering areas e.g. mechanical engineering, electrical engineering or chemistry. For many practical problems the underlying
object of interest is represented by B-splines or NURBS curves due to computer-aided design software. Many properties of such objects of interest depend on the solution of a partial differential
equation (PDE). So far the B-Spline or NURBS based computer model is usually decomposed into finite elements for the analysis. Additionally, this has the consequence that usually the boundary of the
model has to be approximated with polygonal subdomains. In 2005, a new idea came up for such problems, called isogeometric analysis (IgA). The idea in IgA is that the domain for the analysis remains
the same as for the geometry of the object of interest constructed with some computer-aided design program. Although the finite element method (FEM) is a well established method for shape
optimization this new idea seems to be beneficial because on the one hand no conversion of the models is necessary, which can be computationally very costly. On the other hand, because there need not
be a conversion, we have an exact representation of the domain. In this thesis we will investigate IgA for shape optimization problems subject to PDEs.
We will show that the IgA approach has its justification in PDE constrained shape optimization processes. First we are going to investigate a linear model problem in IgA with a well established
standard algorithm and then we will apply a relatively new optimization algorithm to this linear model problem. Finally, we are going to put an electric motor into the IgA framework. We compare our
results with other results obtained with standard FEM to confirm the correctness of our obtained results. | {"url":"http://www.numa.uni-linz.ac.at/Talks/abstract/584/","timestamp":"2024-11-02T21:54:03Z","content_type":"text/html","content_length":"8416","record_id":"<urn:uuid:59eca8b5-0fae-4cc3-91a8-3bbda12612b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00194.warc.gz"} |
[103] Custom Inductors – Applying the ONE Design Equation
Dr. Ray Ridley continues his discussion on the ONE design equation.
ONE Design Equation
In the last article of this series, I introduced one equation that matters for inductor design. This is the equation that you will use when you are creating a custom design from scratch. Please note
that we are talking about the design of an inductor, NOT the analysis process. Mathematical structure analysis can be quite involved and is a popular subject for PhD dissertations. You can also read
many articles on this topic in our Design Center [1].
Figure 1: The Basic Constituents of a Custom-Designed Inductor.
The design equation that matters is as follows:
We have arranged this equation with the magnetics variables of core saturation level, number of turns, and core cross-sectional area (minimum) on the left. The electrical engineering variables are on
the right, and these are usually determined by the circuit designer through simulation, experience, and personal preferences. We will talk more about this in a future article, since the choices of
inductance can vary widely from one designer to another, and from one application to another.
Defining the BEST Inductor
Design engineers think they want the best inductor for an application. But, what does this mean? To the purchasing department, “best” typically means cheapest, and they will work hard to drive the
price down. To the electrical engineer, “best” may mean the most efficient for the application. For the packaging engineer, “best” will mean a smaller size. And for the EMI engineer, “best” may mean
small capacitance, and minimal stray electromagnetic fields from the part.
Satisfying all these definitions simultaneously is not possible. For every design, there will be a tradeoff between allowable size, heat dissipation, impact on the rest of the circuit design, and
cost. This tradeoff will be different for every application, and that is why you will find a very wide range of inductor designs from one power supply to another.
Starting with Arbitrary Choices
Now we come to the heart of the problem – how to start a design? At many companies I work with, there is an almost pathological obsession with finding the equations to be programmed into Mathcad that
will provide the “RIGHT” solution to the problem. Which core should you choose? What material? What is the right answer? Many managers feel that if they can see an equation, their engineers must have
chosen the proper solution.
But, there is NO single right answer. There are an infinite number of solutions, and the choice of which way to go is arbitrary. I am deliberately being extreme in saying this to make a point – if
you want an optimized inductor, stop looking for nonexistent sets of equations and begin exploring engineering options with an unconstrained mind.
There is an important consequence with the ONE design equation above. Let us assume that we have chosen the inductance and the subsequent peak current that results. (more about this will be discussed
in future articles – it is not necessarily a simple choice.) The equation implies that you can use a core of ANY size, and if you put enough turns on the core, the inductor will not saturate. That
means you can start the design anywhere you wish!
If you are doing this for the first time, just pick a number for the core area. Plug it into the design equation and see how many turns you are going to need to make sure the inductor doesn’t
saturate. From that result, it won’t take long to ascertain whether the design is sensible or not. Choosing a number that is too small, for example, will help build our experience base by seeing the
impact on losses in the windings or the core. There is usually more to be learned from making a wrong initial choice that starting with a design that will work comfortably. This is the process we
encourage in our design workshops – iterate quickly with several different options and you learn quickly.
If you are working with a company that already does custom inductor designs, you may choose just pick a core already being used. This will provide purchasing power that can drive a lower price than a
smaller core. It may also fit better with your company’s tooling. There are many factors that will drive the decision, but a single optimizing equation is not one of them.
We will explore the idea of exploiting the design freedom that exists by starting with any size core with examples in later articles. This is also the process that we demonstrate in our design
workshops [3].
But now let’s consider how design guidelines that you find in text books or databooks have the unfortunate consequence of removing design freedom and iteration to arrive at a single solution. This is
counter-productive to the creative process.
Wire Current-Density Equation
Most data books, text books, and magnetics guides take the same fundamental step in overcomplicating the design process. They search for a single solution that cannot possibly exist for every
application. The view is presented that there must be a starting point for the design, and design freedom should not be considered.
For many publications [2], the size of the wire can immediately be locked down according to the arbitrary (but common) current-density equation:
A[Cu] = 500 circular mils/A
Please note that the number 500 is arbitrary. In some books it will be 350, and in others 750. We have deliberately left this expression in one of the absurd units that you will find in many
magnetics texts and handbooks (apologies to non-US engineers). A circular mil is the area of a piece of wire that is 1 mil in diameter, that is 0.001” or 25 microns. I don’t like to choose a more
sensible unit since I don’t want to encourage people to use this guideline. For all of the magnetics I have worked with, I have never known the current density, but I do know the temperature rise
after it has been built and tested.
The equation gives the recommended cross-section of the wire for the rms current in the inductor windings. It seems initially like a reasonable starting point. However, we must ask, where does this
equation come from? It is hard to trace the first usage, but ultimately it is derived from the early days of line-frequency magnetics work. Transformers and inductors had hundreds or thousands of
turns, and they were massive thermal structures with the windings and cores intimately thermally connected. The recommended current density made sense when the frequency was preselected and the
thermal situation was known and well characterized.
However, with high frequency magnetics, we have many compounding factors. Firstly, there may be just a few turns of wire or foil. The thermal situation is profoundly different from multiple layers of
wire in a conventional line-frequency inductor. The windings can be in a single layer, multiple layers, on a bobbin, or part of a PCB. The thermal variations are tremendous. So why apply the same
design guideline?
And, for most high frequency inductors, there are multiple frequency components to the current waveforms. This leads to uneven current distribution in the wire which is usually considerably thicker
than a single skin depth. The current density rule no longer applies, and it is not a good starting point.
Design Tip: Wire Current Density – calculate it if you like, but don’t use it to drive a design. I may never know the current density of the final design. I do, however, want to know how hot it gets.
Window-Area Product Equation
Some texts take a further step in trying to pin down a single “correct” solution by defining a window-area product. This is the product of the area of the core, multiplied by the area of the window
available for the winding. This quantity is then used in a design equation based on the power level of the application [2].
Equations like this give the appearance of providing a unique solution and design path. But no two applications can be the same. One of the primary drivers of a design is the cooling of the part.
This is never the same from one application to another. Some designs have forced air cooling, and at the other extreme, some magnetics must work in a vacuum with zero air flow. The design process
should not be the same for these examples.
Design Tip: Window Area Product – Without full thermal information, it can never be a good guide to design direction.
Gap Length Equation
Some design engineers like to start with a gap length. We ran into this on our LinkedIn group recently with questions asked by new designers in the field [4]. Why? I don’t really know. Perhaps they
have a pre-gapped core on hand and don’t want to order another. Regardless, the gap length is not the way to start your design.
However, when you are DONE with a design, you can use an equation to estimate the gap as follows:
l[g] = µ^0n^2A[e]/L
Figure 2: The Simplest Equation Assumes All Energy is Stored in the Gap.
Is this a good and accurate equation? No, not at all! The accuracy will depend on the size of the gap relative to the core total path length, the permeability of the core, and the gap length compared
to the dimensions of the core cross section. You can read many papers about fringing fields, and how to find a better equation for the gap.
But, in the end, it doesn’t matter. All equations for gap length will be empirically inaccurate. Don’t worry about this, just gap a finished design until you get the desired value of inductance. That
is how inductor manufacturing works. The gap is adjusted to get the inductor with the specified tolerance required.
There are two things that are important about the gap length – one, it should not be too small, or you will be relying on the permeability of the core material, a quantity that can be very variable.
And two, it must not be too large, or excessive fringing and EMI will result. In between these two extremes, the actual value of gap is determined empirically.
Design Tip: Gap length equation– Use it to establish upper and lower bounds, but ultimately the gap will be adjusted empirically for the desired result.
We encourage engineers to experience the freedom in working with just ONE design equation. Try many alternatives, and you will learn quickly through iteration. Most design books constrain designs
simply to arrive at a single solution. This discourages creative thought leading to designs that are suboptimal for most applications.
[1] Magnetics Design Videos and Articles, Ridley Engineering Design Center.
[2] Mag Inc Design Guide: https://www.mag-inc.com/Design/Design-Guides/Transformer-Design-with-Magnetics-Ferrite-Cores
[3] Learn about proximity losses and magnetics design in our hands-on workshops for power supply design www.ridleyengineering.com/workshops.html (The only hands-on magnetics seminar in the world.
[4] Join our LinkedIn group titled Power Supply Design Center. Noncommercial site with over 7800 experienced and helpful industry experts.
[5] Join our Facebook group titled Power Supply Design Center. Advanced in-depth discussion group for all topics related to power supply design | {"url":"https://ridleyengineering.com/design-center-ridley-engineering/39-magnetics/278-103-custom-inductors-%E2%80%93-applying-the-one-design-equation.html","timestamp":"2024-11-03T06:32:05Z","content_type":"application/xhtml+xml","content_length":"41569","record_id":"<urn:uuid:a805d3ac-8db5-462b-887d-50700beeb019>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00778.warc.gz"} |
General Neural Gauge Fields
The recent advance of neural fields, such as neural radiance fields, has significantly pushed the boundary of scene representation learning. Aiming to boost the computation efficiency and rendering
quality of 3D scenes, a popular line of research maps the 3D coordinate system to another measuring system, e.g., 2D manifolds and hash tables, for modeling neural fields. The conversion of
coordinate systems can be typically dubbed as gauge transformation, which is usually a pre-defined mapping function, e.g., orthogonal projection or spatial hash function. This begs a question: can we
directly learn a desired gauge transformation along with the neural field in an end-to-end manner? In this work, we extend this problem to a general paradigm with a taxonomy of discrete and
continuous cases, and develop an end-to-end learning framework to jointly optimize the gauge transformation and neural fields. To counter the problem that the learning of gauge transformations can
collapse easily, we derive a general regularization mechanism from the principle of information conservation during the gauge transformation. To circumvent the high computation cost in gauge learning
with regularization, we directly derive an information-invariant gauge transformation which allows to preserve scene information inherently and yield superior performance. | {"url":"https://fnzhan.com/projects/neural-gauge-fields/","timestamp":"2024-11-05T22:07:32Z","content_type":"text/html","content_length":"17180","record_id":"<urn:uuid:6d970f18-45fe-42ac-b05d-56a69040acd7>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00020.warc.gz"} |
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Difference Between Distance and Displacement: JEE Main 2024
How to differentiate between Distance and Displacement
Distance is the measurement of paths taken by an object. In simple words, distance is something an object covers in a given time ‘t.’ However, displacement is the shortest path taken by an object
during its motion.
Distance and displacement are two physical quantities that we use in our everyday life. So, which point differentiates the two terms even if they have a common word “path” in them?
Also, why do we consider two different terms for the measurement of paths while considering their magnitudes? This page discusses all the differences between distance and displacement in tabular
format. Also, we will go through illustrating examples on the same.
Distance vs Displacement - Tabular Format
Below is the tabular format with underlying differences of distance and displacement:
Now, let us understand in detail distance and displacement.
Distance - Understanding with an Example
One day, Riya decides to go for a long drive. Instead of considering a path, she roams around the city.
Distance = Speedtime and the unit of distance is metres - ‘m’.
Here, what do you understand from this example?
Well, Riya’s car is covering certain points, and let us join two points and other two points her car travelled in the following manner:
(Image will be Uploaded soon)
Here, AB is the first path, and GH is another path. The two paths AB and GH are the distances Riya travelled in time t1 and t2, respectively.
Please note that one thing that Riya does is just “Roaming” around the city, not considering the “types of paths” she took.
Now, let us understand what displacement is.
Displacement - Understanding with an Example
Assume a new scenario where the same person, Riya is heading towards her office hurriedly. As she went for a long drive last night, she was tired and woke up late. She has got a very important
project to do and is getting late.
Now, she looks for a shortcut to reach 30 minutes prior to the daily timings, so what that shortest path is? Well, that shortest path is nothing but the “displacement.”
(Image will be Uploaded soon)
From here, we understood that Riya has to consider the direction of the type of path to reach the office early. Thus, when we consider the type of path, it is displacement.
We can measure the path an object takes and also the direction of the path.
The average velocity of this path = total displacement/total time taken
Here, velocity is calculated in m/s
Time in seconds and
Displacement in m.
Hence, from our examples on distance and displacement, we understand that distance has just magnitude, which is regardless of the direction. However, displacement takes both the magnitude and
direction of the path travelled by an object.
Hence, distance is a scalar quantity and displacement is a vector quantity. Distance is always positive or zero, while displacement can be positive, negative or zero.
Now, let us go through solved examples applying distance and displacement in our real lives.
Solved Examples on Distance and Displacement - Mathematical Application
Example 1: Closed path is travelled by a body. Point A, B, C & D represents the path. Find the displacement and total distance travelled by the body.
(Image will be uploaded soon)
Solution: Distance travelled by the body is
Distance = 2πr
Where r= radius of the body = 3km
Distance = 2 π 3 = 6π km
Displacement of the body is Zero because the body started from point A and came back to its initial position A.
Example 2: Find the displacement and distance travelled by the body if the body moves from point A to B to C to D.
(Image will be Uploaded soon)
Distance travelled is
= \[\frac {3}{4}\] (2πr)
= \[\frac {3}{4}\] (2π3)
= 4.5π km
Since \[\frac {3}{4}\] is the portion travelled by the body so that’s why distance travelled is 4.5π km
Displacement -
By applying Pythagoras Theorem on the given figure we get,
AD2 = AO2 + OD2
=> 32 + 32 = 18
=> AD=3 \[\sqrt {2}\]km
Hence, the displacement covered by an object is 3 \[\sqrt {2}\]km.
From our content, we conclude that distance is a scalar quantity that refers to "how much ground an object has covered" during its motion, without considering a direction of motion. Displacement is a
vector quantity that refers to "how far out of place an object is''; it is the object's overall change in position, i.e., considering a direction. | {"url":"https://www.vedantu.com/jee-main/physics-difference-between-distance-and-displacement","timestamp":"2024-11-04T07:25:10Z","content_type":"text/html","content_length":"228244","record_id":"<urn:uuid:81d9d134-a1db-4c57-920e-4c36f15b53fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00425.warc.gz"} |
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Future Value of an Annuity Calculator | FV of an Investment
Future Value of an Annuity Calculator
Calculate the final value after a series of investments, deposits, or withdrawals.
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An annuity, as used here, is a series of regular, periodic payments to or withdrawals from an investment account. Wikipedia lists these examples of annuities "regular deposits to a savings account,
monthly home mortgage payments, monthly insurance payments, and pension payments." We can classify annuities by the frequency of the cash flow dates. The investor may make deposits (withdrawals,
payments) weekly, monthly, quarterly, yearly, or at any other regular interval of time. This calculator supports eleven frequencies.
The future value of an annuity is the amount the cash flow will be worth as of a future date. Due to the investment gain or interest earned on the principal (the amount deposited), the final value is
greater than the sum of the deposits.
This future value of an annuity (FVA) calculator calculates what the value will be as of any future date. The calculator optionally allows for an initial amount that is not equal to the periodic
deposit. This feature enables the user to calculate the FVA for an existing investment.
If the investment is a new investment set the "Starting Amount (PV)" to 0.
This FVA calculator also calculates the future value after a series of withdrawals. If you start with $1,000,000 and assume it earns 4.0% per year, the calculator will calculate the value after 30
years of $5,000 monthly withdrawals. To indicate a withdrawal, enter a negative amount.
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Instructions for the future value of an annuity calculator
• Starting amount (PV): This is the money you have at the beginning of the annuity period. It could be the initial investment amount or the current value of an existing annuity.
• Periodic amount: This is the amount of money you will withdraw (-) from or contribute (+) to the annuity regularly. The terms of the annuity will determine the amount and frequency.
• Number of periods: This is the number of times the periodic cash flow will occur.
• Annual interest rate: This is the interest rate that the annuity will earn. It is expressed as a percentage per year.
• Start date: This is the present value date (see note below). It could be the date you purchase the annuity or another predetermined date.
• First contribution date: This is when you will make your first contribution (or withdrawal) from the annuity. It could be the same as the start date or a later date.
• Cash flow frequency: This is the frequency with which you will contribute to or withdraw from the annuity. It could be monthly, quarterly, annually, or another interval.
• Monthly compounding: This refers to the frequency with which the interest on the annuity will be compounded. If you do not know the compounding frequency, then set it to match the cash flow
Note: As explained, an annuity is a regular cash flow - either scheduled contributions or withdrawals. However, because this calculator lets the user specify both a first cash flow date and a start
date that may not align with the cash flow, the calculator can accurately calculate the future value. This is the case even if the cash flows don't start until years later.
Future Value Schedule Help
Money, in any form (cash, investments, receivables, etc.) will have a different value tomorrow or next month or next year than it does today. Even money stuffed in a mattress won't have the value in
a year from now as it does today. That value is known as the "future value."
You must enter either a "Starting Amount" (the cash-on-hand) or the "Regular Contribution Amount" or both. Set how often you add to your investment by setting the "Contribution Frequency". If you set
the "Contribution Frequency" to monthly and enter 120 for "Number of Contributions" then the "Future Value" will be for the date 10 years from the "First Contribution Date" (120 monthly contributions
= 10 years).
A note or two about "Compounding Frequency". Selecting he "Exact/Simple" option sets the calculator so it will not compound the interest. Also, the exact number of days between withdrawal dates is
used to calculate the interest for the period. The "Daily" option uses the exact number of days between dates, but daily compounding is assumed. (The interest earned each day is added to the
principal amount each day.) The "Exact/Simple" compounding option is the most conservative setting. That is, using it will result in the lowest future value. Daily compounding will result in nearly
the greatest future value (except for "Continuous Compounding".
The other compounding frequencies are based on periods of time other than days. Each period is assumed to be of equal length for the purposes of interest calculations. That is, assuming a balance of
$10,000, the interest earned for January will be the same interest earned for February given the same interest rate.
NOTE: The future value maybe lower than the value reflected today — think inflation. To reflect that fact, simply use a negative interest.
10 Comments on “Future Value Of An Annuity Calculator”
Join the conversation. Tell me what you think.
Very nice tool. Just wish you had the capability to show negative values in your Balance/FV column. Thanks!
Thanks for the compliment.
Here’s another calculator – the Ultimate Financial Calculator that will probably do what you want (I say probably because I’m not sure what you need besides the negative balance.
If you try it, scroll down the page and you’ll there’s a number of tutorials.
Assuming you have some amount call it "X", and you want to make withdrawals, set the Schedule Type to "savings". Create two rows, the first row as a deposit with value "X" and the second row
with value "Y" for the number of withdrawals you expect. If Rounding (under settings) is set to "Open Balance", the balance will go negative.
Let me know if there are other details, and I’m sure we can work through them.
The Future Value calculator is not calculating values. It only shows 4 months of data even if the selection is more than 4 month period. Can someone look into this. This is a great tool that
provides future projected cash values. Really love it. Hoping to get this fixed.
Hello, That’s really embarrassing.
I recently made a small change that broke some calculators. This morning I released a fix.
Please try again.
If you do not see the change right away, you may have to perform a hard refresh of the page:
Depending on your operating system all you need to do is the following key combination:
○ Windows: ctrl + F5
○ Mac/Apple: Apple + R or command + R
○ Linux: F5
Above, from Refresh Your Cache.
If you don’t mind, please let me know if the problem is resolved for you.
Yes the problem is resolved. I appreciate the quick fix.
Thank you so much
You’re welcome. Thanks for the confirmation.
Is it Available on WordPress widget?
This calculator is not available as a plugin. However, the FC Savings Calculator Plugin is very similar. Perhaps it will meet your needs?
I have a sum invested and I would like to know how much I can draw from that sum every month whilst keeping the inflation adjusted value of the sum the same.
Another way of putting this is my monthly withdrawal should equate to the interest on the sum minus the adjustment for inflation.
Is it as simple as subtracting the monthly rate of inflation from the monthly rate of growth and applying that to the sum invested to get my monthly withdrawal amount?
The Ultimate Financial Calculator is designed for this problem.
Change "Schedule Type" to "Savings."
Click on "Cash Flow Options" for your withdrawal series and select "Percent Step."
Enter the assumed inflation rate as the "Percent change per level."
If you have any questions, please ask them Also note the link on the above page to a number of different tutorials.
Comments, suggestions & questions welcomed... | {"url":"https://accuratecalculators.com/future-value-of-an-annuity-calculator","timestamp":"2024-11-06T21:16:00Z","content_type":"text/html","content_length":"129197","record_id":"<urn:uuid:a17c4878-f21c-4a92-9f0a-e0a68b862184>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00050.warc.gz"} |
Awasome 13+ What Is 25 Of 15 References - collegebeautybuff.comAwasome 13+ What Is 25 Of 15 References
Awasome 13+ What Is 25 Of 15 References
Awasome 13+ What Is 25 Of 15 References. To make it easier to calculate, you may write it as an equation: Prime factorization of 13 and 15 is (13) = 13 1 and (3 × 5) = 3 1 × 5 1 respectively.
Acts 2215 For you shall be his witness to all men of what you have seen and heard. from biblepic.com
Therefore, the larger fraction is 22/25. 15% of 25 is 3.75. The calculator provided automatically converts the input percentage into a decimal to compute the solution.
25% Of 15 Is 3.75 Why This Calculation?
Working out 15% of 25. If you know any two values of the formula, you can calculate the third one. The calculator provided automatically converts the input percentage into a decimal to compute the
13/25 = 0.52 Once We Have The Answer To That Division, We Can Multiply The Answer By 100 To Make It A Percentage:
15% of 25 is 3.75. Expressed as a decimal, 13/15 is equal to 0.86 recurring (that is, 0.86666.) expressed as a decimal, 22/25 is equal to 0.88. 25% of 15 is 3.75.
That In General Which Is To Be.
Two different ways to convert. 25 percent of 15 is 3.75. 0.52 x 100 = 52% and there you have it!
Your Restaurant Bill Is $86.67 And You Want To Leave A 15% Tip.
However, if solving for the percentage, the value returned will be the actual. This is arguably the simplest way to ensure that the fractions have a common denominator. Working out 25% of 15.
Everyone Knows That There Are Arithmetic Operations That We Need On A Daily Basis, And Among These Operations We Find The Process Of Calculating The.
However, in most cases, the solutions to these equations will not appear in simplified form (the. Write 15% as 15 / 100; Lcm of 13 and 15 can be obtained by multiplying prime factors raised to their
respective highest power, i.e. | {"url":"https://collegebeautybuff.com/blog/2022/12/20/13-what-is-25-of-15/","timestamp":"2024-11-10T22:39:48Z","content_type":"text/html","content_length":"56256","record_id":"<urn:uuid:9c83a962-2311-4e94-94d2-c6e908b13463>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00867.warc.gz"} |
Options Greeks: A Simple Guide to Learn
Options Greeks are a set of mathematical functions that describe the price movements of options. They can use them to hedge positions and speculate on changes in the market. This blog post will
provide an overview of how each of the option greeks works and how it influences an option's price movement.
The price movements of options are directly related to five mathematical functions that fluctuate daily. A proper understanding of these can be advantageous, allowing traders to play the greeks in
their favor instead of playing against them. There are five primary greeks: delta, gamma, theta, vega, and rho. Each Greek measures a different aspect of an option's price movement. Inside this blog
post, we will go in-depth on each greek and its effect on the options you might be buying or shorting. This article will also overview specific option strategies and scenarios where you can use the
greeks in your favor.
Each greek has a range from +1 to -1. A positive greek will increase the value of the option contract price when the underlying factor rises. For options traders to know how much they affect the
options price, we will need to multiply each one by 100 because each option contract can buy or sell 100 shares depending on whether you are holding a call option or put option.
The price of an option is calculated through all five greeks, delta, gamma, vega, and rho. An option's value can be split into two different categories that account for all of the current value.
Options value = Extrinsic value + Intrinsic value
Intrinsic value explained
The intrinsic value is the price of an option at the time of expiration which can help traders know how much weight the option has to lose. Intrinsic value can only exist if there is a difference
between your strike and market price on the expiration date. This option would be considered expiring in the money. Therefore, it is the only value left if you were to exercise your option contract
at any given time before or on the time an option expires.
Traders need to know what type of options they are buying because it can affect how much their position will gain or help minimize significant risk based on changes in the market price and how much
it will cost them. If you are buying options, the only greek that matters is delta because its value directly affects your position's overall price changes and, therefore, performance.
On the other hand, if you were to sell an option against a long stock position, then gamma would be vital since it can determine how fast or slow your stock position will move.
Extrinsic value explained
One of the main factors affecting an option's price is its extrinsic value. Extrinsic value can be thought of as the time value of money. The option price's extrinsic value decreases quickly as the
expiration date nears and is extremely large the further away we are from expiration. An option is mainly a vehicle to gain leverage on a particular asset but the downside is paying a premium to hold
this leverage vehicle mainly due to time decay. Time is the most significant value on any option and decreases the value very quickly as expiration nears.
Intrinsic value example
This Facebook option is a price of $4.95 currently has seven days till expiration, giving it a certain amount of extrinsic and intrinsic value. We will break those two values up so you can know the
true risk of holding an option like this.
First, to calculate intrinsic value, we need to know the underlying's price, which is $335.62. Because this number is higher than the strike price, we know these options are considered deep in the
To calculate the intrinsic value, you must subtract the same strike price by the underlying stock price, which is $335.62-$335 = $0.62. This is the actual value of the option, and everything else
over this is considered extrinsic value, which is the premium you are paying to hold this option. So to calculate the extrinsic value, we subtract $0.62 by the option price, which is $4.95, which
means the value is around $4.95-$0.62 = $4.33.
This number is a mix of time decay, volatility of the underlying, and more. For example, if you were to hold this option for the next seven days and the underlying price didn't move at all, you would
be losing all $433 and only have $62 at the time of expiration, which comes down to a $86 loss per day if we exclude the weekend.
This example same option with an extra week till expiration, which has a value of $755, and an additional year till expiration is $4,863. All three of these options have the same intrinsic value of
$62 dollars, but the further your expiration date, the more extrinsic value you add to the option.
Traders or investors will buy these options to speculate on price movements, but the percentage change amount they can make it smaller and smaller the further the expiration date because the price
increases while the amount of leverage doesn't. Option sellers see the opportunity and look to put risk factors like time decay in their favor and hedge portfolios in certain situations.
In the example below, you can see the greeks column on Robinhood at the bottom for the Facebook 12/31 call option with a strike price of $335 and $495 (because we have to multiply it by 100). The
positive greeks will be delta, gamma, vega, and rho. The factors related to these greeks will increase the options price if those factors increase.
For example, Delta is related to the price of the underlying stock. So when the price of the underlying goes up $1 the price of the option should increase as well. The amount it increases will be
0.5154 to the option price, which would be $4.95+$0.5154 =$5.4654 multiplied by 100, = $546.
This means your option increased around 10% in value of $51 because of the $1 move in the price of the underlying. We will go more in-depth on these greeks later, but it is essential to note + greeks
add to the call option value, and negative greeks decrease the value of the option.
The only negative greek in the example here is theta related to value lost per day, which is around $0.34 x 100 = $34. So every single day, this option can expect to lose that much money.
When shorting options, you can take advantage of the negative greeks and the factors where certain greeks decrease the option price. This risk management means going short on anything means you are
borrowing the option with hopes of the value decreasing. To make money from short, the option's value needs to drop.
So while all the greeks are unchanged, staying the same, positive and negative, compared to the previous example, as the option is shorter, you can profit from the time value now. As well as when the
factors affecting gamma, vega, delta, and rho decrease.
In this options trading investment strategy of how much delta changes, If delta increases the option price when the price of the underlying goes up, as the options seller, you would hope for the
price of the underlying to drop in price since the delta also can decrease the value of the option by the same amount.
For example, when the price of the underlying goes down $1 the price of the option decreases as well. The amount it reduces will be 0.5154 to the option price, which would be $4.95-$0.5154 =$4.44
multiplied by 100, = $444. This means your option decreased around 10% in value or $51 because of the $1 move in the price of the underlying.
Delta value is the rate of change in an option's price, given a $1 move in the underlying security, and ranges from +1 to -1. Call options have a positive delta which means the cost of the option
increases with the price of the stock. Put options have negative deltas, which means the option's price increases when the stock price moves downward.
How delta changes with a strike price
The delta of an at the money (ATM) call option is around 0.50, in the money (ITM) call option is above +0.50-1.00, and lastly, out of the money (OTM) call options are between 0-0.5. Lower delta
options can be seen as incredibly risky since they contain much extrinsic value and move very little per dollar in the underlying stock.
As you can see in the example below, at the money options offer the best leverage per dollar move, and the further you move up or down the option chain, the less of a percentage change increase you
get compared to the value of the option.
An option's delta can also be expressed regarding the percentage chance of expiring in or out of the money. So an option with a delta of 0.50 would have a 50% chance of ending up in the money at
expiration (it would increase $0.50 if the underlying asset increased by $100).
As discussed in the example above, the percentage affect delta has gotten smaller as the expiration nears.
Delta to strategy
Some traders recognize the amount of leverage they can receive and try to take advantage of mispriced options that offer a higher intrinsic value than extrinsic value.
In the Facebook example above, the percentage of extrinsic value is around 85%. That is the amount of the option working against you through time value and volatility of the underlying drops. It is
essential to note for risk measures, the higher this percentage, the harder it can be to profit from a trade.
Some traders will look towards low volatile stocks that will soon experience high volatility of the underlying. Currently, at this moment in the market, a very quiet dividend stock called AT&T has
moved around 10%, which doesn't happen often.
If we look a the percentage of extrinsic value from at the money options, we can see the price of the stock is $24.90, and the strike price is $24.5. Subtracting these numbers, we get $0.40, which is
the option's intrinsic value. So the amount of extrinsic value is around 20%.
You are paying much less for the leverage when you are options trading. The option buyer pays around four times more for the same leverage for Facebook at the money options. Some traders will scan to
see price changes and search for these options to give them a better chance to profit from a trade.
3. Theta - time value
Theta measures how much option prices will change given the passing of one day. It is sometimes called time decay or time value because it works against you as days go by (or in favor if you are
shorting options).
Imagine that you have a one-month option with a theta of with a theoretical value -0.05. This means that, on average, the option's price will decrease by $5 each day. If we started with an option
priced at 0.50 and it had a theta of -0.05, it would be worth 0.25, losing 50% of its value in one week. This is because most of the option's value is made up of extrinsic value.
Theta is also expressed in terms of percentage. So an option with a theta of 0.05 would lose (on average) or 10% of the option's value, which only increases closer to expiration (0.05/0.50 = 10%).
The amount of decaying time value has a positive relationship further and further. The options are out of the money and the reverse, where the time decay amount decreases significantly the further
and further options are in the money.
An option that expires out of the money is considered gambling because they are entirely of extrinsic value, which again means they will have no value at the time of expiration. Which again means the
options are going to zero. This is great for option sellers but extremely dangerous for option buyers. A choice that will expire in the money is ideal for options traders.
Minimal decaying time value options
Options with a small amount of time decay as a percentage of the option value can offer a lot less risk to option traders. In the example above, AT&T offered a tremendous amount of intrinsic value at
80% of the option's value.
This is mainly due to how small the time decay is, at -0.0182 per day or around a 3% loss of the value of the option. If this is hard to stomach for holding options overnight, this amount will
decrease as price changes go further and further in the money.
In the example below. we can see the time decay remains similar at about -0.01 per day but as a percentage of the contract, it is almost not noticeable, being around 0.07% loss per day.
Use time decay in your favor
Time decay can be a nightmare or another trader's paradise if utilized properly. By selling options a trader can profit from the daily time decay of options. Certain options can pay a hefty premium
per week if they go straight to zero and those options can be found on BreadAlerts.
BreadAlerts is a software that is an option seller's best friend, it scans the whole market every second to sort and filter for the highest possible premiums to sell. In the example below, if we sort
the premium % column traders will be able to see the stocks containing the largest premiums to sell.
These premiums can range from 5 to even 15% some weeks. These can be considered options with a massive amount of extrinsic value compared to intrinsic value.
In the example below, we can see BreadAlerts top option with the highest premium as a percentage of the capital needed is NKLA. If we head to the option chain on Robinhood we can see the 11.5 strike
price put options is considered out of the money which means 100% of the value is extrinsic which means a large portion if not the whole amount could go to zero at the time of expiration. This option
would be incredibly dangerous to sell because of this fact but the option seller has a huge opportunity if it in fact does expire worthless out of the money.
To sell a put option, the trader would need 100 shares per contract in case they needed to buy the shares for whatever reason. This amount would be $1,150 and the amount the option seller could
receive if the stock price stays the same for the next 7 days is $94.
Which is a possible 8% return on their capital in one week. This example shows the dangers of being an option buyer especially holding options with a large amount of extrinsic value. In these cases,
it is much safer to become an option seller.
Time decay towards expiration
The decaying time value greatly increases as it gets closer to expiration when comparing it to the total percentage of the option value. As you can see in the example below, time decay is almost
non-existent for options with more than 90-120 days till expiration. But the close and close we get towards expiration the faster options decay. This is the big worry holding options overnight with
less than 10 days till expiration. Although this is an option seller's best friend because of how quickly value could be lost. Many option sellers will look to target options 7-30 days out because of
4. Gamma - the rate of change in an option's price as a function of changes in underlying asset price and volatility
Gamma is the second-most important greek after delta. Gamma measures how much an option's Delta will change given a $1 move in the underlying asset's price.
Since delta is the amount the option's price increases in value, a higher delta would mean more money per $1 move. So an increase in delta is very beneficial to an options trader and a higher gamma
will allow that.
For call options, if the price of the stock increases then gamma will be added to the delta. For put options, if the price of the stock decreases then gamma would be subtracted from the delta. Since
delta is negative for put options, this would still mean a higher delta. This is good for an option holder because put options make money when the stock decreases in value.
Gamma Example
In our recent Facebook example, if we were to understand the impact of gamma per dollar move we would look at the greeks on our options trading platform. We can see that the 335 strike price Facebook
option has a gamma of 0.033 which means for a $1 move at this specified price in the stock 0.033 would be added to the delta of 0.5154.
So the resulting delta would be 0.5484, which means the next dollar move you would make $54.84 instead of $51.54. This also means the delta would increase yet again for that second dollar move to
possibly 0.5814. So a third dollar move would yield $58.14, and so on.
Out of the money gamma exposure
Traders will utilize the gamma increases from out of the money options. Out of the money options can be the riskiest to hold but also have the greatest potential to increase massively in price given
a big move in the underlying asset. This increase of gamma is the largest out of the money and does lead to those options moving more as a percentage of at the money options or in the money options.
In the example below, we can see NKLA ran 25% in one day from the underlying stock and the option percent gains were the largest out of the money. The further and further the strike price was in the
money the smaller the percent one day change was. This is a large attraction to the out of the money options and largely due to gamma exposure.
5. Vega - sensitivity to changes in implied volatility
Vega measures an option's sensitivity to changes in implied volatility. If vega is a positive number, a move higher in volatility from the underlying asset increases the price of the option or vice
versa for a negative vega value. We don't really see negative vega value but we know that is the opposite for option sellers. They would be losing money if the option price happened to increase from
Volatile options can be incredibly dangerous to hold if volatility changes suddenly. If we pull up the greeks for NKLA we can see vega is around 0.0084 and the implied volatility is around 116%.
This means that every 1 percentage move in implied volatility higher adds 0.0084 to the option and every 1 percentage drop in IV lowers the option's value by 0.0084. If NKLA happened to experience a
50% in IV, that would be detrimental for an option holder, as the value of the option could plummet 50 multiplied but 0.0084, or 0.42.
This is around 64% loss for option holders, even if the price happens not to move. This is the danger of trading highly volatile options and this is why option sellers will target these. These
options have a large amount of extrinsic value and if they happen to not move for one or two days it could result is huge losses.
6. Rho- sensitivity to the risk-free interest rates and dividends on the stock price
Rho measures an option's sensitivity to changes in the risk-free interest rate. It can be a positive number if the underlying asset is expected to move up and it will increase as the risk-free
interest rate increases or vice versa for a negative Rho value.
An at the money call option has rho of -0.011 while ATM puts have 0.011 because they have positive and negative rho respectively.
An option's Rho can be expressed in terms of percentage (i.e., rho divided by 100) which is simply the Rho divided by Delta. So an option that has a rho of -0.00075 would have its delta change (on
average) -0.000125 for every one percentage point change in the interest rates rise or fall.
Rho has little impact
We are less worried about this option greeks option value as the movement of the fed funds rate has been very minimal for the past 20 years. From 2010 to 2015, the interest rates has been glued to
zero which means there change in the options price for this period of time.
This greek has more an effect on options being held for 1-2 years, called leap options. During the 1980s to 1990s, the interest rates moved from 7 to 20% and then back down. This move in the interest
rates would have had a much greater effect on options, especially ones being held for many months.
Risk free interest rate increase example
Despite the large swings in the interest rates, at the end of the day, Rho is still very small compared to the total option value. In the example below we can see CHPT has a Rho value of 0.07 which
is around 1% value of the option price. The good news is since the interest rates are currently so low, any major increases to the fed fund interest rates would increase all long options being held.
If the interest rates were to increase by 5% the price of CHPT's 18 strike price option would increase by 0.35 to 6.35. This was calculated from 0.07*5 because Rho increases for every percentage the
interest rate goes up. Overall that would increase the price of the options by more than 5% but the chance of that happening in one year would be extremely small. Because the fed is slowly increasing
rates, we could see little movement in rho as this will have an effect on the underlying stock price.
7. Implied move calculations
We concluded that the options market is still very volatile and can be dangerous for traders who are not well versed. This volatility and measure move can be calculated daily from the price of an
option. The implied move calculations are an easy way to see what the expected movement of the stock price is and can be very helpful when making trades. In the previous example, the implied move can
be calculated from the price of the at the money options and multiply it by 2. The example above tells us that NKLA has an expected move of 0.65 * 2 = $1.30. Which is around a 10% move in the stock
alone. This can tell traders what the options market is pricing in and what could be expected. If this move fails to happen many of the options are going to lose a great amount of value.
Conclusion - Wrapping it Up
The Greeks, or greeks as traders often refer to them, can be tricky and intimidating. But understanding how they affect the price of an option is key to knowing what type of options you should trade
for your portfolio. Of course, we recommend consulting a professional before making any decisions about investing in this volatile market!
Trade with us!
If you're looking for a more sophisticated and educational options trading experience, look no further than "Market Moves Premium Options Trading Group." Our exclusive 7-day membership offers swing
trading set-ups, fast text signals, and +100 hours of educational content. Plus, you'll have access to live trading sessions twice per day. So if you're ready to take your options trading to the next
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Financial Disclaimer: Market Moves LLC is a company that provides education in financial and stock market literacy. WE ARE NOT FINANCIAL ADVISORS. In fact, it is illegal for us to provide any
financial advice to you. Under U.S. law, the only persons who can give you financial advice are those who are licensed financial advisors through the SEC. Results shown from Market Moves LLC or
customers who use our product and/or service are individual experiences, reflecting real-life experiences. These are individual results, and results do vary. Market Moves LLC does not claim that they
are typical results that consumers will generally achieve. Past performance does not guarantee future results. You should not rely on any past performance as a guarantee of future investment | {"url":"http://tradewithmarketmoves.com/options-greeks-a-simple-guide-to-learn","timestamp":"2024-11-13T20:57:13Z","content_type":"text/html","content_length":"651208","record_id":"<urn:uuid:5424e00e-b2b8-4c37-a9ed-4aaf4f068eea>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00298.warc.gz"} |
How Long is 6 Months in Hours?
Six months is approximately 4,320 hours. But what does that really mean in terms of everyday activities? Let’s look at some examples to put it into perspective.
Understanding the Calculation
To understand how we get to 4,320 hours in six months, here’s a simple breakdown:
• Days in a Month: For simplicity, assume each month has 30 days.
• Hours in a Day: Each day has 24 hours.
By multiplying these together, we get: 6 months×30 days/month×24 hours/day=4,320 hours
Examples of Activities That Last 4,320 Hours
1. Half-Year Projects: Many work or personal projects are planned to last about six months. This timeframe allows for detailed planning, execution, and review within 4,320 hours.
2. Travel and Sabbaticals: Extended travel plans or sabbaticals often span six months, giving individuals ample time to immerse themselves in new cultures and experiences.
3. Academic Semesters: An academic semester in many universities lasts around six months, including class sessions, study time, and exams.
4. Fitness Goals: Training for major fitness events, such as marathons or bodybuilding competitions, can be planned over six months, using the 4,320 hours for progressive training and preparation.
Breaking Down 6 Months into Hours
Understanding 6 months in terms of hours helps visualize it better.
• Days: 6 months is 180 days (6 months * 30 days per month).
• Hours: 180 days multiplied by 24 hours per day equals 4,320 hours.
So, 6 months is equal to 4,320 hours.
Real-Life Applications
Knowing how many hours are in 6 months can help in planning and managing long-term goals effectively. Here are some practical uses:
• Project Management: For long-term projects, knowing that 6 months is 4,320 hours helps in setting realistic milestones and deadlines.
• Fitness and Health Goals: Planning a fitness regimen or health routine over six months allows for significant progress and achievement. For example, committing 1 hour a day to exercise adds up to
180 hours in six months.
• Financial Planning: Budgeting and saving over a six-month period becomes easier when you understand it spans 4,320 hours. This helps in setting achievable financial goals and tracking progress.
Practical Uses of 4,320 Hours
1. Work and Productivity: If you work 8 hours a day, 5 days a week, you work about 960 hours in six months. This is a fraction of the total hours, giving you plenty of time for rest and personal
2. Learning and Development: Dedicating time each day to learning or personal development can add up significantly over six months. For example, spending 2 hours a day on learning a new skill totals
360 hours in six months.
3. Long-Term Commitments: Six months is a substantial period for various commitments, such as volunteering, internships, or part-time jobs. Knowing it spans 4,320 hours helps in managing and
optimizing this time effectively. | {"url":"https://hours-in.com/6-months-to-hours/","timestamp":"2024-11-04T08:47:02Z","content_type":"text/html","content_length":"45798","record_id":"<urn:uuid:c9a3f149-08fc-4eac-b578-81f4c9a626c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00440.warc.gz"} |
November 8-10, 2015
ROBUST OPTIMIZATION
in applied probability
The aim of this workshop is to bring together leaders whose expertise spans two areas of research: robust optimization & applied probability.
Robust optimization has emerged as a tractable paradigm for analyzing and optimizing models in which certain system parameters are unknown. Alternatively, applied probability has traditionally been a
field yielding insights into how systems behave under uncertainty, when the uncertainty is characterized by known random variables. New modeling and optimization challenges, e.g. those coming from
data-rich environments and arising in the domain of machine learning, give rise to questions best addressed by combining insights from both domains of robust optimization and applied probability. The
aim of this workshop is to bring together leaders whose expertise spans these areas of research, facilitating the transference of ideas, insights, and methodologies to tackle new and exciting
problems at the interface of robust optimization and applied probability.
│David Goldberg │Georgia Tech │
│Dick den Hertog │Tilburg University │
│Johan van Leeuwaarden │TU Eindhoven │
Aharon Ben-Tal Technion
Erick Delage HEC Montreal
Varun Gupta University of Chicago
Bernd Heidergott VU Amsterdam
Dick den Hertog Tilburg University
Nathan Kallus MIT
Ger Koole VU Amsterdam
Ton de Kok TU Eindhoven
Daniel Kuhn EPFL
Henri Lam University of Michigan
Melvyn Sim National University of Singapore
MONDAY November 9
│09.00 - 09.15│Registration │ │ │
│09.15 - 10.15│Tutorial │Dick den Hertog│Tutorial on robust optimization │
│10.15 - 10.30│Break │ │ │
│10.30 - 11.20│Keynote │Daniel Kuhn │Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric │
│11.20 - 11.45│ │Ralf Werner │Consistent uncertainty sets and consistent robust estimators │
│11.45 - 12.10│ │Chaitanya Bandi│Tractable Simulation-Optimization: A Robust Optimization Approach │
│12.10 - 13.30│Lunch │ │ │
│13.30 - 14.20│Keynote │Ger Koole │Robust optimization for integrated call center planning │
│14.20 - 14.30│Break │ │ │
│14.30 - 14.55│ │Erick Delage │The Value of Distribution Information in Distributionally Robust Optimization │
│14.55 - 15.20│ │Henry Lam │ │
│15.20 - 15.45│ │David Goldberg │Distributionally robust inventory control when demand is a martingale │
│15.45 - 16.00│Break │ │ │
│16.00 - 16.50│Keynote │Melvyn Sim │A Practically Efficient Framework for Distributionally Robust Optimization with Recourse │
│16.50 - 17.50│Discussion │Future Research│ │
│19.00 - 22.00│Conference dinner│ │ │
TUESDAY November 10
│09.30 - 10.20│Keynote │Aharon Ben-Tal │Robust Solutions of Uncertain Optimization Problems under Ambiguous Stochastic Data │
│10.20 - 10.45│ │Gideon Weiss │Transient control of queueing networks, using continuous linear programs │
│10.45 - 11.10│Break │ │ │
│11.10 - 12.00│Keynote │Ton de Kok │t.b.a. │
│12.00 - 13.15│Lunch │ │ │
│13.15 - 13.40│ │Nathan Kallus │Robust SAA and the Statistics of DRO │
│13.40 - 14.05│ │Sandeep Juneja │Selecting the best population using large deviations and multi-armed bandit methods │
│14.05 - 14.30│ │Varun Gupta │Online stochastic bin packing │
│14.30 - 14.45│Break │ │ │
│14.45 - 15.10│ │Bernd Heidergott│Statistics, Applied Probability and the Human Factor │
│15.10 - 16.10│Discussion│Future Research │ │
Chaitanya Bandi
Tractable Simulation-Optimization: A Robust Optimization Approach
To understand and optimize the performance of a system under uncertainty, the traditional approach consists of positing a probability distribution over the uncertain parameters, and derive
information on the system performance and optimize it. When the system’s dynamics are complex, computing the above expression analytically becomes challenging, and simulation becomes the only resort.
However, simulation models may take a considerable amount of time for the results to be statistically significant and can often be complex, making it difficult to understand and further optimize key
qualitative insights. Motivated by these challenges, we propose a robust optimization approach to analyzing and optimizing the expected performance of stochastic systems characterized by linear
dynamics, based on a robust optimization framework. Our approach consists of the following steps: (1) We model uncertainty via polyhedral sets inspired by the limit laws of probability. The size of
these sets is characterized by few variability parameters, which controls the degree of conservatism of the model. (2) We then cast the performance analysis problem as a optimization problem, for
instance maximizing a given performance measure subject to the system’s dynamics and the parametric uncertainty sets. The output of this optimization can be thought of as a worst case performance
measure function of the variability parameters. (3) Instead of computing the expected performance of the system via averaging simulation outputs, we assume the variability parameters follow limiting
distributions (derivatives of the normal distribution for light-tailed systems and the stable distribution for heavy-tailed systems) and average the worst case outputs from the optimization over a
discretized space over the variability parameters. (4) To optimize the stochastic system, we propose to cast the problem of finding the optimal design input via a robust optimization problem, e.g.,
minimizing the average or conditional value-at-risk of the worst case performance outputs. This framework allows a significant reduction in the dimension of uncertainty which provides grounds for
tractable analyses. We illustrate the tractability and accuracy of our approach by (a) simulating the transient behavior of multi-server feedforward queueing networks, and (b) determining optimal
base stock policies of generalized supply chain networks. This is joint work with Dimitris Bertsimas and Nataly Youssef.
Aharon Ben-Tal
Robust Solutions of Uncertain Optimization Problems under Ambiguous Stochastic Data
We show how robust optimization can provide tractable safe approximations to probabilistic constrains (chance constraints) even under partial information (ambiguity) on the random parameters. In
particular we address the case where the only available information is on means and dispersion measures. Unlike previous attempts, using the variance as the dispersion measure, here we derive tight
approximations using the MAD (mean absolute deviation). The theory is applied to problems in portfolio selection, inventory control and antenna array design.
Erick Delage
The Value of Distribution Information in Distributionally Robust Optimization
Decisions often need to be made with incomplete knowledge about some parameters of the problem. While formulating a stochastic model that captures accurately this knowledge can be a costly task,
solving a distributionally robust optimization model that makes use of historical data can provide useful guidance. Unfortunately, one might worry of being misled by distributionally robust solutions
and be tempted to invest in the acquisition of more data before committing to a decision. In this work, we propose tractable methods for bounding the value of additional (or more accurate)
distribution information in a two-stage linear program with objective uncertainty.
David Goldberg
Distributionally robust inventory control when demand is a martingale
Demand forecasting plays an important role in many inventory control problems. To mitigate the potential harms of model misspecification, various forms of distributionally robust optimization have
been applied. Although many of these methodologies suffer from the problem of time-inconsistency, the work of Klabjan et al. established a general time-consistent framework for such problems by
connecting to the literature on robust Markov decision processes.
Motivated by the fact that many forecasting models exhibit very special structure, as well as a desire to understand the impact of positing different dependency structures, in this paper we formulate
and solve a time-consistent distributionally robust multi-stage newsvendor model which naturally unifies and robustifies several inventory models with demand forecasting. In particular, many simple
models of demand forecasting have the feature that demand evolves as a martingale (i.e. expected demand tomorrow equals realized demand today). We consider a robust variant of such models, in which
the sequence of future demands may be any martingale with given mean and support. Under such a model, past realizations of demand are naturally incorporated into the structure of the uncertainty set
going forwards.
We explicitly compute the minimax optimal policy (and worst-case distribution) in closed form, by combining ideas from convex analysis, probability, and dynamic programming. We prove that at
optimality the worst-case demand distribution corresponds to the setting in which inventory may become obsolete at a random time, a scenario of practical interest. To gain further insight, we prove
weak convergence (as the time horizon grows large) to a simple and intuitive process. We also compare to the analogous setting in which demand is independent across periods (analyzed previously by
Shapiro), and identify interesting differences between these models, in the spirit of the price of correlations studied by Agrawal et al.
Varun Gupta
Online stochastic bin packing
In online bin packing, a sequence of items is revealed one-at-a-time and must be packed in a feasible bin (from a collection of infinite bins). The goal is to minimize the number of bins used at the
end of the packing horizon, the usual measure of performance being regret against the offline-optimal packing. In the stochastic version of the problem one further assumes that item sizes are i.i.d.
samples from an unknown distribution. We will first propose an algorithm that is the first asymptotically optimal algorithm while being truly distribution oblivious. In addition, we present three
flavors of robustness guarantees for non-i.i.d. sequences.
Bernd Heidergott (short talk)
Statistics, Applied Probability and the Human Factor
In the past decades we have witnessed a paradigm-shift from scarcity of data to abundance of data. How to deal with this data-revolution that is driven mainly by computer science and statistics? In
my opinion the most challenging question is how to marry data science with the wealth of knowledge on models in applied probability. Rather than discarding analytical models and the analysis thereof,
I advocate to build a shell around these models to allow for data dependency in a controlled way. Here, the word ``model'' refers to stochastic models describing a phenomenon such as, for example,
the stationary throughput in a queuing system. An instance of a stochastic model is obtained by choosing the actual values of parameters defining the underlying dynamics of the model. Parameter
insecurity refers to subjective probabilities (inferred by data) and I will argue that behavioral techniques should be incorporated into the analysis, thus bringing the human decision taker into the
picture. For an example from queuing theory I will show how the density of the model outcome under parameter insecurity can be evaluated using the technique of nested derivatives and how this allows
to jointly address the epistemological and aleatoric insecurity.
Dick den Hertog
Tutorial on robust optimization
This tutorial will provide you a basic understanding of practical robust optimization. Optimization problems in practice often contain parameters that are uncertain, due to e.g. estimation or
rounding. The idea of robust optimization is to find a solution that is immune against these uncertainties. The last two decades efficient methods have been developed to find such robust solutions.
The underlying idea is to formulate an uncertainty region for the uncertain parameters, and then require that the constraints should hold for all parameter values in this uncertainty region. It can
be shown that e.g. for linear programming, for the most important choices of the uncertainty region, the robust optimization problem can be reformulated as linear programming or conic quadratic
programming problems, for which very efficient solvers are available nowadays. In this tutorial we restrict ourselves to linear programming; extensions to nonlinear programming are briefly sketched.
We will treat the basics of robust linear optimization, and also show the huge value of robust optimization in (dynamic) multistage problems. Attention will also be given to important modelling
issues in robust optimization. We also shortly introduce distributionally robust optimization, since several talks in this workshop deal with this topic. Different applications of (adjustable) robust
optimization will be given in the tutorial. Robust optimization has already shown its high practical value in many fields: logistics, engineering, finance, medicine, etc. Some state-of-the-art
modelling packages have already implemented robust optimization technology.
Sandeep Juneja (short talk)
Selecting the best population using large deviations and multi-armed bandit methods
Consider the problem of finding a population amongst many with the smallest mean when these means are unknown but population samples can be generated. Typically, by selecting a population with the
smallest sample mean, it can be shown that the false selection probability decays at an exponential rate. Lately researchers have sought algorithms that guarantee that this probability is restricted
to a small d in order log(1/d) computational time by estimating the associated large deviations rate function via simulation. We show that such guarantees are misleading. Enroute, we identify the
large deviations principle followed by the empirically estimated large deviations rate function that may also be of independent interest. Further, we show a negative result that when populations have
unbounded support, any policy that asymptotically identifies the correct population with probability at least 1-d for each problem instance requires more than O(log(1/d)) samples in making such a
determination in any problem instance. This suggests that some restrictions are essential on populations to devise O(log(1/d)) algorithms with 1-d correctness guarantees. We note that under
restriction on population moments, such methods are easily designed. Further, under similar restrictions, sequential methods from multi-armed bandit literature can also be adapted to devise such
Nathan Kallus
Robust SAA and the Statistics of DRO
The sample average approximation (SAA) is the standard approach to data-driven optimization. While it enjoys computational tractability and asymptotic convergence guarantees, it is known to produce
unstable results and to lack strong, general finite-sample guarantees. We study a robustification of SAA via distributionally robust optimization (DRO) with respect to confidence regions of
goodness-of-fit (GoF) tests and explore its computational and statistical properties. We develop theory that intimately links guarantees and convergence of data-driven DRO to newly developed
statistical properties of GoF tests. This, along with considerations of computational tractability, guides our choice of tests in formulating DRO depending on the problem case and leads us to develop
a novel test for multivariate distributions that, when used in DRO, offers tractable optimization with both finite-sample and asymptotic guarantees. Our theory also allows us to analyze existing
data-driven DRO approaches and our statistical testing perspective suggests ways these can be improved in practice via applied statistical tools like the bootstrap. Examples from inventory management
and portfolio allocation demonstrate that this approach produces solutions that are stable and perform well even with little data, while at the same time converging as more data becomes available.
Ton de Kok
Ger Koole
Robust optimization for integrated call center planning
We consider a multi-period staffing problem in a single-shift call center. The call center handles inbound calls, as well as some alternative back-office jobs. The call arrival process is assumed to
follow a doubly non-stationary stochastic process with a random mean arrival rate. The inbound calls have to be handled as quickly as possible, while the back-office jobs, such as answering emails,
may be delayed to some extent. The staffing problem is modeled as a generalized newsboy-type model under an expected cost criterion. Two different solution approaches are considered. First, by
discretization of the underlying probability distribution, we explicitly formulate the expected cost newsboy-type formulation as a stochastic program. Second, we develop a robust programming
Daniel Kuhn
Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric
We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space
of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case
distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly
become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as
finite convex programs - in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample
performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization, uncertainty quantification and machine learning.
Henry Lam (short talk)
Melvyn Sim
A Practically Efficient Framework for Distributionally Robust Optimization with Recourse
We develop a modular and tractable framework to obtain approximate and possibly exact solutions to a distributionally robust optimization problem with recourse, where we minimize the worst-case
expected cost over an ambiguity set of probability distributions. Such a framework has numerous management inspired applications including, among other things, improving healthcare services and
reducing inventory costs. In characterizing uncertainty, we adopt the ambiguity set proposed by Wiesemann et al. (2014) and show that by restricting the recourse decision functions to those with
affine dependency on the actual and auxiliary random variables, we can improve the solutions of distributionally robust optimization problems with recourse. Using an algebraic modeling package we
have developed, we illustrate how our approach can be used to obtain high quality solutions to a class of appointment scheduling problems. This is a joint work with Dimitris Bertsimas and Meilin
Gideon Weiss (short talk)
Transient control of queueing networks, using continuous linear programs
Optimal control of a queueing network over a finite time horizon is an intractable task and can only be achieved approximately. We will survey two approaches: Controlling the fluid approximation of
the network and using a stochastic maximum pressure policy to track the fluid solution (following results of Nazarathy-Weiss), and formulation the problem as a robust optimization problem (following
results of Bertsimas-Nasarabaldi-Paschallidis). In either case this involves a centralized solution of a separated continuous linear program (SCLP). We will describe that nature of the exact solution
of this SCLP, and explain how the nature of the exact solution of the SCLP will help in implementing the optimal control. This supports using a simplex type algorithm to solve the SCLP (following
Weiss and Shindin).
Ralf Werner (short talk)
Consistent uncertainty sets and consistent robust estimators
In statistics one is interested in the asymptotical properties of point estimators. From this perspective we provide a fresh look at robust estimators obtained as solutions to robust optimization
For this purpose we introduce the notion of a consistent uncertainty set. We illustrate that - together with continuity of the solution in the problem parameters - this is the right notion to obtain
strong consistency of the robust estimator.
● Venue
Eurandom, Mathematics and Computer Science Dept, TU Eindhoven,
De Groene Loper 5, 5612 AZ EINDHOVEN, The Netherlands
Eurandom is located on the campus of Eindhoven University of Technology, in the Metaforum building (4th floor)
. The university is located at 10 minutes walking distance from Eindhoven main railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).
Accessibility TU/e campus and map.
Registration is closed.
● Accommodation
For invited spekaers, we will take care of accommodation. Other attendees will have to make their own arrangements.
We have a preferred hotel, which can be booked at special rates. Please email Patty Koorn for instructions on how to make use of this special offer.
For other hotels around the university, please see: Hotels (please note: prices listed are "best available").
More hotel options can be found on the webpages of the Tourist Information Eindhoven, Postbus 7, 5600 AA Eindhoven.
● Travel
For those arriving by plane, there is a convenient direct train connection between Amsterdam Schiphol airport and Eindhoven. This trip will take about one and a half hour. For more detailed
information, please consult the NS travel information pages or see Eurandom web page location.
Many low cost carriers also fly to Eindhoven Airport. There is a bus connection to the Eindhoven central railway station from the airport. (Bus route number 401) For details on departure times
consult http://www.9292ov.nl
The University can be reached easily by car from the highways leading to Eindhoven (for details, see our route descriptions or consult our map with highway connections.
● Conference facilities : Conference room, Metaforum Building MF11&12
The meeting-room is equipped with a data projector, an overhead projector, a projection screen and a blackboard. Please note that speakers and participants making an oral presentation are kindly
requested to bring their own laptop or their presentation on a memory stick.
● Conference Secretariat
Upon arrival, participants should register with the workshop officer, and collect their name badges. The workshop officer will be present for the duration of the conference, taking care of the
administrative aspects and the day-to-day running of the conference: registration, issuing certificates and receipts, etc.
● Cancellation
Should you need to cancel your participation, please contact Patty Koorn, the Workshop Officer.
● Contact
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Last updated 12-11-15,
by PK
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
tel. +31 40 2478100
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Spiral SIRs Form Dual-Band Filter
Bandpass filters are essential components in wireless systems design. While the filtering function is critical in removing interference, spurious, and other unwanted signals, the physical size
required by many filter designs is often a limiting factor for many systems architectures. Fortunately, the authors have developed a complact microstrip bandpass filter (BPF) that is broadband enough
to handle the two bands of wireless-local-area-network (WLAN) systems in the 2.45- and 5-GHz bands. The design features spiral-shaped, quarter-wavelength stepped-impedance resonators (SIRs) in order
to achieve a remarkable small size. To demonstrate the capabilities of the design approach, a second-order BPF was modeled, fabricated, and tested. It measures only 5.95 X 6.2 mm^2 but provides
outstanding performance in the dual WLAN bands.
In modern wireless communications, many end-user facilities, such as personal digital assistants (PDAs), laptop computers, and mobile telephones, operate according to different communications
standards and in different frequency bands. For a system working in two different frequency bands, dual-band BPFs are an important building block in the transceiver circuitry. A number of different
dual-band BPFs have been reported recently.^1-4 For example, one research group proposed a Z-transform technique for the design of single- and dual-band BPFs, in which multiple quarter-wavelength
open and short stubs were used.^1
Although the approach is efficiently systematic and flexible, it can lead to BPFs that are too large for practical application. For miniaturization purposes and for achieving a wider stopband beyond
the desired second passband, stepped-impedance resonators (SIRs), originally proposed in ref. 5, have been widely applied in dual-band BPF designs.^2,3 In ref. 2, the electric and magnetic coupling
schemes were combined to obtain the required coupling coefficients for the two passbands. In ref. 3, the researchers adopted half-wavelength hairpin SIRs to create cross-coupled dual-band BPFs. The
proper coupling coefficients required for the two designed passbands can be obtained by adjusting both the coupling length and the coupling gap between adjacent resonators, as well as by using the
two different SIRs.
For the work presented in refs. 2 and 3, the half-wavelength SIRs lead to a relatively large filter size compared to those based on quarter-wavelength SIRs; the half-wavelength SIRs can also result
in an undesirable third passband very close to the second passband of a dual-band BPF. In ref. 4, a novel dual-behavior-resonator technique with liquid-crystalpolymer (LCP) system-on-package
technology was used to design an asymmetrical dual-band BPF which can be operated in the ISM band from 2.4 to 2.5 GHz and the UNII band from 5.15 to 5.85 GHz. Unfortunately, this design suffered from
inadequate DC blocking and allowed DC signals to freely pass through.
The authors have developed a dualband BPF that overcomes the limitations of this design. It is based on compact, quarter-wave SIRs and aimed at WLAN applications operating at the 2.45- and
5.2-to-5.8-GHz bands. With this essential design, a third passband (i.e., the first spurious response) can be located around a frequency higher than that of the corresponding BPF designed using
half-wavelength SIRs. The resulting filter will have a wider rejection band beyond the desired second passband.
To achieve a high degree of miniaturization, the BPF's quarter-wavelength SIRs are bent into a spiral shape. A quarter-wavelength SIR BPF was fabricated on Duroid 6010 printed-circuitboard (PCB)
material from Rogers Corp. (www.rogerscorp.com) in an area measuring only 5.95 X 6.2 mm^2. Compared to predictions made using the High-Frequency Structure Simulator (HFSS) three-dimensional
electromagnetic (EM) simulation software from Ansoft Corp. (www.ansoft.com), the measured performance is in close agreement.
A quarter-wavelength SIR can be constructed from two transmission-line sections having different line widths (Fig. 1). The line section with one end shorted to the ground through a viahole has a
characteristic impedance of Z[1 ]and an electrical length of θ[1]. The line section with an open end has a characteristic impedance of Z[2 ]and an electrical length of [2]. The parallel resonance
condition of the SIR was derived in ref. 5 as:
R[z ]= the impedance ratio.
For analysis, let the nth resonant frequency of the SIR be denoted by f[n]. In
the dual-passband filter design, the first two resonance frequencies, f[1 ]and f[2], are usually chosen to be the center frequencies of the two desired passbands. Then, the third passband with center
frequency f[3 ]represents the spurious response. With the assumption that these two line sections have the same electrical length, i.e., θ[1]= θ[2]= θ[0], the frequency ratio of f[2 ]to f[1 ]can be
expressed as:^5
and a similar analysis reveals that f[3 ]= f[2 ]+ 2f[1 ]for the quarter-wavelength SIRs. For the dual-band BPF with the center frequencies located at 2.45 and 5.5 GHz, R[z ]is found to be 2.11 for
the quarter-wavelength SIRs, whose corresponding third resonant frequency is f[3 ]= 10.4 GHz. For half-wavelength SIRs, the frequency ratio of f[2 ]to f[1 ]can be found as:^5
An extended analysis indicates that that f[3 ]is related to f[1 ]and f[2 ]by f[3 ]= 2f[2 ]– f[1]. If the BPF for the same values of f[1 ]and f[2 ](2.45 and 5.5 GHz) is designed using half-wavelength
SIRs, its third resonant frequency will be 8.55 GHz, which is lower than that of the BPF using quarter-wavelength SIRs. Hence, besides having the advantage of a smaller circuit size, the filter using
quarter-wavelength SIRs can have an upper rejection band wider than that of the filter using half-wavelength SIRs.
The external quality factor of the tapped quarter-wavelength SIR shown in Fig. 2 can be expressed as:^6
B(f) = the total susceptance of the resonator seen by the feed line at the tapping point,
f[i ]= the ith passband center frequency, and
Y[0 ]= the characteristic admittance of the feed line.
Page Title
This susceptance is the sum of the susceptance values looking from the tapping point toward the open end and the grounded end of the quarter-wavelength SIR. The total susceptance depends on the
tapping position that is physical length t or electrical length away from the via, as does the external quality factors.
For 0 <φ< θ[0], the total susceptance at the tapping point of the quarter-wavelength SIR can be derived as:
From Eqs. 4 and 5, Q[e ]as a function of frequency and the tapping-location related physical length t can be determined.
Using the proposed design approach, a microstrip BPF was designed on 0.635mm-thick Duroid 6010 substrate with a dielectric constant of 10.2 and loss tangent of 0.0023. The quarter-wavelength SIRs are
bent into a spiral shape for compactness. Line segments of the spiral-shaped SIR (Fig. 3) have widths of W[1](W[2]) and characteristic impedance of Z[1 ](Z[2]). Width W[1 ]was preselected to be 1.2
mm for characteristic impedance, Z[1], of 33.1 Ω. Width W[2 ]was then found to be 0.4 mm after performing fine tuning in a computer simulation. Lengths l[1], l[2], l[3], l[4], and l[5 ]were
determined to be 4.9, 1.75, 4.65, 0.85, and 3.55 mm, respectively. For this experiment, the diameter, D, of the viahole was fixed at 0.6 mm.
In designing a second-order dualband BPF with SIRs, the coupling coefficient between the two SIRs for a prescribed filter function can be evaluated using the relationship:^6
g[i ]= the element value of the second-
order low-pass filter prototype, and
Δ[i ]= the fractional bandwidth of the ith passband (defined as the ratio of 3dB bandwidth to the corresponding passband center frequency) of the BPF.
Figure 4 shows the simulated fractional bandwidth (BFW) versus the gap distance (d) between the two SIRs arranged in an anti-parallel coupled-line (APCL) configuration (see the inset).
For the tapped SIR at the input stage, Q[ei ]= g[0]g[1]/Δ[][i ](from ref. 6); at the output stage, Q[ei ]= g[0]g[2]/ [i]. For a given set of and Q[e1 ]and Q[e2], the length t (see Fig. 2) associated
with the tapping location can be determined by solving Eq. 4 with the substitution of Eq. 5. Since parameters θ[0 ]and φvary with frequency, the length t to be determined for f[1 ]is in general
different from that for f[2 ]even if Q[e1 ]is identical to Q[e2]. Thus, the average of the t value computed for f[1 ]and that for f[2 ]can be taken as a compromise. Because Eq. 5 was derived with all
the discontinuity effects neglected, the averaged t value still must be fine tuned in a full-wave simulation (such as HFSS), in which the viaholes and all other discontinuity effects can be taken
into account.
For the purpose of evaluating this design approach, a quarter-wavelength SIR dual-band BPF was designed and fabricated. Figure 5 shows simulated responses for the second-order Butterworth filter. The
filter was designed with Δ[1 ]= 8.5 percent for the 2.45-GHz band and Δ[2 ]= 19 percent for the 5.5-GHz band. The distance d can be determined to be 0.3 mm (see Fig. 4) for both of the designed
passbands. Also, the required Q[e1 ]and
Q[e2 ]values are found to be 16.63 and 7.44, respectively, which leads to the optimized length of t = 2 mm after fine-tuning the values in simulations performed in the full-wave simulator (HFSS). The
measured fractional bandwidths (the percent of the full band over which minimum insertion loss occurs) for the 2.45 and 5.5 GHz bands are 8.16 percent (with 1.11 dB insertion loss) and 19.09 percent
(with 0.78 dB insertion loss), respectively, which agree very well with the simulated data of 8.57 percent (with 0.95 dB insertion loss) and 19.27 percent (with 0.91 dB insertion loss).
The measured center frequency of the third passband is around 10 GHz, which is very close to the frequency of f[3 ]= 10.4 GHz predicted in the previous section. The transmission zero due to the APCL
section (ref. 7) helps enhance the rejection of the stopband that lies between the two passbands. The measured transmission zero (around 3.09 GHz) only slightly deviates from the simulated one (at
3.03 GHz). Figure 6 shows a photograph of the fabricated BPF circuit, which measures only 5.95 X 6.2 mm^2 on the PCB. Data for the simulated and measured dual-band BPF responses are summarized in the
This report has proposed a compact dual-band microstrip BPF design using spiral quarter-wavelength SIRs for the ISM (2.4-to-2.5-GHz) and UNII (5.15 -to-5.85-GHz) frequency bands. The design was
implemented in a size of only 5.95 X 6.2 mm^2 to validate the approach. The first spurious passband around 10.4 GHz was found to be very far away from the second passband; thus, a wide upper stopband
was achieved. The transmission zero produced by the APCL section of the filter was purposely located between the two desired passbands to achieve good rejection between passbands. The measured data
showed not only that the fabricated BPF not only offers satisfactory response, but that those measurements agree closely with the computer-simulated results.
1. L.C. Tsai and C.W. Hsue, "Dual-band bandpass filters using equal-length coupled-serial-shunted lines and Ztransform technique," IEEE Transactions on Microwave Theory & Techniques, Vol. 52, No. 4,
pp. 1111-1117, April 2004.
2. H.M. Lee, C.R. Chen, C.C. Tsai, and C.M. Tsai, "Dual-band coupling and feed structure for microstrip filter design," IEEE MTT-S Digest, Vol. THIF-36, pp. 1971-1974, 2004.
3. J.T. Kuo and H.S. Cheng, "Design of quasi-elliptic function filters with a dual-passband response," IEEE Microwave Wireless Components Letters, Vol. 14, No. 10, pp. 472-474, Oct. 2004.
4. V. Palazzari, S. Pinel, J. Laskar, L. Roselli, and M.-M. Tentzeris, "Design of an asymmetrical dual-band WLAN filter in liquid crystal polymer (LCP) System-On-Package technology," IEEE Microwave
Wireless Components Letters, Vol. 15, No. 3, pp. 165-167, March 2005.
5. M. Makimoto and S. Yamashita, "Bandpass filters using parallel coupled stripline stepped impedance resonators," IEEE Transactions on Microwave Theory & Techniques, Vol. 28, No. 12, pp. 1413-1417,
December 1980.
6. J.S. Hong and M.J. Lancaster, Microstrip filters for RF/Microwave applications, Wiley, New York, 2001.
7. C.-M. Tasi, S.-Y. Lee, and H.-M. Lee, "Transmission-line filters with capacitively loaded coupled lines," IEEE Transactions on Microwave. Theory & Techniques, Vol. 51, No. 5, pp. 1517–1524, May
Sponsored Recommendations | {"url":"https://www.mwrf.com/technologies/components/article/21842995/spiral-sirs-form-dual-band-filter","timestamp":"2024-11-04T15:03:29Z","content_type":"text/html","content_length":"264810","record_id":"<urn:uuid:86fe77b0-0f0c-452e-86a4-724568ed6def>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00857.warc.gz"} |
Heat capacity and Specific heat in Thermodynamics
Chemical Thermodynamics: Heat capacity and Specific heat
Heat capacity is the amount of heat(measured in Joules or Calories) required to raise a unit amount of substance (measured in grams or moles) the temperature of an object by 1°C temperature (measured
in C or K). The heat capacity of an object is
$C\equiv Q/\Delta T.$
This addition of energy or heating can be carried out at constant volume or constant pressure. At constant pressure, some of the heat supplied goes into doing work of expansion and less is available
with the system (to raise its temperature).
Heat capacity at constant Volume (C[V]):
${C}_{V}=\left(\partial E/\partial T{\right)}_{V}$
It is portrayed as the slope of the plot of internal energy with temperature.
Heat capacity at constant Pressure (C[P]):
${C}_{P}=\left(\partial H/\partial T{\right)}_{P}$
It is portrayed as the slope of the plot of enthalpy with temperature.
Water has a high heat capacity (of C[P] = 4186 J/K/mole =1 Cal/C/Kg) and it heats up slowly in comparison to air (with heat capacity, C[P] = 29.07J/K/mole). Consequently, oceans heat up slowly as
compared to the atmosphere.
As T – > 0K, the heat capacity tends to zero. Hence, near 0 Kelvin minimal heat is required to raise the temperature of a sample.
Specific heat is defined as the amount of energy required to raise the temperature of 1 gram of a substance by 1°C under standard pressure. Also, the specific heat capacity is the heat capacity per
unit mass, . Accordingly, heat capacity can be computed for any combination of conditions such as constant V, constant P, constant P&V or other intensive or extensive properties.
Heat capacity is an intrinsic physical property of a substance. It measures the amount of heat required to change that substance’s temperature by a specific amount. In the International System of
Units (SI), heat capacity is denoted as joules per kelvin (J\K). Likewise, heat capacity is also is dependent on the size and the mass of substance under consideration.
There are two derived quantities that specify heat capacity as an intensive property (i.e., independent of the size of a sample) of a substance: the molar heat capacity, which is the heat capacity/
mole of a pure substance.
Molar heat capacity is subjected to pressure or volume respectively and it is often designated as C[P], to denote heat capacity under constant pressure conditions, as well as C[V], to denote heat
capacity under constant volume conditions. The specific heat capacity is also called specific heat, which is the heat capacity per unit mass of a pure substance.
Calculating Specific Heat Capacity
Figure 1: Calculation of specific heat capacity
Some common specific heat and heat capacities:
Substance S (J/g ºC) C (J/ºC) for 100 g
Air 1.01 101
Aluminum 0.902 90.2
Copper 0.385 38.5
Gold 0.129 12.9
Iron 0.450 45.0
Mercury 0.140 14.0
NaCl 0.864 86.4
Ice 2..03 203
Water 4.179 417.9
A derivation of heat capacity and specific heat capacity is associated with the quantity of the substance involved in the reaction. Wherein, the amount of substance is expressed as moles ‘µ’ and we
can define the heat capacity per mole of a substance by:
${C}^{‘}=C/\mu =∆Q/\mu ∆T$
mathrm{is}\mathrm{J}/\left(\mathrm{mol}\bullet \mathrm{K}\right)$ | {"url":"https://www.w3schools.blog/heat-capacity-and-specific-heat-in-thermodynamics","timestamp":"2024-11-05T16:54:18Z","content_type":"text/html","content_length":"173864","record_id":"<urn:uuid:6b857b66-f866-4348-bfa9-e676af59a301>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00826.warc.gz"} |
Shop Information
ASVAB Shop Information Practice Test 345871
Question 3 of 5
Soldering is a low-temperature process by which two or more items (typically metal) are joined together by melting a filler metal (solder) into the joint. An electrically powered soldering iron or
soldering gun is used to melt the solder which is an alloy of lead and tin that has a melting point below the melting point of the items being joined. A chemical cleaning agent called flux is also
used to clean the surfaces before soldering.
solder is an alloy of lead and tin
soldering can only join metals
flux is used to clean surfaces before soldering
solder has a comparatively low melting point | {"url":"https://www.asvabtestbank.com/shop-information/practice-test/345871/q/5/3?c=","timestamp":"2024-11-04T14:55:34Z","content_type":"text/html","content_length":"11048","record_id":"<urn:uuid:a0de6777-c84b-4a3b-9b62-0a063fafd5c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00634.warc.gz"} |
5 - Safari Tour of Modern ML Models
Series: A (Slightly) Technical Intro to AI
1. A Safari Tour of Modern ML Models
The power we now possess with deep neural networks opens many doors. We’ll explore several important variations on this theme — convolutional NNs, recurrent NNs, GANs, reinforcement learning — that
each structure the neural network to take advantage of some inherent aspect of the data or problem at hand. For good measure, we’ll also take a look at some other popular and important, but
non-deep-learning, ML models.
Deep learning rules everything around me
Convolutional neural nets
Convolutional neural networks (CNNs) take advantage of the fact that within a single observation in the data, there may be some inherent structure. For example, the position of facial features in
facial images.
Using our cat/dog image classification example, and with the rough outline of how an actual NN works in your mind, consider what the structure of a NN classifier should look like. The input layer is
an image — square, perhaps, like \(n\times n\) pixels. The output is two values: probability of image being a dog and probability it is a cat.
It turns out to be helpful to gradually winnow down the large input layer through successively smaller hidden layers — this helps us keep track of that inherent structure. For example, in the first
hidden layer, have each node aggregate the input from, say, a 4-pixel subsquare. If the input images are \(32\times 32=1,024\) nodes, the first hidden layer would only need 64 nodes. We could
continue this condensing of the input layer all the way to the output layer. (Actually, we typically allow some overlap in the subsquare patches, and we don’t use the same size patch in each hidden
layer, and images are more often input with a third dimension representing depth or color, but you get the idea.) This act of combining and pooling information is known as convolution, thus the name.
This all gives a network architecture something like this:
Interestingly, we usually observe each hidden layer starting to specialize in different aspects of the image recognition task: the first layer might discriminate what part of the image is the face
vs. the background, while the second layer discriminates facial features, etc. And magically, this happens simply by setting up the network structure (i.e. the mathematical model), giving it some
training data, and executing a parameter estimation technique like stochastic gradient descent.
If you recall our discussion about model complexity and regularization, notice that by forcing a more restrictive structure on our CNN, we’ve automatically decreased the model complexity in a
purposeful way from the naively fully connected NNs in the previous post.
Recurrent neural nets
CNNs are designed for input data where each observation has inherent internal structure. What if successive observations have inherent relationships, like prices over time, or words in a sentence?
This leads us to recurrent neural networks (RNNs).
The big structural idea of an RNN is that each layer passes its values not just forward into the next layer like before, but also backward into previous layers. This allows the network to retain a
sort of memory of past events.
As you might expect, RNNs get applied in many settings with an underlying time sequence (“time series data”), like stock forecasting (although nota bene I am not recommending this). They’re also very
successful in natural language processing (NLP): after all, is a sentence anything more than a sequence of words (inputs), each of which shares inherent relationships with its predecessors? What
comes next in this sentence: “I live in India. I speak _____.” To accurately predict the blank, a model needs a short-term memory to recall the word “India” that came earlier in the phrase, and a
long-term memory to recall that likely fill might be “Hindi.”
Consider this little experiment: we could give the RNN a word, have it predict the next word, feed that word back into the RNN, and repeat … generating entire passages. Here’s an example of doing
this with an RNN that was trained on the works of Shakespeare:
What’s even more amazing about the above example is the network was generating the passage not word by word, but letter by letter.
Generative Adversarial Networks (GANs)
One of the most mesmerizing innovations of NNs to me are Generative Adversarial Networks, or GANs. Here’s an example of the idea: design NN #1 to take some generic image input and output a
convincing, more detailed variation on that image. Then design NN #2 to take an image and classify it as real or fake. Finally, use the feedback of NN #2 to improve the faking ability of NN #1, and
vice versa. You have now placed two powerful ML algorithms in mortal combat to out-optimize each other, and the results can be terrifying.
A fairly benign application of this innovation is to allow computers to generate convincing images or videos for gaming, backgrounds, etc. Here are three faces generated by a GAN, and do not
represent the faces of real people. Pretty convincing.
A more (potentially) nefarious application is to have computers generate convincing false video, for example, of a state leader denouncing a neighbor government — these are termed “deep fakes.” (And
now you know what “deep” really means!) Here’s a video of a GAN superimposing a person’s face on an actor in real time.
Reinforcement learning (RL)
Reinforcement learning (RL) traces its origins to the 1950s, and it is a different animal than the models we’ve discussed so far: it’s not really supervised, since we will give RL models inputs but
only occasionally give it outputs (so we say “semi-supervised”), and it’s not inherently NN-based, although we’ll see its modern popularity has grown after infusing it with NNs. RL has recently beat
the world grandmaster at Go, it has self-taught simulations how to walk, it is at the heart of the algorithms which control autonomous vehicles, and seems to be making inroads in stock trading.
RL is the idea that we can take a (computer) agent, allow it to take actions in some state where it receives rewards for certain actions, and over time, it will learn to act nearly optimally. This
means we don’t need petabytes of data anymore, we just need to be able to enforce the rules of an environment and allow the agent enough time to explore and learn in that environment.
For flavor, let’s try to wrap our heads around a common brand of RL called Q-learning. Imagine you are a computer trying to learn to play chess. For any given board position, you have a few dozen
options of possible moves. You have no idea which move is better than any other, so you just play randomly and lose a lot of games. Each time you lose, you go back along the trail of decisions that
got you there and mark them as bad, being less pessimistic the further back you go, since your first few moves contributed less to your loss than your final moves, and are therefore less bad.
Finally, you win a few games, and update those trails as good. You might even allow some intermediate rewarding of actions if nice things happen like capturing the queen or setting up a really juicy
fork. (This is the idea behind dynamic programming and more specifically, Bellman’s equation, the essential piece of math that makes RL work.)
You are a computer, so you work at this tirelessly for several million games. You record your lessons learned in a table, called a Q table: on the left, all the millions of board positions you have
encountered, along on the top, all the possible actions one may take in chess, and in each cell, the “goodness” of that action from that position in terms of the final reward (win/lose). We call this
“goodness” a q-value. If we play enough games (our training step), these values will allow us to make pretty good decisions and win games (our predict step).
But …. chess is too complicated to try listing all possible moves and board positions, much less train on all of them. This limitation of Q-learning kept it mostly on the shelf, despite its discovery
in 1989.
Enter Google’s DeepMind research laboratory, which in 2013 applied deep neural networks to Q-learning and started turning heads. First it learned to play Atari at human levels, and later it was a
piece of the AlphaGo algorithm that defeated Lee Sedol at Go.
Remember our Q table? Think of it as a function, or black box: input a state and an action, output a q-value. As we know, neural networks are very good at representing very complex functions.
DeepMind’s insight was to do Q-learning, but train a deep NN to mimic the Q-table along the way, so that when training is complete, even though you haven’t seen a certain state-action pair, your NN
can give you an approximate value for it. Just like our linear model in Part 2 had never seen our ad spending for the upcoming summer, it could provide a prediction.
RL is a huge, active field, and deep Q-learning is just one variant, and it has broad, powerful applications as mentioned earlier. RL is also perhaps the most compelling of the algorithms we’ve
explored because it seems so humanlike. AlphaZero has learned to beat its predecessor by training purely onself-play. Recently, DeepMind used RL to propose a new understanding of the reward
mechanisms of our brain. One might even trace RL’s heritage back to experiments in the 1950s to replicate animal learning with computers. Often, when people begin breathlessly speculating about the
future potential of AI, a deep-RL algorithm has made a recent impression on their mind.
Other Hall of Famers
Not all of what you hear about in modern ML falls in the category of deep learning, however. Let’s try to get the flavor of a few other popular models.
Ensemble learning
Random forests
Imagine you are trying to develop a classification model to help decide what applicants to hire for your company, based on historical data of successful hires. This has some tricky ethical
considerations we’ll discuss later, but for now, consider the simplest possible classification model: take their work experience, and if it’s more than 5 years, hire them, if not, don’t. Based on
this single decision, you could examine past data and assign some sort of accuracy score to this model, like what percentage of people with more than 5 years experience were successful hires. You
could extend this idea and come up with additional criteria: what level education do they have, and so on. This type of model is called a decision tree, and unsurprisingly, we can have a machine
learn what these questions and thresholds should be to optimize the accuracy, just as with all our previous models.
Instead of just one decision tree, with large and complex data it is often better to create many, many decision trees, each focused on a random different subset of input variables, and then average
their outputs. This is called a random forest (get it? lots of trees make a forest??). The technique of re-using parts of a dataset in different ways is called bootstrapping, and the averaging at the
end is called aggregating, so random forests are a type of ensemble learning called bootstrap aggregating, or “bagging.”
Like NNs however, the cost of increasing accuracy from a decision tree to a random forest is a loss in interpretability.
Boosted methods
Another type of ensemble learning is boosting. The idea here is to train a sequence of small decision trees, and in each one, focus on the part of the dataset which the previous trees had the hardest
time accurately classifying. Although this method can be sensitive to outliers, it is generally considered one of the strongest “out of the box” models, requiring very little experienced tweaking to
get top-shelf results.
Bayesian methods
Lumping all of “Bayesian methods” into a subsection is a bit disingenuous, since there are Bayesian versions of nearly every model we’ve discussed so far, so in a sense Bayes has been with us all
Bayes’ theorem, named after the Reverend Thomas Bayes, says that if you start off believing a coin is fair, but then you observe a few dozen heads flipped in a row, you no longer believe it’s very
fair. Well, his theorem says this in a much more general, probabilistic way, but the idea is: we have some prior belief about the probability of a certain hypothesis \(H\), we have some likelihood of
some event (or evidence) \(E\) happening given that prior, and we have the posterior probability of \(H\) being true after having observed \(E\). In the language of probability,
\[P(H|E) \propto P(E|H)\cdot P(H)\]
where the notation \(P(H\vert E)\) reads “probability of \(H\) given \(E\)” and the little \(\propto\) means “proportional to.”
(Notice we started talking about “belief”. I thought this was math! There is a whole tribal rivalry between so-called “Bayesians” and “frequentists” that leads to different interpretations of Bayes’
theorem and the implications it has on statistics and life itself. It is fundamentally important to everything we’ve been discussing, and so naturally I will completely bypass here.)
Anyway, one big idea in Bayesian approaches is to incorporate a prior belief in your model: in our linear model, we just let our parameter values be whatever they wanted, but in a Bayesian approach
we would specify in advance that we would much prefer the coefficients on the large degree terms to be quite small. This is the Bayesian way of regularizing model complexity, and in many cases,
certain priors lead to identical results as the non-Bayesian approach.
Another big idea in Bayesian approaches is to never discard information as you’re building and training and testing a model. Specifically, you specify a probability distribution for any unknown
value, like a parameter, which describes how likely different values are (the classic “bell curve” is a type of probability distribution). Your prior, your posterior, all need distributions, and the
posterior’s gets “updated” as you examine more data (what we have been referring to as “training”). Instead of your final model being a single line, your final model is a distribution of possible
lines, weighted by which ones suit the data better. And if you need your prediction to be, like, a single number so you can actually take an action, Bayesians say, ok fine, if you must, use the mean
(average) of the distribution.
Unfortunately, these ideas, while beautiful, lead to extremely difficult calculations when we apply them to Reverend Bayes’ little theorem. As a result, Bayesian approaches usually involve lots of
heavy computations and simulations — techniques like Gibbs sampling, or Markov chain Monte carlo (MCMC), or variational inference.
Bayesian approaches tend to have better performance in many applications, and are often less used only because of these mathematical and computational challenges, not any philosophical prejudices.
(As an aside, a non-philosophical objection I have read to Bayesianism is that it relies on well-behaved distributions so that the posterior will converge to something meaningful after a reasonable
amount of evidence/data — this is fine for well-behaved things like language processing, but a flawed assumption when extremal events are more likely, like the stock market.)
Dimension reduction
Most datasets are high dimensional, that is, many variables for each datapoint. A hundred years ago, a dataset of human measurements would probably contain height and weight for each person. Now, a
laser scan could provide us hundreds of measurements per person, from upper torso length to circumference of right wrist. Although, there won’t be much difference between the “right leg length” and
“left leg length” values, do we really lose much by replacing both with an average, and reducing the dimension of our problem by one?
There are principled ways to approach this dimension reduction problem. Possibly the best way is low-tech: use domain expertise to manually select or create a better set of features out of the
available raw data. But there are automated ways as well: the averaging approach is a simplistic version of a family of techniques that project the dataset down into a lower dimensional subspace
(think of the projection like a 2-D shadow of a 3-D object).
For example, principal component analysis is a technique from classic statistics that finds a flatter representation of a dataset which maximizes the amount of variation still explained by the
smaller dataset.
In the image above (taken from Elements of Statistical Learning), a set of points in 3-D (left) is projected onto a plane, resulting in new coordinates in 2-D (right).
More recent methods take different approaches: for example, t-SNE tries to find a low-dimension representation of the data such that, roughly, if two points are similar (close) in the original
dataset, they are likely to be close in the smaller one.
In the image above (credit: Nicola Pezzotti), t-SNE is applied to a high-dimensional dataset consisting of images of handwritten digits (the famous MNIST dataset). In the resulting low (2)
dimensional image, the digits naturally separate into homogeneous groupings, which are colored to demonstrate the stark groupings. The somewhat magical part is that t-SNE produces this without access
to the image labels, only the raw images themselves.
This brings us to the final category of ML models we’ll get familiar with: clustering methods. The essential idea is to group a set of datapoints in such a way that datapoints in the same group (or
“cluster”) are more similar to each other than datapoints in other clusters. Similar to dimension reduction, this is an unsupervised learning task because we are not associating the data with any
sort of label or target value — in fact, we want to discover the labels ourselves, through some intrinsic pattern that we believe is hiding in the data.
There are two essential tasks here: first, define what “similar” means and second, figure out a way to find a good grouping of objects without exhaustively trying every possible combination. One
example of a similarity measure is the squared difference that we were using to measure error in the least squares model (this is called the Euclidean distance). One example of finding good groupings
is to start by finding all the closest pairs, then pair those pairs, etc., and stop joining groups together when the distances become unreasonably big, whatever that means for you (this is called
hierarchical clustering).
Applications of this idea abound: market research (finding groupings in a customer base, or finding customers with similar tastes to make product recommendations), bioinformatics (automating
genotyping), social networks (finding social network structure), image segmentation (like border detection), and on and on.
Here’s a playful clustering I did a couple years ago of NFL wide receivers based on similar distributions of yardage gains.
Notice how “explosive players” like Randall Cobb and Doug Baldwin get grouped together, while players used more for checkdown plays like Tavon Austin or Cole Beasley get a different group. (I know,
it’s ridiculous.)
Okay, you’ve made it through all the technical bits of this series, let’s close with some nice, non-technical pontificating about the limitations of ML and where the experts think the future lies.
1. A Safari Tour of Modern ML Models
Written on April 5th, 2020 by Steven Morse | {"url":"https://stmorse.github.io/journal/ai-5.html","timestamp":"2024-11-03T19:55:36Z","content_type":"text/html","content_length":"31545","record_id":"<urn:uuid:1017cde2-4d3d-4670-9895-6102b0d6b5b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00306.warc.gz"} |
One Step Addition Equations - Tessshebaylo
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This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://www.tessshebaylo.com/one-step-addition-equations/","timestamp":"2024-11-12T16:10:26Z","content_type":"text/html","content_length":"59225","record_id":"<urn:uuid:e308c963-b0f9-447f-b4aa-8a9338f2bdac>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00418.warc.gz"} |
Re: How F Ratio Value is determined for Brown-Forsythe and Levene's test for Equal VarianceRe: How F Ratio Value is determined for Brown-Forsythe and Levene's test for Equal Variance
I had evaluate the JMP output for Brown-Forsythe and Levene's test for Equal Variance under Fit X-Y analysis module. The values from the JMP Fit X-Y are F-Ratio, DOF for factor, DOF for sample, and
Prop>F. I performed Levenes and Brown-Forsythe test in excel using the single factor ANOVA module in the data analysis add-on to complete the Levenes and Brown-Forsythe analysis. The F-ratio from JMP
and the excel F values are different. If the F-ratio value is different from the F value, can the F-ratio be translated into an F-value? If F-ratio and F value are the same, why are there
discrepancies between excel and JMP values? Any insight would be greatly appreciated. | {"url":"https://community.jmp.com/t5/Discussions/How-F-Ratio-Value-is-determined-for-Brown-Forsythe-and-Levene-s/m-p/783260","timestamp":"2024-11-09T07:12:36Z","content_type":"text/html","content_length":"681182","record_id":"<urn:uuid:7e1a260a-9343-4665-a89f-ac83f07cddbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00574.warc.gz"} |
Nonstochastic Bandits with Composite Anonymous Feedback
Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni, Claudio Gentile, Yishay Mansour; 23(277):1−24, 2022.
We investigate a nonstochastic bandit setting in which the loss of an action is not immediately charged to the player, but rather spread over the subsequent rounds in an adversarial way. The
instantaneous loss observed by the player at the end of each round is then a sum of many loss components of previously played actions. This setting encompasses as a special case the easier task of
bandits with delayed feedback, a well-studied framework where the player observes the delayed losses individually. Our first contribution is a general reduction transforming a standard bandit
algorithm into one that can operate in the harder setting: We bound the regret of the transformed algorithm in terms of the stability and regret of the original algorithm. Then, we show that the
transformation of a suitably tuned FTRL with Tsallis entropy has a regret of order $\sqrt{(d+1)KT}$, where $d$ is the maximum delay, $K$ is the number of arms, and $T$ is the time horizon. Finally,
we show that our results cannot be improved in general by exhibiting a matching (up to a log factor) lower bound on the regret of any algorithm operating in this setting. | {"url":"https://www.jmlr.org/papers/v23/21-1443.html","timestamp":"2024-11-13T21:47:49Z","content_type":"text/html","content_length":"6473","record_id":"<urn:uuid:04b221e7-de6f-4492-a4a9-ae87690d4032>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00232.warc.gz"} |
DEC2BIN() Formula in Google Sheets
Converts a decimal number to signed binary format.
Common Questions about the DEC2BIN Formula:
What is the DEC2BIN formula used for? How do I use the DEC2BIN formula? How do I convert decimal to binary using the DEC2BIN formula?
How can the DEC2BIN Formula be used appropriately?
The DEC2BIN formula can be used to convert a decimal number to a binary number. To use it properly, you must use the correct parameters, with the decimal number in the first argument followed by a
zero in the second argument.
How can the DEC2BIN Formula be commonly mistyped?
The DEC2BIN formula is commonly mistyped as DEC2BINN or DEC2BIIN, or the arguments might be typed incorrectly.
What are some common ways the DEC2BIN Formula is used inappropriately?
The DEC2BIN formula might be used inappropriately if the parameters are not entered correctly, or if the parameters are not numeric (for example if the decimal number is to be converted to a text
What are some common pitfalls when using the DEC2 BIN Formula?
One of the most common pitfalls when using the DEC2BIN formula is not providing the correct parameters. If the decimal number is not provided in the first argument, or if a zero is left out of the
second argument, the formula will return an incorrect result.
What are common mistakes when using the DEC2BIN Formula?
Common mistakes when using the DEC2BIN formula include mistyping the formula, entering the wrong parameters, or leaving out the second parameter.
What are common misconceptions people might have with the DEC2BIN Formula?
One of the common misconceptions people may have with the DEC2BIN formula is that it can be used to convert binary numbers to decimal numbers, when it can only be used to convert decimal numbers to
binary numbers." | {"url":"https://www.bettersheets.co/formulas/dec2bin","timestamp":"2024-11-14T07:15:37Z","content_type":"text/html","content_length":"31900","record_id":"<urn:uuid:2780eb50-ea76-4716-b76b-b8fdc4ee4e4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00053.warc.gz"} |
The Ultimate Speedsolving.com Cubing Cup // 29/30 competitors // 3x3 R1!!!
Actually, on Tuesday. (That’s when I come back from vacation)
Feb 5, 2024
Actually, on Tuesday. (That’s when I come back from vacation)
Wasn't skewb next? But I don't mind, so that I can actually practice it even more until then
Jan 18, 2024
Wasn't skewb next? But I don't mind, so that I can actually practice it even more until then
Oh yeah right, but if you can practice Skewb and improve, I’ll take that in later,
so you can win
Sep 16, 2024
Here are the scrambles:
1. F R U R U2 F' U' R2 U'
2. F U2 R' F R U F' U2 R2
3. R2 U F' U R2 U F2 U R'
4. U R2 U2 R' F U2 R F2 R2
5. R' U2 R' F R U2 F' R U
@yes cubing @Cubist-Guitar @Sombi McJunior @2x2MasterYT @EvanCuber1 @ifpigscouldfly54 @BVCuber13
Good luck!
Idk somebody else calculate it I’m lazy
I love this event
Cubedrop on 3/5 solves
Jan 5, 2024
Oh yeah right, but if you can practice Skewb and improve, I’ll take that in later, so you can win
And so I can get 2nd
Sep 3, 2024
jeez I forgot about this comp, would it be possible to do that optional round late to get into the finals? I’m busy with school sorry
Jan 18, 2024
jeez I forgot about this comp, would it be possible to do that optional round late to get into the finals? I’m busy with school sorry
That won’t work, sorry, because BVCuber13 and yes cubing, the eliminated ones, got in.
I'll compete later today when I come home from school
Jan 18, 2024
Jan 18, 2024
We need these four people to submit their times and then we can finalize 2x2 and start 3x3!
oh I forgot ill submit later
Jan 5, 2024
2. 1.74
3. 1.65
4.( 2.33 )
5. (1.52)
1.88 avg
Jan 7, 2024
We need these four people to submit their times and then we can finalize 2x2 and start 3x3!
2.66 ao5
2. 1.74
3. 1.65
4.( 2.33 )
5. (1.52)
1.88 avg
2x2SkullMasterYT for the win!1!1!
Last edited:
Jan 5, 2024
Feb 16, 2024
Not today
Generated By csTimer on 2024-10-13
avg of 5: 1.85
Time List:
4138. 1.78 U' R2 U' R2 F R2 F U2 F2 U'
4139. 1.59 F R2 U2 R' F U' F R2 U2
4140. 2.18 U2 F' R' U' F' R' U R2 U'
4141. (1.46) R2 U2 R2 U' F2 R2 F' R' U'
4142. (2.50) F R2 F U' R U F2 U2 R'
I’m the real 2x2SkullMaster here
Jan 5, 2024
Not today
Generated By csTimer on 2024-10-13
avg of 5: 1.85
Time List:
4138. 1.78 U' R2 U' R2 F R2 F U2 F2 U'
4139. 1.59 F R2 U2 R' F U' F R2 U2
4140. 2.18 U2 F' R' U' F' R' U R2 U'
4141. (1.46) R2 U2 R2 U' F2 R2 F' R' U'
4142. (2.50) F R2 F U' R U F2 U2 R'
I’m the real 2x2SkullMaster here
Jan 5, 2024
what do you mean(totally didn’t edit it)
Oh, Umm sorry must've been a mistake | {"url":"https://www.speedsolving.com/threads/the-ultimate-speedsolving-com-cubing-cup-29-30-competitors-3x3-r1.93354/page-11","timestamp":"2024-11-10T00:24:26Z","content_type":"text/html","content_length":"234363","record_id":"<urn:uuid:1f1181e3-6ca7-4c81-8407-195c8810e48b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00251.warc.gz"} |
Parallel usage of vertex_to_dof_map
I have an issue which I can’t figure out why it might be happening. The problem is as follows:
In an MPI parallelised code (For simplicity, lets says 2 processes P_0,P_1), suppose I have an array on each process A_0, A_1 which will always have a length matching the number of dofs and vertices
on that process. (I will only be using CG1 elements so this should always hold true).
I also have two corresponding arrays which store the local vertex at which each value in A_k should lie on. Lets call these v_k.
Essentially, now I want to map the values of A_k onto a function u, but using v_k to ensure that they are mapped to the correct vertices.
If I am not mistaken, I can use vertex_to_dof_map to do this. However, each time I have tried the vertex_to_dof_map it gives me dof indices which are higher than that of the number of dofs owned by a
process k.
Consider the MWI below:
from fenics import *
import numpy as np
# MPI communicator
comm = MPI.comm_world
rank = comm.Get_rank()
# Generate a mesh and functionspace
grid = RectangleMesh(Point(0,0),Point(10,10),10,10,"crossed")
V = FunctionSpace(grid,'CG',1)
u = Function(V)
# Connectivity
dofs = V.dofmap().dofs()
v2d = vertex_to_dof_map(V)
# Generate a random array on each process
A = np.random.uniform(0,1,len(dofs))
# Vertex mapping array
v = np.linspace(0,len(dofs)-1,len(dofs)).astype(int)
# Assign to the function vector
u.vector()[:] = A[v2d[v]]
The above code returns the errors (running with 2 processes):
IndexError: index 113 is out of bounds for axis 0 with size 109
IndexError: index 114 is out of bounds for axis 0 with size 112
If i investigate the ranges of the dofmap using:
I get:
Process: 0 117 109 (0,109)
Process: 1 118 112 (109,221)
Clearly, some dof indices returned from vertex_to_dof_map exceed that of the number of dofs on any process k - Am i missing something regarding the mapping used here? As far I as know, they should
all be process (local) indices here and should work fine.
If run with 1 process, it all works out fine.
Any help would be greatly appreciated,
I would check the documentation to see if you’re handling the ghost entries correctly.
HI nate, thanks for your reply.
I had a look, and if I understand correctly - In a parallel setting, dofs which lie on the boundary between processes are marked as shared and are owned by both processes? Thus they both have a local
index essentially mapping to the same global dof?
I cannot, however, find where the indexing scheme comes from - it appears to me that shared dofs have a different numbering system when they show up on a local process?
How would one go about assigning (in parallel) a value to a shared dof?
Please correct me if my understanding is incorrect,
Since dolfin stores parts of the mesh on each process, dofs on process boundaries are owned by one process, and shared with the other(s). If you want to assign a value to only the local dofs, they
are the first n dofs in V.dofmap().dofs().
The following code shows you how to get the number of dofs owned by each process, and how to get the global number of each degree of freedom not owned by the process that are in v2d:
from dolfin import *
import numpy as np
grid = RectangleMesh(Point(0,0),Point(10,10),1,1,"crossed")
V = FunctionSpace(grid,'CG',1)
u = Function(V)
# Connectivity
dofs = V.dofmap().dofs()
ownership_range = V.dofmap().ownership_range()
num_dofs_local = ownership_range[1] - ownership_range[0]
global_unowned = V.dofmap().local_to_global_unowned()
v2d = vertex_to_dof_map(V)
print(MPI.comm_world.rank, f"Num owned dofs {num_dofs_local}, Num dofs on process {len(v2d)} Unowned dofs (global) {global_unowned}")
which gives
1 Num owned dofs 4, Num dofs on process 4 Unowned dofs (global) []
0 Num owned dofs 1, Num dofs on process 4 Unowned dofs (global) [4 3 1]
I would recommend only assigning data to dofs owned by the process
2 Likes
Hi @dokken,
Thanks so much for this insight - everything makes sense now, and I got my application working. Cheers!
@dokken If I’m looking at dof_to_vertex_map(V) how do I know which indices correspond to which dofs in V.dofmap().dofs()? Because there are fewer local dofs than there are indices in the map, it’s
not clear to me how the map corresponds to the dofs. I want to initialize the values of a function with data that I have, but I have to map the correct data to the proper vertices. I’ve been using
dof_to_vertex_map(V) to do this but I’m struggling to get it to work in parallel:
V = fenics.FunctionSpace(mesh, 'CG', 1)
loc0, loc1 = V.dofmap().ownership_range()
dofs = V.dofmap().dofs()
d_to_vert = dof_to_vertex_map(V)
# T_0 is an array of raster data.
T_local = T_0.flatten()[loc0:loc1][d_to_vert]
d_to_vert is too large and has indices that are too large to be used as a mask here. How do I adapt the mask to only map the local dofs to their vertices?
dof_to_vertex_map includes ghosted dofs, as easily illustrated by:
from dolfin import *
n = 10
mesh = UnitSquareMesh(n, n)
V = FunctionSpace(mesh, "Lagrange", 1)
u = Function(V)
loc0, loc1 = V.dofmap().ownership_range()
d_to_vert = dof_to_vertex_map(V)
num_ghosts = len(V.dofmap().local_to_global_unowned())
f"Number of local dofs {loc1-loc0}, Number of dofs (including ghosts) {len(d_to_vert)}, Number of ghosts {num_ghosts}")
returning the following when executed on three processes
Number of local dofs 40, Number of dofs (including ghosts) 48, Number of ghosts 8
Number of local dofs 39, Number of dofs (including ghosts) 45, Number of ghosts 6
Number of local dofs 42, Number of dofs (including ghosts) 47, Number of ghosts 5
Thus you should only access the loc1-loc1 first entities in dof_to_vertex_map(V) if you only want to work with local entries.
2 Likes
@dokken How does that work exactly? If I access the first loc1-loc0 entries in dof_to_vertex_map(V) then I’ll still have indices that are out of bounds for something of the size of dofs. I guess I’m
interpreting what you’re saying to be something like verts = dofs[d_to_vert[:loc1-loc0]], but that obviously doesn’t work with the indexing issue. I feel like I’m still missing something basic here.
You do not need the last part of your function,
should be verts=d_to_vert[:loc1-loc0]
This gives you the local vertex indices on the current process that are associated with the dofs owned by this process.
1 Like
Thanks a ton! However, I still don’t understand how that vertex indexing works. If I want to initialize a vector with data, I would think I need to do something like this:
local_data = data[loc0:loc1]
local_vert_idxs = [d_to_vert][:loc1-loc0]
But local_vert_idxs here has indices that are out of range for local_data.
Thanks for being patient with me; I’m really trying to get this.
The vertices you get out is using local numbers.
See the following code on how to map global data to the local dofs:
from dolfin import *
from mpi4py import MPI as _MPI
import numpy as np
n = 10
mesh = UnitSquareMesh(n, n)
V = FunctionSpace(mesh, "Lagrange", 1)
u = Function(V)
num_dofs = V.dim()
num_vertices = mesh.num_entities_global(0)
assert num_dofs == num_vertices
global_data = np.arange(num_dofs, dtype=np.int32)
loc0, loc1 = V.dofmap().ownership_range()
d_to_vert = dof_to_vertex_map(V)
global_vertex_numbers = mesh.topology().global_indices(0)
global_vertices = global_vertex_numbers[d_to_vert[:loc1-loc0]]
f"Process {MPI.comm_world.rank} Num vertices with owned dofs: {len(global_vertices)}")
print(f"Process {MPI.comm_world.rank} Total number of dofs {MPI.comm_world.allreduce(len(global_vertices), op=_MPI.SUM)}")
local_data = global_data[global_vertices]
assigned_data = MPI.comm_world.gather(local_data, root=0)
if MPI.comm_world.rank == 0:
all_data = np.hstack(assigned_data)
unique_data = np.unique(all_data)
print(f"Unique vertices: {len(unique_data)}")
1 Like
@dokken Thanks! I’ve been trying to work with this, but I’m not sure how to recover the data with a vert_to_d mask or something. I’m initializing data as below:
loc0, loc1 = V.dofmap().ownership_range()
global_vertex_numbers = mesh.topology().global_indices(0)
global_vertices = global_vertex_numbers[d_to_vert[:loc1-loc0]]
T_0 = 2d_np_array_spatial_data
T_local = T_0.flatten()[global_vertices]
Then in a for loop I step through time solving for new T values and at certain times I want to save the numpy array of T values. How can I convert back to the original spatial coordinates to save the
new values? I’ve been trying to play around with the global_vertices you showed before to be able to something like
result = T_n.vector().gather_on_zero()
if mpi_rank == 0:
vert_to_d = ?
perimeters[j] = result[vert_to_d]
but everything I’ve tried has failed and I don’t understand how I should go about this. Any help appreciated!
I would also add, I had a reason for using dolfin instead of dolfinx, but it no longer applies, so if this is easier to do/advisable I’ll happily upgrade to dolfinx and begin learning.
Here is an example on how to map from a function to the mesh geometry:
from dolfin import *
from mpi4py import MPI as _MPI
import numpy as np
n = 3
mesh = UnitSquareMesh(n, n)
V = FunctionSpace(mesh, "Lagrange", 1)
u = Function(V)
u.interpolate(Expression("2*x[0]", degree=1))
num_dofs = V.dim()
num_vertices = mesh.num_entities_global(0)
assert num_dofs == num_vertices
global_data = np.arange(num_dofs, dtype=np.int32)
loc0, loc1 = V.dofmap().ownership_range()
d_to_vert = dof_to_vertex_map(V)
global_vertex_numbers = mesh.topology().global_indices(0)
global_data = np.zeros(num_vertices, dtype=np.float64)
] = u.vector().get_local()[:loc1-loc0]
# print(V.tabulate_dof_coordinates()[:loc1-loc0])
global_data = MPI.comm_world.allreduce(global_data, op=_MPI.SUM)
global_vertices = np.zeros((num_vertices, mesh.geometry().dim()))
vertices = MPI.comm_world.gather(mesh.coordinates(), root=0)
vertex_order = MPI.comm_world.gather(mesh.topology().global_indices(0), root=0)
if MPI.comm_world.rank == 0:
for vertex, order in zip(vertices, vertex_order):
global_vertices[order] = vertex
for coordinate, value in zip(global_vertices, global_data):
assert(2*coordinate[0] == value)
2 Likes
That’s perfect! Thanks so much for all your help! | {"url":"https://fenicsproject.discourse.group/t/parallel-usage-of-vertex-to-dof-map/6420","timestamp":"2024-11-03T14:08:19Z","content_type":"text/html","content_length":"51017","record_id":"<urn:uuid:ce721e12-9556-41dd-8cd7-740c27f17dcc>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00289.warc.gz"} |
regarding the original edit in #1, I have now:
• fixed wording and typesetting of the definition,
• added the actual original references
• added an Idea-section
diff, v4, current
One could even consider adding this example to the entry
I have added a 2-categorical description of the HNN-extension. See https://mathoverflow.net/q/352894/117693 for a discussion (Should the MO post be referenced?) Sorry if I made anything wrong, this
is my first submission.
Lukas Heger
diff, v2, current
I have created a stub HNN-extension. I have been wondering how to link in the connection between homotopy colimits and graphs of groups (see Fiore, Luck and Sauer), any ideas? Perhpas it will have to
wait until there is a graphs of groups and a complexes of groups entry
Regarding the edit from #2:
I have made some substantial edits to the new section “As a 2-colimit” (here) in order to bring out more carefully (maybe: pedantically, but this is important) the fact that the delooping groupoid
$\mathbf{B}(-) \;\colon\; Grps \to Grpds$
is not fully-faithful (only regarded as landing in pointed objects among groupoids does it become a full embedding).
Of course this is essentially the point that makes the intended construction in this section work in the first place, but it still important to make the notational distinction.
diff, v4, current
also, I have now tried to harmonize (and fix) the notation: Now, throughout, the ambient group is denoted “$G$” and the given subgroup “$H$”, as usual.
But maybe double-check that I didn’t miss some occurrence.
diff, v4, current | {"url":"https://nforum.ncatlab.org/discussion/3724/hnnextension/?Focus=105410","timestamp":"2024-11-15T00:51:31Z","content_type":"application/xhtml+xml","content_length":"47524","record_id":"<urn:uuid:cf8e133c-fc3d-4237-a4ca-d5a886c1d916>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00642.warc.gz"} |
Back to Papers Home Back to Papers of School of Physics
Paper IPM / P / 9011
School of Physics
Title: Non-Relativistic CFT and Semi-Classical Strings
Author(s): 1. A. Akhavan
2. M. Alishahiha
3. A. Davody
4. A. Vahedi
Status: Published
Journal: JHEP
Vol.: 03
Year: 2009
Pages: 053
Supported by: IPM
We study different features of 3D non-relativistic CFT using gravity description. As the corresponding gravity solution can be embedded into the type IIB string theory, we study semi-classical closed
/open strings in this background. In particular we consider folded rotating and circular pulsating closed strings where we find the anomalous dimension of the dual operators as a function of their
quantum numbers. We also consider moving open strings in this background which can be used to compute the drag force. In particular we find that for slowly moving particles, the energy is lost
exponentially and the characteristic time is given in terms of the temperature, while for fast moving particles the energy loss goes as inverse of the time and the characteristic time is independent
of the temperature.
Download TeX format
back to top | {"url":"https://www.ipm.ac.ir/ViewPaperInfo.jsp?PTID=9011&school=Physics","timestamp":"2024-11-04T17:36:15Z","content_type":"text/html","content_length":"42064","record_id":"<urn:uuid:e9efceb1-af6f-47c7-b6cf-20b9b213288c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00021.warc.gz"} |
Tutorial 15: First principles calculation of
Tutorial 15: First principles calculation of \(U\) and \(J\)
Davide Sarpa
Aug 2024
The goal of the tutorial is to provide a working example on how it is possible to compute the \(U\) and \(J\) parameters from first principles. We will work on Hematite +--+ antiferromagnetic
configuration as you should already be familiar with it, if not, refer to tutorial 9.
The reason behind computing the parameters via first principles is because they directly correct the spurious localised self-interaction error (\(U\)) and static correlation error (\(J\)) and hence
the physics of the system. While choosing and empirical \(U\) and \(J\) might give a better description of a specific property of the material, it does not guarantee that these errors are
consistently corrected.
Theoretical background
We start by defining the response function \(\chi\), which describes how the occupation of localised orbitals changes with respect to a shift in the potential acting on these orbitals: The linear
response method determines the Hubbard \(U\) parameter by comparing the response of the system to a perturbation in standard DFT and DFT+\(U\) frameworks.
We define the response function \(\chi\) as:
\[\chi = \frac{dn^{I\sigma}}{d\alpha}\]
where \(n\) is the occupation matrix of the localised orbitals and \(\alpha\) is a potential shift applied to these orbitals.
We compute two response functions:
• \(\chi_0\): the bare Kohn-Sham (KS) response (without \(U\))
• \(\chi\): the interacting response (with \(U\))
These are related by:
\[U = \chi^{-1} - \chi_0^{-1}\]
which allows us to compute \(U\).
The perturbation is applied by shifting the potential of the localised orbitals:
\[V_{\text{ext}}^{p} = V_{\text{ext}} + \alpha \sum_{m,m'}\lvert\varphi_{m'}^{(I)}\rangle\langle\varphi_m^{(I)}\rvert\]
This is the conventional linear response and its done in a supercell as the perturbation should not interact with its periodic images. Another approach to compute \(U\) and \(J\) is known as minimum
tracking method [Moynihan2017] [Linscott2018].
Minimum Tracking Method
The minimum tracking method is based on a reformulation of the response matrices based on the ground state density of the perturbed system [Moore2024] . We can redefine the interacting and
noninteracting response matrices as (in practice we’ll be using simpler yet equivalent formulae)
\[\chi_{IJ} = \frac{dn^I}{dv_\text{ext}^J},\]
\[(\chi_0)_{IJ} = \left[\frac{dn}{dv_\text{KS}}\left(\frac{dv_\text{KS}}{dv_\text{ext}}\right)^{-1}\right]_{IJ}\]
This allows us to work around the practical issues from the conventional linear response. This approach can also be extended to include the \(J\) exchange term In practice this is done by modifying
the perturbation by including an additional term (spin-splitting):
\[V_{\text{ext}}^{p} = V_{\text{ext}} + \beta \sum_{m,m'}\lvert\varphi_{m'}^{(I\uparrow)}\rangle\langle\varphi_m^{(I\uparrow)}\rvert-\lvert\varphi_{m'}^{(I\downarrow)}\rangle\langle\varphi_m^{(I\
Setting up the calculations
We will configure a set (9 total) of bulk hematite single-point calculations to compute \(U\) and \(J\) for the Fe \(3d\) orbitals. We apply distinct labels to Fe atoms, enabling us to assign
different parameters to spin-up and spin-down Fe atoms. We will be using a 4x4x1 supercell generated from the conventional cell.
Tutorial files
All the files needed for the simulations can be downloaded from
Practical calculation
The step by step approach to compute \(U\) and \(J\) is:
1. add hubbard_calculating_u : T in the input file,
2. choose an atom for the atom type we want to compute \(U\) or \(J\) for, and label it differently. In our case you can see from the input file that we have labelled this single atom Fe1U. It
does not matter whether we choose a spin up or spin down atom for an AFM material.
3. apply the perturbation to this atom only and perform single-points calculations,
4. compute \(U\) and \(J\) with the following formulas:
\[U = \frac{1}{2} \frac{\delta v^\uparrow_{\text{Hxc+local}} + \delta v^\downarrow_{\text{Hxc+local}}}{\delta(n^\uparrow + n^\downarrow)}\]
\[J = -\frac{1}{2} \frac{\delta v^\uparrow_{\text{Hxc+local}} - \delta v^\downarrow_{\text{Hxc+local}}}{\delta(n^\uparrow - n^\downarrow)}\]
where \(\delta v^\uparrow_{\text{Hxc}}\) and \(\delta v^\downarrow_{\text{Hxc}}\) represent the derivative of the Hxc+local potential with respect to the applied potential (either \(\alpha\) to
compute \(U\) or \(\beta\) to compute \(J\)) and \(\delta(n^\uparrow + n^\downarrow)\) and \(\delta(n^\uparrow - n^\downarrow)\) represent the derivative of the total occupation \(n^\uparrow + n^\
downarrow\) with respect to \(\alpha\) and of \(n^\uparrow - n^\downarrow\) with respect to \(\beta\).
How and where to apply the perturbation
Looking at the input file provided you can see we activated the hubbard_calculating_u functionality and in the Hubbard block we have
Fe1 2 0.0 0.0 -10.0 0.0 0.0
Fe1U 2 0.0 0.0 -10.0 0.0 0.0
Fe2 2 0.0 0.0 -10.0 0.0 0.0
where the columns of the hubbard block are described as follows:
1. Species Label
The species to apply the DFT+\(U\) correction to.
2. Angular Momentum: \(l\)
The angular momentum of the projectors which the Hubbard correction is applied to. In this example \(l=2\) which corresponds to d orbitals
3. Hubbard \(U\) value
The value of the Hubbard \(U\) for this sub-shell, in electron-volts. We are computing it so we can choose 0 as its value
4. Hund’s exchange \(J\) value
The value of the Hund’s exchange \(J\) for this sub-shell, in electron-volts. We are computing it so we can choose 0 as its value
5. Effective Charge \(\mathbf{Z}\) and Projectors type The default projectors are NGWFs. For other possibility, refer to the DFT+ \(U\) documentation
6. The \(\alpha\) prefactor
The perturbation term needed to compute \(U\)
7. The spin-splitting factor \(\beta\)
The perturbation term needed to compute \(J\).
To compute \(U\) you need to change the \(\alpha\) value while keeping \(\beta\) equal to 0. To compute \(J\) you need to change the \(\beta\) value while keeping \(\alpha\) equal to 0.
We have provided you only 1 input file – the one corresponding to 0 for both \(\alpha\) and \(\beta\), you need to generate the remaining 8 files.
The \(\alpha\) and \(\beta\) values you need to use for the \(U\) calculation are = -0.2, -0.1, 0.0, 0.1, 0.2.
Why these values? We want to apply a big enough perturbation to see an effect and to be able to compute derivatives but also remain in the linear regime. It is not necessary to use 5 datapoints to
obtain a good value but it’s highly recommended.
Evaluating the outputs
In order to compute \(U\) and \(J\) we need the values of the \(v^\uparrow_{\text{Hxc}}\) and \(v^\downarrow_{\text{Hxc}}\),which can be found in the following block:
DFT+U information on Hubbard site 72 of species Fe1U and spin 1
The average Hxc+local potential is -100.04043423 eV.
The average Hubbard potential is -0.10000000 eV.
DFT+U information on Hubbard site 72 of species Fe1U and spin 2
The average Hxc+local potential is -96.03296381 eV.
The average Hubbard potential is -0.10000000 eV.
Note that we are looking only at the values for Fe1U atom which is the only atom we have applied the perturbation to. There are multiple instances of this block and we are only interested in the last
Next, we need to look at occupation of the Hubbard manifold \(n^\uparrow + n^\downarrow\) and \(n^\uparrow - n^\downarrow\),which can be found in the following block:
DFT+U information on atom 1 of Hubbard species Fe1U
Occupancy matrix of Hubbard site 72 and spin 1 is
m_l = -2 -1 0 1 2
0.98583311 0.01105739 0.00017283 0.00149346 -0.00039754
0.01106973 0.98239066 -0.00021203 0.00037893 0.00244851
0.00017266 -0.00021405 0.99296562 0.00030517 0.00069962
0.00149451 0.00037878 0.00029134 0.98210951 -0.01203475
-0.00039830 0.00244943 0.00069122 -0.01204334 0.98340592
WARNING: DFT+U ENERGY of Hubbard site 72 and spin 1 is negative.
Occupancy matrix of Hubbard site 72 and spin 2 is
m_l = -2 -1 0 1 2
0.32009924 -0.06393836 -0.00012245 -0.01033413 -0.00070413
-0.06400973 0.33409081 -0.00029354 0.00034179 -0.01142806
-0.00012106 -0.00027777 0.19025018 -0.00114325 0.00745246
-0.01034138 0.00034159 -0.00106271 0.33014982 0.06774687
-0.00070499 -0.01143070 0.00762074 0.06779446 0.29199808
WARNING: DFT+U ENERGY of Hubbard site 72 and spin 2 is negative.
Total occupancy of Hubbard site 72 is 6.39329292 e
Local magnetic moment of Hubbard site 72 is 3.46011669 mu_B
DFT+U energy of Hubbard site 72 is -0.02349492 Ha
The total occupancy of Hubbard site is the \(n^\uparrow + n^\downarrow\), while the local magnetic moment of Hubbard site is the \(n^\uparrow - n^\downarrow\). We now have all the data we need to
compute \(U\) and \(J\).
Step by step to compute \(U\) :
□ Calculate the slope of \(v^\uparrow_{\text{Hxc}}\) and \(v^\downarrow_{\text{Hxc}}\) with respect to \(\alpha\), these are the \(\delta v^\uparrow_{\text{Hxc}}\) and \(\delta v^\downarrow_{\
text{Hxc}}\) that appear in the formula to compute \(U\)
□ Calculate the slope of the \(n^\uparrow + n^\downarrow\) with respect to \(\alpha\) this is the denominator appearing in the formula to compute \(U\)
□ Compute \(U\) using the formula provided above.
To compute \(J\) follow similar procedure but the derivatives are with respect to \(\beta\).
IMPORTANT: The actual \(\beta\) values in the calculations are half of the one specified in the input file.
To compute the slope, we first plot the Hxc+local for spin 1 and spin 2 as well as the occupation number against the values of \(\beta\), the same should be done with values of \(\beta\) to compute \
You can see from the plots that while the changes of the occupation numbers are perfectly linear at all \(\alpha\) values, this is not the case for the Hxc+local potential where a degree of
non-linearity is present at a value of \(\alpha=0\), this is VERY important as if we were to include this data point in our calculation of \(U\), we would obtain a wrong value as our perturbation
goes beyond the linear response regime.
If you discard the non-linear data point, you should obtain the following values.
• \(U\) = 5.158 eV
• \(J\) = 0.604 eV
What to do next
The tutorial is now complete, but you could still move forward. What can you do next?
• Compute \(U\) for oxygen p states as this is commonly done in transition metal oxides, it’s usually large. For more information [Moore2024]
[Moore2024] (1,2)
G. C. Moore, M. K. Horton, E. Linscott, A. M. Ganose, Ma. Siron, D. D. O’Regan, K. A. Persson Phys. Rev. Materials 8, 014409 (2024). https://doi.org/10.1103/PhysRevMaterials.8.014409
G. Moynihan, G. Teobaldi, and D. D. O’Regan, A self-consistent ground-state formulation of the first-principles Hubbard U parameter validated on one-electron self- interaction error (2017), | {"url":"https://tutorials.onetep.org/T15_hematite_linear_response.html","timestamp":"2024-11-11T23:58:25Z","content_type":"text/html","content_length":"33386","record_id":"<urn:uuid:24b1c020-df6d-4f57-9a1b-34f2a1891b92>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00696.warc.gz"} |
class snowflake.core.stage.StageResource(name: str, collection: StageCollection)¶
Bases: SchemaObjectReferenceMixin[StageCollection]
Represents a reference to a Snowflake stage.
With this stage reference, you can drop, list files, put files, get files, and fetch information about stages.
delete() None¶
The delete method is deprecated; use drop instead.
download_file(stage_path: str, file_folder_path: str) None¶
The download_file method is deprecated; use get instead.
drop(if_exists: bool | None = None) None¶
Drop this stage.
if_exists (bool, optional) – Check the existence of this stage before suspending it. Default is None, which is equivalent to False.
Dropping a stage using its reference:
>>> stage_reference.drop()
fetch() Stage¶
Fetch the details of a stage.
Fetching a reference to a stage to print its name:
>>> my_stage = stage_reference.fetch()
>>> print(my_stage.name)
get(stage_location: str, target_directory: str | PathLike, *, parallel: int = 4, pattern: str | None = None) None¶
Download the specified files from a path in the stage to a local directory.
References: Snowflake GET command.
○ stage_location (str) – A directory or filename on a stage, from which you want to download the files. e.g. /folder/file_name.txt or /folder
○ target_directory (str, PathLike) – The path to the local directory where the files should be downloaded. If target_directory does not already exist, the method creates the
○ parallel (int, optional) – Specifies the number of threads to use for downloading the files. The granularity unit for downloading is one file. Increasing the number of threads might
improve performance when downloading large files. Valid values: Any integer value from 1 (no parallelism) to 99 (use 99 threads for downloading files).
○ pattern (str, optional) – Specifies a regular expression pattern for filtering files to download. The command lists all files in the specified path and applies the regular
expression pattern on each of the files found. Default: None (all files in the specified stage are downloaded).
Getting file from stage:
>>> stage_reference.get("/folder/file_name.txt", "/local_folder")
Getting files with a specific pattern:
>>> stage_reference.get("/folder", "/local_folder", pattern=".*.txt")
list_files(*, pattern: str | None = None) Iterator[StageFile]¶
List files in the stage, filtering on any optional ‘pattern’.
pattern (str, optional) – Specifies a regular expression pattern for filtering files from the output.
Listing all files in the stage:
>>> files = stage_reference.list_files()
Listing files with a specific pattern:
>>> files = stage_reference.list_files(pattern=".*.txt")
Using a for loop to retrieve information from iterator:
>>> for file in files:
... print(file.name)
put(local_file_name: str | PathLike, stage_location: str, *, parallel: int = 4, auto_compress: bool = True, source_compression: str = 'AUTO_DETECT', overwrite: bool = False) None¶
Upload local files to a path in the stage.
References: Snowflake PUT command.
○ local_file_name (str, PathLike) – The path to the local files to upload. To match multiple files in the path, you can specify the wildcard characters * and ?.
○ stage_location (str) – The prefix where you want to upload the files. e.g. /folder or /
○ parallel (int, optional) –
Specifies the number of threads to use for uploading files. The upload process separates batches of data files by size:
■ Small files (< 64 MB) are staged in parallel as individual files.
■ Larger files are automatically split into chunks, staged concurrently, and reassembled in the target stage. A single thread can upload multiple chunks.
Increasing the number of threads can improve performance when uploading large files. Supported values: Any integer value from 1 (no parallelism) to 99 (use 99 threads for uploading
○ auto_compress (boolean, optional) – Specifies whether Snowflake uses gzip to compress files during upload. Default is True.
○ source_compression (str, optional) –
Specifies the method of compression used on already-compressed files that are being staged.
Values can be AUTO_DETECT, GZIP, BZ2, BROTLI, ZSTD, DEFLATE, RAW_DEFLATE, NONE, default is AUTO_DETECT.
○ overwrite (boolean, optional) – Specifies whether Snowflake will overwrite an existing file with the same name during upload. Default is False.
Putting file on stage and compressing it using the stage’s reference:
>>> stage_reference.put("local_file.csv", "/folder", auto_compress=True)
upload_file(file_path: str, stage_folder_path: str, *, auto_compress: bool = True, overwrite: bool = False) None¶
The upload_file method is deprecated; use put instead. | {"url":"https://docs.snowflake.com/en/developer-guide/snowflake-python-api/reference/latest/_autosummary/snowflake.core.stage.StageResource","timestamp":"2024-11-11T05:09:48Z","content_type":"text/html","content_length":"259751","record_id":"<urn:uuid:a04ddeaf-f56e-4818-aeb5-3eda739ba681>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00084.warc.gz"} |
Plus and Minus Signs in Betting Odds Explained
For someone only starting out in sports betting, the terms and formats it has can be overwhelming to learn. The American odds are particularly "unique" in this regard. While any other odds formats
are pretty self-explanatory, the American odds introduce a bunch of things that aren't immediately obvious. The "+" and "-" (plus and minus signs) are the weirdest characteristics of them.
Thankfully, it's very simple - let's take a look at what they mean.
This article will cover one aspect of betting odds. For a more comprehensive overview, check out our betting odds guide via the link!
How to Read American Odds
Bettors located in North America (or those who just prefer this particular way of displaying odds) should all know this format. It's common that we see something like this (odds taken from
Houston Rockets +160 -217 Cleveland Cavaliers
Sometimes called Moneyline odds, this odds format represents the stake amount of $100 dollars bet on either team. It's a bit more complex than other odds formats but still easy. All you have to
remember is:
• The positive odds (+) represent the potential profit in the case of a $100 bet.
• The negative odds (-) represent the amount of money required to be put down to get a $100 profit.
If we apply this logic here, we'll see that we would get $160 if we put down $100 on the Rockets, but we'll need to put down $217 for a chance to win $100 in the case of the Cavaliers. We can also
tell that the Cavaliers are the favorites of the event, and the Rockets are the underdog - more about these terms via the link!
Additionally, remember that in American odds, the odds number doesn't include the initial wager. So, if you win a bet on the Rockets, you'd get $160 (as indicated by the odds) plus your initial stake
of $100, for a total of $260.
What is +200 in Betting?
Now we know what the common odds number of +200 represents - an underdog bet in American odds.
Calculating Odds and Winnings
Now we see how easy calculating winnings is. But there's an even easier way - like using an odds calculator! Calculate between all the popular odds types and see your winnings straight away with our
easy-to-use calculator
Betting calculator
DecimalAmericanFractionalHong Kong
Plus and Minus Symbols In Spreads
Although they're called moneyline odds, they don't leave us even in spreads! They denote a similar thing as in moneyline bets here, too. Other betting odds formats also use plus and minus signs to
denote point spreads - but with minor variations.
Spread bets, also known as handicaps, are wagers that aim to make betting on underdogs more accessible by giving them a virtual advantage in the bet.
In spreads, the minus sign means the point spread (or handicap) on the favorite, and the plus sign represents the point spread wager on the underdog. If we want to bet on point spreads on an American
bookie, we would most likely see something like this (odds taken from betus.com). Note that in ice hockey betting, spread bets are called the "puck line."
Just looking at these odds, we can tell that the Buffalo Sabres are the favorites, while the Detroit Red Wings are the runner-ups (underdogs), although the match is pretty damn close! We see both
forms of plus and minus symbols in this case. For instance, betting on Red Wings with a +1 1/2 handicap will be a high-chance wager, but the payout would be a measly $41 for every $100 bet. | {"url":"https://tips.gg/betting/plus-minus/","timestamp":"2024-11-10T09:13:54Z","content_type":"text/html","content_length":"63504","record_id":"<urn:uuid:489d2c93-067d-4d5b-a230-9aea8f020cb3>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00637.warc.gz"} |
This structure represents a 3D ray represented by two points. More...
#include <include/reactphysics3d/mathematics/Ray.h>
Ray (const Vector3 &p1, const Vector3 &p2, decimal maxFrac=decimal(1.0))
Constructor with arguments.
Vector3 point1
First point of the ray (origin) in world-space.
Vector3 point2
Second point of the ray in world-space.
decimal maxFraction
Maximum fraction value.
This structure represents a 3D ray represented by two points.
The ray goes from point1 to point1 + maxFraction * (point2 - point1). The points are specified in world-space coordinates.
The documentation for this struct was generated from the following file:
• include/reactphysics3d/mathematics/Ray.h | {"url":"https://reactphysics3d.com/documentation/structreactphysics3d_1_1_ray.html","timestamp":"2024-11-14T17:05:09Z","content_type":"application/xhtml+xml","content_length":"9113","record_id":"<urn:uuid:9eeed33b-1816-46b0-a105-a8188ed79429>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00092.warc.gz"} |
Game Theory: Decision-Making and Public Spending
Key Topics
• An Overview of Game Theory for Understanding
□ Laying the Groundwork for Regular Form Games
□ Game Theory-Based Analysis of Public Spending
□ Nash Equilibrium: Looking for Consistency in Approaches
□ Streamlining Decisions through the Iterated Elimination of Dominant Strategies
• Conclusion
In economics, choices have a big impact on how things turn out. Game theory, which emerged from the fusion of mathematics and economics, provides a powerful framework for analyzing strategic
interactions and decision-making procedures. Understanding game theory, particularly through the lens of normal-form games, proves to be both enlightening and empowering for students dipping their
toes into the fascinating world of economics. This article explores game theory with a focus on normal-form games. It sheds new light on government spending, a topic important to economics students'
assignments on game theory. The art of decision-making has a significant influence on how outcomes are shaped in the economic environment. Game theory, which has its roots in the fusion of economics
and mathematics, proves to be a powerful toolkit for analyzing tactical interactions and the processes that underlie decision-making. Understanding the nuances of game theory, especially when seen
through the lens of normal-form games, bestows a dual gift of enlightenment and empowerment on students making their first forays into the fascinating field of economics. In the discussion that
follows, we straddle the fine line between game theory and normal-form games. In the course of this investigation, we present a novel viewpoint on public spending, a subject that is particularly
pertinent to the game theory homework problems that economics students frequently encounter. To complete your game theory homework effectively, understanding the nuances of public spending in the
context of game theory is crucial.
An Overview of Game Theory for Understanding
Game theory serves as a flexible toolkit that sheds light on the complex process by which people make decisions that are intertwined with the decisions of their counterparts. This discipline, which
emerged in the middle of the 20th century, has developed into a cross-disciplinary lighthouse, finding applications not only in economics but also in fields as diverse as political science, biology,
and philosophy. Game theory focuses on the interaction of decision-makers, also known as players, who are motivated by the opposing forces of rationality and self-interest. This analytical framework
breaks down the dynamics of strategic decisions, illuminating the motives that lead people and things in situations where their fates are intertwined.
Laying the Groundwork for Regular Form Games
The cornerstone of game theory is the normal form game, also known as the strategic form game. These video games represent the complex dance of strategic decision-making as a compelling abstraction.
They accomplish this by providing a well-organized matrix that opens up the range of potential play-styles and the resulting rewards. The payoff in this matrix echoes the fruits reaped from the
collective interplay of strategies selected by all players, while each player's strategy takes the shape of a deliberate course of action. With this framework, you can get a bird's-eye view of the
competitive or cooperative dynamics at work, where logical actors carefully consider the effects of their decisions in a web of choices that ripples throughout the matrix. Consider the Prisoner's
Dilemma as a classic example to help illustrate. Two suspects are detained in this game after being suspected of committing a crime, and each has the choice to cooperate (remain silent) or to
confess. The payoff matrix displays the number of years in prison that each person would serve based on their collective decisions. The best outcome for both parties results from cooperation, but
individual incentives frequently result in betrayal, highlighting the tension between individual rationality and collective optimality.
Game Theory-Based Analysis of Public Spending
The area of public spending is one where strategic choices weigh societal welfare and economic advancement heavily. It is a complex area of economics. Game theory emerges as a potent lens, offering a
new perspective on the forces that influence public spending decisions, as we peel back the layers of this complex landscape Decisions about public spending are frequently entangled in a web of
interdependencies, having an effect on not only the government but also on people, businesses, and external stakeholders. Game theory enters this complex ballet and begins to unravel the webs of
strategic interactions that support these choices. Imagine a situation where a government must divide its funds between healthcare and education. Each choice has effects that spread throughout the
economy. A payoff matrix that incorporates societal benefits, financial burdens, and even political popularity can be created using game theory. The equilibrium points in this matrix show not only
the best option for the government but also potential responses from citizens and interest groups .The idea of a "public good dilemma"—a circumstance in which individual self-interest may result in
less than ideal collective outcomes—is emphasized by game theory. For instance, the environmental degradation brought on by numerous governments seeking economic growth through extensive
industrialization. This situation necessitates strategic coordination, and game theory provides decision-makers with a methodical way to deal with such difficulties. Assignments dealing with
hypothetical public spending situations can profit from a game-theoretic perspective as students delve deeper into the field of economics. Students can analyze the potential effects of various policy
decisions and spot equilibriums that balance societal and economic interests by modeling the players, strategies, and payoffs involved.Game theory essentially acts as a compass for decision-makers,
guiding them toward choices that balance immediate and long-term gains while taking other players' actions into account. Students of economics who adopt this strategic framework can not only
comprehend the complex dance of public spending but also help to shape the policies that promote societal well-being and economic prosperity. Public spending transcends simple budget allocations when
viewed strategically through the lens of game theory, becoming a canvas on which the strokes of sensible choices paint a picture of a better future.
Nash Equilibrium: Looking for Consistency in Approaches
The idea of Nash Equilibrium emerges as a guiding star in the grand theater of game theory, where rational actors choreograph their movements to optimize outcomes. This equilibrium, which bears the
name of Nobel laureate John Nash, depicts stability in strategic decision-making and provides insights into the equilibrium that develops when players' choices interact.The fundamental idea behind
Nash Equilibrium is that if the other players maintain their current course of action, no player will have an incentive to change their chosen course of action. It is a situation of mutual impasse in
which each player's decision is the best response to the decisions of the others. The strategic dynamics of the game are anchored by this equilibrium, which resembles a gravitational center. Consider
a scenario in which two businesses compete with one another on pricing. If both decide to undercut one another, a price war results in lower profits for both parties. However, profits might rise if
both businesses decide to keep their higher prices. The Nash Equilibrium is embodied in this precarious balance, where neither firm gains from a unilateral deviation.Beyond its theoretical beauty,
Nash Equilibrium has a significant impact on many disciplines, such as politics, biology, and economics. This equilibrium directs players toward tactical decisions that avoid unintended consequences
in decision matrices that depict strategic interactions and potential outcomes.Understanding the fundamentals of Nash equilibrium is like using a compass in uncharted territory for students just
entering the field of game theory. It not only clarifies the strategic landscape but also equips these aspiring economists to crack the complex decision-making code that determines outcomes. With
knowledge of Nash equilibrium, students can approach their game theory homework with a critical eye, looking for stability among tactical decisions and uncovering the intriguing stories woven within
the matrix of strategies and payoffs.
Streamlining Decisions through the Iterated Elimination of Dominant Strategies
Iterated Elimination of Dominated Strategies (IEDS) is a key concept in the fascinating field of game theory, where strategic interactions and decision-making choreograph a delicate dance. This
approach acts as a clarifying lens for examining complex situations, like a sculptor chipping away extra stone to reveal the masterpiece within. IEDS is essentially an iterative process of continual
improvement. It entails methodically eliminating tactics that, regardless of the decisions made by the rival players, hold no appeal for rational players. The decision landscape is reduced by
eliminating these dominated strategies, leaving only the strategies that genuinely enhance the artistic quality of the game. Imagine a battle of strategies between businesses deciding whether to
raise prices to increase profit margins or to maintain lower prices to gain a competitive edge. Through IEDS, the analysis would expose the equilibrium points where each player's choice maximizes
their payoff while being constrained by the choices of the other players.This approach elevates strategic thinking, in addition to streamlining decisions. With a more focused set of options, players
can concentrate their logical faculties on the tactics with the best chance of success. IEDS thus refines the very essence of strategic play rather than merely eliminating strategies. Iterated
Elimination of Dominated Strategies shines as a beacon of clarity in the game theory mosaic, where complexity can occasionally obscure the fundamental dynamics. It offers a methodical way to navigate
the maze of options, enabling a deeper comprehension of the strategic interplay that characterizes the game. Embracing IEDS enables them to extract brilliance from complexity and approach game theory
assignments with increased analytical finesse as economics students navigating the complex dance of strategic decisions.
Students of economics can use the game theory's insights into strategic decision-making as a strong framework to analyze a variety of scenarios. Public spending decisions are used as an example to
show how normal-form games offer a structured way to comprehend the dynamics of interactions, strategies, and payoffs. Students can understand the complexities of decision-making and develop a fresh
understanding of how rational agents negotiate strategic interactions by embracing ideas like Nash equilibrium and iterated elimination of dominated strategies. The use of game theory in economic
assignments can offer a novel method for comprehending and resolving challenging issues. As you begin your "game theory assignment," keep in mind that each scenario has a matrix of potential
strategies and payoffs that is just waiting to be discovered. These insights could reshape how we view strategic decision-making in both theoretical and real-world settings.
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Lesson 7: Practical Deep Learning for Coders 2022
[Jeremy Howard]
Lesson 7: Inside a Neural Net
Right, welcome to lesson seven, the penultimate lesson of Practical Deep Learning for Coders part one. And today we’re going to be digging into what’s inside a neural net. We’ve already seen what’s
inside a kind of the most basic possible neural net, which is a sandwich of fully connected layers or linear layers and and relu’s. And so we built that from scratch. But there’s a lot of tweaks that
we can do. And so most of the tweaks actually that we probably care about are the tweaking the very first layer or the very last layer. So that’s where we’ll focus. But over the next couple of weeks,
we’ll look at some of the tweaks we can do inside as well.
Patty, Rice Patty Competition
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different types of basic pre-processing. We added test time augmentation. And then we scaled that up to larger images and rectangular images. And that got us into the top 25% of the competition.
So that’s part two of the so-called road to the top series, which is increasingly misnamed. Since we’ve been presenting these notebooks, more and more of our students have been passing me on the
leaderboard. So currently, first and second place are both people from this class, Kurian and Nick. Go to hell, you’re in my target, and leave my class immediately. And congratulations, good luck to
Scaling Up Models
So in part three, I’m going to show you a really interesting trick, a very simple trick for scaling up these models further. What you’ll discover if you’ve tried to use larger models, so you can
replace the word small with the word large in those architectures, and try to train a larger model.
A larger model has more parameters. More parameters means it can find more tricky little features. And broadly speaking, models with more parameters therefore ought to be more accurate. Problem is
that those activations, or more specifically, the gradients that have to be calculated, choose up memory on your GPU. And your GPU is not as clever as your CPU at kind of sticking stuff it doesn’t
need right now into virtual memory on the hard drive. When it runs out of memory, it runs out of memory. And it also doesn’t do such a good job as your CPU at kind of shuffling things around to try
and find memory. It just allocates blocks of memory, and it stays allocated until you remove them. So if you try to scale up your models to bigger models, unless you have very expensive GPUs, you
will run out of space.
And you’ll get an error. Something like CUDA, out of memory error.
CUDA Out of Memory Error
So if that happens, first thing I’ll mention is it’s not a bad idea to restart your notebook, because they can be a bit tricky to recover from otherwise. And then I’ll show you how you can use as
large a model as you like. Almost as, you know, basically you’ll be able to use a X large model on Kaggle. So let me explain. Now, I want, when you run something on Kaggle, like actually on Kaggle,
you’re generally going to be on a 16 gig GPU. And you don’t have to run stuff on Kaggle. You can run stuff on your home computer or paper space or whatever. But sometimes you’ll have, if you want to
do Kaggle competitions, sometimes you’ll have to run stuff on Kaggle, because a lot of competitions are what they call code competitions, which is where the only way to submit is from a notebook that
you’re running on Kaggle.
And then a second reason to run stuff on Kaggle is that, you know, your notebooks will appear, you know, with the leaderboard score on them, and so people can see which notebooks are actually good.
And I kind of like, even in things that aren’t code competitions, I love trying to be the person who’s number one on the notebook score leaderboard, because that’s something which, you know, you
can’t just work at NVIDIA and use a thousand GPUs and win a competition through a combination of skill and brute force. Everybody has the same nine hour timeout to work with. So I think it’s a good
way of keeping the, you know, things a bit more fair. Now, so my home GPU has 24 gig.
So I wanted to find out what can I get away with, you know, in 16 gig. And the way I did that is, I think, a useful thing to discuss, because again, it’s all about fast iteration. So I wanted to
really quickly find out how much memory will a GPU, will a model use. So there’s a really quick hacky way I can do that, which is to say, okay, for the training set, let’s not use, so here’s the
value counts of labels, so the number of each disease. Let’s not look at all the diseases. Let’s just pick one, the smallest one, right? And let’s make that our training set. Our training set is the
bacterial panicle blight images. And now I can train a model with just 337 images without changing anything else. Not that I care about that model, but then I can see how much memory it used. It’s
important to realize that, you know, each image you pass through is the same size, each batch size is the same size.
So training for longer won’t use more memory. So that’ll tell us how much memory we’re going to need. So what I then did was I then tried training different models to see how much memory they used
up. Now, what happens if we train a model, so obviously ConfNec small doesn’t use too much memory. So here’s something that reports the amount of GPU memory just by basically printing out CUDA’s GPU
processes. And you can see ConfNec small took up four gig. And also, this might be interesting to you, if you then call Python’s garbage collection, gc.collect, and then call PyTorch’s empty cache,
that should basically get your GPU back to a clean state of not using any more memory than it needs to when you can start training the next model without restarting the kernel.
So what would happen if we tried to train this little model and it crashed with a CUDA out of memory error? What do we do?
Gradient Accumulation
We can use a cool little trick called gradient accumulation. What’s gradient accumulation? So what’s gradient accumulation? Well, I added this parameter to my train method here. So my train method
creates my data loaders, creates my learner, and then depending on whether I’m fine-tuning or not, either fits or fine-tunes it. But there’s one other thing it does. It does this gradient
accumulation thing. What’s that about? Well, the key step is here. I set my batch size, so that’s the number of images that I pass through to the GPU all at once, to 64, which is my default, divided
by, slash slash means integer divide in Python, divided by this number.
So if I pass two, it’s going to use a batch size of 32. If I pass four, it’ll use a batch size of 16. Now, that obviously should let me cure any memory problems, use a smaller batch size, but the
problem is that now the dynamics of my training are different, right? The smaller your batch size, the more volatility there is from batch to batch. So now your learning rates are all messed up. You
don’t want to be messing around with trying to, you know, find a different set of kind of optimal parameters for every batch size for every architecture. So what we want to do is find a way to run
just, let’s say, accum is two, accumulate equals two. Let’s say we just want to run 32 images at a time through. How do we make it behave as if it was 64 images?
Well, the solution to that problem is to consider our training loop. This is basically the training loop we used from a couple of lessons ago, the one we created manually. We go through each x, y
pair in the data loader. We calculate the loss using some coefficients based on that x, y pair, and then we call backward on that loss to calculate the gradients, and then we subtract from the
coefficients the gradients times the learning rate, and then we zero out the gradients. I’ve skipped a bit of stuff like the with torch dot no grad thing. Actually, no, I don’t need that because I’ve
got dot data. No, that’s it. That should all work fine. I’ve skipped out printing the loss. That’s about it. So here is a variation of that loop where I do not always subtract the gradient times the
learning rate. Instead, I go through each x, y pair in the data loader.
I calculate the loss. I look at how many images are in this batch. So initially, I start at zero, and this count is going to be 32, say, if I’ve divided the batch size by two. And then if count is
greater than 64, I do my gradient, my coefficients update. Well, it’s not. So I skip back to here, and I do this again. And if you remember, there was this interesting subtlety in PyTorch, which is
if you call backward again without zeroing out the gradients, then it adds this set of gradients to the old gradients. So by doing these two half size batches without zeroing out the gradients
between them, it’s adding them up. So I’m going to end up with the total gradient of a 64 image batch size, but passing only 32 at a time.
If I used accumulate equals four, it would go through this four times, adding them up before it subtracted out the coefficients dot grad times learning rate and zeroed it out. If I put in a cube
equals 64, it would go through into a single image one at a time. And after 64 passes through, eventually count would be greater than 64, and we would do the update. So that’s gradient accumulation.
It’s a very simple idea, which is that you don’t have to actually update your weights every loop through for every mini-batch. You can just do it from time to time. But it has quite significant
implications, which I find most people seem not to realize, which is if you look on, like, Twitter or Reddit or whatever, people could say, oh, I need to buy a bigger GPU to train bigger models.
But they don’t. They could just use gradient accumulation. And so given the huge price differential between, say, a RTX 3080 and an RTX 3090 Ti, huge price differential, the performance is not that
different. The big difference is the memory. So what? Just put in a bit smaller batch size and do gradient accumulation. So there’s actually not that much reason to buy giant GPUs. John?
Are the results with gradient accumulation numerically identical?
[Jeremy Howard]
They’re numerically identical for this particular architecture. There is something called batch
normalization, which we will look at in part two of the course, which keeps track of the moving average of standard deviations and averages, and does it in a mathematically slightly incorrect way as
a result of which if you’ve got batch normalization, then it could, it basically will introduce more volatility, which is not necessarily a bad thing, but because it’s not mathematically identical,
you won’t necessarily get the same results. ConvNext doesn’t use batch normalization, so it is the same. And in fact, a lot of the models people want to use really big versions of, which is NLP ones,
transformers, tend not to use batch normalization, but instead they use something called layer normalization, which, yeah, doesn’t have the same issue. I think that’s probably fair to say. I haven’t
thought about it that deeply.
In practice, I found adding gradient accumulation for ConvNext has not caused any issues for me. I don’t have to change any parameters when I do it. Any other questions on the forum, John?
Gradient Accumulation Clarifications
Tamori asking, shouldn’t it be count greater than equal to 64 if BS equals 64? I haven’t.
[Jeremy Howard]
No, I don’t think so. Oh, yeah. So we start at zero, then it’s going to be 32, then it’s going to be, yeah, yeah, probably. You can probably tell I didn’t actually run this code.
Madhav is asking, does this mean that LRFind is based on the batch size set during the data block?
[Jeremy Howard]
Yeah, so LRFind just uses your data loader’s batch size.
Edward is asking, why do we need gradient accumulation rather than just using a smaller batch size?
And follows up with, how would we pick a good batch size?
[Jeremy Howard]
Well, just if you use a smaller batch size, here’s the thing, right? Different architectures have different amounts of memory, you know, which they take up. And so you’ll end up with different batch
sizes for different architectures, which is not necessarily a bad thing, but each of them is going to then need a different learning rate and maybe even different weight decay or whatever. Like the
kind of the settings that’s working really well for batch size 64 won’t necessarily work really well for batch size 32. And, you know, you want to be able to experiment as easily and quickly as
possible. I think the second part of your question was how do you pick optimal batch size? Honestly, the standard approach is to pick the largest one you can. Just because it’s faster that way,
you’re getting more parallel processing going on.
Although to be honest, I quite often use batch sizes that are quite a bit smaller than I need, because quite often it doesn’t make that much difference. But yeah, the rule of thumb would be, you
know, pick a batch size that fits in your GPU. And for performance reasons, I think it’s generally a good idea to have it be a multiple of eight. Everybody seems to always use powers of two, I don’t
know, like, I don’t think it actually matters.
Learning Rate Scaling
And look, there’s one other just a clarification or a check if the learning rate should be scaled according to the batch size.
[Jeremy Howard]
Yeah, so generally speaking, the rule of thumb is that if you divide the batch size by two, you divide the learning rate by two. But unfortunately, it’s not quite perfect. Did you have a question,
Nick? If you do, you can. Okay, cool.
Yeah. Now that’s us all caught up.
Gradient Accumulation in Fast AI
Thanks, Jeremy.
[Jeremy Howard]
Good questions. Thank you.
So gradient accumulation in fast AI is very straightforward. You just divide the batch size by however much you want to divide it by. And then you add something called a callback. And a callback is
something which changes the way the model trains. This callback is called gradient accumulation. And you pass in the effective batch size you want. And then you say, when you create the learner, you
say, these are the callbacks I want. And so it’s going to pass in gradient accumulation callback. So it’s going to only update the weights once it’s got 64 images. So if we pass in a Qm equals one,
it won’t do any gradient accumulation. And that uses four gig. If we use Qm equals two, about three gig. Qm equals four, about two and a half gig.
And generally, the bigger the model, the closer you’ll get to a kind of a linear scaling, because models have a kind of a bit of overhead that they have anyway.
Training Different Models
So what I then did was I just went through all the different models I wanted to try. So I wanted to try ConvNex large, add a 320 by 240, VIT large, SWIN V2 large, SWIN large. And on each of these, I
just tried running it with a Qm equals one. And actually, every single time for all of these, I got a amount of memory error. And then I tried each of them independently with Qm equals two. And so it
turns out that all of these worked with a Qm equals two. And it only took me 12 seconds each time. So that was a very quick thing for me to then know, okay, I now know how to train all of these
models on a 16 gigabyte card. So I can check here, they’re all in less than 16 gig. So then I just created a little dictionary of all the architectures I wanted.
And for each architecture, all of the resize methods I wanted and final sizes I wanted. Now, these models, VIT, SWIN V2 and SWIN are all transformers models, which means that, well, most transformers
models, nearly all of them have a fixed size. This one’s 192, this one’s 224. So I have to make sure that my final size is a square of the size required. Otherwise, I get an error. There is a way of
working around this. But I haven’t experimented with it enough to know when it works well and when it doesn’t. So we’ll probably come back to that in part two. So for now, we’re just going to use the
size that they asked us to use. So with this dictionary of architectures and for each architecture, kind of pre-processing details, we switch the training path back to using all of our images.
And then we can loop through each architecture and loop through each item transforms and sizes and train the model. And then the training script, if you’re fine-tuning, returns the TTA predictions.
So I append all those TTA predictions for each model, for each type, into a list. And after each one, it’s a good idea to do this garbage collection and empty cache, because otherwise I find what
happens is your GPU memory kind of, I don’t know, I think gets fragmented or something. And after a while, it runs out of memory, even when you thought it wouldn’t. So this way, you can really do as
much as you like without running out of memory.
Ensemble of Models
So they all train, train, train, train. And one key thing to note here is that in my train script, my data loaders does not have the seed equals parameter. So I’m using a different training set every
time. So that means that for each of these different runs, they’re using also different validation sets. So they’re not directly comparable, but you can kind of see they’re all doing pretty well, 2.1
percent, 2.3 percent, 1.7 percent, and so forth. So why am I using different training and validation sets for each of these? That’s because I want to ensemble them. So I’m going to use bagging, which
is I am going to take the average of their predictions.
Now, I mean, really, when we talked about random forest bagging, we were taking the average of, like, intentionally weak models. These are not intentionally weak models. They’re meant to be good
models, but they’re all different. They’re using different architectures and pre-processing approaches. And so in general, we would hope that these different approaches, some might work well for some
images and some might work well for other images. And so when we average them out, hopefully we’ll get a good blend of kind of different ideas, which is kind of what you want in bagging. So we can
stack up that list of different, of all the different probabilities and take their mean. And so that’s going to give us 3,469 predictions. That’s our test set size. And each one has 10 probabilities,
the probability of each disease.
And so then we can use argmax to find which probability index is the highest. So that’s going to give us our list of indexes. So this is basically the same steps as we used before to create our CSV
submission file. So at the time of creating this analysis, that got me to the top of the leaderboard. And in fact, these are my four submissions and you can see each one got better. Now you’re not
always going to get this nice monotonic improvement, right? But you want to be trying to submit something every day to kind of like try out something new, right? And the more you practice, the more
you’ll get a good intuition of what’s going to help, right? So partly I’m showing you this to say, it’s not like purely random as to whether things work or don’t. Once you’ve been doing this for a
while, you know, you will generally be improving things most of the time.
So as you can see from the descriptions, my first submission was our confnex small, the 12 epochs with TTA. And then an ensemble of confnex. So it’s basically this exact same thing, but just
retraining a few with different training subsets. And then this is the same thing again. This is the thing we just saw, basically. The ensemble of large bottles with TTA. And then the last one was
something I skipped over, which was the VIT models were the best in my testing. So I basically weighted them as double in the ensemble. And pretty unscientific, but again, it gave it another boost.
And so that was, that was it.
K-Fold Cross-Validation
All right, John. Uh, yes, thanks, Jeremy. Uh, so in no particular order, Kurian is asking, would trying out cross-validation with k-folds with the same architecture make sense?
[Jeremy Howard]
Okay, so a popular thing is to do k-fold cross-validation. So k-fold cross-validation is something very, very similar to what I’ve done here. So what I’ve done here is I’ve trained a bunch of models,
um, with, uh, different training sets. Each one is a different random 80% of the data. Um, five-fold cross-validation does something similar, but what it says is rather than picking, like, say, five
samples out with different random subsets, in fact, uh, instead, uh, first, like, do all except for the first 20% of the data, and then all but the second 20%, and then all but the third, third, and
so forth. And so you end up with five subsets, each of which have non-overlapping validation sets. Um, and then you’ll ensemble those.
Um, you know, in a, in theory, maybe that could be slightly better because you’re kind of guaranteed that every row is, appears four times, you know, effectively. Um, it also has a benefit that you
could average those five validation sets because there’s no kind of overlap between them to get a, uh, a cross-validation. Personally, I generally don’t bother. Um, and the reason I don’t is because,
um, this way, I can add and remove models very easily. Um, I don’t, you know, I, I can just, you know, add another architecture and whatever to my ensemble without trying to find a different
overlapping, non-overlapping subset. So, um, yeah, cross-validation is therefore something that I use probably less than most people or almost, or almost never.
Drawbacks of Gradient Accumulation
Awesome. Thank you. Um, are there any, just going back to gradient accumulation, any other kind of drawbacks or potential gotchas with gradient accumulation?
[Jeremy Howard]
No, not really. Um, yeah, like amazingly, um, it doesn’t even really slow things down much, you know, going from a batch size of 64 to a batch size of 32. By definition, you had to do it because your
GPU’s full. So you’re obviously giving a lot of data. So it’s probably going to be using its processing speed pretty effectively. So yeah, um, no, it’s just, it’s just a good technique that we should
all be buying cheaper graphics cards with less memory in them and using, you know, have, like, I don’t know the prices. I suspect, you could probably buy like two 3080s for the price of one 3090 Ti
or something. Um, that would be a very good deal.
GPU Recommendations
Uh, yes, clearly you’re not on the Nvidia payroll.
So look, this is a good segue then. We did have a question about sort of GPU recommendations and there’s been a bit of chat on that as well. I bet. Um, so any, any, you know, commentary, any
additional commentary around GPU recommendations?
[Jeremy Howard]
No, not really. I mean, obviously, at the moment, Nvidia is the only game in town, you know, if you buy, if you’re trying to use a, you know, Apple M1 or M2 or, or an AMD card, you’re basically in
for a world of pain in terms of compatibility and stuff and unoptimized libraries and whatever. Um, the, the Nvidia, um, consumer cards. So the ones that start with RTX are much cheaper, um, but are
just as good as the expensive, um, enterprise cards. So you might be wondering why anybody would buy the expensive enterprise cards.
And the reason is that there’s a licensing issue that Nvidia will not allow you to use an RTX consumer card in a data center, which is also why cloud computing is more expensive than it kind of ought
to be because everybody selling cloud computing GPUs is selling these cards that are like, I can’t remember. I think they’re like three times more expensive for kind of the same features. Um, so
yeah, if you do get serious about deep learning to the point that you’re prepared to invest, you know, a few days in administering a box and, you know, I guess it depends, you know, prices hopefully
will start to come down, but currently a thousand or $2,000, a thousand or $2,000 on buying a GPU, then, you know, that’ll probably pay you back pretty quickly.
Great. Thank you. Um, let’s see, another one’s come in.
Teacher-Student Models
Uh, if you have a, back on models, not hardware, if you have a well-functioning but large model, can it make sense to train a smaller model to produce the same final activations as the larger model?
[Jeremy Howard]
Oh yeah, absolutely. I’m not sure we’ll get into that this time around, but, um, yeah, um, we’ll cover that in part two, I think, but yeah, basically there’s a teacher student models and model
distillation, which broadly speaking, there, there are ways to make inference faster by training small models that work the same way as large models. Great. Thank you. All come up.
Road to the Top Conclusion
All right. So that is the actual real end of road to the top, because beyond that, we don’t actually cover how to get closer to the top.
Multi-Target Model
You’d have to ask Kurian to share his techniques to find out that or Nick to get the second place in the top. Um, part four is actually, um, something that I think is very useful to know about for,
for learning.
And it’s going to teach us a whole lot about how the last layer of a neural networks. And specifically what we’re going to try to do is we’re going to try to build a model that doesn’t just predict
the disease, but also predicts the type of rice. So how would you do that?
Data Loader with Two Dependent Variables
So here’s the data loader we’re going to try to build. It’s going to be something that for each image, it tells us the disease and the type of rice. I say disease, sometimes normal, I guess some of
them are not diseased. So to build a model that can predict two things. The first thing is you’re going to need data loaders that have two dependent variables. And that is shockingly easy to do in
fast AI, um, thanks to the data block. So we’ve seen the data block before.
We haven’t been using it for the patty competition so far because we haven’t needed it. We could just use a image data loader dot from folder. So that’s like the highest level API, the simplest API.
If we go down a level deeper into the data block, we have a lot more flexibility. So if you’ve been following the walkthroughs, you’ll know that as I built this, the first thing I actually did was to
simply replicate the previous notebook, but replace the image data loader dot from folders with a data block to try to do first of all, exactly the same thing. And then I added the second dependent
variable. So if we look at the previous image data loader from folders thingy, here it is, we are passing in some item transforms and some batch transforms. And we had something saying what
percentage should be the validation set.
So in a data block, if you remember, we have to pass in a blocks argument saying what kind of data is the independent variable and what is the dependent variable. So to replicate what we had before,
we would just pass in image block comma category block because we’ve got an image as our independent variable and a category, one type of rice, as the dependent variable. So the new thing I’m going
to show you here is that you don’t have to only put in two things. You can put in as many as you like. So if you put in three things, we’re going to generate one image and two categories. Now
fast.ai, if you’re saying I want three things, fast.ai doesn’t know which of those is the independent variable and which is the dependent variable. So the next thing you have to tell it is how many
inputs are there, number of inputs. And so here I’ve said there’s one input. So that means this is the input. And therefore, by definition, two categories will be the output.
Because remember, we’re trying to predict two things, the type of rice and the disease. Okay, this is the same as what we’ve seen before. To find out, to get our list of items, we’ll call
getImageFiles. Now here’s something we haven’t seen before. getY is our labeling function. Normally we pass to getY a single thing, such as the parent label function, which looks at the name of the
parent directory, which remember is how these images are structured. And that would tell us the label. But getY can also take an array. And in this case, we want two different labels. One is the name
of the parent directory, because that’s the disease. The second is the variety.
getVariety Function
So what’s getVariety? getVariety is a function. So let me explain how this function works. So we can create a data frame containing our trainings, our training data that came from Kaggle.
So for each image, it tells us the disease and the variety. And what I did is something I haven’t shown before. In pandas, you can set one column to be the index. And when you do that, in this case
imageId, it makes this series, this data frame, kind of like a dictionary. I can index into it by saying, tell me the row for this image. And to do that, you use the lock attribute, the location. So
we want in the data frame, the location of this image. And then you can also say optionally what column you want, this column. And so here’s this image. And here’s this column. And as you can see, it
returns that thing. So hopefully now you can see it’s pretty easy for us to create a function that takes a row, sorry, a path, and returns the location in the data frame of the name of that file.
Because remember, these are the names of files for the variety column. So that’s our second gateway. Okay. And then we’ve seen this before, randomly split the data into the 20% and 80%. And so we
just switch them all to 192, just for this example. And then use data augmentation to get us down to 128 square images, just for this example. And so that’s what we get when we say show batch. We get
what we just discussed. So now we need a model that predicts two things.
Model Predicting Two Things
How do we create a model that predicts two things?
Well, the key thing to realize is we never actually had a model that predicts two things. We had a model that predicts 10 things before. The 10 things we predicted is the probability of each disease.
So we don’t actually now want a model that predicts two things. We want a model that predicts 20 things. The probability of each of the 10 diseases and the probability of each of the 10 varieties. So
how could we do that? Well, let’s first of all try to just create the same disease model we had before with our new data loader. So this is going to be reasonably straightforward. The key thing to
know is that since we told Fast.ai that there’s one input, and therefore by definition there’s two outputs, it’s going to pass to our metrics and to our loss functions three things instead of two.
Metrics and Loss Functions
The predictions from the model and the disease and the variety. So we can’t just use error rate as our metric anymore because error rate takes two things. Instead, we have to create a function that
takes three things and return error rate on the two things we want, which is the predictions from the model and the disease. Okay, so there’s predictions in the model. This is the target. So that’s
actually all we need to do to define a metric that’s going to work with our new data set, with our new data loader. This is not going to actually tell us anything about variety. First, it’s going to
try to replicate something that can do just disease. So when we create our learner, we’ll pass in this new disease error function.
Okay, so we’re halfway there. The other thing we’re going to need is to change our loss function. Now, we never actually talked about what loss function to use, and that’s because VisionLearner
guessed what loss function to use. VisionLearner saw that our dependent variable was a single category, and it knows the best loss function that’s probably going to be the case for things with a
single category, and it knows how big the category is. So it just didn’t bother us at all. It just said, okay, I’ll figure it out for you. So the only time we’ve provided our own loss function is
when we were kind of doing linear models and neural nets from scratch. And we did, I think, mean squared error. We might also have done mean absolute error. Neither of those work when the dependent
variable is a category. Now, how would you use mean squared error or mean absolute error to say how close were these 10 probability predictions to this one correct answer?
So in this case, we have to use a different loss function.
Cross-Entropy Loss
We have to use something called cross-entropy loss. And this is actually the loss function that Fast.ai picked for us before without us knowing. But now that we are having to pick it out manually,
I’m going to explain to you exactly what cross-entropy loss does. Okay? And, you know, these details are very important indeed. Like, remember I said at the start of this class, the stuff that
happens in the middle of the model, you’re not going to have to care about much in your life, if ever. But the stuff that happens in the first layer and the last layer, including the loss function
that sits between the last layer and the loss, you’re going to have to care about a lot. Right? This stuff comes up all the time. So you definitely want to know about cross-entropy loss.
And so I’m going to explain it using a spreadsheet. This spreadsheet’s in the course repo. And so let’s say you were predicting something like a kind of a mini image net thing, where you’re trying to
predict whether something, an image is a cat, a dog, a plane, a fish, or a building. So you set up some model, whatever it is, a convex model, or just a big bunch of linear layers connected up, or
whatever. And initially you’ve got some random weights, and it spits out at the end five predictions. Right? So remember to predict something with five categories, your model will spit out five
probabilities. Now it doesn’t initially spit out probabilities. There’s nothing making them probabilities. It just spits out five numbers. Could be negative, could be positive. Okay? So here’s the
output of the model. So what we want to do is we want to convert these into probabilities.
And so we do that in two steps. The first thing we do is we go exp, that’s e to the power of. We go e to the power of each of those things. Like so. Okay? And so here’s the mathematical formula we’re
using. This is called the softmax, what we’re working through. We’re going to go through each of the categories. So these are our five categories. So here k is five. We’re going to go through each of
our categories. And we’re going to go e to the power of the output. So zj is the output for the jth category. So here’s that. And then we’re going to sum them all together. Here it is, sum up
together. Okay? So this is the denominator.
And then the numerator is just e to the power of the thing we care about. So this row. So the numerator is e to the power of cat on this row, e to the power of dog on this row, and so forth. Now if
you think about it, since the denominator adds up all the e to the power ofs, then when we do each one divided by the sum, that means the sum of these will equal one, by definition. Right? And so now
we have things that can be treated as probabilities. They’re all numbers between zero and one. Numbers that were bigger in the output will be bigger here. But there’s something else interesting,
which is because we did e to the power of, it means that the bigger numbers will be like pushed up to numbers closer to one.
Like we’re saying, like, oh, really try to pick one thing as having most of the probability. Because we are trying to predict, you know, one thing. We’re trying to predict which one is it. And so
this is called softmax. So sometimes you’ll see people complaining about the fact that their model, which they said, let’s say, is it a teddy bear or a grizzly bear or a black bear? And they feed it
a picture of a cat. And they say, oh, the model’s wrong, because it predicted grizzly bear. But it’s not a grizzly bear. As you can see, there’s no way for this to predict anything other than the
categories we’re giving it. We’re forcing it to that. Now we don’t, if you want, like, there’s something else you could do, which is you could actually have them not add up to one. Right? You could
instead have something which simply says, what’s the probability it’s a cat? What’s the probability it’s a dog? What’s the totally separately? And they could add up to less than one.
And in that situation, you can, you know, or more than one, in which case you could have, like, more than one thing being true or zero things being true. But in this particular case, where we want to
predict one and one thing only, we use softmax.
The first part of the cross-entropy formula, the first part of the cross-entropy formula, in fact, let’s look it up. nn dot cross-entropy loss. The first part of what cross-entropy loss in PyTorch
does is to calculate the softmax. It’s actually the log of the softmax, but don’t worry about that too much. It’s just a slightly faster to do the log. Okay. So now for each one of our five things,
we’ve got a probability.
The next step is the cross-entropy calculation, which is we take our five things, we’ve got our five probabilities, and then we’ve got our actuals. Now, the truth is the actual, you know, the five
things would have indices, right? Zero, one, two, three, or four. And the actual turned out to be the number one. But what we tend to do is we think of it as being one-hot encoded, which is we put a
one next to the thing for which it’s true, and a zero everywhere else. And so now we can compare these five numbers to these five numbers, and we would expect to have a smaller loss if the softmax
was high where the actual is high. Okay. And so here’s how we calculate, this is the formula, the cross-entropy loss.
We sum up, so switch to m this time for some reason, but it’s the same thing. We sum up across the five categories, so m is five. And for each one, we multiply the actual target value, so that’s
zero. So here it is here, the actual target value. And we multiply that by the log of the predicted probability, the log of red, the predicted probability. And so, of course, for four of these, that
value is zero. Because see here, yj equals zero, by definition, for all but one of them, because it’s one-hot encoded. So for the one that it’s not, we’ve got our actual times the log softmax. Okay.
Cross-Entropy Loss Calculation
And so now actually you can see why PyTorch prefers to use log softmax, because then it kind of skips over having to do this log at all. So this equation looks slightly frightening, but when you
think about it, all it’s actually doing is it’s finding the probability for the one that is one and taking its log. Right? It’s kind of weird doing it as a sum, but in math, it can be a little bit
tricky to kind of say, oh, look this up in an array, which is basically all it’s doing. But yeah, basically, at least in this case, a single result with softmax, this is all it’s doing. It’s finding
the 0.87 where it’s 1, 4, and taking the log, and then finally negative. So that is what cross-entropy loss does. We add that together for every row.
So here’s what it looks like if we add it together over every row. Right? So n is the number of rows. And here’s a special case. This is called binary cross-entropy. What happens if we’re not
predicting which of five things it is, but we’re just predicting, is it a cat? So in that case, if you look at this approach, you end up with this formula, which this is identical to this formula,
but for just two cases, which is you either are a cat or you’re not a cat. Right? And so if you’re not a cat, it’s 1 minus you are a cat. And same with the probability. You’ve got the probability you
are a cat, and then not a cat is 1 minus that. So here’s this special case of binary cross-entropy. And now our rows represent rows of data. Okay? So each one of these is a different image, a
different prediction.
And so for each one, I’m just predicting, are you a cat? And this is the actual. And so the actual, are you not a cat, is just 1 minus that. And so then these are the predictions that came out of the
model. Again, we can use Softmax or its binary equivalent. And so that will give you a prediction that you’re a cat. And the prediction that it’s not a cat is 1 minus that. And so here is each of the
part, yi times log of pyi. And here is, why did I subtract? That’s weird. Oh, because I’ve got minus of both, so I’ll just do it this way, avoids parentheses. Yeah, minus the, are you not a cat,
times the log of the prediction of are you not a cat.
And then we can add those together. And so that would be the binary cross-entropy loss of this data set of five cat or not cat images.
Binary Cross-Entropy
Now, if you’ve got an eagle eye, you may have noticed that I am currently looking at the documentation for something called nn.crossentropy loss. But over here, I had something called f.crossentropy.
Basically, it turns out that all of the loss functions in PyTorch have two versions. There’s a version which is a class. This is a class, which you can instantiate, passing in various tweaks you
might want. And there’s also a version which is just a function. And so if you don’t need any of these tweaks, you can just use the function.
Loss Function Versions
The functions live in a, I kind of remember what the sub-module called, I think it might be like torch.nn.functional, but everybody, including the PyTorch official docs, just calls it capital F. So
that’s what this capital F refers to. So our loss, if we just care about disease, we’re going to be past the three things. We’re just going to calculate cross-entropy on our input versus disease. All
right. So that’s all fine. So now when we create a vision learner, you can’t rely on fast.ai to know what loss function to use because we’ve got multiple targets. So you have to say, this is the loss
function I want to use. This is the metrics I want to use. And the other thing you can’t rely on is that fast.ai no longer knows how many activations to create, because again, there’s more than one
target. So you have to say the number of outputs to create at the last layer is 10.
So this is just saying, what’s the size of the last matrix? And once we’ve done that, we can train it and we get basically the same kind of result as we always get, because this model at this point
is identical to our previous convex small model. We’ve just done it in a slightly more roundabout way.
Multi-Target Model
So finally, before our break, I’ll show you how to expand this now into a multi-target model. And the trick is actually very simple. And you might have almost got the idea of it when I talked about
it earlier. Our vision learner now requires 20 outputs. We now need that last matrix to produce 20 activations, not 10. 10 of those activations are going to predict the disease, and 10 of the
activations are going to predict the variety.
So you might be then asking like, well, how does the model know what it’s meant to be predicting? And the answer is, with the loss function, you’re going to have to tell it. So for example, disease
loss, remember, it’s going to get the input, the disease, and the variety. This is now going to have 20 columns in. So we’re just going to decide, all right, we’re just going to decide the first 10
columns, we’re going to decide the prediction of what the disease is, the probability of each disease. So we can now pass to cross-entropy the first 10 columns and the disease target. So the way you
read this, colon means every row, and then colon 10 means every column up to the 10th. So these are the first 10 columns. And that will, that’s a loss function that just works on predicting disease
using the first 10 columns.
For variety, we’ll use cross-entropy loss with the target of variety. And this time we’ll use the second 10 columns. So here’s column 10 onwards. So then the overall loss function is the sum of those
two things, disease loss plus variety loss. And that’s actually it. That’s all the model needs to basically, it’s now going to, if you kind of think through the manual neural nets we’ve created, this
loss function will be reduced when the first 10 columns are doing a good job of predicting the disease probabilities, and the second 10 columns are doing a good job of predicting the variety
probabilities. And therefore the gradients will point in an appropriate direction that the coefficients will get better and better at using those columns for those purposes.
Error Rate for Disease and Variety
It would be nice to see the error rate as well for each of disease and variety. So we can call error rate passing in the first 10 columns in disease, and then variety the second 10 columns in
variety. And we may as well also add to the metrics the losses. And so now when we create our learner, we’re going to pass in as the loss function the combined loss. And as the metrics, our list of
all the metrics, and n out equals 20. And now look what happens when we train. As well as telling us the overall train in valid loss, it also tells us the disease and variety error, and the disease
and variety loss. And you can see our disease error is getting down to similar levels it was before. It’s slightly less good, but it’s similar. It’s not surprising it’s slightly less good, because
we’ve only given it the same number of epochs, and we’re now asking it to try to do more stuff, which is to learn to recognize what the rice variety looks like, and also learns to recognize what the
disease looks like.
Multi-Target Model Performance
Here’s the counterintuitive thing though. If we train it for longer, it may well turn out that this model, which is trying to predict two things, actually gets better at predicting disease than our
disease-specific model. Why is that? Like, that sounds weird, right? Because we’re trying to get it to do more stuff, and it’s the same size model. Well, the reason is that quite often it’ll turn out
that the kinds of features that help you recognize a variety of rice are also useful for recognizing the disease. You know, maybe there are certain textures, right? Or maybe some diseases impact
different varieties in different ways.
So it’d be really helpful to know what variety it was. I haven’t tried training this for a long time, and I don’t know the answer. In this particular case, does a multi-target model do better than a
single-target model at predicting disease? But I just wanted to let you know sometimes it does. For example, a few years ago, there was a Kaggle competition for recognizing the kinds of fish on a
boat, and I remember we ended up doing a multi-target model where we tried to predict a second thing. I can’t even remember what it was. Maybe it was a type of boat or something, and it definitely
turned out in that Kaggle competition that predicting two things helped you predict the type of fish better than predicting just the type of fish. So there’s at least, you know, there’s two reasons
to learn about multi-target models. One is that sometimes you just want to be able to predict more than one thing. So this is useful. And the second is that sometimes this will actually be better at
predicting just one thing than a just-one-thing model.
Reasons to Learn Multi-Target Models
And of course, the third reason is it really forced us to dig quite deeply into these loss functions and activations in a way we haven’t quite done before. So it’s okay. It’s absolutely okay if this
is confusing. The way to make it not confusing is, well, the first thing I do is, like, go back to our earlier models where we did stuff by hand on, like, the Titanic data set and built our own
architectures. And maybe you could try to build a model that predicts two things in the Titanic data set. Maybe you could try to predict both sex and survival or something like that, or class and
survival. Because that’s going to kind of force you to look at it on very small data sets. And then the other thing I’d say is run this notebook and really experiment at trying to see what kind of
outputs you get.
Like, actually look at the inputs and look at the outputs and look at the data loaders and so forth.
All right. Let’s have a six-minute break. So I’ll see you back here at ten past seven.
Collaborative Filtering Deep Dive
Okay. Welcome back. Oh, before I continue, I very rudely forgot to mention this very nice equation image here is from an article by Chris Sedd called Things That Confused Me About Cross Entropy. It’s
a very good article. So I recommend you check it out if you want to go a bit deeper there. There’s a link to it inside the spreadsheet. So the next notebook we’re going to be looking at is this one
called Collaborative Filtering Deep Dive.
Movie Lens Data Set
And this is going to cover our last of the four major application areas, collaborative filtering. And this is actually the first time I’m going to be presenting a chapter of the book largely without
variation. Because this is one where I looked back at the chapter and I was like, oh, I can’t think of any way to improve this. So I thought I’ll just leave it as is. But we have put the whole
chapter up on Kaggle. So that’s the way I’m going to be showing it to you. And so we’re going to be looking at a data set called the Movie Lens data set, which is a data set of movie ratings. And
we’re going to grab a smaller version of it, 100,000 record version of it.
And it comes as a CSV file, which we can read in. But it’s not really a CSV file, it’s a TSV file. This here means a tab in Python. And these are the names of the columns. So here’s what it looks
like. It’s got a user, a movie, a rating, and a timestamp. We’re not going to use the timestamp at all. So basically three columns we care about. This is a user ID. So maybe 196 is Jeremy, and maybe
186 is Rachel, and 22 is John, I don’t know. Maybe this movie is Return of the Jedi, and this one’s Casablanca, this one’s LA Confidential. And then this rating says, how did Jeremy feel about Return
of the Jedi? He gave it a three out of five. That’s how we can read this data set. This kind of data is very common.
Collaborative Filtering Data
Any time you’ve got a user and a product or service, and you might not even have ratings, maybe just the fact that they bought that product. You could have a similar table with zeros and ones. So for
example, Radek, who’s in the audience here, is now at NVIDIA doing basically just this, right? Recommendation systems. So recommendation systems, it’s a huge industry. And so what we’re learning
today is a really key foundation of it. So these are the first few rows. This is not a particularly great way to see it. I prefer to kind of cross-tabulate it like that, like this. This is the same
information. So for each movie, for each user, here’s the rating. So user 212, never watched movie 49.
Now if you’re wondering why there’s so few empty cells here, I actually grabbed the most watched movies and the most movie watching users for this particular sample matrix. So that’s why it’s
particularly full. So yeah, so this is what kind of a collaborative filtering data set looks like when we cross-tabulate it.
Filling in the Gap
So how do we fill in this gap? So maybe user 212 is Nick and movie 49. What’s a movie you haven’t seen, Nick, and you’d quite like to, maybe not sure about it? The new Elvis movie. Baz Luhrmann, good
choice. Australian director. Filmed in Queensland. Yeah. Okay. So that’s movie number 49. So is Nick going to like the new Elvis movie?
Predicting User Preferences
Well, to figure this out, what we could do ideally, we’d like to know for each movie, what kind of movie is it? Like what are the kind of features of it? Is it like action-y, science fiction-y,
dialogue-driven, critical acclaimed, you know? So let’s say for example, we were trying to look at The Last Skywalker. Maybe that was the movie that Nick’s wondering about watching. And so if we like
had three categories being science fiction, action, or kind of classic old movies, we’d say The Last Skywalker is very science fiction. Let’s see, this is from like negative one to one. Pretty
action, definitely not an old classic, or at least not yet. And so then maybe we then could say like, okay, well, maybe like Nick’s tastes in movies are that he really likes science fiction, quite
likes action movies, and doesn’t really like old classics.
Right? So then we could kind of like match these up to see how much we think this user might like this movie. To calculate the match, we could just multiply the corresponding values. Use a one times
Last Skywalker and add them up. 0.9 times 0.98 plus 0.8 times 0.9 plus negative 0.6 times negative 0.9. That’s going to give us a pretty high number, right? With a maximum of three. So that would
suggest Nick probably would like The Last Skywalker. On the other hand, the movie Casablanca, we would say definitely not very science fiction, not really very action, definitely very old classic. So
then we’d do exactly the same calculation and get this negative result here.
So you probably wouldn’t like Casablanca. This thing here, when we multiply the corresponding parts of a vector together and add them up, is called a dot product in math. So this is the dot product
of the user’s preferences and the type of movie. Now the problem is, we weren’t given that information. We know nothing about these users or about the movies. So what are we going to do?
Latent Factors
We want to try to create these factors without knowing ahead of time what they are. We wouldn’t even know what factors to create. What are the things that really matters when people decide what
movies they want to watch? What we can do is we can create things called latent factors. Latent factors is this weird idea that we can say, I don’t know what things about movies matter to people, but
there’s probably something.
And let’s just try like using SGD to find them.
Latent Factors in Excel
And we can do it in everybody’s favorite mathematical optimization software, Microsoft Excel. So here is that table. And what we can do, let’s head over here actually, here’s that table. So what we
could do is we could say for each of those movies, so let’s say for movie 27, let’s assume there are five latent factors. I don’t know what they’re They’re just five latent factors. We’ll figure them
out later. And for now, I certainly don’t know what the value of those five latent factors for movie 27.
So we’re going to just chuck a little random numbers in them. And we’re going to do the same thing for movie 49. Pick another five random numbers. And the same thing for movie 57. Pick another five
numbers. And you might not be surprised to hear, we’re going to do the same thing for each user. So for user 14, we’re going to pick five random numbers for them. And for user 29, we’ll pick five
random numbers for them. And so the idea is that this number here, 0.19, is saying, if it was true, that user ID 14 feels not very strongly about the fact that for movie 27 has a value of 0.71. So
therefore in here, we do the dot product. The details of why don’t matter too much, but well, actually, you can figure this out from what we’ve said so far.
Matrix Product and Dot Product
If you go back to our definition of matrix product, you might notice that the matrix product of a row with a column is the same thing as a dot product. And so here in Excel, I have a row and a
column. So therefore I say matrix multiply that by that. That gives us the dot product. So here’s the dot product of that by that, or the matrix multiply, given that row and column. The only other
slight quirk here is that if the actual rating is 0, is empty, I’m just going to leave it blank. I’m going to set it to 0 actually. So here is everybody’s rating, predicted rating of movies.
Stochastic Gradient Descent
I say predicted, of course, these are currently random numbers, so they are terrible predictions. But when we have some way to predict things, and we start with terrible random predictions, we know
how to make them better, don’t we?
We use stochastic gradient descent. Now to do that, we’re going to need a loss function. So that’s easy enough. We can just calculate the sum of x minus y squared divided by the count. That is the
mean squared error. And if we take the square root, that is the root mean squared error. So here is the root mean squared error in Excel between these predictions and these actuals. And so now that
we have a loss function, we can optimize it.
Optimizing the Loss Function
Data, solver, set objective, this one here, by changing cells, these ones here, and these ones here, solve.
Okay, and initially our loss is 2.81. So we hope it’s going to go down. And as it solves, not a great choice of background color, but it says 0.68. So this number is going down. So this is using,
actually in Excel it’s not quite using stochastic gradient descent, because Excel doesn’t know how to calculate gradients. There are actually optimization techniques that don’t need gradients. They
calculate them numerically as they go, but that’s a minor quirk. One thing you’ll notice is it’s doing it very, very slowly. There’s not much data here, and it’s still going. One reason for that is
that if it’s, because it’s not using gradients, it’s much slower, and the second is Excel is much slower than PyTorch. Anyway, it’s come up with an answer, and look at that. It’s got to 0.42. So it’s
got a pretty good prediction. And so we can kind of get a sense of this.
For example, looking at the last three, user 14 likes, dislikes, likes. Let’s see somebody else like that. Here’s somebody else. This person likes, dislikes, likes. So based on our kind of approach,
we’re saying, okay, since they have the same feeling about these three movies, maybe they’ll feel the same about these three movies. So this person likes all three of those movies, and this person
likes two out of three of them. So, you know, you kind of, this is the idea, right? As if somebody says to you, I like this movie, this movie, this movie, and you’re like, oh, they like those movies
too. What other movies do you like? And they’ll say, oh, how about this? There’s a chance, good chance, that you’re going to like the same thing. That’s the basis of collaborative filtering, okay?
And mathematically, we call this matrix completion.
Matrix Completion
So this matrix is missing values. We just want to complete them. So the core of collaborative filtering is, it’s a matrix completion exercise. Can you grab a microphone?
[Audience Member]
Cosine Similarity and Correlation
My question was, is with the dot products, right? So if we think about the math of that for a minute, is, yeah, if we think about the cosine of the angle between the two vectors, that’s going to
roughly approximate the correlation. Is that essentially what’s going on here in one sense?
[Jeremy Howard]
So is the cosine of the angle between the vectors much the same thing as the dot product? The answer is yes. They’re the same once you normalize them. So, yep. Is that still on?
[Audience Member]
It’s correlation, what we’re doing here at scale as well.
[Jeremy Howard]
Yeah, you can, yeah, you can think of it that way.
PyTorch Implementation
Now, this looks pretty different to how PyTorch looks. PyTorch has things in rows, right? We’ve got a user, a movie rating. User, movie rating, right? So how do we do the same kind of thing in
PyTorch? So let’s do the same kind of thing in Excel, but using the table in the same format that PyTorch has it, okay?
Excel Implementation with PyTorch Format
So to do that in Excel, the first thing I’m going to do is I’m going to see, okay, this, I’ve got to look at user number 14, and I want to know what index, like how far down this list is 14, okay? So
we’ll just match means find the index. So this is user index one. And then what I’m going to do is I’m going to say these five numbers is basically I want to find row one over here.
And in Excel, that’s called offset. So we’re going to offset from here by one row. And so you can see here it is 0.19, 0.63, 0.19, 0.63, et cetera, right? So here’s the second user, 0.25, 0.83, et
cetera. And we can do the same thing for movies, right? So movie 417 is index 14. That’s going to be 0.75, 0.47, et cetera. And so same thing, right? But now we’re going to offset from here by 14 to
get this row, which is 0.75, 0.47, et cetera. And so the prediction now is the dot product is called some product in Excel.
Dot Product in Excel
This is some product of those two things.
So this is exactly the same as we had before, right? But when we kind of put everything next to each other, we have to like manually look up the index. And so then for each one, we can calculate the
error squared prediction minus rating squared. And then we could add those all up. And if you remember, this is actually the same root mean squared error we had before we optimized before, 2.81,
because we’ve got the same numbers as before. And so this is mathematically identical. So what’s this weird word up here?
Embedding. You’ve probably heard it before, and you might have come across the impression it’s some very complex, fancy mathematical thing. But actually it turns out that it is just looking something
up in an array. That is what an embedding is. So we call this an embedding matrix.
And these are our user embeddings and our movie embeddings. So let’s take a look at that in PyTorch.
Embedding in PyTorch
And you know, at this point, if you’ve heard about embeddings before, you might be thinking that can’t be it. And yeah, it’s just as complex as the rectified linear unit, which turned out to be
replace negatives with zeros. Embedding actually means look something up in an array. So there’s a lot of things that we use as deep learning practitioners to try to make you as intimidated as
possible so that you don’t wander into our territory and start winning our Kaggle competitions. And unfortunately, once you discover the simplicity of it, you might start to think that you can do it
yourself. And then it turns out you can. So yeah, that’s what basically it turns out pretty much all of this jargon turns out to be.
Learning Latent Factors in PyTorch
So we’re going to try to learn these latent factors, which is exactly what we just did in Excel. We just learned the latent factors.
Data Loaders
All right. So if we’re going to learn things in PyTorch, we’re going to need data loaders. One thing I did is there is actually a movies table as well with the names of the movies. So I merged that
together with the ratings so that then we’ve now got the user ID and the actual name of the movie. We don’t need that, obviously, for the model, but it’s just going to make it a bit more fun to
interpret later. So this is called ratings. We have something called collab data loaders, so collaborative filtering data loaders. And we can get that from a data frame by passing in the data frame.
And it expects a user column and an item column. So the user column is what it sounds like, the person that is rating this thing.
And the item column is the product or service that they’re rating. In our case, the user column is called user, so we don’t have to pass that in. And the item column is called title, so we do have to
pass this in. Because by default, the user column should be called user, and the item column will be called item. Give it a batch size. And as usual, we can call show batch. And so here’s our data
loaders, a batch of data loaders, or at least a bit of it. And so now that we’ve dealt with the names, we actually get to see the names, which is nice.
User and Movie Factors
All right, so now we’re going to create the user factors and movie factors, i.e. this one and this one.
So the number of rows of the movie factors will be equal to the number of movies, and the number of rows of the user factors will be equal to the number of users. And the number of columns will be
whatever we want, however many factors we want to create. John?
Choosing the Number of Factors
This might be a pertinent time to jump in with a question. Any comments about choosing the number of factors?
[Jeremy Howard]
Not really. We have defaults that we use for embeddings in Fast.ai. It’s a very obscure formula, and people often ask me for the mathematical derivation of where it came from. But what actually
happened is I wrote down how many factors I think is appropriate for different size categories on a piece of paper at a table, or actually in Excel, and then I fitted a function to that, and that’s
the function. So it’s basically a mathematical function that fits my intuition about what works well.
But it seems to work pretty well. I’ve seen it used in lots of other places now. Lots of papers will be like, using Fast.ai’s rule of thumb for embedding sizes, here’s the formula.
Cool. Thank you.
[Jeremy Howard]
Training Speed
It’s pretty fast to train these things. You can try a few. So we’ve got to create, so the number of users is just the length of how many users there are. Number of movies is the length of how many
titles there are. So create a matrix of random numbers of users by five, and movies of movies by five. And now we need to look up the index of the movie in our movie latent factor matrix.
Embedding as Matrix Multiplication
The thing is, when we’ve learned about deep learning, we learned that we do matrix multiplications, not look something up in a matrix, in an array.
So in Excel, we were saying offset, which is to say find element number 14 in the table, which, that’s not a matrix multiply. How does that work? Well, actually it is. It actually is for the same
reason that we talked about here, which is we can represent, find the element number one thing in this list, is actually the same as multiplying by a one-hot encoded matrix. So remember how, if we,
let’s just take off the log for a moment. Look, this is returned 0.87. And particularly if I take the negative off here, if I add this up, this is 0.87, which is the result of finding the index
number one thing in this list.
But we didn’t do it that way. We did this by taking the dot product of this, sorry, of this and this. But that’s actually the same thing, right? Taking the dot product of a one-hot encoded vector
with something is the same as looking up this index in the vector. So that means that this exercise here of looking up the 14th thing is the same as doing a matrix multiply with a one-hot encoded
One-Hot Encoded Vector
And we can see that here. This is how we create a one-hot encoded vector of length n users, in which the third element is set to 1 and everything else is 0.
And if we multiply that, so at means, do you remember, matrix multiply in Python? So if we multiply that by our user factors, we get back this answer. And if we just ask for user factors number
three, we get back the exact same answer. They’re the same thing.
Embedding as a Computational Shortcut
So you can think of an embedding as being a computational shortcut for multiplying something by a one-hot encoded vector. And so if you think back to what we did with dummy variables, right, this
basically means embeddings are like a cool math trick for speeding up doing matrix multipliers with dummy variables. Not just speeding up, we never even have to create the dummy variables. We never
have to create the one-hot encoded vectors. We can just look up in an array.
Collaborative Filtering Model
All right, so we’re now ready to build a collaborative filtering model.
Creating a Model from Scratch
And we’re going to create one from scratch. And as we’ve discussed before, in PyTorch, a model is a class. And so we briefly touched on this, but I’m going to touch on it again.
Creating a Class in Python
This is how we create a class in Python. You give it a name, and then you say how to initialize it, how to construct it. So in Python, remember, they call these things dunder, whatever, this is
dunder init. These are magic methods that Python will call for you at certain times.
Magic Methods
The method called dunder init is called when you create an object of this class. So we could pass it a value. And so now we set the attribute called a equal to that value. And so then later on, we
could call a method called say that will say hello to whatever you passed in here. And this is what it will say. So for example, if you construct an object of type example, passing in sylva, self.a
now equals sylva. So if you say, use the dot method, the dot say method, nice to meet you. x is now nice to meet you. So it will say hello, sylva, nice to meet you. So that’s kind of all you need to
know about object-oriented programming in PyTorch to create a model.
Object-Oriented Programming in PyTorch
Oh, there is one more thing we need to know, sorry, which is you can put something in parentheses after your class name, and that’s called the super class.
Super Class
It’s basically going to give you some stuff for free, give you some functionality for free. And if you create a model in PyTorch, you have to make module your super class.
Module Super Class
This is actually Fast.ai’s version of module, but it’s nearly the same as PyTorch’s. So when we create this dot product object, it’s going to call dunder init.
dunder init Method
And we have to say, well, how many users are going to be in our model? And how many movies? And how many factors? And so we can now create an embedding of users by factors for users and an embedding
of movies by factors for movies. And so then PyTorch does something quite magic, which is that if you create a dot product object like so, then you can treat it like a function.
Treating a Model as a Function
You can call it and calculate values on it. And when you do that, this is really important to know, PyTorch is going to call a method called forward in your class.
forward Method
So this is where you put your calculation of your model. It has to be called forward. And it’s going to be past the object itself and the thing you’re calculating on. In this case, the user and movie
for a batch. So this is your batch of data. Each row will be one user and movie combination, and the columns will be users and movies. So we can grab the first column, right? So this is every row of
the first column, and look it up in the user factors embedding to get our users embeddings.
So that is the same as doing this. Let’s say this is one mini batch. And then we do exactly the same thing for the second column, passing it into our movie factors to look up the movie embeddings.
And then take the dot product. Dim equals one, because we’re summing across the columns for each row. We’re calculating a prediction for each row.
Training the Model
So once we’ve got that, we can pass it to a learner, passing in our data loaders and our model. And our loss function means squared error. And we can call fit. And away it goes.
And this, by the way, is running on CPU. These are very fast to run. So this is doing 100,000 rows in 10 seconds, which is a whole lot faster than our few dozen rows in Excel. And so you can see the
loss going down. And so we’ve trained a model.
Model Limitations
It’s not going to be a great model. And one of the problems is that, let’s see if we can see this in our Excel one. Look at this one here. This prediction’s bigger than five. But nothing’s bigger
than five. So that seems like a problem. We’re predicting things that are bigger than the highest possible number. And in fact, these are very much movie enthusiasts.
Movie Enthusiasts
Nobody gave anything a one. Yeah, nobody even gave anything a one here. So do you remember when we learned about sigmoid, the idea of squishing things between zero and one?
Sigmoid Function
We could do stuff still without a sigmoid. But when we added a sigmoid, it trained better, because the model didn’t have to work so hard to get it kind of into the right zone. Now, if you think about
it, if you take something and put it through a sigmoid, and then multiply it by five, now you’ve got something that’s going to be between zero and five. Used to have something that’s between zero and
one. So we could do that. In fact, we could do that in Excel. I’ll leave that as an exercise to the reader. Let’s do it over here in PyTorch.
Sigmoid Range
So if we take the exact same class as before, and this time we call sigmoid range. And so sigmoid range is something which will take our prediction and then squash it into our range.
And by default, we’ll use a range of zero through to 5.5. So it can’t be smaller than zero, can’t be bigger than 5.5. Why didn’t I use five? That’s because a sigmoid can never hit one, right? And a
sigmoid times five can never hit five. But some people do give things movies five. So you want to make it a bit bigger than our highest. So this one got a loss of 0.8628. Oh, it’s not better. Isn’t
that always the way? All right, didn’t actually help, doesn’t always, so be it.
Improving the Model
Let’s keep trying to improve it. Let me show you something I noticed. Some of the users, like this one, this person here just loved movies.
User Bias
They give nearly everything a four or five. Their worst score is a three, right? This person, oh here’s a one, this person’s got much more range. Some things are twos, some ones, some fives. This
person doesn’t seem to like movies very much considering how many they watch. Nothing gets a five. They’ve got discerning tastes, I guess. At the moment, we don’t have any way in our kind of
formulation of this model to say this user tends to give low scores and this user tends to give high scores. There’s just nothing like that, right? But that would be very easy to add. Let’s add one
more number to our five factors, just here, for each user.
Adding User Bias to the Model
And now, rather than doing just the matrix multiply, let’s add, oh it’s actually the top one, let’s add this number to it, h19.
And so for this one, let’s add i19 to it. Yeah, so I’ve got it wrong. This one here, so this this row here, we’re going to add to each rating. And then we’re going to do the same thing here.
Movie Bias
Each movie’s now got an extra number here that, again, we’re going to add a 26. So it’s our matrix multiplication plus, we call it the bias, the user bias plus the movie bias. So effectively, that’s
like making it so we don’t have an intercept of zero anymore.
Training with Bias
And so if we now train this model, data, solver, solve. So previously we got to 0.42, okay, and so we’re going to let that go along for a while. And then let’s also go back and look at PyTorch
version. So for PyTorch now, we’re going to have a user bias, which is an embedding of n users by one, right. Remember there was just one number for each user. And movie bias is an embedding of n
movies also by one. And so we can now look up the user embedding, the movie embedding, do the dot product, and then look up the user bias and the movie bias and add them. Chuck that through the
Let’s train that, see if we beat 0.865. Wow, we’re not training very well, are we?
Still not too great, 0.894. I think Excel normally does do better though. Let’s see. Okay, Excel. Oh, Excel’s done a lot better. It’s gone from 0.42 to 0.35. Okay, so what happened here? Why did it
get worse? Well, look at this. The valid loss got better, and then it started getting worse again. So we think we might be overfitting, which, you know, we have got a lot of parameters in our
Weight Decay
So how do we avoid overfitting?
So a classic way to avoid overfitting is to use something called weight decay, also known as L2 regularization, which sounds much more fancy.
Weight Decay in the Loss Function
What we’re going to do is, when we compute the gradients, we’re going to first add to our loss function the sum of the weights squared. Now this is something you should go back and add to your
titanic model, not that it’s overfitting, but just to try it, right? So previously, our gradients have just been, and our loss function, has just been about the difference between our predictions and
our actuals, right? And so our gradients were based on the derivative of that with respect to the, the derivative of that with respect to the coefficients. But we’re saying now, let’s add the sum of
the square of the weights times some small number.
So what would make that loss function go down? That loss function would go down if we reduce our weights. For example, if we reduce all of our weights to zero, I should say we reduce the magnitude of
our weights. If we reduce them all to zero, that part of the loss function will be zero, because the sum of zero squared is zero. Now problem is, if our weights are all zero, our model doesn’t do
anything, right? So we’d have crappy predictions. So I would want to increase the weights, so that’s actually predicting something useful. But if it increases the weights too much, then it starts
overfitting. So how is it going to actually get the lowest possible value of the loss function? By finding the right mix. Weights not too high, right?
But high enough to be useful at predicting. If there’s some parameter that’s not useful, for example, say we asked for five factors and we only need four, it can just set the weights for the fifth
factor to zero, right? And then problem solved, right? It won’t be used to predict anything, but it also won’t contribute to our weight decay part.
Weight Decay in PyTorch
So previously, we had something calculated in the loss function, so now we’re going to do exactly the same thing, but we’re going to square the parameters, we’re going to sum them up, and we’re going
to multiply them by some small number, like 0.01 or 0.001. And in fact, we don’t even need to do this, because remember, the whole purpose of the loss is to take its gradient, right?
Time to print it out. The gradient of parameters squared is two times parameters. It’s okay if you don’t remember that from high school, but you can take my word for it. The gradient of y equals x
squared is 2x. So actually, all we need to do is take our gradient and add the weight decay coefficient, 0.01 or whatever, times two times parameters. And given this is just number, some number we
get to pick, we may as well fold the two into it and just get rid of it. So when you call fit, you can pass in a wd parameter, which does, adds this times the parameters to the gradient for you. And
so that’s going to ask the model, it’s going to say to the model, please don’t make the weights any bigger than they have to be.
Reducing Overfitting
And yay, finally, our loss actually improved. Okay, and you can see it getting better and better. In fast AI applications like Vision, we try to set this for you appropriately, and we generally do a
reasonably good job, just the defaults are normally fine. But in things like tabular and collaborative filtering, we don’t really know enough about your data to know what to use here. So you should
just try a few things. Let’s try a few multiples of 10, start at 0.1, and then divide by 10 a few times, you know, and just see which one gives you the best result.
So this is called regularization. So regularization is about making your model model no more complex than it has to be, right? It has a lower capacity.
And so the higher the weights, the more they’re moving the model around, right? So we want to keep the weights down, but not so far down that they don’t make good predictions. And so the value of
this, if it’s higher, will keep the weights down more, it will reduce overfitting, but it will also reduce the capacity of your model to make good predictions. And if it’s lower, it increases the
capacity of model and increases overfitting.
Next Time
All right. I’m going to take this bit for next time. Before we wrap up, John, are there any more questions?
Yeah, there are. There’s some from back at the start of the collaborative filtering. So we had a bit of a conversation a while back about the size of the embedding vectors.
Hyperparameter Search
And you talked about your fast AI rule of thumb. So there was a question if anyone has ever done a kind of a hyperparameter search and exploration.
[Jeremy Howard]
I mean, people often will do a hyperparameter search for sure. People will often do a hyperparameter search for their model, but I haven’t seen any other rules other than my rule of thumb.
Right. So not productively to your knowledge.
[Jeremy Howard]
Productively for an individual model that somebody’s built.
And then there’s a question here from Zakir, which I didn’t quite wrap my head around.
Recommendation Systems Based on Averages
So Zakir, if you want to maybe clarify in the chat as well, but can recommendation systems be built based on average ratings of users experience rather than collaborative filtering?
[Jeremy Howard]
Not really. Right. I mean, if you’ve got lots of metadata, you could. Right. So if you’ve got lots of information about demographic data, about where the user’s from and what loyalty scheme results
they’ve had and blah, blah, blah.
And then for products, there’s metadata about that as well. Then sure, averages would be fine. But if all you’ve got is kind of purchasing history, then you really want granular data. Otherwise, how
could you say like, they like this movie, this movie and this movie. Therefore, they might also like that movie or you’ve got is like, oh, they kind of like movies. There’s just not enough
information there. Yep. Great. That’s about it.
Thanks. Okay, great. All right. Thanks everybody. See you next time for our last lesson. | {"url":"https://learnspacy.com/python/lesson-7-practical-deep-learning-for-coders-2022/","timestamp":"2024-11-12T23:59:04Z","content_type":"text/html","content_length":"142664","record_id":"<urn:uuid:4c4e6bf9-33f9-43e9-ac8f-4090f08207d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00422.warc.gz"} |
Verification of Far-Field Array Pattern Using Superposition with Embedded Element Patterns
This example shows that the far-field radiation pattern of a fully excited array can be recreated from the superposition of the individual embedded patterns of each element. The pattern
multiplication theorem in array theory states that the far-field radiation pattern of an array is the product of the individual element pattern and the array factor. In the presence of mutual
coupling, the individual element patterns are not identical and therefore invalidates the result from pattern multiplication. However, by computing the embedded pattern for each element and using
superposition, we can show the equivalence to the array pattern under full excitation.
Set up Frequency and Array parameters
Choose the design frequency to be 1.8 GHz, which happens to be one of the carrier frequencies for 3G/4G cellular systems. Define array size using number of elements, N and inter-element spacing, dx.
fc = 1.8e9;
vp = physconst('lightspeed');
lambda0 = vp/fc;
N = 4;
dx = lambda0/2;
Design Antenna Element and Create the Array
For this example, we design a reflector backed half-wavelength dipole antenna. The reflector is half-wavelength in length along the x-axis and a quarter-wavelength in width, along the y-axis.
r = design(reflector,fc);
r.GroundPlaneLength = lambda0/2;
r.GroundPlaneWidth = lambda0/4;
Use the reflector backed dipole as the individual element for the linear array. Use the NumElements property to change the linear array to have 4 elements instead of the default of 2. Change the
element spacing to be half-wavelength.
lA = linearArray;
lA.Element = r;
lA.ElementSpacing = dx;
lA.NumElements = N;
Calculate and Plot the 3D Array Pattern
By default all four elements in this array are excited with a voltage of 1V at a phase of 0 deg. Compute the far-field directivity pattern of this uniformly excited array at the center frequency.
E and H-Plane Pattern Variation of the Fully Excited Array
The array being situated in the x-y plane results in most of the radiation being directed towards the zenith. The array pattern variations along the elevation angles can be captured along two
orthogonal azimuth slices; at azimuth of 0° and at 90°. Visualize the directivity variation with elevation angle in these two planes is using the polarpattern function.
az = 0:5:360;
el = -180:1:180;
pE = polarpattern('gco');
pH = polarpattern('gco');
Calculate Embedded Element Complex Far-Fields
The embedded element pattern refers to the pattern of a single element embedded in the finite array, that is calculated by driving the central element in the array and terminating all other elements
into a reference impedance [1]-[3]. The pattern of the driven element, referred to as the embedded element, incorporates the effect of coupling with the neighboring elements. In the Antenna Toolbox™,
an ideal voltage source is used as excitation. To recreate the far-field pattern from superposition of the complex far-fields, use a very small value of resistance to terminate the remaining
elements. Secondly, the superposition must be done on the complex far-field. Use the EHfields function to calculate the complex electric and magnetic fields at different points in space due to each
excited element. For this example, choose a spherical arrangement of points in the E and H-plane angles defined earlier. The far-field points are computed at a radius of 100 $\lambda$.
R = 100*299792458/min(fc);
phi1 = az;
theta1 = 90-el;
[theta, phi] = meshgrid(theta1, phi1);
phi = phi(:);
theta = theta(:);
X = R.*sind(theta).*cosd(phi);
Y = R.*sind(theta).*sind(phi);
Z = R.*cosd(theta);
Points = [X';Y';Z'];
N = lA.NumElements;
E = zeros(3,size(Points,2),N);
for i = 1:N
E(:,:,i) = EHfields(lA,fc,Points,'ElementNumber',i,'Termination',1e-12);
Superposition of Embedded Element Pattern Fields
Combine the individual embedded element electric field patterns in the far-field. For the sake of comparison with the pattern of the fully excited array, compute the magnitude. This will be used to
calculate the total directivity in the E and H-plane respectively.
arrayEfieldpat = sum(E,3);
MagEsquare = dot(arrayEfieldpat, arrayEfieldpat);
MagE = sqrt(MagEsquare);
MagE = reshape(MagE,length(az),length(el));
Compute Directivity of Array
Directivity is a measure of the power projection ability of an antenna or array as a function of different angles in space. It defines the overall shape of the power projection capability of the
radiating structure. To calculate this, find the radiation intensity in particular directions and divide it by the total radiated power from the structure over all directions. The total radiated
power is computed as a product of the radiation efficiency and the input power. Each element of the array is assumed to be excited by a 1 Volt excitation source for computing the input power. The
radiation efficiency of the array is computed using the efficiency function.
RadEff = efficiency(lA,fc);
InputPower = sum(0.5*real(1./conj(impedance(lA,fc))));
RadiatedPower = RadEff*InputPower;
eta = sqrt(1.25663706e-06/8.85418782e-12);
U = R^2*MagE.^2/(2*eta);
D = 10*log10(4*pi*U/RadiatedPower);
Comparison of Patterns
Overlay the directivity result from the superposition of the embedded element patterns on the result from the computation for the fully excited array.
idphi0 = find(az==0);
idphi90 = find(az==90);
Dphi = D(idphi0,:);
Dphi90 = D(idphi90,:);
pE.LegendLabels = {'Full-wave','Embedded superposition'};
pE.MagnitudeLim = [-40 20];
pE.Marker = {'+','.'};
pE.TitleTop = 'Elevation Slice @ az = 0 deg';
pH.LegendLabels = {'Full-wave','Embedded superposition'};
pH.MagnitudeLim = [-40 20];
pH.Marker = {'+','.'};
pH.TitleTop = 'Elevation Slice @ az = 90 deg';
The use of superposition on the complex far-fields produced by the individual elements of an array generates the same pattern as the one from the uniformly excited array.
See Also
Modeling Mutual Coupling in Large Arrays Using Embedded Element Pattern
[1] R. J. Mailloux, 'Phased Array Antenna Handbook', Artech House, 2nd edition, 2005.
[2] W. Stutzman, G. Thiele, 'Antenna Theory and Design', John Wiley & Sons Inc., 3rd Edition, 2013.
[3] R. C. Hansen, Phased Array Antennas, Chapter 7 and 8, John Wiley & Sons Inc.,2nd Edition, 1998. | {"url":"https://it.mathworks.com/help/antenna/ug/verification-of-far-field-array-pattern-using-superposition-with-embedded-element-patterns.html","timestamp":"2024-11-01T19:13:16Z","content_type":"text/html","content_length":"84027","record_id":"<urn:uuid:47e72ced-e9e6-496b-b62d-6c10775b63c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00069.warc.gz"} |
Module 2 Fundamental data analysis_checked Flashcards
<p>What two types of errors are uncertainties caused by?</p>
<p>Random and systematic</p>
<ol><li>What are systematic errors?</li><li>How are they caused?</li><li>How easy are they to spot?</li><li>What is their effect?</li></ol>
<ol><li>Systematic errors [including zero errors] are the same every time you repeat the experiment, They shift all the values by the same amount.</li><li>They may be caused by the equipment you’re
using or how it’s set up e.g. you're not lining up a ruler correctly when measuring extension the of a spring.</li><li>Systematic errors are really hard to spot.</li><li>Systematic errors affect the
accuracy of your results. It is always worth checking your apparatus at the start of an experiment e.g. measure a few known masses to check that a mass meter is calibrated properly.</li></ol>
<p>Describe random errors. How can you reduce their effect?</p>
<p>Random errors make the results a bit different each time you repeat an experiment. If you measured the length 20 times, the chances are you'd get a slightly different value each time e/g/ due to
your head being in a slightly different position when reading the scale. It could be that you just can’t keep controlled variables exactly the same throughout the experiment.<br></br><br></br>
Repeating measurements can also reduce the effects of random errors. Using equipment with a higher resolution means that the equipment can detect smaller changes. This can reduce random error and
make the results more precise.</p>
<p>How do you find the uncertainty in the value of the gradient of a graph?</p>
<p>The uncertainty in the gradient is given by the difference between the best gradient and the worst gradient. </p>
<p>An alternative method using gradients is:</p>
<p>Uncertainty = [max gradient - min gradient] / 2</p>
<p>How do you find the uncertainty in the y-intercept of a graph?</p>
<p>Draw the worst lines through the uncertainty bars. The uncertainty is the difference between the best and worst intercepts vertically for an uncertainty bar</p>
<p>How do you calculate the angle of a circle arc in radians?</p>
<p>angle (in radians) = arc length (in m) / radius (in m)<br></br>think L = r * (theta)</p>
<p>How do you calculate percentage uncertainty?</p>
<p>% uncertainty = [abs uncertainty in reading/ actual reading] * 100%</p>
<p>How do you calculate uncertainties when <strong>adding or subtracting</strong> quantities</p>
<p>When <strong>adding or subtracting </strong>quantities, you <strong>add</strong> the absolute uncertainties</p>
<ol><li>How do you calculate percentage uncertainty when<strong> multiplying or dividing</strong> quantities, </li><li>How to use this to find <strong>absolute</strong> uncertainty</li></ol>
<ol><li>When <strong>multiplying or dividing</strong> quantities you <strong>add</strong> the <strong>percentage</strong> uncertainties </li><li><strong>Multiply the final %uncertainty</strong> by
the final quantity value to find its absolute uncertainty</li></ol>
<ol><li>How do you calculate percentage uncertainty when raising a quantity to a power</li><li>How do you calculate absolute uncertainty from that?</li></ol>
<ol><li>When you a raise a quantity to a power, n, you <strong>multiply</strong> the % uncertainty of that quantity <strong>by n</strong>,</li><li>Multiply the <strong>final %uncertainty</strong> by
<strong>the final quantity value</strong> to find its absolute uncertainty</li></ol>
<p>How do you calculate spread from range?</p>
<p>spread = 0.5 * range<br></br>spread is the uncertainty in a reading.</p>
<p><i>When working with dot plots be careful of anomalous values</i></p>
<p>What is the line of worst fit and how do you calculate it?</p>
<p>This is essentially the maximum gradient or the minimum gradient.</p>
<p>This is the least acceptable straight line through the data points. after you have drawn your uncertainty bars at each point. Start from the bottom of the first uncertainty bar and get to top of
the last uncertainty bar i.e. the maximum or minimum gradient</p> | {"url":"https://www.brainscape.com/flashcards/module-2-fundamental-data-analysis_check-10840135/packs/19333830","timestamp":"2024-11-06T10:50:00Z","content_type":"text/html","content_length":"99003","record_id":"<urn:uuid:8b822eb6-4631-462d-b936-d88e9bca0589>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00486.warc.gz"} |
The Stacks project
Remark 98.27.18. The proof of Theorem 98.27.17 uses that $X'$ and $W$ are separated over $S$ in two places. First, the proof uses this in showing $\Delta : F \to F \times F$ is representable by
algebraic spaces. This use of the assumption can be entirely avoided by proving that $\Delta $ is representable by applying the theorem in the separated case to the triples $E'$, $(E' \to V)^{-1}Z$,
and $E'_{/Z} \to E_ W$ found in Remark 98.27.7 (this is the usual bootstrap procedure for the diagonal). Thus the proof of Lemma 98.27.14 is the only place in our proof of Theorem 98.27.17 where we
really need to use that $X' \to S$ is separated. The reader checks that we use the assumption only to obtain the morphism $x' : V' \to X'$. The existence of $x'$ can be shown, using results in the
literature, if $X' \to S$ is quasi-separated, see More on Morphisms of Spaces, Remark 76.43.4. We conclude the theorem holds as stated with “separated” replaced by “quasi-separated”. If we ever need
this we will precisely state and carefully prove this here.
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Paper IPM / P / 16747
School of Physics
Title: A new nonlinear electrodynamics and electrically charged regular black holes
Author(s): 1. M.B. Jahani Poshteh
2. N. Riazi
Status: Published
Journal: Int. J. Mod. Phys. D
No.: 11
Vol.: 30
Year: 2021
Pages: 2150079
Supported by: IPM
A regular static, spherically symmetric electrically charged black hole solution of general relativity coupled to a new theory for nonlinear electrodynamics is presented. This theory has the
interesting feature that, at far distances from the black hole, in the weak field limit, the theory reduces to Maxwell Lagrangian with Heisenbergâ Euler correction term of quantum electrodynamics.
The singular center of the black hole is replaced by flat, de Sitter, or anti de Sitter space, if the spacetime in which the black hole is embedded is asymptotically flat, de Sitter, or anti de
Sitter, respectively. Requiring the correspondence to Heisenbergâ Euler Lagrangian at large distances, in the weak field limit, we find that (i) a minimum mass is required for the formation of an
event horizon for the regular static, spherically symmetric solution of the theory, and, (ii) the mass of the solution must be quantized. We also study the basic thermodynamic properties of the black
hole solution and show that they are qualitatively similar to those of Reissnerâ Nordström black hole.
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Convex Formulation of the Upper Bound Approach with Noise in all Images
Convex Formulation of the Upper Bound Approach with Noise in all Images
In this section, we propose a convex formulation of the principles sketched in section 7.2 that, compared to (144), accounts for noise in both the template and the input images. We can express this
in terms of image-plane measurements. As in (164,166), our approach is formulated as an SOCP problem. However, contrary to (164,166), our approach is a point-wise method that does not require us to
tune the relative influence of minimizing the reprojection error and maximizing the depths.
Let us first remark that the basic principles explained in section 7.2 can be formulated as SOCP problems. In this first formulation, the noise is only accounted for in the template image. The
inextensibility constraint can be written:
Including the maximization of the depths, we obtain this SOCP problem:
where , and is a set of pairs of points to which the inextensibility constraints are applied.
Noise in Both the Template and the Input Images
Let us now suppose that the inaccuracies are expressed in terms of image-plane measurements. Suppose that points are measured in the image with a maximum error of , i.e.
Since we are searching for the true 3D position of the point , we say that:
Equation (7.3) can thus be rewritten:
We finally add the inextensibility constraints and the maximization of the depths (which are given by ) and we obtain the following SOCP problem:
where is the concatenation of the 3D points , for .
Contributions to Parametric Image Registration and 3D Surface Reconstruction (Ph.D. dissertation, November 2010) - Florent Brunet
Webpage generated on July 2011
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Confidence Intervals
Episode #3 of the course Introduction to statistics by Polina Durneva
Good morning!
Today, you’ll learn about confidence intervals. But before that, we need to have a clear understanding of the difference between parameters and statistics.
Parameters vs. Statistics
Parameters are used to describe population, while statistics are used to describe samples from population. In Lesson 1, we used an example of a fast food restaurant in which we wanted to find the
proportion of customers who buy burgers and fries and the proportion of customers who buy chicken wings. If we were to ask each single customer about their choice of food, we would have an exact
value that demonstrates how many customers buy one type of meal or another. This value would be our parameter, as it describes the entire population of our fast food restaurant.
However, it would be quite a tedious and tiring process to survey each customer, and therefore, we agreed to choose a sample of random customers and ask them about their choice of meals. We would
have a random sample that would provide us with a statistic, which approximates the parameter. Statistics are used to provide us with the estimation of parameters.
Margin of Error
To have a better estimation of a parameter with respect to a statistic, statisticians come up with a margin of error. Using a margin of error, we can have an interval that provides us with a range of
values, one of which is a parameter. A margin of error is calculated using multiple samples: More different samples provide us with a lower value of an error and a more accurate estimation of a
For instance, let’s assume that our estimated statistic for customers who prefer chicken wings is 30%, meaning that 30% of the clients in our fast food restaurant prefer chicken wings to burgers with
fries. An estimated margin of error is 5%, meaning that our parameter would be any value between 25% and 35%.
Confidence Interval
We can also add some features to the aforementioned interval, which is based on a statistic and a margin of error, and redefine it as a confidence interval, which is a range of plausible values.
However, sometimes we might get a statistic that is biased and does not capture a parameter even with a margin of error. To evaluate that, statisticians came up with something called the confidence
level. The confidence level is the probability of the confidence interval to capture a population parameter.
Most of the time, statisticians use a 95% confidence interval. To better understand it, look at the graph below:
The graph illustrates a bell-shaped distribution: 95% of the distribution roughly lies within two standard deviations of the center of the distribution. Therefore, it is assumed that a population
parameter lies within two standard deviations—or standard errors, in this case—within the center. The formula for a 95% confidence interval would be: (Statistic – 2 * (Standard Error)); (Statistic +
2 * (Standard Error)).
For example, let’s say that we want to estimate the percentage of high school seniors who got accepted to college in state X. After getting a random sample of high school students, we get 76%. We
also know that the standard error is 4.5%. Thus, our confidence interval would be, (76% – 2 * 4.5%; 76% + 2 * 4.5%) = (67%; 85%), meaning that we can be 95% sure that between 67% and 85% of high
school seniors in state X got accepted to college.
That’s it for today. Tomorrow, we will discuss hypothesis tests.
See you,
Recommended book
An Introduction to Statistical Learning: with Applications in R by Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani
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Antigravity Could Replace Dark Energy as Cause of Universe's Expansion
Antigravity Could Replace Dark Energy as Cause of Universe’s Expansion
Since the late 20th century, astronomers have been aware of data that suggest the universe is not only expanding, but expanding at an accelerating rate. According to the currently accepted model,
this accelerated expansion is due to dark energy, a mysterious repulsive force that makes up about 73% of the energy density of the universe. Now, a new study reveals an alternative theory: that the
expansion of the universe is actually due to the relationship between matter and antimatter. According to this study, matter and antimatter gravitationally repel each other and create a kind of
“antigravity” that could do away with the need for dark energy in the universe.
Massimo Villata, a scientist from the Observatory of Turin in Italy, began the study with two major assumptions. First, he posited that both matter and antimatter have positive mass and energy
density. Traditionally, the gravitational influence of a particle is determined solely by its mass. A positive mass value indicates that the particle will attract other particles gravitationally.
Under Villata’s assumption, this applies to antiparticles as well. So under the influence of gravity, particles attract other particles and antiparticles attract other antiparticles. But what kind of
force occurs between particles and antiparticles?
To resolve this question, Villata needed to institute the second assumption – that general relativity is CPT invariant. This means that the laws governing an ordinary matter particle in an ordinary
field in spacetime can be applied equally well to scenarios in which charge (electric charge and internal quantum numbers), parity (spatial coordinates) and time are reversed, as they are for
antimatter. When you reverse the equations of general relativity in charge, parity and time for either the particle or the field the particle is traveling in, the result is a change of sign in the
gravity term, making it negative instead of positive and implying so-called antigravity between the two.
Villata cited the quaint example of an apple falling on Isaac Newton’s head. If an anti-apple falls on an anti-Earth, the two will attract and the anti-apple will hit anti-Newton on the head;
however, an anti-apple cannot “fall” on regular old Earth, which is made of regular old matter. Instead, the anti-apple will fly away from Earth because of gravity’s change in sign. In other words,
if general relativity is, in fact, CPT invariant, antigravity would cause particles and antiparticles to mutually repel. On a much larger scale, Villata claims that the universe is expanding because
of this powerful repulsion between matter and antimatter.
What about the fact that matter and antimatter are known to annihilate each other? Villata resolved this paradox by placing antimatter far away from matter, in the enormous voids between galaxy
clusters. These voids are believed to have stemmed from tiny negative fluctuations in the primordial density field and do seem to possess a kind of antigravity, repelling all matter away from them.
Of course, the reason astronomers don’t actually observe any antimatter in the voids is still up in the air. In Villata’s words, “There is more than one possible answer, which will be investigated
elsewhere.” The research appears in this month’s edition of Europhysics Letters.
76 Replies to “Antigravity Could Replace Dark Energy as Cause of Universe’s Expansion”
1. Something is not right here. I looked at the paper and will try to read this to see what is wrong. There is a germ of something here though.
Here is the problem. I just use Newton’s second law of motion F = ma and consider the force as Newtonian gravity F = -Gmm’/r^2. We let m be mass and m’ be anti-mass. Let us consider the motion of
the anti-mass m’ a = -Gm/r^2,
where it turns out the sign of m’ is irrelevant to the acceleration of m’. The means the anti-mass is attracted to the mass. Now consider the acceleration of the mass m,
ma = -Gmm’/r^2 – -> a = -Gm’/r^2,
and since m’ a = -Gm’/r^2,
where the sign of the anti-mass considered is irrelevant, but the other antimass flips the sign of the gravity acceleration. Consequently two anti-masses would accelerate apart from each other.
To think about general relativity with respect to anti-mass, consider Hawking radiation from a black hole. A black hole has this event horizon, and for a modest black hole the horizon may have
sufficient curvature that a little bit of wave scattering by curvature occurs. Consider the case where the wave scattered is the Dirac field. The Dirac field is the “square root” of the momentum
interval in special relativity E^2 – p^2 = m^2 in quantized form. This means the spinor fields have positive and negative energy solutions. The energy-momentum of the field upon scatter changes
its root value. This is the Dirac sea, and the curvature of spacetime is perturbing the Dirac sea of negative energy states and the particle states of positive energy. This means that Dirac
particles of negative energy can be scattered out of the Dirac sea by the spacetime curvature into a real particle state, which is Hawking radiation
Negative energy in this Dirac sea, which is a form of the quantum vacuum, enters a black hole and cancels a unit of mass in the black hole where this mass appears outside the black hole. This is
standard Dirac sea logic, where a quanta of anti-energy E + mc^2 (both negative) interact with some positive mass E to generate a positive mass particle with mass m and some kinetic energy. Due
to other quantum numbers, such as charge, which are conserved the input energy has to be sufficient to generate a particle plus anti-matter particle. The negative mass particle has opposite
values of other quantum numbers, where only the anti-mass is flipped to mass.
Now let us consider the Dirac vacuum as a set of occupation states (all filled) with negative energy (anit-mass). The above analysis with Newtonian gravity indicates that this vacuum is
self-repelling. Now the Dirac vacuum has particles with mass m existing within a momentum light cone with energy E = E’ + m < 0 which is arbitrarily large. So there is something which cancels
this out. It is the supersymmetric partner with these fermionic particles. When the quantum numbers are computed (which has not been done completely) the vacuum energy might indeed be negative.
Or maybe better put, it has some negative aspect to it. This is a part of the “sign problem” with understanding the Fermi-Dirac field. One might then say that this over all negative component of
the vacuum state is self-repelling and this is maybe then an aspect of why the universe accelerates away.
1. I reposted this. Unfortunately this has a problem with carot signs, which means less than symbols have the effect of killing off parts of things. I have encountered this before elsewhere,
which is extremely annoying.
Something is not right here. I looked at the paper and will try to read this to see what is wrong. There is a germ of something here though.
Here is the problem. I just use Newton’s second law of motion F = ma and consider the force as Newtonian gravity F = -Gmm’/r^2. We let m be mass and m’ be anti-mass. Let us consider the
motion of the anti-mass 0 > m’
m’a = -Gmm’/r^2 – -> a = -Gm/r^2,
where it turns out the sign of m’ is irrelevant to the acceleration of m’. The means the anti-mass is attracted to the mass. Now consider the acceleration of the mass m,
ma = -Gmm’/r^2 – -> a = -Gm’/r^2,
and since 0 > m’ the sign of the acceleration is changed. This means the mass accelerates away from the anti-mass. This means that if the mass m and the anti-mass m’ have the same magnitude,
so the total mass m + m’ = 0, then the mass and the anti-mass will accelerate away in the same direction. This is a funny situation, but since the total mass is zero this amounts to the
acceleration away of nothing in total.
What happen if both m and m’ are anti-masses? Well the second law of motion is
ma = -Gmm’/r^2 – -> a = -Gm’/r^2,
where the sign of the anti-mass considered is irrelevant, but the other antimass flips the sign of the gravity acceleration. Consequently two anti-masses would accelerate apart from each
To think about general relativity with respect to anti-mass, consider Hawking radiation from a black hole. A black hole has this event horizon, and for a modest black hole the horizon may
have sufficient curvature that a little bit of wave scattering by curvature occurs. Consider the case where the wave scattered is the Dirac field. The Dirac field is the “square root” of the
momentum interval in special relativity E^2 – p^2 = m^2 in quantized form. This means the spinor fields have positive and negative energy solutions. The energy-momentum of the field upon
scatter changes its root value. This is the Dirac sea, and the curvature of spacetime is perturbing the Dirac sea of negative energy states and the particle states of positive energy. This
means that Dirac particles of negative energy can be scattered out of the Dirac sea by the spacetime curvature into a real particle state, which is Hawking radiation
Negative energy in this Dirac sea, which is a form of the quantum vacuum, enters a black hole and cancels a unit of mass in the black hole where this mass appears outside the black hole. This
is standard Dirac sea logic, where a quanta of anti-energy E + mc^2 (both negative) interact with some positive mass E to generate a positive mass particle with mass m and some kinetic
energy. Due to other quantum numbers, such as charge, which are conserved the input energy has to be sufficient to generate a particle plus anti-matter particle. The negative mass particle
has opposite values of other quantum numbers, where only the anti-mass is flipped to mass.
Now let us consider the Dirac vacuum as a set of occupation states (all filled) with negative energy (anit-mass). The above analysis with Newtonian gravity indicates that this vacuum is
self-repelling. Now the Dirac vacuum has particles with mass m existing within a momentum light cone with energy 0 > E = E’ + m which is arbitrarily large. So there is something which cancels
this out. It is the supersymmetric partner with these fermionic particles. When the quantum numbers are computed (which has not been done completely) the vacuum energy might indeed be
negative. Or maybe better put, it has some negative aspect to it. This is a part of the “sign problem” with understanding the Fermi-Dirac field. One might then say that this over all negative
component of the vacuum state is self-repelling and this is maybe then an aspect of why the universe accelerates away.
1. Wasn’t the Dirac sea abandoned because it is an awfully finetuned model introducing infinite energies (having infinite positive energy making up for the infinite negative sea)?
I peeked into Wikipedia, and what do you know: they mention that, and also reminded me/pointed out that
– The Dirac sea was based on fermions for state separation. We don’t have that here, I think, as the graviton is a force mediating boson. I’m not sure it helps that gravity is coupled to
vacuum energy density, if the balance is based on the net of graviton states. (But maybe it isn’t and I just can’t see how.)
If we don’t have a sea, particles would be shedding energy indefinitely. Anti-particles would try to shoot out of any containment trying to get away from gravity fields ASAP (thus
achieving even higher negative energy, unless I’m mistaken)! That isn’t observed.
– You can always find new particles that can be inserted into the sea. The concept may not be self-consistent. “Captin, I think we have a wee bit of a problem here.”
I guess my question, aside from any technical difficulties, is if a Dirac sea can be reintroduced for one type of particle alone? That doesn’t seem to be a likely outcome.
2. No. Unlikely.
(Is that comment short enough for you, Nancy?).
3. The title is very misleading.
4. huh? antigravity?
5. Could it be both?
6. so: an aggregated mass of antimatter will cause either a positive or negative “gravometric” field.
(gravity or antigravity)
if antigravity: then the remaining antimatter from the big bang is causing expansion.
if gravity: then the antimatter is expanding with all the rest of the matter.
third posit: the entire universe is expanding into a larger region of lesser “gravometric” density. the asymmetric production of matter and antimatter was resolved by initial annihilation- the
result being the current intermix. the universe is not so much being driven to expand from within as being pulled apart from without.
this points to the universe having been the result of a hypermassive collision in a much
larger (infinite) void of near zero “gravometric” density.
the vector sum of the collision being in the direction of the “great attractor.”
what we need here is enough antimatter to see if it flies away!
7. novice here, but aspects of the idea sound more ‘logical’ and comprehensible to me than some of the convoluted explanations of how the mysterious dark matter works, or multiple dimensions
intersecting, etc.
8. The word “gravity” is used, so I expect some nutjobs here.
The word “anti-gravity” is used so I expect even more nutjobs than the nutjobs that claim gravity does not exist.
9. WOW!
According to this study, matter and antimatter gravitationally repel each other
What evidence is there that they repel.
Also I think that if they repel each other and form dense locations of only anti-matter, then these regions should be visible the same way as normal matter regions. They have the same properties
as normal matter with one small exception, so you would have normal planets and stars out there.
1. I really miss some editing feature here, put the closing italics at the wrong place.
2. Just experimenting…
3. D’oh. Let’s see if this works.
1. Apprently not Maybe one of the admin can fix it.
2. Fixed.
4. That would make some pretty cool science fiction eh, they were from two worlds, two parts of the universe …one matter, one anti-matter …they could never meet …but their love would bring them
together …lol, or some really bad space soap opera!
10. Matter repelling anti-matter, is something the smells funny.
But I am wondering if matter has the property to attract other matter.
And anti-matter could have the property to repel other anti-matter.
No scientific claim here, just loud thinking.
11. BLACK HOLES, EXPANSION, AND DARK ENERGY
In the continuum of space and time, exists the dichotomy of matter and energy. All things exist as both matter and energy, but are experienced as one or the other.
As energy, all things exist as wave patterns. Most wave patterns are interferences of simpler wave patterns. The simplest wave forms are those that do not interfere with other waves. These
simplest wave forms hold their shape as they propagate. There are three such wave forms.
The rest of this comment has been deleted as it is in violation of Universe Today’s comment policy.
1. Consider the torus as a universe.
No. The universe is unlikely to be a torus.
1. The universe is not a torus, or at least not likely to be. Of course this does touch a bit on the problem of negative energy, for multiply connected spacetimes such as a torus has a
stress-energy T^{00} which is negative. Also such a universe is a type of time machine with closed timelike curves. This is of course a problem with the whole idea of negative energy, it
tends to give spacetimes which have pathological causal conditions.
2. L.C. Is it reasonable to think of the Universe as a globule of electromagnetic energy?
3. No, but it may have emerged from some gauge field vacuum which tunnelled out of another cosmology. Electromagnetism is the simplest case of a YM gauge field.
2. Is it me? Or is this a lot of scientific sounding words mixed together that sounds impressive but in reality means nothing.
3. There used to be a don’t advertise your stuff warning about comments.
This is spamming or I’ve never seen any.
All things exist as both matter and energy,
That is an open question, as I understand it.
Fields are made up of particles, but according to string theory not everything needs to approximate fields. There have been research into “particle less sectors” of string theory, fairly
decently accepted by that community I believe.
As for the wave function, it is complicated. Particle wavelets in QM aren’t simple when they travel without interaction (free as opposed to confined); all types of persistent solitons are
more complicated still (because they _do_ interact with the environment).
12. Well you wait just one doggone minute here…
I thought it was the universe (space-time) that is expanding, and specifically not merely just the bits of matter (and anti-matter) moving away from each other…
— Yosemite Sam…
13. Oh, Olaf screwed the layout 😀
We need Ivan3man, the master of the italics, immediately. 😉
I am neither an expert on GR nor on particle physics (had some courses on it, though). However, matter and anti-matter repelling each other sounds strange to me.
Matter and anti-matter annihilates (btw: why is that?). If we put an electron and a positron close together, they will attract each other.
Since the attraction is orders of magnitude stronger than the repulsion, they wikk destroy each other in the end. (Ok, this does not contradict the idea.)
So, the repulsion could only be at work, if we have neutral particles, like atoms. An Atom and an anti-atom should repel each other according to the idea.
However, in the first 300.000 years after the Big Bang there were only charged particles and no atoms. This is quite enough time to annihilate due to the much stronger attraction.
As it seems, this would only work if the universe was cold right after its “creation”. Sounds a bit like the ideas of Alfvèn, about an infinite universe where matter is put in “one place” and
anti-matter in “another”.
These are my thoughts at the moment….
1. Oops did I do this?
14. I will make a much simpler argument as to why this other attractive argument is likely inaccurate. There no evidence of anti-matter structures which would be generating the necessary anti-gravity
proportedly causing these voids to expand. Such structures should be colossal in size and in fact quite visible. Unless you want to go down the path that the source of anti-gravity in the voids
is dark anti-matter or enormous anti-black holes. If this is the case than perhaps the sparse galaxies we see in these voids are anti-matter galaxies. I don’t think that the physics of this will
pan out considering there is just no evidence for anything in these voids that would be providing anti-gravity.
15. I believe I have figured out what the problem is. Before charging into that, there is a simple argument which illustrates how this is flawed without going into gravity. Suppose you have an
electron and a positron. These are carefully brought together so there is not much energy involved with the conditions leading up to their interaction. This is how positronium is made, which is a
hydrogen atom-like system of a +/- positron/electron system. This decays and produces gamma ray photons which have a total energy equal to 2mc^2, for m the mass of the electron and positron. This
is an experiment done long ago and consistently works. Now suppose the positron has a negative mass. This means the net mass-energy is (m – m)c^2 = 0 and there should be no photons created.
Matter and anti-matter would annihilate each other by producing nothing. This is not observed in nature
Here is the flaw in the author’s paper. The author uses the geodesic equation
d^2x^a/ds^2 + Gam^a_{bc)U^bU^c = 0
to make his case. Let us concentrate on the second term. In the PT reversal Gam^a_{bc}, which is the connection term, transforms into its negative. In a weak gravity field Gam^a_{00} = -GMx^a/r^
2, which is a Newtonian force in the vector direction x^a. The PT operation will reverse M – -> -M. So far so good. The spacetime velocities U^b = dx^b/ds are reversed by x^b – -> -x^b and s – ->
-s, so these remain the same. Everything is fine up to this point. However, the author then performs the same with the first term with x^a – -> -x^a and s^2 – -> s^2 and gets a sign reversal
there as well.
It is with this second part that things go wrong. CPT operations act on fields. For some field f = f(q, x, t) one performs the operation CPT*f(q, x, t) = f(-q, -x, -t). So if I have an equation F
= ma, the first part of this “F” is a dynamical field effect, as is F/m. The mass m is a scalar quantity. The acceleration is a geometric quantity and is the unknown part of the equation. So if I
have an unknown on one side of an equation and I want to know how it transforms under a symmetry operation on the other side, I perform that operation and see how the unknown transforms. I don’t
transform both sides, and CPT operations are preformed on fields.
In the case of the geodesic equation the fundamental thing which is transformed is the covariant derivative of the metric. A metric term g_{00} = 1 – 2GM/rc^2 under a derivative gives the Gam^a_
{00} = -GM/r^2. This is the field that is transformed by CPT. From there you compute what the dynamics are. As a result, the argument I made with just Newton’s laws holds. Anti-mass particles
would repel each other, mass particles attract and mass + anti-mass system runs away.
Negative energy is a horrible thing really. In what I wrote about the Dirac equation it has some applicability to the Boulware vacuum across black hole horizons. But a general negative energy in
the universe, in particular negative mass particles, results in catastrophes.
Ah yes, another paper demolished! Sorry for the long posts on this, but when I see something like this I know there is a “bug in the program” and I have to find out what it is.
1. I made a mistake in my argument above. I wrote Gam^a_{00} = -GMx^a/r^2, which should be
Gam^a_{00} = -GMx^a/r^3
1. It’s all Greek to me! 😉
2. That is what I like, peer review 🙂
3. “Now suppose the positron has a negative mass.” A reply above has “Let us consider the motion of the anti-mass 0 > m’”.
The article summary above says “First, he posited that both matter and antimatter have positive mass and energy density.”
Mind you, I would guess that his claim is bollocks. I don’t have anywhere near the physics or math backgrounds to follow the rest of the arguments made. I just wanted to point out that the
rebuttals don’t seem to address the claims as summarized here.
The details of his claim I don’t know. Maybe he would posit a new multiplicand in the equation F = G*m1*m2/r^2, or rather the relativistic replacement? Or maybe he would throw in the absolute
value of the mass in other equations where needed?
1. The article basically invokes negative mass in the way the CPT is applied. On page 5 there is a statement to this effect. In that way there is a funny apparent inconsistency here. The PT
operation on the geodesic equation changes the sign of the connection term, but all this ends up saying is that the geodesic is examined in a time reversed manner. So the dynamics of
gravity repulsion between two anti-masses is the same as the time reversed viewing of the attraction between two masses.
2. Hi LC
Unfortunately I can’t speak about this at such a high level of mathematical understanding, but would the fact that the observed direction of time runs backwards for anti-matter be a
problem? A reversal in polarity of every property of matter could include the effects it has on the space-time surrounding it.
Even if anti-matter does produce an anti-gravity effect, I’m not quite sure why or how any expansion might be explained by this guys theory anyway, as any possible distribution of matter/
anti-matter throughout the universe would result in something quite different from what we are seeing! But I am glad people are looking for alternatives to dark energy and dark matter, as
pretty as it sounds. I think there are explanations of our observed ‘expansion’ to do with other factors, involving shifting densities of space-time, in and around dense pockets of matter
(like a galaxy for instance), and that the concepts of dark matter and dark energy are not required for this.
3. Oops, ok I just realised anti-matter is supposed to repel anti-matter too…
4. According to Stephen Hawking +ve matter is attractive and will form Universes.
-ve matter is repulsive and will disperse and not form Universes.
5. If I understand what +ve and –ve means then this reference to Stephen Hawking is correct, which is seen in my simple calculations on the previous page to this blog post. Of course we have
to separate out our meaning of antimatter and anti-mass. Antimatter is well understood, antiprotons are produced in the LHC to generate hadron collisions with zero net quantum number,
such as charge, baryon number etc. The generation of anti-hydrogen has been used to determine if antimatter has anti-mass, and if I remember right the test indicated it has positive mass.
A world of negative mass would be very strange. Freeman Dyson wrote a little paper where he asked what would happen if electric charge were imaginary, e – -> ie for i = sqrt{-1}. The
electric potential U = -ke*e’/r would not be negative, but positive due to i^2 = -1. The result would be the vacuum filled with electron and positron pairs would produce them with
enormous abandon. Rather than attracting each other they repel and the vacuum is unstable. In effect a world with anti-mass is similar to this. For m a mass and m’ an antimass (less than
zero) then m*m’ is less than zero. This is equivalent to Dyson’s transformation to imaginary charge, but here we just say that both m and m’ are imaginary. So this has a certain
relationship to the tachyon state. Tachyons are fields which are cancelled on the vacuum in string theory, which is a long story there I can’t go into here. The runaway situation of a
mass and anti-mass is a funny situation which is similar to a tachyon, where the tachyon is not cancelled on the vacuum state.
4. Congratiolations for demolishing the paper! You should send your bulldozer to Europhysics Letters.
In your first post you say:
“if the mass m and the anti-mass m’ have the same magnitude, so the total mass m + m’ = 0, then the mass and the anti-mass will accelerate away in the same direction”
If m’ is larger than m though, then the mass and the anti-mass will accelerate apart from each other if I understand it correctly, because -Gm’/r^2 > -Gm/r^2.
Villata places antimatter far away from matter, in the voids between galaxy clusters. If we assume that that is true, then is there a problem with the newtondynamic aspect of Villata’s theory
in case there would be more antimatter in the voids than matter in the galaxy clusters? (Not considering relativity.) Matter-matter pulls together, antimatter-antimatter repel, and in this
case antimatter-matter would repel as well.
1. If this anti-matter existed in these voids there should then be gravitational lensing of more distant objects.
2. The antimatter would be diffusely spread across the voids, it wouldn’t coagulate because of its self-repelling character. Therefore gravitational lensing could be not very distinct.
I am sure you demolished Villata’s theory correctly. I’m just wondering if there is a problem with newtondynamics if the voids between clusters would contain antimatter, in similar or
larger amounts than the amount of matter in the clusters. (Not considering relativity or any issues other than newtondynamics.) Could that model match the expansion of the universe the
way it is observed?
16. Attempting italic fix…
i {font-style: normal}
1. Test test. Is this none-itallic?
2. Scheiße!
1. The bloody HTML filter won’t let me use the appropriate tags to fix the italics!
Nice try to surround the swear-word filter — just use another language!
17. Wow the itallics are gone!
18. I solved this in May 2007 by adding a third body to the equation to be the negative gravitational force. Say where there is no negative gravitational force or zero, Newton’s second law is not
So, F=(G(m1*m2/r^2))+(-G(m1*m3/r^2))
where -G is the negative gravitational force and m3 is the mass of the antimatter body (antimatter will have negative mass).. Also please read about Mr. Forward:
Mr. Robert L. Forward in a link
called “Exotic Matter” by wikipedia:
“Ever since Newton first formulated his theory of gravity, there have been at least three conceptually distinct quantities called mass. However, these
three—inertial mass, active gravitational mass, and passive gravitational mass—have so far always been found to be equivalent. When considering
hypothetical particles with negative mass, it is important to consider which of these concepts of mass are negative. From Newton’s law:
F=m(sub i)a
Thus it can be seen that an object with negative inertial mass would be expected to accelerate in the opposite direction to that in which it was pushed, which is arguably a strange concept. If
one were to treat inertial mass m(sub i), passive gravitational mass m(sub p), and active gravitational mass m(sub a)
distinctly, then Newton’s law of universal gravitation would take the form
F=(-G(m(sub p)m(sub a)/r^2))
Thus objects with negative gravitational mass (both passive and active), but with positive inertial mass, would be expected to be repelled by positive
active masses, and attracted to negative active masses. If all such negative matter were like this, then gravity would work similarly to the electric force except that like masses would attract
and unlike masses would repel.” endquote
I use the example of the Pioneer Anomaly. Say this is being affected by antigravity in a system that has a large gravity body and a large antigravity body. Call this a mirror system that has a
balance between being affected by either body. But for my example, the Pioneer has found itself affected by the antigravity body “a little”.
Use the formula:
where the mass of the Pioneer 10 is affected by the antimatter monopole as it approached the mirror system. The second term becomes positive (from the two negatives) and the force becomes
increasingly stronger against the Pioneer 10 as it traverses the mirror system toward the antigravity body. This in turn causes a deceleration as it approaches the antigravity body..
Now let’s assume that the Pioneer 10 was an antimatter body traversing our solar system. The equation calculates to a negative force being a repulsive force:
so these formulae allow for no violation of Newton’s Law and at the same time, allow for variances and if we were privy to an antigravitational system, this variance would be more prolific.
I do believe that dark energy is a negative gravitational force and we cannot observe it directly due to the scattering of light against the antimatter body. In our observeable universe, we see
what is not obscured by the affects of antimatter. Take two magnets opposite poles and image one of those as a light ray and the other antimatter. Unless the light ray is repulsed back directly
into our observeable universe, we would not see it. Any reflected light we do see would be from an object that is not an antimatter body. Ergo, we would not “see” the dark energy or antimatter
body directly and may not even know it’s there.
I just wanted to say my 2 cents and am glad to see that Universe Today is more open-minded now.
Okay I’ll go back to just observing again.
Todd Coolen
1. The behavior of a negative mass is indeed to be repelled by any gravity field, whether from positive or negative mass. The author of this paper has clearly made a mistake in concluding
anti-mass would attract anti-mass.
The inflationary pressure is due to positive energy. The gravity field is due to the quantum vacuum, and this defines an effective stress-energy tensor T^{ab} with components T^{00} =
const*e, for e and energy density 0 = time coordinate index, and T^{ij} = const*pu^iu^j, for i and j running over spatial coordinates u^i velocity and p a pressure density. For the de Sitter
spacetime the energy density and pressure satisfies a state p = w*e where w = -1. So the pressure in effect is what is stretching out space and frame dragging galaxies with it. There is no
need for a negative energy density or exotic matter.
Negative energy density or negative mass fields have serious pathologies. Principally since they are due to quantum mechanics the negative eigen-energy states have no lower bound. This then
means the vacuum for these fields is unstable and would descend to ever lower energy levels and produce a vast amount of quanta or radiation. I don’t believe this happens.
As for Pioneer 10 fogettaboudit. I think it likely this is due to some sort of leaking pressure tank, sublimation of frost, or …, more or less prosaic processes.
2. The Pioneer anomaly is now predicted from first principles of thermal radiation, so it is an “anomaly” (read: “‘E’s off the twig! ‘E’s kicked the bucket”).
Unless the research is wrong, but that risk is slim to none: earlier work removed 1/3 of the initial anomaly.
1. Oops, sorry: just to be clear, I didn’t mean to evaluate the rest of the comment. It was a fact of the context (that recently circulated).
19. No way… that’s far too easy a solution! Matter/anti matter annihilation creating all those humongous magnetic fields and particles and herding all that matter into neutrally charged accretion
disks around galaxies and stars… no way.
1. Here I feel I must note the recently detected gamma ray fountains at either pole of the Milky Way…
20. ok, now you’ll all have to excuse my lack of university degree (i’m in the process 😉 ), but it is not strange for the microscopic and the macroscopic species of our universe to behave alike, so
why would it be wrong to hypothesize a migration of mass, comparable to osmosis in gas particles, with respect to space-time itself. It would thin out, desiring a less dense region of.. well..
emptiness to occupy ?? feel free to disprove, but spare me some dignity. 🙂
Antigravity Could Replace Dark Energy as Cause of Universe’s Expansion
No. [Hey, shorter than HSBC!]
– There is no mechanism in GR for antigravity (but for negative pressure and/or diminishing what spacetime curvature there is). I don’t think Villata has managed to change GR. Conveniently for me
I’m short on time and LC has studied GR anyway, so I can defer this for now. 😉
– Gravitational mass = inertial mass in GR, and that has been tested many times over. And tests put antiproton inertial mass = proton inertial mass to 10 significant digits or so.
– Large scale structures. I agree on what has been said above, and I think it is the most damning prediction that this idea fails.
22. seems to me that gravity is charge agnostic.
if it is then antimatter stellar systems could possibly form.
they would not radiate antiphotons but be indistinguishable from normal stars.
or not…
23. Picking up on Todd’s point regarding the three types of mass, it seems qualitatively straightforward:
As far as any mass is concerned, it simply moves into its future following the GR equivalent of a straight line which is a geodesic as long as there is no force acting on it. This follows from
symmetry since, without an external force, there is nothing to identify a preferred direction in which the particle would deviate from the geodesic. This means that anti-matter should follow the
same path as matter under the influence only of the gravitational effect of other ordinary matter. “Passive gravitational mass” is really a pseudo-effect created by coordinate rotation and thus
must be positive.
Inertial mass can be determined from the action of an electric or magnetic field on an anti-particle. We know it is accelerated in the oppposite direction from ordinary matter and tis could be
attributed either to the particle having the opposite charge polarity or having negative inertial mass. If total charge is to be conserved, the polarities must be opposite hence the inertial mass
must be positive.
That leaves the active gravitational mass. If that were negative, it would cause geodesics to curve away from anti-matter rather than towards, but as has been said above, both matter and
anti-matter must follow those geodesics thus either anti-matter has positive active gravitational mass like ordinary matter or it must repel anti-matter as well as matter.
In conclusion, the idea that anti-matter could form galaxies which would repel normal matter galaxies doesn’t appear feasible either way, either anti-matter flies apart or it attracts matter.
On more general note, surely if say 1% of the universe were some form of exotic mass which generated “repulsive gravity”, surely that would only reduced the net expansion by 2% at all times. For
the effect to grow with time, it needs to be stronger at long range and weaker at short, or evolving in time in some way.
1. Your conclusion is similar to mine. Antimatter does not have anti-mass. Besides all the anti-mass states in the Dirac theory compose the vacuum state as the so called Dirac sea.
2. I haven’t commented here, but I agree.
The only thing really special about antimatter, is that the nucleus has a negative charge (anti-protons) and positive electrons (positrons). They are essentially the same particles but or
reverse in their polarity.
The other problem is if there were pockets or regions of antimatter still existing in the universe, the boundaries between matter and antimatter regions would be a blaze with energy and gamma
rays; but there does not seem to be an astrophysical phenomena that would support that view.
Another issue would be with magnetic fields and jets whose positrons colliding with electrons along the field lines could easily travel significant distances and trigger astrophysical
observable phenomena (which we do not see.) Also, if there was any repulsion, we would see either the matter jets antimatter jets travelling in a straight line, then at the boundary of the
matter and antimatter a ultra bright gamma ray ‘star’ where the annihilations would occur. (Again not seen in nature.)
Finally, if the Big Bang is correct, I though that the energy seen in the universe was created by matter and antimatter, and hat the reason why the universe is one kind of matter, is that
there was a slight excess of one for over another. This net energy drives and continues the expansion.
If the force was repulsive, wouldn’t the universe after all this time be like little clumps of matter and antimatter regions scattered everywhere? Instead we see galaxies distributed along
surfaces akin to many many lathered soap bubbles. Also, would not the ‘centres’ of these bubbles (void) there should be anti-matter galaxies? f so, you would expect something to be observed
there. (as far as I’ve read, there is no observational evidence to support this view!
1. As I showed yesterday, anti-mass (antimatter with negative mass) repels itself. I think in general we would be living in a different sort of world than the one we observe.
2. Lawrence
I am not as clued up with this subject as I should be. While I was surprised with this story popping up, I came across (admittedly from investigating our mutual ‘friend’s’; Hunter, J.H.
Jr., “On the cosmology of Alfvén and Klein”, MNRAS, 137, 271 (1967).
This has some interesting ideas and discussion on matter / antimatter issues (pg.271) and the kinds of astrophysical objects that might be expected. Whilst the conclusions may have now
been mostly rejected by astrophysics, this referenced article in this story has some quite interesting parallels.
After reading this article, it seems just another different way of trying to bring antimatter into the cosmos equation.
As for you saying; “I think in general we would be living in a different sort of world than the one we observe.” is truly the point. I.e.
“We are the way we are because the Universe is the way that it is… and no vice-versa.”
3. Antimatter clearly plays a role in the universe. The high energy universe is likely CP invariant. This means given a wave function Y_q(x) that CP Y_q(x) = Y_{-q}(-x) and CP invariance
means this returns the same wave function. CP discrete symmetry is broken at lost energy and this gave rise to an excess of matter over anti-matter in the colder low energy universe.
Antimatter states are due to the occurrence of sufficient positive mass-energy on the Dirac negative momentum-energy states which are filled up and define the Fermi-Dirac vacuum. This
means that a negative mass virtual particle state with quantum number opposite those of the positive mass-energy particles now exist with positive energy.
The Dirac equation is the spinorial form of the square root of the Klein-Gordon equation. The KG equation is a quantized form of the special relativistic momentum interval
(mc^2)^2 = E^2 – (pc)^2.
Going into the spinor mathematics of the Dirac equation is a bit beyond the scope of UT, and further requires some graphic math-tools not available here. So looking this up, even on
wikipedia, is advised. However, the square root of an equation has two roots, and just as y = x^2 has positive and negative x’s (and recall the binomial equation) the same happens with
the Dirac equation.
The physics of CP violations is a big issue, and Fermilab has been looking hard at CP violations with the T-quark, which follows the Desy results on the B-quark factory results. The T and
B quarks are in the highest mass doublet of QCD.
4. Funny you mention the CP violations. I read today on the New Scientist website “Lonely, spun-out proton reveals magnetic secret”, which talks about the g-factor.
According to this, there is a possible experiment to verify if the g-factor has the same value between protons and antiprotons.
In this story, if the fields of either are of different strengths, then it would pose an additional problem for astrophysical phenomena and even nucleosynthesis / stellar evolution. (This
linked article has the arXiv paper attached with it.) Again, this is a possible broken symmetry.
5. I would be surprised if the Lande g-factors differed between matter and antimatter. The factor is g = 2.0023318416 for
mu = -geS/2m
for the magnetic moment. The straight forwards calcuation gives g = 2. It requires QED to get g – 2, where physics related to the Lamb shift give the departure. There are expected
departures for the muon g factor, where there can be a virtual transition to the neutralino state.
24. {Violation of comment policy: text deleted. There is a thread on this topic in BAUT’s Against the Mainstream section.}
1. “For example, the electron and the positron have the plus mass, although the positron is an antiparticle in relation to the electron. However, this is a very large problem, which is outside
the framework of this chapter.”
Why quote the chapter then? The suggestion that the positron (for example) would have negative active gravitational mass is the topic being discussed. You wouldn’t be trying to publicise your
book, would you?
2. I think we have a copyright issue here with such a big excerpt that clearly is a copy and paste of a book.
25. here is a tidbit of some salience:
where are the antineutrons?
1. Anti-neutrons are produced in anti-proton factories but last I heard they had no way to slow them down. You can’t use a moderator (e.g. graphite) of ordinary matter obviously and techniques
like laser slowing only work well (if at all) on charged particles.
1. A high energy event with particles can generate a proton plus anti-neutron plus a positron and an antineutrino.
2. Oops, sorry I wrote too fast. I meant an anti-neutron plus proton plus and electron and neutrino.
26. Recently I started to realize that a lot of people have a big wrong concept of what gravity is.
I think that a lot of conspiracy theorists think that gravity is a surface effect only. For example the moon pulls up the surface and thus cause tension inside the interior.
Reality is that gravity acts on every atom and in the complete Earth pulling all in the same direction. This includes the back side, the inner side, the left and right side and the front side.
The resulting vectors is basically near zero in gravitational difference between the backside and the front-side. Earth does not get stretched like a big balloon but moves in the orbit as one
whole thing.
27. The “effect” of gravity is canceled (zero G) at the center of a mass which is located at the bottom of the space/time well it itself created. This center will be offset in the direction of the
centers of other nearby masses (n-body physics).
This is what I came away with the many times this subject was tackled here on UT.
28. I’m 10 std student,i cant able to understand all.But i know it is very difficult to explore the universe when expands | {"url":"https://www.universetoday.com/84934/antigravity-could-replace-dark-energy-as-cause-of-universes-expansion/","timestamp":"2024-11-11T21:39:01Z","content_type":"text/html","content_length":"309517","record_id":"<urn:uuid:e40386c1-c25d-40c1-87a4-e237df933445>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00506.warc.gz"} |
What s the square root of 104
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Cell Calculations • Genstat v21
A single value calculation can be entered into a numerical cell in the spreadsheet by starting the calculation with an equals sign (=), and then following with a calculation using standard Genstat
syntax. These calculations can also be entered in the edit spreadsheet cell dialog opened by double-clicking a cell or by using the F8 key.
These calculations are not stored, and are lost once the result has been calculated. These are only provided to enable quick in-cell calculations and the calculations menu should be used for other
column calculations. Cell-wise calculations are considered to be both a major strength and weaknesses of any spreadsheet. You have the flexibility to perform easy calculation but can easily end up
with inconsistent formulae within a column.
The length of the cell calculation is limited by the character buffer size of the column, for example, 45 characters for numerical columns. To enter formulae that are greater than 45 characters you
can use the Edit Spreadsheet Cell dialog.
Not all Genstat functions are supported, but in general, functions that give scalar results with scalar input are supported. The list of functions supported are:
abs, sqrt, sin, cos, tan, arcsin, arccos, arctan, sinh, cosh, tanh, exp, log, log10, factorial, ncombinations, int, modulo, radians, degrees, logit, alogit, date and constants.
Note the arguments of date are date(day,month,year).
These functions can be abbreviated to 4 letters, with the exception of constants, which can be abbreviated to c. Function arguments are separated by semicolons (;).
The operators supported are ** (power), * (multiplication), / (division), + and –. Parentheses can be used to change the order of operations. Numbers can also be entered using the e notation for
powers of 10 (i.e. 7e+5 = 700000, 7e-5 = 0.00007). The Boolean operators ==, /= = are not supported.
The constants pi (3.14159265…) and e (2.71828182…) can be entered using the constants function: c(‘pi’) and c(‘e’)
On an illegal calculation (e.g. log(0) or sqrt(-1), 1/0, misspelled function or syntax error), a dialog will appear which lists the fault and the term that it occurs in. You can either correct the
calculation, and then retry it, or use the escape key (Esc) to abort the calculation.
The cell calculation can just be typed into any numerical cell:
Pressing return results in the calculation being replaced by its result:
The following are some cell calculations that could be used:
=sqrt(2-4) (This contains an error, giving the error dialogs):
Note: This error dialog only occurs for domain errors in functions.
The cell calculator is based on a code provided by Mark Morley, Victoria, Canada (morley@camosun.bc.ca).
See also
Calculations menu
CALCULATE directive | {"url":"https://genstat21.kb.vsni.co.uk/knowledge-base/cell-calculator/","timestamp":"2024-11-02T12:11:16Z","content_type":"text/html","content_length":"42386","record_id":"<urn:uuid:00bee219-fa87-4fac-aff8-297245449e66>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00520.warc.gz"} |
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories assure
that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are trivially true for the empty set.
Null set was once a common synonym for "empty set", but is now a technical term in measure theory.
Read more about Empty Set: Notation, Properties
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How to work out your rate of Service Pension
This page is to be used if you wish to work out your rate of Service Pension.
The rates on this page are effective from 20 September 2024.
Back to top
When not to use this
This page will not be accurate if you:
• are paid under transitional arrangements
• have dependent children
• are blind for Service Pension purposes
• are paying rent and are eligible to receive rent assistance
• are eligible to receive rent assistance or remote area allowance
• receive a War Widow(er)’s Pension or a Wholly Dependent Partner’s Pension under the Military Rehabilitation and Compensation Act 2004 (MRCA) and your Service Pension is more than $346.20.
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How is your pension worked out?
The rate of your Service Pension is based on your income and assets.
Separate tests are applied to your income and your assets.
For the income test, your income is added up and compared to the income free area. If your income is below the income free area you will receive the maximum rate of pension. If your income is over
the income free area, your pension will be reduced from the maximum rate by 50 cents for every $1 it is in excess of the income free area. A work bonus may apply if you are over qualifying age and
have employment earnings.
A similar process is applied to your assets in the assets test. Your total assessable assets are compared to the assets value limit. If your total assets are less than this limit you will receive the
maximum rate of pension. If your assets are over this limit, your pension will be reduced by 75 cents for every $250 they are in excess of the limit.
The test that results in the lower rate of pension is the one that is used to calculate your pension.
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Step 1 Calculate your financial assets
Use the table below to add up your financial assets.
Financial assets do not include things such as vehicles, property etc. Refer to Deeming and Financial Assets.
Type of financial assets Asset value
Financial Institutions (banks, building societies and credit union accounts)*
Asset-tested short term income streams and superannuation account-based income streams subject to deeming
Bonds and Debentures
Shares **
Managed Investments **
Gifts in excess of $10,000 in a financial year or in excess of $30,000 in a rolling 5-year period ***
Cash on hand in excess of $500
Total financial assets
* Proceeds from the sale of your home, even if they have been put aside for the purpose of purchasing a new home, should be included in this calculation.
** Calculate the value of shares and managed investments by multiplying the number of shares and units by the share or unit price. You can check the unit prices that are being used to calculate your
pension by contacting DVA.
*** A rolling 5-year period in the current financial year and the previous 4 financial years.
If you are under pension age, do not include any superannuation products, such as roll over funds, in your calculation of financial assets. For more information on pension age refer to Managed
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Step 2 Calculate your deemed income from your financial assets
Use the table and the instructions below to calculate your deemed income from your total financial assets calculated in step 1.
1. After you have worked out your total financial assets in step 1, write the total at (A).
2. The low deeming rate applies up to the first $62,600 of your financial assets. Write the amount of your financial assets that attract this deeming rate at (B).
3. If you have more than $62,600 in financial assets, write the amount that is in excess of $62,600 at (C).
4. Multiply (B) by 0.0025 and write the result at (D).
5. Multiply (C) by 0.0225 and write the result at (E).
6. Work out your total deemed income per year by adding (D) and (E) together. Write this total at (F).
7. Divide (F) by 26 and write this figure at (G). This is your deemed income per fortnight.
Total Financial Assets from Step 1 = (A)
Amount up to $62,600 to be deemed at 0.25% (B) x 0.0025 = (D)
Balance to be deemed at 2.25% (C) x 0.0225 = (E)
Total deemed income per year (D + E) = (F)
Divide by 26 ÷ 26
Deemed income per fortnight = (G)
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Step 3 Calculate your total assessable income and assets
Fill in the table below to work out the totals for your assets and income.
Type of income or asset Asset value Income per fortnight *
Financial Assets (from Step 1) and Deemed Income (from Step 2)
Gross Superannuation1*
Gross Salary/Wages1*
Gross Foreign Pensions2*
Purchased Annuities and Pensions not included in financial assets
Other Income
Household Contents
Business (including private trusts and private companies)
Other Assets (e.g. collectibles, valuables, life insurance policies)
Less Work Bonus 3*
Less Maintenance Paid to Ex Spouse 4*
Less Deductible Assets (e.g. Proceeds from sale of home; mortgages on property)
Total assessable income and assets
1* If you receive income but only know the annual amount, divide the annual amount by 26 to work out the income per fortnight. The amount of superannuation that has been applied to reduce a Special
Rate Disability Pension (SRDP) under MRCA is not counted as income.
2* Excluding foreign Disability Pension paid to a veteran for a war caused injury or disease, or an amount paid under the Compensation for Nationalist Socialist Republic Policy of Austria or Germany.
3* If you are over qualifying age, take the first $300.00 off your income if earned through active participation in the workforce (rather than through passive activities such as managing your own
investment portfolio). This amount is your work bonus. You may also have a work bonus bank balance that may be applied. For more details refer to Work Bonus.
4* Excludes child maintenance.
Back to top
Step 4 Calculate your daily rate of Service Pension using the income test and the assets test
To work out your rate of Service Pension you must calculate your rate under both the income test and the assets test.
Complete part (a) and then go on to part (b). When you have calculated your rates under both tests, go to step 5.
a) Use the following instructions and the table below to calculate your Service Pension under the income test.
1. Write your total income from step 3 in the box provided.
2. Subtract the income free area ($212.00) from your total income.
3. Multiply this figure by 0.50.
4. Subtract this figure from the maximum rate of single Service Pension ($1,144.40 - includes pension supplement).
5. Divide by 14 to establish daily entitlement rounded to 4 decimal places.
The pension instalment paid each fortnight will be made up of an amount for each day for a period of 14 days. These daily amounts may be calculated using different rates of pension for different
days. The total amount is rounded to the nearest cent.
1. Total income (from step 3)
2. Less income free area - $212.00
Total excess =
3. Multiply by 0.50 x 0.50
Total amount to be deducted =
Maximum rate of pension $1,144.40
4. Less total amount to be deducted -
= *
5. Divide by 14 ÷ 14
Daily entitlement rounded to 4 decimal places =
* $58.90 is the minimum amount payable per fortnight. If your result for fortnightly pension is less than $58.90 but greater than zero, the amount on this line should be $58.90.
b) Now use the instructions and the table below to calculate your Service Pension under the assets test.
1. Write your total assets from step 3 in the box provided.
2. Subtract the assets value limit from your total assets. If you are a home owner the assets value limit is $314,000. If you are a non-home owner the assets value limit is $566,000.
3. Round down to the nearest multiple of $250.
4. Divide the amount of assets in excess of the limit by $250.
5. Multiply this total by 0.75.
6. Subtract this from the maximum rate of pension ($1,144.40 - includes pension supplement).
7. Divide by 14 to establish daily entitlement rounded to 4 decimal places.
The pension instalment paid each fortnight will be made up of an amount for each day for a period of 14 days. These daily amounts may be calculated using different rates of pension for different
days. The total amount is rounded to the nearest cent.
1. Total value of your assets (from step 3)
2. Less asset value limit ($314,000 for home owners, $566,000 for non-home owners) -
3. Total excess (rounded down to nearest multiple of $250) =
4. Divide by ÷ $250
5. Multiply by x 0.75
Total to be deducted =
Maximum rate of pension $1,144.40
6. Less total amount to be deducted -
= *
7. Divide by 14 ÷ 14
Daily entitlement rounded to 4 decimal places =
* $58.90 is the minimum amount payable per fortnight. If your result for fortnightly pension is less than $58.90 but greater than zero, the amount on this line should be $58.90.
Back to top
Step 5 Rate of pension payable
Your rate of pension under the income test $
Your rate of pension under the assets test $
The test that pays the lower rate of pension will be the one that is applied.
Example: If the result is $100 from the income test and $110 from the assets test, then the lower of the 2 amounts ($100) is the rate of pension payable.
If you find there is a difference of at least $10 between your result in step 5 and your actual pension, you should contact DVA.
Back to top | {"url":"https://www.dva.gov.au/get-support/financial-support/income-support/service-pension/how-work-out-your-rate-service-pension","timestamp":"2024-11-05T05:35:22Z","content_type":"text/html","content_length":"75255","record_id":"<urn:uuid:e8bea7ee-2327-405c-a250-be931726371e>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00742.warc.gz"} |
f(x)=x−72x+3 के फलन का प्रात ज्ञात... | Filo
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Question Text के फलन का प्रात ज्ञात
Updated On Sep 13, 2023
Topic Trigonometry
Subject Mathematics
Class Class 11
Answer Type Video solution: 2
Upvotes 238
Avg. Video Duration 3 min | {"url":"https://askfilo.com/user-question-answers-mathematics/ke-phln-kaa-praat-jnyaat-35353330343937","timestamp":"2024-11-09T13:04:40Z","content_type":"text/html","content_length":"228648","record_id":"<urn:uuid:e8d363d6-aab4-4009-8b8b-c58f471b7906>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00538.warc.gz"} |
Category: algorithms Component type: function
Partial_sort_copy is an overloaded name; there are actually two partial_sort_copy functions.
template <class InputIterator, class RandomAccessIterator>
partial_sort_copy(InputIterator first, InputIterator last,
RandomAccessIterator result_first,
RandomAccessIterator result_last);
template <class InputIterator, class RandomAccessIterator,
class StrictWeakOrdering>
partial_sort_copy(InputIterator first, InputIterator last,
RandomAccessIterator result_first,
RandomAccessIterator result_last, Compare comp);
Partial_sort_copy copies the smallest N elements from the range [first, last) to the range [result_first, result_first + N), where N is the smaller of last - first and result_last - result_first. The
elements in [result_first, result_first + N) will be in ascending order.
The two versions of partial_sort_copy differ in how they define whether one element is less than another. The first version compares objects using operator<, and the second compares objects using a
function object comp.
The postcondition for the first version of partial_sort_copy is as follows. If i and j are any two valid iterators in the range [result_first, result_first + N) such that i precedes j, then *j < *i
will be false. The corresponding postcondition for the second version is that comp(*j, *i) will be false.
The return value is result_first + N.
Defined in the standard header algorithm, and in the nonstandard backward-compatibility header algo.h.
Requirements on types
For the first version:
• InputIterator is a model of InputIterator.
• RandomAccessIterator is a model of Random Access Iterator.
• RandomAccessIterator is mutable.
• The value types of InputIterator and RandomAccessIterator are the same.
• RandomAccessIterator's value type is LessThan Comparable.
• The ordering relation on RandomAccessIterator's value type is a strict weak ordering, as defined in the LessThan Comparable requirements.
For the second version:
• InputIterator is a model of InputIterator.
• RandomAccessIterator is a model of Random Access Iterator.
• RandomAccessIterator is mutable.
• The value types of InputIterator and RandomAccessIterator are the same.
• StrictWeakOrdering is a model of Strict Weak Ordering.
• RandomAccessIterator's value type is convertible to StrictWeakOrdering's argument type.
• [first, last) is a valid range.
• [result_first, result_last) is a valid range.
• [first, last) and [result_first, result_last) do not overlap.
Approximately (last - first) * log(N) comparisons, where N is the smaller of last - first and result_last - result_first.
int A[] = {7, 2, 6, 11, 9, 3, 12, 10, 8, 4, 1, 5};
const int N = sizeof(A) / sizeof(int);
vector<int> V(4);
partial_sort_copy(A, A + N, V.begin(), V.end());
copy(V.begin(), V.end(), ostream_iterator<int>(cout, " "));
// The printed result is "1 2 3 4".
See also
partial_sort, sort, stable_sort, binary_search, lower_bound, upper_bound, less<T>, StrictWeakOrdering, LessThan Comparable
Copyright © 1999 Silicon Graphics, Inc. All Rights Reserved. TrademarkInformation | {"url":"http://seanborman.com/STL_doc/partial_sort_copy.html","timestamp":"2024-11-14T23:48:21Z","content_type":"text/html","content_length":"7330","record_id":"<urn:uuid:b0c40f50-4ebf-4c5a-867c-7e3e0d804edf>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00723.warc.gz"} |
Sudo Null - Latest IT News
January 23, 2013 at 14:17
Inside regular expressions
Regular expressions (RVs) are a very convenient form of writing so-called regular or automaton languages. Therefore, PBs are used as an input language in many systems processing chains. Consider
examples of such systems:
• The grep command of the Unix operating system or similar commands to find conversations that can be found in Web browsers or text formatting systems. In such systems, PBs are used to describe
patterns that the user is looking for in a file. Various search engines convert RV into either a deterministic finite state machine (DFA) or a non-deterministic finite state machine (NKA) and
apply this automaton to the file in which the search is performed.
• Generators of lexical analyzers. Lexical analyzers are a compiler component; they break the source program into logical units (tokens), which can consist of one or several characters and have a
certain meaning. The generator of lexical analyzers receives formal descriptions of tokens, which are essentially RV, and creates a DFA, which recognizes which of the tokens appears at its input.
• RV in programming languages.
In this article, we will first familiarize ourselves with finite state machines and their types (DFA and NKA), and then consider an example of constructing a minimal DFA by regular expression.
State machines
A finite state machine (KA) is a converter that allows you to map the corresponding output to an input, and this output may depend not only on the current input, but also on what happened earlier, on
the history of the state machine. Even human behavior, and not just artificial systems, can be described using spacecraft. For example, your reaction to your neighbor listening to loud music at night
will be one after the first such event and completely different after several such cases. There can be an infinite number of such histories; the question arises: what kind of memory does the
spacecraft have to behave differently for each histogram? It is clear that it is impossible to store an infinite number of histories. Therefore, the automaton, as it were, breaks up all possible
histories into equivalence classes. Two stories are equivalent, if they equally affect the behavior of the automaton in the future. The equivalence class to which the automaton related its current
background is also called the internal state of the automaton.
Consider an example of a primitive spacecraft:
This spacecraft consists of:
• the tape represented by the input chain.
• reader device.
• a control unit that contains a list of transition rules.
The reader can move in one direction, usually from left to right, thereby reading the characters of the input chain. For each such movement, it can count one character. Further, the read symbol is
transmitted to the control unit. The control unit changes the state of the machine based on the transition rules. If the list of transition rules does not contain a rule for the character read, then
the machine "dies."
Now consider the ways in which the spacecraft can be defined. They can be specified as graphs or as control tables. In the form of a graph, the spacecraft is defined as follows:
• the vertices of the graph correspond to the states of the spacecraft.
• directed edges correspond to transition functions (the symbol along which the transition is performed is indicated near each such edge).
• a vertex with an edge entering it that does not exit from more than one state corresponds to the initial state.
• final states of the spacecraft are marked in bold.
In the form of a control table, like this:
• spacecraft states are located in the rows of the table.
• Recognized language characters are in columns.
• at the intersection, the state into which you can get from this state by this symbol is indicated.
An example of a spacecraft in the form of a graph and in the form of a control table will be presented below.
DKA and NKA
The main difference between DKA and NKA is that DKA in the process of work can only be in one state, and NKA in several states at the same time. An example of the work of the NCA is the idea of the
American physicist
Hugh Everett
from the fact that any event divides the world into several worlds, in each of which this event ended in its own way. For example, in one world, Hitler won the Second World War, in another - Newton
went into business instead of physics, and the discovery of the laws of classical mechanics had to be postponed for 50 years. In order to draw any conclusions from the work of the automaton, one
should study all the “worlds”. After the entire input chain has been read, we believe that the NCA admits this chain if it completed work in an accepting state in at least one of the many “worlds”.
Accordingly, the machine rejects the chain if it completed the work in an inadmissible state in each "world". A DFA accepts a chain, this is obvious if after reading the entire input chain it is in a
valid state.
In most cases, building an NKA is much easier than a DKA. But, despite this, using NKA for modeling is not a good idea. Fortunately, for each NFA, it is possible to construct a DFA that admits the
same input language. In this article, we will not give an algorithm for constructing a DFA from an NFA, but consider this algorithm based on a clear example below.
Building a minimal DCA by regular expression
To begin with, we give a list of RV operations used in this article, in order of priority:
• iteration (wedge closure) using the symbol "*"
• concatenation is specified using a space or an empty string (for example: ab)
• union using the character "|"
Consider an example, given a regular expression:
xy * (x | y *)
ab (x | y *)
(x | a *) (x | y *)
It is necessary to construct a minimal DFA from a regular expression and demonstrate recognition of the correct and incorrect chains.
To begin with, we simplify this RV, using the right-hand distribution law of concatenation with respect to union, we obtain the following RV:
(xy * | ab | (x | a *)) (x | y *)
Now we construct an automaton according to this RV:
According to the concatenation transformation rule (we will not to give the rules for converting PBs into spacecraft, since they are quite obvious), we obtain the following automaton:
By the union transformation rule:
By the concatenation transformation rule:
And at the end, we apply the closure transformation rule and get εНКА. It should be noted here that εNKA is an NKA that contains ε-transitions. An ε-transition, in turn, is a transition in which the
automaton does not take into account the input chain or, in other words, a transition by an empty symbol.
We get rid of ε-transitions (the “final state” is indicated by an “asterisk”):
In this SCA, the states s3 and s5 are equivalent, since δ (s3, x) = δ (s5, x) = s1 and δ (s3, y) = δ ( s5, y) = s5, s7. Rename the states s6 -> s5 and s7 -> s6: we
construct a DFA according to the NFA:
In this DFA, the states p1 and p5 are equivalent, since
δ (p1, x) = δ (p5, x) = p4 and δ (p1, y) = δ (p5, y) = p5. Rename the states p6 -> p5 and p7 -> p6:
This automaton is the minimum DFA.
Let δ be the transition function, then the extended transition function constructed from δ is denoted by δ ', and ω is the input chain.
Suppose that the chain ω = aaax is fed into the input, we expect the automaton to be in one of the admissible states.
δ '(p0, ε) = p0
δ' (p0, a) = δ (δ '(p0, ε), a) = δ (p0, a) = p3
δ' (p0, aa) = δ (δ ' (p0, a), a) = δ (p3, a) = p5
δ '(p0, aaa) = δ (δ' (p0, aa), a) = δ (p5, a) = p5
δ '(p0 , aaax) = δ (δ '(p0, aaa), x) = δ (p5, x) = p4
p4 is an admissible final state, so the chain aaax is correct for this automaton.
Now suppose that ω = xyyb:
δ '(p0, ε) = p0
δ' (p0, x) = δ (δ '(p0, ε), x) = δ (p0, x) = p1
δ' (p0 , xy) = δ (δ '(p0, x), y) = δ (p1, y) = p1
δ '(p0, xyy) = δ (δ' (p0, xy), y) = δ (p1, y) = p1
δ '(p0, xyyb) = δ (δ' (p0, xyy), b) = δ (p1, b) = ∅
Here we see that if the symbol b is input to the automaton when it is in state p1, then this automaton will die, therefore the chain xyyb is incorrect.
PS In this article, the algorithm for constructing DFA on RV was considered, but there are more convenient algorithms, in particular for programming, but this is a topic for another article ... | {"url":"https://sudonull.com/post/131907-Inside-regular-expressions","timestamp":"2024-11-13T23:01:53Z","content_type":"text/html","content_length":"18759","record_id":"<urn:uuid:6554213e-22ba-4ce7-9f3f-07a0a91ac541>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00310.warc.gz"} |
What is the Difference Between ARIMA and ARIMAX? - Varsha Saini
Frequently Asked Questions
What is the Difference Between ARIMA and ARIMAX?
The only difference between ARIMA and ARIMAX is the addition of an exogenous (external) variable. The ARIMA model works on a single time series data (univariate) whereas ARIMAX uses multiple
variables to include the external feature.
Equation of ARIMAX
Δr(t)= c + φ Δr(t-1) + θ ε(t-1) + ε(t) + β x
• Δr(t)= r(t)-r(t-1) , difference in consecutive period.
• ε(t),ε(t-1) = current error term and one period ago.
• c = baseline constant factor.
• φ = value coefficient, what part of the last period value is relevant in explaining the current value.
• θ = error coefficient, what part of the last period value is relevant in explaining the current error value.
• β = coefficient for the external variable.
• x = value of the external variable.
Equation of ARIMA
The equation of the ARIMA model is the same as ARIMAX except the external factor.
Δr(t)= c + φ Δr(t-1) + θ ε(t-1) + ε(t)
Other Popular Questions | {"url":"https://varshasaini.in/questions/what-is-the-difference-between-arima-and-arimax/","timestamp":"2024-11-07T01:05:53Z","content_type":"text/html","content_length":"186245","record_id":"<urn:uuid:dca39182-fd96-41ee-9c19-1716303bd5f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00280.warc.gz"} |
The Electric Field As A Web
UY1: The electric field as a web
Consider the mutual repulsion of two positively charged bodies A and B(at point P). How does each one know the other is there?
As a result of the charge that body A carries, the properties of the space around it are modified. Body A produces an electric field at point P (where B is at).
$$\vec{E} = \lim_{q_{0} \to 0} \frac{1}{q_{0}} \vec{F_{0}}$$
Important note: You might think that as q[0] goes to 0, the whole expression will go to infinity. It is correct in the mathematical point of view. But, as physicists are rather sloppy in mathematical
notations, the above expression actually means that the point charge (q[0]) must be small.
As a result of the charge that “particle” B carries B senses how space has been modified at P.
$$ \vec {F_{0}} = q_{0} \vec{E}$$
Analogy: You can think of the electric field as a spider web. The spider (Body A) sets up a web (electric field). A fly (Body B) encounters the spider web (electric field) and felt a force.
Next: Electric Field Of A Point Charge
Previous: Electric Charge & Coulomb’s Law
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Stellar - Live Stellar price and market cap
Stellar is a decentralized platform that aims to connect banks, payments systems, and people. Integrate to move money quickly, reliably, and at almost no cost. Supported by a nonprofit, Stellar's
goal is to bring the world together by increasing interoperability between diverse financial systems and currencies.
Stellar is a technology that enables money to move directly between people, companies and financial institutions as easily as email. This means more access for individuals, lower costs for banks and
more revenue for businesses. | {"url":"https://surfbtc.com/en/currencies/stellar","timestamp":"2024-11-03T11:59:47Z","content_type":"text/html","content_length":"529752","record_id":"<urn:uuid:d941da65-d18d-43f4-a021-5573af34db62>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00402.warc.gz"} |
How Many Meters per second Is 79.7 Feet per second?
79.7 feet per second in meters per second
How many meters per second in 79.7 feet per second?
79.7 feet per second equals 24.293 meters per second
Unit Converter
Conversion formula
The conversion factor from feet per second to meters per second is 0.3048, which means that 1 foot per second is equal to 0.3048 meters per second:
1 ft/s = 0.3048 m/s
To convert 79.7 feet per second into meters per second we have to multiply 79.7 by the conversion factor in order to get the velocity amount from feet per second to meters per second. We can also
form a simple proportion to calculate the result:
1 ft/s → 0.3048 m/s
79.7 ft/s → V[(m/s)]
Solve the above proportion to obtain the velocity V in meters per second:
V[(m/s)] = 79.7 ft/s × 0.3048 m/s
V[(m/s)] = 24.29256 m/s
The final result is:
79.7 ft/s → 24.29256 m/s
We conclude that 79.7 feet per second is equivalent to 24.29256 meters per second:
79.7 feet per second = 24.29256 meters per second
Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 meter per second is equal to 0.041164866938684 × 79.7 feet per second.
Another way is saying that 79.7 feet per second is equal to 1 ÷ 0.041164866938684 meters per second.
Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that seventy-nine point seven feet per second is approximately twenty-four point two nine three
meters per second:
79.7 ft/s ≅ 24.293 m/s
An alternative is also that one meter per second is approximately zero point zero four one times seventy-nine point seven feet per second.
Conversion table
feet per second to meters per second chart
For quick reference purposes, below is the conversion table you can use to convert from feet per second to meters per second | {"url":"https://convertoctopus.com/79-7-feet-per-second-to-meters-per-second","timestamp":"2024-11-05T00:13:12Z","content_type":"text/html","content_length":"34764","record_id":"<urn:uuid:979f728d-bc1e-4a85-adec-214b602707c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00860.warc.gz"} |
OpenStax College Physics, Chapter 14, Problem 8 (Problems & Exercises)
(a) The number of kilocalories in food is determined by calorimetry techniques in which the food is burned and the amount of heat transfer is measured. How many kilocalories per gram are there in a
5.00-g peanut if the energy from burning it is transferred to 0.500 kg of water held in a 0.100-kg aluminum cup, causing a $54.9^\circ\textrm{C}$ temperature increase? (b) Compare your answer to
labeling information found on a package of peanuts and comment on whether the values are consistent.
Figure 14.a Peanut nutrition label
Question by
is licensed under
CC BY 4.0
Final Answer
a. $5.73 \textrm{ kcal/g}$
Note: at the end of the calculation for part (a) in the video I mistakenly wrote the units for the final answer as $\textrm{kcal/kg}$ but it should be per gram. The units for the final answer to
part (a) should be $\textrm{kcal/g}$.
b. According to the peanuts nutrition label in Figure 14.a above, peanuts have $6.1\textrm{ kcal/g}$. This agrees with our calculation to within 1 significant figure.
Solution video
OpenStax College Physics, Chapter 14, Problem 8 (Problems & Exercises)
vote with a rating of votes with an average rating of.
Video Transcript
This is College Physics Answers with Shaun Dychko. This peanut is burned and all of the heat energy is captured in this water and the aluminum container holding the water and based on knowing how
much heat energy this is, we'll say that that's the total energy in this peanut and we'll figure out how much energy it has per gram in units of food calorie's per gram or kilocalories per gram. So
there's a 0.100 kilogram aluminum cup holding the water and the aluminum has a specific heat of 0.215 kilocalories per kilogram per Celsius degree— that's what we look up in table [14.1]— and there
is 0.500 kilograms of water which has a specific heat of 1 kilocalorie per kilogram per Celsius degree and the change in temperature of both the water and the aluminum cup together is 54.9 Celsius
degrees. So the total amount of energy that's added to this water and aluminum combination is the mass of aluminum times the specific heat times the change in temperature plus the mass of the water
times the water's specific heat times its change in temperature but the change in temperature is the same for both substances and so it doesn't need a subscript and it can be factored out. So we have
ΔT times the mass and specific heat of the aluminum and water, respectively. So that's 54.9 Celsius degrees— change in temperature— times 0.100 kilograms of aluminum times 0.215—specific heat— plus
0.500 kilograms of water times 1.000 kilocalorie per kilogram per Celsius degree and we get 28.6304 kcal of energy. Now to figure out how many kilocalories there are per gram, we can take that amount
of heat energy and divide it by 5.00 grams of the peanut and then that's 5.73 kilocalories per kilogram. Now on a peanut nutrition label that I found on Amazon, it says there are 170 Calories with a
'C' in every 28 grams and calories with a capital 'C' is the same as kilocalories. So 170 divided by 28 is 6.1 kilocalories per gram and yes, to one significant figure, this would be 6 if you rounded
it and this is 6 if you rounded it to the ones place... these numbers agree and there we go!
5.00g = .005kg. thus, 5726 kcal/kg ... yes? Otherwise, state as 5.73 kcal/g
Hi blue, thank you so much for noticing this. I have fixed the typo in the final answer and made a note about how kcal/g is what should have been written in the video.
All the best, | {"url":"https://collegephysicsanswers.com/openstax-solutions/number-kilocalories-food-determined-calorimetry-techniques-which-food-burned-and","timestamp":"2024-11-08T18:33:25Z","content_type":"text/html","content_length":"186575","record_id":"<urn:uuid:0dc516c2-9f46-4475-82fc-5e39806b4e73>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00643.warc.gz"} |
NumPy Exponential: Using the NumPy.exp() Function • datagy
In this tutorial, you’ll learn how to use the NumPy exponential function, np.exp(). The function raises the Euler’s constant, e, to a given power. Because Euler’s constant has many practical
applications in science, math, and deep learning, being able to work with this function in meaningful ways is an asset for any Python user!
By the end of this tutorial, you’ll have learned:
• What the np.exp() function does
• How to apply the function to a single value and to NumPy arrays
• How to use the function to graph exponential arrays
Understanding the np.exp() Function
The NumPy exp() function is used to calculate the exponential of all the elements in an array. This means that it raises the value of Euler’s constant, e, to the power all elements of an array, or a
single element, passed into the function. Euler’s constant is roughly equal to 2.718 and has many practical applications such as calculating compound interest. To learn more about Euler’s constant in
Python, check out my in-depth tutorial here.
The exponential function is commonly used in deep learning in the development of the sigmoid function. Let’s take a look at the function:
# Understanding the np.exp() Function
import numpy as np
x=, # Input values
out=None, # Location to store
where=True # Condition to broadcast over input
In most cases, you’ll see the function applied only with the x argument supplied. Let’s take a look at how we can run the function with a single value passed in:
# Running the np.exp() Function with a Single Value
import numpy as np
# Returns: 2.718281828459045
The function call above is the same as calling e^1. The real value of the function comes into play when its applied to entire arrays of numbers. This is what you’ll learn in the next section.
How to Apply the np.exp() Function to a 2-Dimensional Array
In this section, you’ll learn how to apply the np.exp() function an array of numbers. Applying the function to an array works the same as applying it to a scalar, only that we pass in an array.
Because numpy works array-wise, the function is applied to each element in that array.
Let’s take a look at an example:
# Applying the np.exp() Function to a 2-d Array
import numpy as np
arr = np.arange(1,6)
# Returns: [ 2.71828183 7.3890561 20.08553692 54.59815003 148.4131591 ]
In the example above, we use the np.arange() function to create the values from 1 through 5. We then pass this array into the np.exp() function to process each item.
The function also works for multi-dimensional arrays, as shown in the next section.
How to Apply the np.exp() Function to a Multi-Dimensional Array
Similar to working with two-dimensional arrays, the np.exp() can be applied to multi-dimensional arrays. The function will be broader to each value in the array, despite its dimensionality. Let’s
take a look at an example:
# Applying the np.exp() Function to a Multidimensional Array
import numpy as np
arr = np.arange(4).reshape((2, 2))
# Returns:
# [[ 1. 2.71828183]
# [ 7.3890561 20.08553692]]
In the example above, we reshape the values of 0 through 3 into a 2×2 array. We then pass this array into the np.exp() function.
How to Graph the np.exp() Function Using Matplotlib
In this final section, we’ll learn how to plot the resulting arrays of the np.exp() function to see how it behaves. We can create a finely spaced array using the np.linspace() function to create a
linear space, which we can pass into the function.
Let’s take a look at how we can do this:
# Graphing the np.exp() Function
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 10, 1000)
y = np.exp(x)
plt.plot(x, y)
In the example above, we create an evenly-spaced array of numbers from 0 through 10 with 1000 values. We then pass this array into the np.exp() function. This returns the following plot:
Plotting the np exp Function in Matplotlib
This shows the distribution of the exponential function.
In this post, you learned how to use the np.exp() function. You learned how the function is commonly applied in machine learning and deep learning. Then, you learned how to use the function on a
scalar, a 2-dimensional array, and a multi-dimensional array. Finally, you learned how to plot the function using Matplotlib.
Additional Resources
To learn more about related topics, check out the tutorials below: | {"url":"https://datagy.io/numpy-exp-exponential/","timestamp":"2024-11-13T01:59:53Z","content_type":"text/html","content_length":"147152","record_id":"<urn:uuid:774b1d29-a501-4b37-acec-947fc23c2dce>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00431.warc.gz"} |
Overview of fertility estimation methods based on the P/F ratio
Almost all methods of estimating fertility indirectly have their origins in the P/F ratio method first proposed by Brass (1964). In addition, the interpretation of the results from other methods (for
example, cohort-period fertility rates) and some of the diagnostic tools used to assess the quality of the data when estimating child mortality also rely on the intrinsic logic of the P/F ratio
approach. Thus, while the method in its original and modified forms has been superseded by the relational Gompertz model and its variants, it is useful to present the essential logic of the method
here. The interested reader is referred to Manual X (UN Population Division 1983) for a full exposition of the approach.
The Brass P/F ratio method
The foundation of the method rests on the observation that if fertility has been constant for an extended period of time, cohort and period measures of fertility will be identical. In other words,
under conditions of constant fertility, the cumulated fertility of a cohort of women up to any given age will be the same as the cumulated fertility up to that same age in any given period.
If we assume that there are no appreciable mortality differentials by the fertility of mother, so that surviving women do not have materially different levels of childbearing from deceased women, the
cumulated fertility of a cohort of women up to any given age is the same as the average parity in that cohort. (This assumption is not very important as even if there are differentials in the
fertility of living and deceased women, in most populations the magnitude of female mortality in the reproductive ages is very small and the effect of differential survival will therefore be small.)
Brass defined P to be the average parity (cumulated lifetime fertility) of a cohort of women up to a given age, and F to be closely related to the cumulated current (period) fertility up to that same
age. The P/F ratio method expresses these two quantities in relation to each other in the form of a ratio for each age group.
The derivation of F is a little more complicated than suggested above for two reasons. First, any comparison of cohort and period fertility has to deal with the probable shifting of the data on
recent fertility brought about by the question being based on the age of the mother at the time of the inquiry rather than her age at the time of her most recent birth. Second, while the cumulation
of period fertility to any given age will reflect the fertility experience of all women up until that age, the average parities typically calculated reflect those of women in 5-year age groups and
hence reflect (approximately) the average parity of women aged at the midpoint of that age group. The method formulated by Brass addresses both these aspects.
It follows that if fertility has been constant in a population for an extended period of time, and if the data are free of error, the P/F ratio would equal 1 in every age group. If fertility has been
falling, however, cumulated life time fertility would be greater than cumulated current fertility. In this case (in the absence of errors in the data) the P/F ratio would depart from unity
systematically with increasing age of mother.
The corollary to this observation is that one would expect the P/F ratio to be fairly close to unity at the youngest ages because even by women’s mid-twenties one would not expect significant
deviation of cumulated period fertility from cumulated lifetime cohort fertility as most of the births to women in that cohort would have happened fairly recently. It is from this observation that
the P/F ratio derived from women aged 20-24 at the time of a survey is held to be the most reliable indicator of the quality of the fertility data collected. Conveniently, the supposition is that the
average parities of younger women are usually fairly accurately reported, at least relative to those of older women.
It is this characteristic pattern of departure from unity with age of mother that forms the basis for many diagnostic investigations into the nature and quality of data drawn from questions based on
recent and lifetime fertility.
Diagnostics based on the P/F ratio
In reality the data are never free from error, and so the hypothetical pattern of departure of the P/F ratio from unity is confounded and obfuscated by underlying errors in the data.
As discussed on the sections on evaluation of recent fertility data and evaluation of lifetime fertility data, two errors typically affect these data. The first is that reports on lifetime fertility
– that is, cumulated cohort fertility – become increasingly inaccurate with age of the respondent, with older women tending to under-report their lifetime fertility. Errors of this kind will
therefore tend to depress the numerator of the P/F ratio, particularly at the older ages. If such errors occur in the data, the ratio will tend to be closer to unity than it might truly be.
The second kind of error frequently encountered is that women tend to under-report recent births, regardless of their age. Errors of this type will result in the reported level of recent fertility
being somewhat lower than anticipated, thereby causing the P/F ratio to be inflated.
The P/F ratio method seeks to correct the second problem by applying the P/F ratio applicable to younger women (for the reasons set out above) to the directly observed fertility schedule as a scaling
Summary of methods based on the P/F ratio method
A number of methods described here were originally presented in Manual X as extensions of the P/F ratio method. The relational Gompertz model can be thought of as an improved and more versatile
version of the Brass P/F ratio method. The model uses the same input data (and makes the same assumptions about errors that affect fertility data) as its precursor. Importantly, however, the method
does not require an assumption that fertility has been constant in the past. Nonetheless, the comparison of lifetime and period fertility lies at the heart of the method.
Most of the extensions to the Brass P/F ratio method presented in Manual X have been recast as extensions to the relational Gompertz model. These extensions include those methods that make use of the
data on parity increments from two censuses to estimate fertility; methods that use parity increments in conjunction with a schedule of intercensal fertility rates (the synthetic relational Gompertz
model); and indirect methods that make use of data from vital registration systems. Cohort-period fertility rates derived from survey data also rely on the logic of the P/F ratio method to shed light
on longer-term trends and dynamics in fertility.
Brass W. 1964. Uses of census or survey data for the estimation of vital rates. Paper prepared for the African Seminar on Vital Statistics, Addis Ababa 14-19 December 1964. Document No. E/CN.14/CAS.4
/V57. New York: United Nations. https://repository.uneca.org/handle/10855/9560
UN Population Division. 1983. Manual X: Indirect Techniques for Demographic Estimation. New York: United Nations, Department of Economic and Social Affairs, ST/ESA/SER.A/81. https://www.un.org/
Suggested citation
Moultrie TA. 2013. Overview of fertility estimation methods based on the P/F ratio. In Moultrie TA, Dorrington RE, Hill AG, Hill K, Timæus IM and Zaba B (eds). Tools for Demographic Estimation.
Paris: International Union for the Scientific Study of Population. https://demographicestimation.iussp.org/content/overview-fertility-estimation-methods-based-pf-ratio. Accessed 2024-11-12. | {"url":"https://demographicestimation.iussp.org/content/overview-fertility-estimation-methods-based-pf-ratio","timestamp":"2024-11-11T23:23:21Z","content_type":"text/html","content_length":"56908","record_id":"<urn:uuid:4f09e3d5-493a-4daf-8f12-3a690cd3c106>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00317.warc.gz"} |
Question: Why Not Use Other Types of Control Charts? Why Not Use Standard Deviation? - Measures of Success - By Mark Graban
Question: Why Not Use Other Types of Control Charts? Why Not Use Standard Deviation?
A reader from Hong Kong asks a question that has been asked by others:
“There are many types of control charts under the six sigma framework, depending on the data type (continuous/attributes). Do we need to consider it in crafting Process Behavior Charts?”
Yes, university statistics courses and Six Sigma programs teach a number of control charts:
• When counting “defectives”
• When counting “defects”
Process Behavior Charts are another name for the XmR Chart.
You can read chapter fourteen of Don Wheeler‘s book Making Sense of Data for more on this subject, as I cite that below.
Here is one table from that chapter that compares the chart types:
Wheeler teaches that the four charts in the bulleted list are “special cases of the X Chart, and since the XmR chart provides a universal way of placing count data on a process behavior chart,” we
generally don’t need to use the other chart types.
Wheeler also writes:
“The XmR Chart gives limits that are empirical — they are based upon the variation that is present in the data (as measured by the moving ranges). The np-chart, the p-chart, the c-chart, and the
u-chart all use a specific probability model to construct theoretical limits…
If you are not sure about when to use a particular probability model, then you may still use the empirical approach of the XmR chart. Remember, the objective is to take the right action, rather
than to find the “right” number.”
The np-chart makes the assumption that the chance of an error, defect, etc. that’s being plotted is the SAME for each opportunity. That’s an assumption that’s unlikely to hold true in the real world.
Does every patient have the exact same probability of getting a hospital-acquired infection? Probably not. So, the XmR chart might be a better choice. The c-chart makes the same assumption about
constant probabilities of an event.
The p-chart makes the same assumptions about constant probabilities and requires additional work of essentially calculating different upper and lower limits for each data point (as illustrated here
from this page). In my experience, this form of limits mainly serves to really confuse people. It’s easier (and arguably more valid) to use the XmR chart with their consistent limits that stay the
same unless there’s a signal of a process shift.
Here is an example of a p-chart for the percentage of calls that go unanswered in different time periods:
Can we really assume that the probability of each individual call going unanswered is exactly the same? Queuing theory tells us NO.
Here is a PBC with that same data:
The PBC is easier to calculate and gives basically the same answer… without the bad assumption.
The c-chart and u-chart make also bad assumptions about the uniformity of what’s being measured and u-charts have the same varying-limits confusion as is caused by the p-charts.
Wheeler calls the XmR chart the “Swiss Army knife of control charts.” This website has a graphical flow chart for choosing the control type chart to use — it’s based on a diagram from Wheeler’s book.
When I have taught my workshop, I had one session where a Six Sigma Master Black Belt talk to me afterward and he said, basically, “My executives get lost and their eyes glaze over when I try
explaining the various types of control charts. It’s great to have a single method that does the job well enough in all circumstances.”
What matters more is the thinking around our Process Behavior Charts. Can we stop from reacting to every up and down in a chart? Can we learn to filter out noise so we can find possible signals?
Wheeler also adds:
“The only instance of count data which cannot be reasonably placed on an XmR chart is that of very rare items or very rare events: data for which the average count per sample falls below 1.0.”
Wheeler addresses how to address this “chunky” data in his book.
For my readers, I address this in my book by offering a method of counting, for example, the days between rare events, like employee injuries or patient falls.
Below is the chart with an average <1:
And here is a chart of the days between infections:
That addresses the “rare events” situation for many, if not all, cases.
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wu :: forums - tough number theory prob.
wu :: forums
putnam exam (pure math) (Moderators: Grimbal, SMQ, Eigenray, towr, william wu, Icarus) « Previous topic | Next topic »
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Author Topic: tough number theory prob. (Read 3027 times)
Grimbal Re: tough number theory prob.
wu::riddles Moderator « Reply #25 on: Oct 18^th, 2008, 11:54pm » Quote Modify
The way I see it is that the correct notation is:
and it should be understood as
So, "(mod n)" is not part of the expression, but gives a context to it.
The equivalence
a = b + k·n for some k
Gender: which implies
Posts: 7527 rem(a,n) = rem(b,n)
This has nothing to do with the binary operator 'mod' that you can see in some computer languages.
In mathematics you use "
Michael Dagg Re: tough number theory prob.
Senior Riddler « Reply #26 on: Nov 7^th, 2008, 12:23pm » Quote Modify
Nice discussion and as pointed out, this is an interesting
problem as well as ones like it. It is crucial in many problems
to know the group of units in an algebraic number field.
(For example, when "ideals", then viewed as "ideal numbers" were
first invented by Kummer, it was in the context of proving Fermat's
Last Theorem. He got as far as calculating that some important
ideal would be principal, but then had to wrestle with the possible
ways that the generator could be presented; the options differ
from one another by multiplication by a unit. So it was very
Gender: important to know the group of units in those algebraic number
Posts: 500 fields.)
A summary of what happens is that the group of units in the
ring of integers in an algebraic number field F is a
finitely-generated abelian group, (that's not obvious!) hence (easy)
it's isomorphic to a finite group T times a free abelian group Z^r .
The finite subgroup T is (easy) the group of roots of unity that
lie in the field, and that' not usually hard to determine; at worst
you have to try factoring a finite list of cyclotomic polynomials
like x^n - 1 over the field. The torsion-free part can be shown
to be of rank s1+s2 where s1 is the number of embeddings of the
field into the real numbers, and s2 is the number of complex-conjugate
pairs of non-real embeddings. (These numbers make s1 + 2 s2 = [ F : Q ] .)
The calculation of this value of r is a little bit clever but not
hard to follow. The hardest part of all is to actually compute a set
of r generators of this group; typically one can find a set of r
independent units, i.e. one can find a group of units which is of
finite index in the full group of units, but then you have to decide
whether for example any simple combination of the units you know is
the square of another unit you didn't previously know about.
I like the book Number Theory by Borevich and Shafarevich (despite its
numerous misprints and lousy index) because it treats all these topics.
Definitely one to add to your bookshelf.
« Last Edit: Nov 7^th, 2008, 12:24pm by Michael Dagg »
Michael Dagg
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Who can provide explanations for MyMathLab statistics inferential statistics? | Hire Someone To Take My Statistics Assignment
Who can provide explanations for MyMathLab statistics inferential statistics? Posted by: jdottari3h pct_3d0 PCT, 2003) [INFO] ———————————————————————- How much of a problem the most difficult is
computing? I’m really looking for simple computations this should provide: How long does the number of tests increase above 5? How often does the arithmetic and subtraction need to get on time? How
frequently does the subtraction need to get off the ground to approximate? How many iterations should the test take? I’m using the base 10th power of 10, compute Matlab profiler to find the number
read tests the total number of tests should use (from standard Matlab output) I think Matlab is going to give you a hint: [NAME] Is there a way to compute math and statistics inferential statistics
using each analysis. Alternatively, it may be more straightforward to just use one tool rather than the entire total. One of our intuitions is that number of tests increases dramatically with the
number of analyzed statements. Assume n is an integer, then we have a number n tests (where n 0 = 5). Are you looking for matlab? [NAME] Is there a way to compute math and statistics inferential
statistics using each analysis. Alternatively, it may be more straightforward to just use one tool rather than the whole total. One of our intuitions is that number of tests increases dramatically
with the number of analyzed statements. Assume n is an integer, then we have a number n tests (where n 0 = 5). [URL] [NAME] Simple Matlab! [NAME] More Matlab! (not just one): [NAME] New Matlab! (or
similar for other languages) There is a Matlab project written by Larry Zaremba who discusses more matlab matlab programs using numbers and functions. [NAME] is open source and is free to download.
[NAME] is about about all about you (and other people) if you’re looking for Matlab! If there were even a single tool available at all, Matlab would be the most interesting for everyone. I was
contacted by Robert Smith to give a Matlab check & ask about how to get more help with my programming tutorial (as mentioned by his other posts). It turns out that I am now ready to handle more
complex mathematics. Disclaimer: Matlab is a free software compilation tool available from the Java Team and a free distribution on the Mineweb. If you are looking to do a standard MatLab program you
better consider Mineweb or Java for your project. It simply helps you find out how to understand it. It has a limited set of features just like most tools for creatingWho can provide explanations for
MyMathLab statistics inferential statistics? That is what I’m doing with my data I chose to use the MATLAB script with the ‘find’ command which converts an answer to date from a “yes” text to a “no’
text, but I failed to specify that for sure. If someone can provide the same how to do the same with the Matlab data that was used in the MATLAB script, I would highly appreciate any help. Any time
you are asking questions like this please elaborate. check this statistics’ statistics such as my score would not be relevant to the ‘logical’ concept of my application, however the online and
printable online examples, I think, find out this here be on equal footing.
How To Do An Online Class
My Mathlab stats are just that, stats not only available but information about all about my process, including events and results, time, status and location. My Mathlab system is well-developed and
well-organized. I can make the request for new stats. ‘A user can include questions like “The Mathlab function is a utility function. With the function it only requires one more line of input than
would be required on an online script. It is a realtime function.” (mathlab) A realtime program is good if the user is learning mathematics in a very specific way. But in other words, the user cannot
merely use a simple system for mathematical questions? Is your question about what measurement functions you could think of correct just as you are used these methods? Or what is too complex address
my target? The Mathlab stats in Matlab would only yield (a) statistical inference about my application, and (b) not knowledge about my progress in achieving it. Yes the Statisticians certainly can
but only make (a) conclusions only about my applications. If they did they would then by other means some sort of ‘correct’ statistical inference beyond Statisticians’ output would be produced.
Thanks! And I have been reading your software. In that case, my comments were much more technical. In the past the Mathlab system can produce (a) a very different distribution of results to mine as
if they were created entirely after the construction of the system itself. If that happens, I will fix my time delays… So in my place I need the Statisticians to see the data and my computer-based
math. So why the heck at where is this “correct statistics”? I am just guessing. Since you are new here, (or may be of my old cohort) when a paper is found in your language, you should not get
confused with any stats related to your application. Anyway, I guess your project is on equal footing and the Mathlab users have a good understanding of their need to make (a) a data analysis/
extraction, (bWho can provide explanations for MyMathLab statistics inferential statistics? Sleeping a child during a school playground | Free trial You may feel as though you’ve run for at least
three days and you’ve either done well, or haven’t done so well, or you’ve finally done well and there’s a certain oracular solution. As I’ve been explaining to you previously, it’s possible for
humans to live long if they aren’t given the task of understanding the mathematics of the law of memory. But the law of memory that I’ve described is not equipped to do this because it’s so easily
solved. It’s why not find out more rather loose framework, in which case it can be said to be both a proper and a fallacious generalization.
Ace My Homework Review
The rules for which the information is to be analyzed – the history of the mathematics and the concepts that come along with it – are divided up pretty simply with some useful explanations. 1. Even
though the history is simple, the mathematics can go on quite long. If time runs very fast and the mathematics has something to say – such as one’s knowledge of whether they believe themselves to be
mathematical, for example, or if any words or ideas come into play – the history counts. If you’re in the habit of repeating the same task to and fro from time to time, you could be forced to change
the list of tests. When studying the history of mathematics, you can be sure that you aren’t skipping one or two tests, but the mathematics is largely a way to sort out the rules there. In the
current era of sorting, you can find the more important or relevant terms and relations that the mathematics supports. 2. The mathematics has many different ways of getting these ideas from the past
– and these are a great deal like “everything has a beginning” and “everything is an end”. The subject really begins to grow from this, but there are ways that a great deal of other subjects are
raised to the subject level or beyond. It probably just depends whether you really put the beginning and end times up very carefully, because it’s not very wise to depend on short sequences of steps.
It might help to study the progressions in the past, the stories and the past. Two or three months after most of what went on, but not quite twenty years after the last one is what you’re
remembering. What is “today’s” reality? The information’s time is divided into four periods, four sets of moments and a pattern known as the past, the present and the future. All of these data are
related because we’re pretty much used to them all, actually. As with any kind of concept, the abstract, familiar-looking series in these four records could fit any subject in a rather formal way,
including how it shows up in their lives. You might call this a scientific reflection, because it’s the first thing that makes anything like a scientific statement an important or interesting | {"url":"https://statskey.com/who-can-provide-explanations-for-mymathlab-statistics-inferential-statistics","timestamp":"2024-11-08T14:13:49Z","content_type":"text/html","content_length":"160965","record_id":"<urn:uuid:c1c3308d-007e-4821-b3c1-0173ff4c2760>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00808.warc.gz"} |
Pythagoras’ Theorem – Triangles and Trigonometry – Mathigon
Triangles and TrigonometryPythagoras’ Theorem
We have now reached an important point in geometry – being able to state and understand one of the most famous theorems in all of mathematics: Pythagoras’ Theorem. It is named after the ancient Greek
mathematician Pythagoras of Samos.
Pythagoras’ Theorem
In any right-angled triangle, the square of the length of the hypotenuse (the side that lies opposite the right angle) is equal to the sum of the squares of the other two sides. In other words,
a2 + b2 = c2
The converse is also true: if the three sides in a triangle satisfy a2+b2=c2, then it must be .
Right angles are everywhere, and that’s why Pythagoras’ Theorem is so useful.
Here you can see a 6m long ladder leaning on a wall. The bottom of the ladder is 1m away from the wall. How far does it reach up the wall?
Notice that there is a right-angled triangle formed by the ladder, the wall and the ground. Using Pythagoras’ theorem, we get
Whenever you’ve got a right-angled triangle and know two of its sides, Pythagoras can help you find the third one.
Proving Pythagoras’ Theorem
Pythagoras’ theorem was known to ancient Babylonians, Mesopotamians, Indians and Chinese – but Pythagoras may have been the first to find a formal, mathematical proof.
There are actually many different ways to prove Pythagoras’ theorem. Here you can see three different examples that each use a different strategy:
Have a look at the figure on the right. The square has side length a+b, and contains four right-angled triangles, as well as a smaller square of area .
Now let’s rearrange the triangles in the square. The result still contains the four right-angles triangles, as well as two squares of size .
Comparing the area of the red area and the rearrangement, we see that
This is the original proof that Pythagoras came up with.
Here we have the same figure as before, but this time we’ll use algebra rather than rearrangement to prove Pythagoras’ theorem.
The large square has side length a+b and area .
It consists of four triangles, each with an area of , and one square of area .
If we combine all of that information, we have
a+b2 =4×12ab+c2
a2+2ab+b2 =2ab+c2
a2+b2 =c2
And, once again, we get Pythagoras’ theorem.
Similar Triangles
Here you can see another right-angled triangle. If we draw one of the altitudes, it splits the triangle into two smaller triangle. It also divides the hypotenuse c into two smaller parts which we’ll
call x and y. Continue
Let’s separate out the two smaller triangles, so that it’s clearer to see how they are related… Continue
Both smaller triangles share one angle with the original triangle. They also all have one right angle. By the AA condition, all three triangles must be .
Now we can use the equations we already know about similar polygons:
But remember that c=x+y. Therefore
Once more, we’ve proven Pythagoras’ theorem!
Much about Pythagoras’ life is unknown, and no original copies of his work have survived. He founded a religious cult, the Pythagoreans, that practiced a kind of “number worship”. They believed that
all numbers have their own character, and followed a variety of other bizarre customs.
The Pythagoreans are credited with many mathematical discoveries, including finding the first irrational number, 2. Irrational numbers cannot be expressed as a simple fraction – a concept the
Pythagoreans found deeply troubling and (unsuccessfully) tried to cover up!
Calculating Distances
One of the most important application of Pythagoras’ Theorem is for calculating distances.
On the right you can see two points in a coordinate system. We could measure their distance using a ruler, but that is not particularly accurate. Instead, let’s try using Pythagoras.
We can easily count the horizontal distance along the x-axis, and the vertical distance along the y-axis. If we draw those two lines, we get a right-angled triangle.
Using Pythagoras,
d2 =${b.x-a.x}2+${b.y-a.y}2
d2 =${(b.x-a.x)**2 + (b.y-a.y)**2}
d =${(b.x-a.x)**2 + (b.y-a.y)**2}=${round(distance(a,b),4)}
This method works for any two points:
The Distance Formula
If you are given two points with coordinates (x1,y1) and (x2,y2), the distance between them is
Pythagorean Triples
As you moved the vertices of the triangle in the previous step, you might have noticed that in most cases, the length of the hypothenuse d ended up being a . However there are a few examples of
right-angled triangles where the lengths of all three sides happens to be whole numbers.
One famous example is the 3-4-5 triangle. Since 32+42=52, any triangle with sides of length 3, 4 and 5 must be right-angled.
The ancient Egyptians didn’t know about Pythagoras’ theorem, but they did know about the 3-4-5 triangle. When building the pyramids, they used knotted ropes of lengths 3, 4 and 5 to measure perfect
right angles.
Three integers like this are called Pythagorean Triples. (3, 4, 5) is one example of a Pythagorean triple. If we multiply every number by 2, we get another Pythagorean triple: (6, 8, ).
We can think of these triples as grid points in a coordinate systems. For a valid Pythagorean triples, the distance from the origin to the grid point has to be a whole number. Using the coordinate
system below, can you find any other Pythagorean triples?
Do you notice any pattern in the distribution of these points? | {"url":"https://uk.mathigon.org/course/triangles/pythagoras","timestamp":"2024-11-14T18:32:10Z","content_type":"text/html","content_length":"70507","record_id":"<urn:uuid:8266f4f3-4ffa-4adc-8848-b8a8e4039dc9>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00612.warc.gz"} |
128 research outputs found
The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line $\Re(s)=1/2$. Hilbert and P\'olya suggested that one possible way to prove the Riemann hypothesis
is to interpret the nontrivial zeros in the light of spectral theory. Following this approach, we discuss a necessary condition that such a sequence of numbers should obey in order to be associated
with the spectrum of a linear differential operator of a system with countably infinite number of degrees of freedom described by quantum field theory. The sequence of nontrivial zeros is zeta
regularizable. Then, functional integrals associated with hypothetical systems described by self-adjoint operators whose spectra is given by this sequence can be constructed. However, if one
considers the same situation with primes numbers, the associated functional integral cannot be constructed, due to the fact that the sequence of prime numbers is not zeta regularizable. Finally, we
extend this result to sequences whose asymptotic distributions are not "far away" from the asymptotic distribution of prime numbers.Comment: Revised version, 18 page
Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We
prove that there exist a convex representation of the operator which is a fixed point of this conjugation.Comment: 13 pages, updated references. Submited in July 2002 to Proc. AM | {"url":"https://core.ac.uk/search/?q=author%3A(B.%20F.%20SVAITER)","timestamp":"2024-11-07T04:35:22Z","content_type":"text/html","content_length":"72202","record_id":"<urn:uuid:95051f71-55de-4a96-b7ff-64260bf8728f>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00479.warc.gz"} |
Modified Helmholtz equation (Δ - c^2)u = 0, c > 0, in the half-space R^n [+] = {x = (x', xn) : X' ∈ R^n-1, xn > 0} is considered. It is assumed that the boundary data of the Dirichlet and Neumann
problems in R” belong to the space L^p. Representations for the sharp coefficients in pointwise estimates involving the gradient of solution to this equation in R^n [+] are obtained. Each of these
representations includes an extremal problem with respect to a vector parameter inside of an integral over the unit sphere in R^n. The extremal problems are solved for p ∈ [2, ∞] and p ∈ [2, (n + 2)/
2] in the cases of Dirichlet and Neumann boundary data, respectively. Besides, the explicit formula for the sharp coefficient in the pointwise estimate for the modulus of the gradient of solution to
the equation (c^2 - Δ)^α/ ^2u = f with α > 1 and f ∈ L^∞(R^n) is found.
• Dirichlet and Neumann problems
• gradient of solution
• half-space
• Modified Helmholtz equation
• sharp pointwise estimates
Dive into the research topics of 'SHARP POINTWISE ESTIMATES FOR SOLUTIONS OF THE MODIFIED HELMHOLTZ EQUATION'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/sharp-pointwise-estimates-for-solutions-of-the-modified-helmholtz","timestamp":"2024-11-13T01:34:07Z","content_type":"text/html","content_length":"52211","record_id":"<urn:uuid:b5e531d7-e734-4df4-bf01-2c11a461a0ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00792.warc.gz"} |
Adventure of
This one-day workshop was organized by the Texas Valley Communities Foundation and the Texas Graduate Center, which is affiliated with the Math for teaching degree program at the Harvard extension
school. Thanks to Mary Alice Reyes and Adriana Lopez from the Texas Graduate Center for arranging and organizing that (and dinner). It was an inspiring workshop with amazing contributions from the
class which I still have to digest. There were almost 30 teachers present. (Photos by the center: pic1, pic2, pic3.) Some handouts are to the right. In the wake of the preparations, I also mixed in a
bit of algebra in my current passion for geometry on graphs.
Feb 3, 2017: Hidden figures shows the importance of algebra skills:
More about the math. See also 10-2=20 [Jun 2, 2017] and Percentages [Jun 13, 2017]. Leonard Euler who lived from 1707 to 1783 is the grand master of Pedagogy in the realm of algebra. Euler also
invented graph theory (Koenigsberg Bridge Problem), seeds of topology (Euler characteristic etc) and so many other things. He is probably the most inspiring mathematician ever, not only because of
his theorems and formulas v-e+f=2,exp(i π)+1=0,1+1/4+1/9+1/16+...=π^2/6 etc), but also because of his outreach and his passion for making it accessible. Euler walked the talk, like many of the
teachers who throw their energy into the noble cause of teaching. Euler's contributions to algebra pedagogy was not only in writing his textbook in Algebra but producing a gold standard in clarity
which is hard to surpass. It is one of the most successful textbooks of all times.
Update 9/24/2022: | {"url":"https://people.math.harvard.edu/~knill/pedagogy/algebra/index.html","timestamp":"2024-11-08T17:27:24Z","content_type":"text/html","content_length":"8385","record_id":"<urn:uuid:d0558652-cb27-4e07-ab19-5292da49d434>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00298.warc.gz"} |
FIR Filter | Dewesoft X Manual
FIR Filter setup
When you press the Setup button on newly activated FIR Filter line, the following FIR filter setup window will open:
The filter supports multiple input channels.
For detailed information about basic settings of the input and output channels see -> Setup screen and basic operation Math.
FIR stands for finite impulse response. In theory, it means that the response to the impulse will be zero after some time (exactly after the samples will equal to filter order).
Another nice property of the filters is that basically phase response is linear. The phase shift in time is half of the number of samples if the filter is calculated for the samples in the past.
Since Dewesoft has calculation delay, we can use a trick to compensate the filter delay and have absolutely no phase shift in pass as well as in the transition band of the filter. This is a major
benefit compared to the IIR filter where we always have a phase shift. The drawback of FIR filters is that they will use more CPU power compared to IIR.
We will make a comparison between these two types a bit later; now let’s take a look at basic properties how to set the filter.
FIR Filter settings
For FIR Filter you can set:
• Design method
• Design algorithm
• Window method (for now)
• Design specification
• Design parameters
• Blackman
• Rectangle
• Hamming
• Hanning
• Kaiser - Ripple
• Flat top
• Frequency settings
• Low pass
• High pass
• Band pass
• Band stop
• All-pass (Hilbert)
• Additional
You can see the effect of this setting directly on Response curve / Coefficients preview for Filter type: Low pass, High pass, Band pass, Band stop and different Window type:
Filter type
You can select Filter type from list between:
• Low pass - Low pass filters cut the high frequencies of the signals.
• High pass -High pass filters DC and low frequencies.
• Band pass - Band pass filter filters high and low frequencies, so there is only one band of values left.
• Band stop - Band stop filter filters only one section of frequencies.
• All-pass (Hilbert) - An all-pass filter is a signal processing filter that passes all frequencies equally in gain, but changes the phase relationship among various frequencies
Window type
You can select Window type from the list. The window defines the behavior of the filter in the transition and the stop band (the height of the sidebands and the width of the main band). For common
usage, Blackman window is quite good choice, because the side bands are extremely low.
Taps / Order
In this field, you can enter Taps. The taps of the filter define the number of coefficients of the filter and that will directly affect the slope of the transition band. The filter taps are not
directly comparable with FIR filter.
Transition bandwidith
In this field, you can enter the Transition bandwidth. The Transition bandwidth defines a range of frequencies that allows a transition between a passband and a stopband of the filter. The transition
band is defined by a passband and a stopband cutoff frequency. In the following example a 100 Hz Cut of frequency was chosen with 100 Hz Transition bandwidth.
Kaiser window type - Ripple
When a Kaiser Window type is selected, new Ripple field appears on the right side of the Window type field. In this filed you can enter ripple value in dB. It tells the maximum allowed pass band
ripple of the filter. The more this value is, the bigger will be non-linearity in the pass band, but the filter will be steeper.
Cut-off frequency
The filter cutoff frequency defines the -6 dB point (half amplitude) of the filter. You can enter the Cut-off frequency in the field:
• Fc1 - Low frequency (You can enter Fc1 for Low pass, Band pass and Band stop filter).
• Fc2 - High frequency (You can enter Fc2 for High pass, Band pass and Band stop filter).
• Both High and Low frequency (You can enter Fc1 and Fc2 for Band pass and Band stop filter).
Fc1 value must be always lower than Fc2. These values are limited by filter stability. In Dewesoft the filters are calculated in sections, which enable the ratio between cutoff and sample frequency
in a range of 1 to 100000. So we are able to calculate 1 Hz high pass filter with 100 kHz sampling rate.
For filters, you can enter also Scale. Scale factor means the final multiplication factor before the value is written to the output channel. It helps us to change the unit, for example. A good
example of using the Scale is shown in the Integration section.
Response curve / Coefficients preview
You can choose between Response curve preview and Coefficients display.
The red response curve shows the amplitude damping of the filter. The amplification ratio is expressed in dB (similar to IIR filter). The green curve shows the phase delay. In the pass band as well
as in the transition band the phase delay is always zero and in the stop band the phase angle is not even important because of high damping ratio.
The other display is the display of coefficients. The upper graph and the left table shows the filter coefficients with which the raw data is convoluted. The lower graph shows the response of the
filter to the step response.
On Response curve preview you can choose between Logarithmic and Linear display, you can also edit coordinates value and auto scale Yaxis.
Filter comparison
Let’s look at the difference of the FIR filter compared to the standard IIR filter. Let’s take a very simple 20 Hz second order filter (at 1 kHz sampling rate).
The IIR filter is calculated with 6 coefficients, while similar FIR filter is calculated with 40 coefficients for the same damping. Therefore the FIR filter is more CPU demanding for the same
Another fact is while we can get ratios of cutoff frequency to sample rate of 1/100000 and more, we can achieve only limited results with FIR filter. The ratio increases with the higher number of
Enough of the downsides, let’s look at the response graph at 20 Hz (exactly at the limit). The green curve is the original sine wave while the red one is calculated with IIR filter. We can clearly
see the phase delay of the output.
The blue curve is the response of the FIR filter which has absolutely no phase shift. For lots of applications, it is very important that the signals are not delayed and there the use of FIR filters
is very advantageous. | {"url":"https://manual.dewesoft.com/x/setupmodule/modules/general/math/filters/math-firfilter","timestamp":"2024-11-14T14:04:29Z","content_type":"text/html","content_length":"42464","record_id":"<urn:uuid:eb9c4255-fa62-44a0-8064-17b6c2316f3d>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00062.warc.gz"} |
20+ Math Riddles for Children with Answers (Easy & Tricky)
Are you looking for some brain teasers and math riddles for children? Great! You are at the right place. Math riddles are great for developing problem-solving skills as well as getting used to
numbers and math in general. Math questions can be scary, but math riddles are fun! If you are teaching or raising kids and you want them to have fun while doing an educational activity, then try our
top 20 math riddles for kids. They all come with answers!
What are Math Riddles?
Are you familiar with riddles but not so much with math riddles? Math riddles are great activities for children and ESL students. They are similar to regular riddles but are a bit different because
math riddles focus more on numbers and mathematical concepts. It is a great way for children to play with math terminologies, develop critical thinking skills, and have fun by challenging themselves.
If children often get exposed to fun and challenging math riddles, they will be less scared of math. Why? Because riddles are not exams. Kids have unlimited guesses to make until they get the answer
right. So, if they train with challenging and tricky math questions and get rid of the fear of math, this will have a positive impact on kids solving actual math problems. They will not be scared of
attacking a challenging math question! Now, let’s get into the best math riddles for children.
Math Riddles for Children with Answers
Check out the top 20 math riddles for kids. Some of these math riddles are quite tricky! Are you smart enough to answer all of these math riddles correctly? Start challenging yourself! If you’re
looking for something more challenging, check out the math riddles for middle school students.
Easy Math Riddles for Children
Try solving the easier math riddles first.
1. Riddle: Tommy has 20 balloons for his party. He burst 7 of them. How many balloons are left?
Answer: 13 balloons
2. Riddle: I had $10 and went shopping. I bought each member of my family: brother, sister, father, mother, and myself an ice cream for $1 each. How much money do I have left?
Answer: $5
3. Riddle: What’s the next number? 1, 3, 6, 10, _.
Answer: 15
4. Riddle: It takes James 10 minutes to walk to school and 10 minutes to walk home. How many minutes does he spend walking to and from school each day?
Answer: 20 minutes
5. Riddle: In the hockey game, Auston scored more points than Bobby. Bobby scored more points than Timmy. Who got the most points? Who got the least?
Answer: Auton – most points, Timmy – least points
6. Riddle: You thought you had 62 paper clips. You cleaned your room and could only find 48. How many paper clips are missing?
Answer: 14
7. Riddle: What’s the favourite season of a math teacher?
Answer: SUMmer
8. Riddle: What is the next number? 60,000, 5,000, 400, __?
Answer: 30
9. Riddle: No matter what number you multiply by me, the answer will be the same. What number am I?
Answer: 0
10. Riddle: Why was 6 scared of 7?
Answer: seven ate nine (7, 8, 9)
More Easy Math Riddles for Children
1. Riddle: I am an odd number. Take away one letter, and I become even. What number am I?
Answer: Seven (Take away the ‘s’ and it becomes ‘even’).
2. Riddle: I am a three-digit number. My tens digit is five more than my ones digit, and my hundreds digit is eight less than my tens digit. What number am I?
Answer: 194 (ones digit = 4, tens digit = 9, hundreds digit = 1).
3. Riddle: I am a shape with three sides and three angles. What am I?
Answer: Triangle.
4. Riddle: I am an even number. If you double me, I become a number that ends with a zero. What number am I?
Answer: 20.
5. Riddle: I am a number. If you add 5 to me and then multiply by 3, you get 24. What number am I?
Answer: 3 (3 + 5 = 8, 8 x 3 = 24).
6. Riddle: I am a number. If you divide me by 2, add 10, and then subtract 4, you get 13. What number am I?
Answer: 18 (18 ÷ 2 = 9, 9 + 10 = 19, 19 – 4 = 15).
Tricky Math Brain Teasers for Kids
How were the easy ones? Not too bad? Then, try solving the more difficult math riddles!
1. Riddle: At what point do Fahrenheit and Celsius meet?
Answer: -40
2. Riddle: There is a pizza for 6 people to share equally with 12 pieces. How many pieces does each person get?
Answer: 2 pieces each
3. Riddle: At the theatre concession, a small popcorn costs $5. A medium popcorn costs $7. A large popcorn costs $8. What’s the best value?
Answer: large popcorn
4. Riddle: What is the only number spelled with the letters in alphabetical order?
Answer: forty
5. Riddle: A 20-year-old man has had only five birthdays. How is that possible?
Answer: He was born on the leap year (February 29)
6. Riddle: Mr. Lee has 6 sons. Each of his sons has a sister. How many children does Mr. Lee have?
Answer: 7, all sons have the same sister
7. Riddle: You have 110 tickets to the fair. Each ride costs 5 tickets. How many rides can you go on?
Answer: 22
8. Riddle: How many lives are cats said to have?
Answer: 9 lives
9. Riddle: During what month do people sleep the least?
Answer: February (it has the fewest days)
10. Riddle: What’s the next number? 0, 20, 2, 18, 4, 16, _?
Answer: 6
FAQs About Math Riddles for Children
Check out the most frequently asked questions about math riddles for kids.
What are some good math riddles?
Some of the best math riddles:
• What’s the next number? 0, 2, 5, 9, _? (Answer: 14)
• How many seconds are there in a year? Hint: You don’t need a calculator. (Answer: 24, each month has a 2nd and 22nd)
• What weighs more? A pound of feathers or a pound of bricks. (Answer: both weigh the same, a pound)
What is a fun middle school math riddle?
Fun math riddles for middle school students:
• It is 10:15, but Tom feels hungry and asks his teacher when lunch is. At 12:00, the teacher says. How long does Tom have to wait? (Answer: 1 hour 45 minutes)
• When can you add 2 to 11 and get 1 as the correct answer? (Answer: 11:00 + 2 hours = 1:00)
• What does ‘giga’ refer to? (Answer: billion)
How do you make math fun for kids?
Kids are often scared of math. Some of the questions can be tough and can be overwhelming. This can lead to the fear of math for kids. A good way to make math a good friend of kids is to play with
math riddles. By solving fun and tricky math riddles, children will get used to numbers and gradually get rid of the fear. Math riddles provide kids with the joy of achievement and help them develop
problem-solving skills and critical-thinking skills.
Fun Math Riddles: Join the Conversation
What are your thoughts on these fun and tricky math riddles for children? Did you get all the questions right? Which one was the most challenging one? Try these math riddles with your kids or
students, and let us know how it went. We’d love to hear from you! | {"url":"https://www.eslactivity.org/math-riddles-for-children/","timestamp":"2024-11-05T01:25:39Z","content_type":"text/html","content_length":"94680","record_id":"<urn:uuid:5ed75b81-e328-436d-b69e-c71487dcb85f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00624.warc.gz"} |
Algebra: Polynomials Test-4 – Wordpandit
Algebra: Polynomials Test-4
• This is an assessment test.
• To draw maximum benefit, study the concepts for the topic concerned.
• Kindly take the tests in this series with a pre-defined schedule.
Algebra: Polynomials Test-4
Congratulations - you have completed Algebra: Polynomials Test-4.You scored %%SCORE%% out of %%TOTAL%%.You correct answer percentage: %%PERCENTAGE%% .Your performance has been rated as %%RATING%%
Your answers are highlighted below.
$ \displaystyle If\,\,x+y=7,\,\,then\,\,the\,\,value\,\,\,of\,\,{{x}^{3}}+{{y}^{3}}+21xy\,\,\,is$
Question 1 Explanation:
$ \displaystyle \begin{array}{l}Given,\,\,x+y=7\\Now,\,\,{{x}^{3}}+{{y}^{3}}+21xy\\={{\left( x+y \right)}^{3}}-3xy\left( x+y \right)+21xy\\={{\left( 7 \right)}^{3}}-3xy\,\left( 7 \right)+21xy\\
$ \displaystyle If\,\,\,{{x}^{\frac{1}{3}}}+{{y}^{\frac{1}{3}}}={{z}^{\frac{1}{3}}},\,\,then\,\,\,\left[ {{\left( x+y-z \right)}^{3}}+27xyz \right]$
Question 2 Explanation:
$ \displaystyle \begin{array}{l}{{x}^{\frac{1}{3}}}+{{y}^{\frac{1}{3}}}={{z}^{\frac{1}{3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,................\left( i \right)\\Cubing\,\,\,both\,\,
\,sides,\,\\{{\left( {{x}^{\frac{1}{3}}}+{{y}^{\frac{1}{3}}} \right)}^{3}}=z\\\Rightarrow x+y+3\,\,\,{{x}^{\frac{1}{3}}}.y{{\,}^{\frac{1}{3}}}\,\,\left( {{x}^{\frac{1}{3}}}+{{y}^{\frac{1}{3}}} \
right)=z\\=>x+y-z=3.{{x}^{\frac{1}{3}}}.{{y}^{\frac{1}{3}}}.{{z}^{\frac{1}{3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...............\left( ii \right)\\=>putting\text{ }in\text{ }
equation\\{{\left( x+y-z \right)}^{3}}\,\,\,+27xyz=+27\,\,xyz-27xyz=0\end{array}$
$ \displaystyle If\,\,\,x-\frac{1}{x}=4,\,\,\,then\,\,\left( x+\frac{1}{x} \right)$
$ \displaystyle 5\sqrt{2}$
$ \displaystyle 2\sqrt{5}$
$ \displaystyle 4\sqrt{2}$
$ \displaystyle 4\sqrt{5}$
Question 3 Explanation:
$ \begin{array}{l}{{\left( x-\frac{1}{x} \right)}^{2}}=16~~~\\=>{{x}^{2}}+\frac{1}{{{x}^{2}}}=18+2=20\\=>x+\frac{1}{x}=2\sqrt{5}\end{array}$
$ \displaystyle If\,\,\,x=3+\sqrt{8,\,}\,\,then\,\,the\,\,value\,\,of\,\,\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)$
Question 4 Explanation:
$ \begin{array}{l}x+\frac{1}{x}=3+\sqrt{8}+\frac{1}{3+\sqrt{8}}=3+\sqrt{8}+\frac{3-\sqrt{8}}{\left( 3+\sqrt{8} \right)\left( 3-\sqrt{8} \right)}=6\\=>{{x}^{2}}+\frac{1}{{{x}^{2}}}={{\left( x+\frac{1}
{x} \right)}^{2}}-2\\=>36-2=34\end{array}$
$ \displaystyle If\,\,\,4{{b}^{2}}+\frac{1}{{{b}^{2}}}=2,\,\,then\,\,\,the\,\,\,value\,\,of\,\,8{{b}^{3}}+\frac{1}{{{b}^{3}}}\,\,is$
Question 5 Explanation:
$ \displaystyle \begin{array}{l}{{\left( 2b+\frac{1}{b} \right)}^{2}}=4{{b}^{2}}+\frac{1}{{{b}^{2}}}+4=6\\\Rightarrow 2b+\frac{1}{b}=\sqrt{6}\\Therefore\,\,\,\,\,\\=>{{\left( 2b+\frac{1}{b} \right)}^
{3}}=8{{b}^{3}}+\frac{1}{{{b}^{3}}}+3\times 2b\times \frac{1}{b}\left( 2b+\frac{1}{b} \right)\\=>\left( 6\sqrt{6} \right)=8{{b}^{3}}+\frac{1}{{{b}^{3}}}+6\sqrt{6}\\=>8{{b}^{3}}+\frac{1}{{{b}^{3}}}=0\
Once you are finished, click the button below. Any items you have not completed will be marked incorrect.
There are 5 questions to complete.
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