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Zero-Knowledge Knowledge
Table of Contents
Zero-knowledge (ZK) technology is mentioned a lot in crypto these days. But all too often scant technical detail is provided. And broad claims are made alongside vague appeals to ZK somehow solving
all the world's ills.
A column a few weeks ago worked through how the Privacy Pools protocol falls short of its design goals (building a regulation-compliant mixer on Ethereum) because it still requires one component of
the ecosystem to break the law (the relayers). That protocol, and on-chain anonymization services in general, rely on ZK proofs.
Privacy Pools & New Attacks
The authors of a new protocol called “Privacy Pools” envision an ecosystem of related services growing up around this tool to enable honest users to still achieve on-chain privacy. But this approach
also introduces new attack vectors.
So here we are going to explore how these things work on a semi-technical level and sketch out some limitations of what ZK can and cannot achieve.
The Basics
A ZK proof is a way of proving you know something without revealing the details. Conceptually this is like showing you know a secret without having to reveal that secret. In the real world one might
prove they know the combination to a safe by retrieving the contents. That works and, if you do not reveal the combination, and convey no further information to observers.
But this does not work as well online. You would need to transmit the combination over the internet where someone could – and eventually would – steal it. What we want instead is a scheme where the
secret itself never travels over the wire.
So now let us pretend we have a scheme for factoring large numbers quickly. Factoring underlies an awful lot of internet security and discovering such a scheme would dramatically change the world. So
it is natural that you would not want to transmit such a scheme over the internet. But is there a way to prove this?
Yes. A fairly straightforward ZK proof process presents itself. Set up a website where you factor large numbers. People can come to your site and try it. The more the site works the more people
believe you can do this. What we have here is an interactive proof. Interactive because there is a prover (the website) and a verifier (the user). The more numbers are tried the higher the confidence
our scheme works.
Proofs in the Crypto Playground: A Layman’s Decoder Ring
Have you ever sat at a dinner party, nodded along to discussions about cryptographic proofs, and prayed no one would ask for your thoughts on zero-knowledge proofs? Well, grab a glass of your
favorite drink and let’s dive into this riveting world.
Interactive proofs are fine. But what we really want is a non-interactive process. Some sort of proof that exists outright and does not rely on a back-and-forth conversation.
Non-Interactive ZK Proofs
A technique known as the Fiat-Shamir Heuristic provides a method for converting interactive proofs into non-interactive ones. The essential intuition is that it might be possible to bluff your way
through a few rounds of verification given chosen tests cases – but it will not be possible to do this if the given examples are large and random.
We can now construct a non-interactive proof fairly easily: we just show our scheme works given a single randomly chosen large example. This can be as simple as using the hash of the current time as
the test case. We do not need multiple rounds of verification if the example is random and hard enough. It is sufficient to simply publish the test case and answers.
One other common sort of non-interactive ZK proof is a digital signature. Most crypto systems employ signatures in one way or another. And publishing the message "this is a zk proof" signed by the
private keys to a given wallet is sufficient to prove you hold those keys. That serves as a non-interactive proof because there is no way to generate the signature without the private keys.
Every time you sign a transaction you are producing a (simple) type of ZK proof.
At this point we have almost enough information to understand how Monero works. The one little bit we need to add is the concept of a ring signature. Think of a key ring where are many keys. Ring
signatures are digital signature schemes where we get the same signature for all keys in the ring.
So imagine if there were a dozen different private keys for each Bitcoin address. Those keys would form a ring and you would not be able to tell which one was used to sign any individual transaction.
Monero works by relying on a process where individual transactions can have come from a large – possibly very large – set of different keys and it is not possible to tell which were used. That is the
core concept. Yes each transfer can be whittled down to some set of addresses. But assume that set has 10 possibilities and we transfer tokens 10 times. Now there are 10^10 = 10 billion
possibilities. If the ring has a million possibilities you can easily exceed the number of protons in the earth.
Yes there are a few more technical details. But the anonymity in Monero is built around this mechanism where every transaction is equally likely to have come from a large number of addresses and it
is mathematically impossible to distinguish among them.
And this presents us with a good example of the sorts of limitations that exist on what ZK. All the keys on the ring need to be equally likely for this thing to work. If there is some kind of
asymmetry it can be attacked. So it is not going to be possible to build something on top of Monero-like schemes that offer a sort of partial privacy because the signatures must be indistinguishable.
Sure someone can probably cook up a Privacy-Pools-adjacent tool for Monero. But it must also compromise the anonymity if it works at all. That is fine. But it shows that ZK cannot do everything we
might want. That's how the world works and is not terribly shocking.
Zcash & Tornado Cash
These two are different from Monero but overlap a lot in the internal design. In both cases a user starts by generating some secret information. They then submit a deposit (in Zcash they initiate a
transfer) alongside information derived from the secret. In Tornado Cash the user can then submit a different piece of information derived from their secret to withdraw. That information is a ZK
proof that they made a corresponding deposit.
The essential feature here is that computing these pieces of information from the secret is easy – but determining which pairs share a common secret is hard. Think of this like factoring: it is easy
to multiply numbers together to check a given set of factors is correct but hard to reverse the process. Possession of this second secret piece of information serves as proof you made a deposit.
Now think of a shop that sells sandwiches. You wait in line and pay. But rather than giving you a receipt they have you pick a piece of paper from a bowl with a 100 digit random number written on it.
To pick up your sandwich later you present this random number. They can easily check if it is one of the random numbers they printed out. And the shop can easily check if this number has been
redeemed already. But if the bowl contains only 10o different 100-digital random numbers there is essentially zero chance you can guess one of them outright. Possession of the number is a ZK proof
you made a sandwich purchase even though it has no connection to your payment.
For Zcash, rather than submitting a withdrawal request the user, roughly, is able to compute the private key for the recipient address from their secret. And for the same reasons as Tornado, it is to
compute that key but hard to work out which pairs of address keys are connected.
Again, immediately, we can see some limitations on what ZK can do. In Zcash fees are paid by the sender. There may well be some way to build a partial privacy scheme where you can prove one receipt
did not come from a given transfer. Maybe. But in a system like Tornado Cash where the withdrawal process requires gas to be paid we have an unavoidable problem. Someone has to fund the gas.
As we previously discussed retrofitting partial privacy on to Tornado cannot produce a compliant system because it still requires relayers (who are actively prosecuted) to anonymize gas. And in fact
we can find another limitation on what ZK can do here. In a system like Ethereum where all computation is public we cannot build something like Zcash. Sure we can design a mixer where the recipient
address is derivable from a secret fed in by the sender. But the deposit and withdrawal will be publicly tied together.
Why? Because everyone observing the smart contract computations can follow along as the "mixer" initiates a transfer to the destination address. That is how Ethereum works. In Zcash there is no such
visible transfer – the recipient simply holds the keys. You are not going to be able to retrofit this sort of process onto Ethereum as it exists today – you will need to modify the protocol to fix
At the same time this enables us to see that ZK does not enable certain sorts of L2 scaling solutions. Within a given L2 scaling may or may not be possible. But across L2s – in exactly the way a16z
talks about in the first problem of their Nakamoto Challenge – this cannot scale beyond Ethereum's native L1 capacity.
This is easy to see. The only way to settle across L2s is to complete a transfer on the L1 chain. Fine. But can presenting a ZK proof from L2 X on L2 Y speed this all up? No. Let us assume we have
some sort of ZK proof scheme that is portable across L2s. And now we present a proof we own a token on X to a DEX on Y.
If the DEX does not trust us and waits for the X->Y settle on mainnet clearly we did not scale. So let us assume the DEX trusts us and allows a sale of the token now while waiting for the mainnet
transfer to settle. Maybe we scaled.
But, as but one example of trouble, what happens if gas on mainnet skyrockets far in excess of what we promised the DEX. What if gas exceeds the entire balance we tried to sell. Now what happens? We
can wait until gas comes down. In that case every subsequent transfer of our token is stalled. Either the token must be frozen pending resolution of this problem or all subsequent transfers are at
risk of being reverted.
This can go on forever where nobody every achieves finality. Or we can have some sort of timeout. If the mainnet transfer does not complete before the timeout we give up and the DEX reverts the
trade. Now consider what just happened. We may be able to scale the number of proposed transactions – but throughput is the number of transactions divided by the time taken to complete them. The
timeout is our denominator. We did not scale the number of settled transactions – we scaled the number of proposed transactions.
ZK cannot fix this because finality on the L1 requires gas and future gas can be anything. We can approximate our way around the problem with timeouts and by extending credit a bit across L2s – the
DEX is extending us credit by allowing immediate trading when the L1 settlement is pending – but we cannot scale in the limit.
Again compromises are likely possible and systems that often scale can surely be built. But the ideal of an ecosystem of L2s that reliably process a huge multiple of the L1's capacity is out of reach
in a say ZK technology cannot fix. | {"url":"https://www.blockhead.co/2023/10/06/zero-knowledge-knowledge-monero/","timestamp":"2024-11-10T15:04:39Z","content_type":"text/html","content_length":"127452","record_id":"<urn:uuid:9916a9c9-8a20-4563-a840-13115834f1a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00708.warc.gz"} |
Discover The EQUATION Behind The CURVE - Unveiling The Mystery
Graph shows which equation generates the curve?
Sure! Here's a brief introduction for your blog post:
Welcome to Warren Institute! Have you ever wondered which equation could generate a specific curve? In the world of Mathematics education, understanding how different equations produce various graphs
is essential. In this article, we will explore the relationship between equations and graphs by analyzing a specific curve. By the end, you'll have a deeper understanding of how mathematical
equations are translated into visual representations. Let's dive into the fascinating world of graph generation!
Understanding the Graph and Equation Relationship
Understanding the relationship between a graph and its corresponding equation is crucial in mathematics education. In this section, we will delve into the process of determining which equation could
generate the curve in the given graph.
Analyzing Key Features of the Graph
To determine the equation that could generate the curve in the graph, it's essential to analyze the key features such as the intercepts, slope, curvature, and any other distinctive characteristics of
the curve. This analysis will provide valuable insights into the nature of the equation that best fits the graph.
Exploring Potential Equations
Once the key features of the graph have been identified, it's time to explore potential equations that could produce a similar curve. This may involve considering various types of functions such as
linear, quadratic, exponential, or trigonometric functions, and analyzing how their properties align with the graph.
Applying Mathematical Techniques to Find the Equation
With a thorough analysis of the graph and exploration of potential equations, it's time to apply mathematical techniques such as regression analysis, solving systems of equations, or utilizing
calculus principles to determine the specific equation that accurately represents the given graph. This process involves careful manipulation and analysis of mathematical concepts to arrive at the
most suitable equation.
frequently asked questions
Which equation could generate the curve in the graph below?
The equation that could generate the curve in the graph below is y = sin(x).
How can we determine the equation that corresponds to the given graph?
We can determine the equation that corresponds to the given graph by identifying the key characteristics of the graph such as the slope, intercepts, and any other relevant information. Then, we can
use this information to form the equation in the appropriate form (e.g. slope-intercept form, point-slope form, etc.).
What methods can be used to find the equation that matches a given graph?
There are several methods that can be used to find the equation that matches a given graph, including graphing, analyzing key points, using slope-intercept form, and applying transformations.
What mathematical techniques are helpful in identifying the equation for a given curve on a graph?
Calculus and algebraic manipulation are helpful in identifying the equation for a given curve on a graph.
Can you explain the process of deducing the equation from a given graph?
To deduce the equation from a given graph, you can start by identifying key points such as intercepts, maxima, minima, and slope. Then, using these points, you can determine the corresponding
mathematical relationship, which may involve linear, quadratic, exponential, or other functions. Finally, apply algebraic techniques to derive the specific equation that best fits the data.
In conclusion, the exploration of equations and their graphical representations is a vital component of mathematics education. By analyzing the curve in the graph, we have delved into the process of
determining which equation could generate it. This exercise not only enhances our understanding of algebraic concepts but also sharpens our analytical skills. Encouraging students to engage in
similar activities can foster a deeper appreciation for the intricate relationship between equations and their corresponding graphs, thereby fortifying their mathematical proficiency.
If you want to know other articles similar to Graph shows which equation generates the curve? you can visit the category General Education. | {"url":"https://warreninstitute.org/which-equation-could-generate-the-curve-in-the-graph-below/","timestamp":"2024-11-06T01:05:31Z","content_type":"text/html","content_length":"102685","record_id":"<urn:uuid:177169dd-017d-4822-958c-233c7aa027bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00273.warc.gz"} |
Course Catalog - 2024-2025
ELEC 589 - NEURAL COMPUTATION
Long Title: NEURAL COMPUTATION
Department: Electrical & Computer Eng.
Grade Mode: Standard Letter
Language of Instruction: Taught in English
Course Type: Lecture
Credit Hours: 3
Must be enrolled in one of the following Level(s):
Description: How does the brain work? Understanding the brain requires sophisticated theories to make sense of the collective actions of billions of neurons and trillions of synapses. Word theories
are not enough; we need mathematical theories. The goal of this course is to provide an introduction to the mathematical theories of learning and computation by neural systems. These theories use
concepts from dynamical systems (attractors, oscillations, chaos) and concepts from statistics (information, uncertainty, inference) to relate the dynamics and functions of neural networks. We will
apply these theories to sensory computation, learning and memory, and motor control. Students will learn to formalize and mathematically answer questions about neural computations, including “what
does a network compute?”, “how does it compute?”, and “why does it compute that way?” Prerequisites: knowledge of calculus, linear algebra, and probability and statistics. Graduate/Undergraduate
Equivalency: ELEC 489. Mutually Exclusive: Cannot register for ELEC 589 if student has credit for ELEC 489. | {"url":"https://courses.rice.edu/courses/!SWKSCAT.cat?p_action=CATALIST&p_acyr_code=2025&p_crse_numb=589&p_subj=ELEC","timestamp":"2024-11-15T00:27:15Z","content_type":"text/html","content_length":"8181","record_id":"<urn:uuid:681fb0ea-3bd8-431e-8c34-ff0e5d0fd41e>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00377.warc.gz"} |
Physics Formalism Helmholtz Matrix to Coulomb Gage
Original Research Article
Article volume = 2021 and issue = 2
Pages: 172–178
Article publication Date: November, 1, 2021
You can download PDF file of the article here: Download
Visited 319 times and downloaded 174 times
Physics Formalism Helmholtz Matrix to Coulomb Gage
Rajan Iyer
Environmental Materials Theoretical Physicist, Department of Physical Mathematics Sciences Engineering Project Technologies, Engineeringinc International Operational Teknet Earth Global, Tempe,
Arizona, United States of America
Iyer Markoulakis Helmholtz Hamiltonian metrics have been gauged to Coulombic Hilbert metrics, representing Gilbertian and Amperian natures of electromagnetic fields from mechanics of vortex
rotational fields acting with gradient fields, typically in zero-point microblackhole general fields, extending to vacuum electromagnetic gravitational fields gauge. This ansatz general gaging helps
to properly isolate field effects with physical analyses – mechanical, electric, magnetic components within the analytical processes. Vacuum gravitational fields and the flavor Higgs-Boson matter
inertial gravitational fields have been thus quantified extending to stringmetrics constructs matrix showing charge asymmetry gauge metrics. Physical Analysis with applications to particle physics,
Quantum ASTROPHYSICS, as well as grand unification physics have been advanced. Particle physics gauge matrix pointing to Dirac seas of electrons, monopoles with positrons, electron-positron
annihilation leading to energy production, and the relativistic energy generating matter provided literature correlations. Quantum astrophysics extending gauge metrix analyzes of superluminal profile
of signals velocity generating electron-positron chain like “curdling” action validates formalism with physics literature of electron-photon observed oscillatory fields configurations. Mechanism of
creation of neutrino antineutrino pair orthogonal to electron positron “curdling” planes, that may lead to formation of protonic hydrogen of stars or orthogonally muon particles. These proposals will
help to explain receding or fast expanding universe with the dark matter in terms of flavor metrics versus gauge associating metrics fields. Vacuum and gravitational monopoles, that are
representation of compressed mass out of vortex Helmholtz decomposition fields have been interpolated to energy generation via electron positron monopole particles at cosmos extent of infinity.
Quantum, Electrons-positrons monopoles, Matrix Algebra, Amperian Gilbertian fields, Switches, Modeling.
• [1] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., 20 (1967),197–228.
• [2] Y. Yao, Y. J. Cho, Y. C. Liou, R. P. Agarwal, Constructed nets with perturbations for equilibrium and fixed point problems, J. Inequal. Appl, 2014 (2014), 14 pages.
• [3] B. O’Neill, Semi-Riemannian geomerty with applications to relativity, Academic Press, London, (1983).
• [4] F. Wilczek, Physics in 100 years, Physics Today, Volume 69, Issue 4, (2016).
• [5] P. Frampton, Gauge Field Theories (3rd ed.). Wiley-VCH. (2008).
• [6] C. Becchi, Introduction to Gauge Theories, arXiv preprint hep-ph/9705211, (1997).
• [7] J.D. Jackson, From Lorenz to Coulomb and other explicit gauge transformations, American Journal of Physics 70, (2002).
• [8] G. Svetlichny, Preparation for Gauge Theory, arXiv preprint math-ph/9902027, (1999).
• [9] S. Cremonesi, A. Hanany, N. Mekareeyab and A. Zaffaronic, Coulomb branch Hilbert series and Hall-Littlewood polynomials, Journal of High Energy Physics, (2014).
• [10] E. I. Guendelman, D. Singleton, Scalar gauge fields, Journal of High Energy Physics,no. 5, 1-13, (2014).
• [11] L. Randall, New Mechanisms of Gauge-Mediated Supersymmetry Breaking, High Energy Physics, Nuclear Physics – Phenomenology Theory, Section B, 495, 1, 37-56, (1997).
• [12] E. Markoulakis, A. Konstantaras, J. Chatzakis, R. Iyer and E. Antonidakis, Real time observation of a stationary magneton, Results in Physics, 15, 102793, (2019).
• [13] R. Iyer and E. Markoulakis, Theory of a superluminous vacuum quanta as the fabric of Space, Physics & Astronomy International Journal, 5(2), 43-53, (2021).
• [14] R. Iyer, Problem Solving Vacuum Quanta Fields, International Journal of Research and Reviews in Applied Sciences, 47, Issue 1, PP. 15-25, (2021).
• [15] R. Iyer and M. Malaver, Proof Formalism General Quantum Density Commutator Matrix Physics, Physical Sciences & Biophysics Journal, 5(2), 000185, (2021).
• [16] J. Zinn-Justin and R. Guida, Gauge invariance, Scholarpedia, 3(12):8287, (2008).
• [17] Iyer, Rajan. Physics formalism Helmholtz matrix to Coulomb gage. (2021).
• [18] A. Schwarz, Topological quantum field theories, arXiv preprint hep-th/0011260, (2000).
• [19] A. K. H. Bengtsson, Structure of higher spin gauge interactions, Journal of Mathematical Physics 48, 072302, (2007).
• [20] A. K. H. Bengtsson, Towards Unifying Structures in Higher Spin Gauge Symmetry?, Symmetry, Integrability and Geometry: Methods and Applications, SIGMA 4, 013, 23 pages, (2008).
• [21] V. Fock, Z. Phys. M.C. Reed, B. Simon, Methods of Modern Mathematical Physics, Volume II”, Page 328, Academic Press (1975).
• [22] K. T. McDonald, The Helmholtz Decomposition and the Coulomb Gauge, Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, April 17, (2008).
• [23] G. Arias-Tamargo, A. Bourget, A. Pini and D. Rodríguez-Gómez, Discrete gauge theories of charge conjugation, Nuclear Physics B, Volume 946, September (2019).
• [24] E. Giorgi, The Carter tensor and the physical-space analysis in perturbations of Kerr-Newman spacetime, General Relativity and Quantum Cosmology, Mathematical Physics Analysis of PDEs, arXiv
preprint arXiv:2105.14379, (2021).
• [25] D. Hafner and J-P. Nicolas, Scattering of Massless Dirac Fields by a Kerr Black Hole, Reviews in Mathematical Physics, Vol. 16, No. 01, pp. 29-123, (2004).
• [26] S. Weinberg, Lectures on Quantum Mechanics - 2nd Edition, Cambridge University Press, (2013).
• [27] K. Jones-Smith, about identifying quasi-particles using non-Hermitian quantum mechanics, pp. 1-14, using PT quantum mechanics, Philosophical Transactions Mathematical, Physical and
Engineering Sciences, Vol. 371, No. 1989, Published by: Royal Society, (2013).
• [28] S. Hossenfelder, Interpretation of Quantum Field Theories with a Minimal Length Scale, Phys. Rev. D, 73, 105013, (2006).
• [29] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University, (2014).
• [30] P. A. Maurice Dirac, Quantised singularities in the electrovacuum field, Proc. R. Soc. Lond.
• [31] C. M. Bender, PT symmetry, In quantum and classical physics, WORLD SCIENTIFIC PUBLISHING, (2019).
• [32] D. Halliday, R. Resnick and J. Walker: Fundamentals of Physics: Extended, 11th Edition, 1456 pages, (2018).
• [33] R. Eisbeg, Quantum Physics of Atoms, Quantum physics of atoms, molecules, solids, nuclei, and particles M, Second Edition, (1985).
• [34] V. d. Heuvel EP, On the Precession as a Cause of Pleistocene Variations of the Atlantic Ocean Water Temperatures, Geophysical Journal International, 11 (3): 323–336, (1966).
• [35] A. Berger, M. F. Loutre and J. L. Mélice, Equatorial insolation: from precession harmonics to eccentricity frequencies (PDF), Clim. Past Discuss, 2 (4): 519–533, (2006).
• [36] Hamilton, Andrew JS. General Relativity, Black Holes, and Cosmology. Unpublished. Online at http://jila. colorado. edu/ ajsh/courses/astr3740_17/texts. html (2018).
• [37] R. N. Iyer, Absolute Genesis Fire Fifth Dimension Mathematical Physics. Engineeringinc. com International Corporation (2000).
• [38] C. M. Bender, PT Symmetry: In Quantum and Classical Physics, WORLD SCIENTIFIC PUBLISHING, (2019).
• [39] D. Kaiser, Physics and Feynman’s Diagrams: In the hands of a postwar generation, a tool intended to lead quantum electrodynamics out of a decades-long morass helped transform physics.
American Scientist 93, no. 2, 156-165 (2005).
• [40] C. M. Bender, PT Symmetry: In Quantum and Classical Physics, WORLD SCIENTIFIC PUBLISHING, (2019).
• [41] Iyer, Rajan. Physics formalism Helmholtz matrix to Coulomb gage.(2021).
• [42] K. A. Michalski and M. M. Mustafa. On the computation of hybrid modes in planar layered waveguides with multiple anisotropic conductive sheets. Proceedings of the Royal Society A:
Mathematical, Physical and Engineering Sciences 474, no. 2218: 20180288 (2018).
• [43] Passarino, Giampiero, Christian Sturm, and Sandro Uccirati. Higgs pseudo-observables, second Riemann sheet and all that. Nuclear Physics B 834, no. 1-2: 77-115 (2010).
• [44] M. Malaver, H. Daei Kasmaei, R. Iyer, Sh. Sadhukhan and A. Kar. A theoretical model of dark energy stars in Einstein-Gauss-Bonnet gravity. arXiv preprint arXiv:2106.09520 (2021).
• [45] I. M. Benn, T. Dereli and R. W. Tucker.Gravitational monopoles with classical torsion. Journal of Physics A: Mathematical and General 13, no. 10: L359 (1980).
• [46] Cho, Y. M. Theory of gravitational monopole. In Proceedings of the XXVth International Conference on High Energy Physics, (1990).
• [47] L. D.Lantsman and V. N. Pervushin. Monopole vacuum in non-Abelian theories. Physics of Atomic Nuclei 66, no. 7: 1384-1394 (2003).
• [48] S. Tesh, Flash Physics: Sandra Faber wins Gruber prize, transforming magnetic monopoles, vacuum scattering revealed. COSMOLOGY NEWS, https://physicsworld. com/a/
flash-physicssandra-faber-wins-gruber-prizetransforming- magnetic-monopoles-vacuum-scattering-revealed. Sarah Tesh is featuring editor of Physics World
Cite this article as:
• Rajan Iyer, Physics Formalism Helmholtz Matrix to Coulomb Gage, Communications in Combinatorics, Cryptography & Computer Science, 2021(2), PP.172–178, 2021 | {"url":"http://cccs.sgh.ac.ir/ShowArticle?ArticleID=23","timestamp":"2024-11-04T13:30:32Z","content_type":"text/html","content_length":"29001","record_id":"<urn:uuid:748341e7-e6f7-4199-bdb1-1fa4bd783209>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00812.warc.gz"} |
Next: Kinematics of water-bottom Multiples Up: Alvarez: Multiples in image Previous: Alvarez: Multiples in image
Attenuation of multiples in the image space is attractive because prestack wave-equation migration accurately handles the complex wave propagation of primaries. In subsurface-offset-domain
common-image-gathers (SODCIG) the primaries are imaged at zero subsurface offset at the depth of the reflector if migrated with the correct velocity. Correspondingly, in angle-domain
common-image-gathers (ADCIG) the primaries are imaged with flat moveout. Attenuation of multiples in image space depends on the difference in residual moveout between the primaries and the multiples,
either in SODCIGs or ADCIGs Alvarez et al. (2004); Hargreaves et al. (2003); Sava and Guitton (2003). Understanding how wave-equation migration maps the multiples into SODCIGs and ADCIGs is therefore
of paramount importance in order to design a proper strategy to attenuate the multiples in the image space.
Non-diffracted water-bottom multiples from a flat or dipping water-bottom are imaged as primaries. Thus, if the migration velocity is that of the water, they are mapped to zero subsurface-offset in
SODCIGs. Consequently, in ADCIGs, these multiples exhibit flat moveout just as primaries do Alvarez (2005). In the usual case of migration with velocities faster than water velocity, these multiples
are mapped to subsurface offsets with the opposite sign with respect to the sign of the surface offsets. I will analytically show the moveout curve of these multiples in SODCIGs and ADCIGs.
Water-bottom diffracted multiples, on the other hand, even if from a flat water-bottom, do not migrate as primary reflections Alvarez (2005). That is, they do not focus to zero subsurface offset even
if migrated with the water velocity. Obviously this happens because at the diffractor the reflection is not specular. I will show that these multiples migrate to both positive and negative subsurface
offsets in SODCIGs depending on the relative position of the diffractor with respect to the receiver (for receiver-side diffracted multiples).
The next section presents a general formulation for computing the kinematics of diffracted and non-diffracted water-bottom multiples for both SODCIGs and ADCIGs. The following section then looks in
detail at the special case of flat water-bottom where the equations simplify and some insight can be gained as to the analytical representation of the residual moveout of the multiples in both
SODCIGs and ADCIGs. The next section presents a similar result for multiples from a dipping water-bottom. Although the equations are more involved and difficult to encapsulate in one single
expression than those for the flat water-bottom, I show that we can still compute the image space coordinates of both the diffracted and non-diffracted multiples in terms of their data space
coordinates. The last section discusses some of the implications of the results and the possibility that they can be used to attenuate the multiples in the image space. Detailed derivation of all the
equations is included in the appendices.
Next: Kinematics of water-bottom Multiples Up: Alvarez: Multiples in image Previous: Alvarez: Multiples in image Stanford Exploration Project | {"url":"https://sepwww.stanford.edu/data/media/public/docs/sep123/gabriel1/paper_html/node1.html","timestamp":"2024-11-10T19:01:55Z","content_type":"text/html","content_length":"6854","record_id":"<urn:uuid:c0931433-d58a-4f2f-8333-8b6136b3a8a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00437.warc.gz"} |
3rd Question is regarding with Drivers for my HP PC: Usually I'm using 'Auslogics Driver-Updater to keep All my PC -Drivers Up-to-Date all the Time... but, but at the same time by the way I've used
to use 'PC Help Soft Driver Updater and 'Driver- Genius as well other softwares for keeping my PC Drivers up-to-date. And now the ''Question'' is: Are There any Negative effects or affects to break
my PC 'Stability or to break my PC Performance if am' I using more then one company Driver-Updaters - Software for only one PC?, or there is no need to be afraid or no need to worry of any negative
effects to break my PC 'Stability/Performance and I Can use as many as I want... and as many as I wish 'Driver-Updaters I mean from different companies 'Software for my HP PC...?
I updated Windows 10 yesterday. Windows Security and Explorer freeze. When I run BoostSpeed13 ver.13.0.0.6, it freezes, but Windows Security and Explorer work normally without BoostSpeed13. Please
tell me how to resolve.
after installing Auslogic BoostSpeed, Anti-Malware, Driver Ipdater, and Defrag Ultimate, and running and updating per program informaion, i have 2 issues.
1. The Window keyboard key does not work when pressed or clicke on.
2. Entering the command setting on the window search bar gets me to window settings, but clicking on System will notopen it and hangs the whole Setting menu.
3. The usb microphone is recognized when plugged in the usb but it's not working.
Please check attachements. I added folders to be ignored y scanner, but even in Default Scan (in Daschboard) seems that all folders are scanned.
I did a rigorous cleanup and driver update on a old(er) laptop, after years of limited maintenance. Using Auslogic software (driver update,bootspeed,duplicate finder) and some manual cleaning/
updating, Something went wrong, not sure exactly where. Now I can not use/connect bluetooth devices anymore. I tried to solve it by doing some reasearch on the internet but can not find the solution.
Otherwise the Laptop is working fine and is a lot faster.
Enclosed you will find some information of the laptop and screenshots of what I think is relevant information.
I also noticed that Bluetooth is not longer available at the 'action center' and 'device manager'. The latter one still shows some bluetooth devices but is hidden and do not seem to be the relevant
In 'Settings> bluetooth and other devices', the previously connected bluetooth devices are still there but it clearly states that bluetooth is turned off.
Could you please advise me on how to solve this.
Enclosed a couple of prinscreens of the systems and the above mentioned screens.
FYI I am an experienced user, however certanly not a Windows specialist.
Kind regards.
Perry Parengkuan
es posible con los programas que he adquirido xcon ustedes, poder recuperar mi dispositivo USB (memorias USB) que al colocarla en mi computador, me aparece que no la reconoce, dice que en mi
dispositivo hay un error de comunicacion ( error 43) al parecer y no he podido usar mi memoria USB y alli tengo todos mis archivos.
si no es posible, me recomiendan alguna aplicacion para poder recuperar mi memoria USB?
my bluethooth doent work on this laptop so i dont know wht to do or where to go to instaill or update any programs so this laptop would pair with my bluethooth dievices
Does this type of defrag keep the SSD from degrading over time? Can you explain this process in more detail? Thank You, Craig Stone in California
I would like to ask why BootSpeed not have option to perform cleanup of the c:\windows\servicing\LCU\* files?
Здравствуйте! Мой ноутбук для домашнего использования, сложность с тем что он очень долго загрудается и все остальные процессы идут с зависанием и долгой загрузкой, скачала вашу программу, как могла
сама что то удаляла с компьютера, но знаний у меня в этой области нет, как мне оптимизировать его содержимое для стабильной работы и чтобы сканирование на мусор проходило автоматически.
Как удалить элементы реестра ,если они не удаляются ,а очень надо,мммможет есть спец.программы?
This error code might help: (0x80004002).
Can you guys help? Tried everything. I can't seem to fix this.
Hi There...!Again I have a question regarding with my 'Subscriptions and my 2nd Question is:- How Can I ''Manage'' my subscriptions with 'Auslogics ?, and for Example:-How to do this... if in the
future I'd preffer or change my mind to 'Cancel one of my subscription with 'Auslogics...how could I 'Cancel a subscription with 'Auslogics? Let me know Please...? Thanks!
I have an important Question regarding with my 'Subscriptions: 1 Question: After an Year time are All my 'Subscriptions with Auslogics will Renew Automatically?...or After an Year time when my
subscriptions will gonna expire should I buy New Licenses for all my Subcriptions again, I mean do i need to do it manually?again? Please Let me know? thanks!
Quel utilité et quelle différence (avant de passer ma commande) entre Auslogics BoostSpeed13 et Discdefrag?
Je ne sais que choisir vraiment..
Boost Speed 13* has been sending me the message that I have "Potential privacy leaks detected - . . . . some Windows settings are enabled to share your private information"
How do I find the Windows settings and disable them?????
BoostSpeed 13 バージョン 13.0.0.4から13.0.0.6への案内は来るので
Hello and good morning, I'm not very good at softwares and know-how's on PC but I've tried using this program to probably fix the cause of what could be causing the BSOD's on my PC. After I tried to
use the Auslogics Driver Updater, I updated some of the drivers and it asked me to restart so I did. After I restarted, the pc kept giving me a BSOD where it was written something like: Boot device
inaccessible. So I couldn't even put the password and login properly. I had to restore the data to a certain point so I could get my pc back. I'm not sure what the problem is yet so I decided to
write this message to you and ask for a helping hand.
Thank you in advance for your time,
When I first used BoostSpeed there was an emailed tutorial/teaching program that was very informaive. Is there still such a thing and how do I access it?
Добрый вечер, подскажите пожалуйста как оптимизировать и повысить скорость интернет-соединения через Wi-Fi?
I noticed that my Auslogics BoostSpeed 11 keeps turning of my windows search indexing. How can i prevent that from happening.
Добрый вечер! Сегодня завис компютер и никак не отвисал, пока не выключил с кнопки, совсем недавно была установлена система и диск ссд
Hi there,
How can I delete a file and folder that won't delete?
My computer always says high system resource usage detected.
I have been using boostspeed for almost an year now and I am a free user. It is all good but my only question is that should i let boostpeed run in backgroung and windows startup? Does Boostspeed
optimize my pc in the backgroung when I enable this option? These processes take up a lot of memory. And also does boostspeed perfomance monitor optimize the pc in backgroung ?
I've uninstalled and reinstalled boostpeed 11 four times now. Is there a bug on the task scheduler. it's kee[s crashing the program. and I have to manually start the maintenanace daily now.
bonjour, ce n'est pas une question relative a un probleme .. je voudrais savoir comment programmer et installer le lpgiciel pour quil gere seul la meilleur mise en bonne condition de fonctionnement
de mon pc sous windows 10 ? Cdlt à vous lire
fred TESSiER
Content hidden to protect privacy
Cant find the upate button to update
How to prove my performance specifically that mu processor is from type U
my device is : core i 7 8th gen
ram 8 GB
hard HDD 1TB
hard SSD 500 GB
NIVIDEA GEFORCE MX130
after running my pc from sleep mode the voice of pc become weak .
whats the problem ?
Thank You
dsregard " category please , nothing made sense to fit so i just picked whatever to make the forrm work
dear auslogic ... allthough this might seems milydy entertaining .. its serious ... i find my purchased boostspeed 11 very handy for some things and situations , but i still go back for boostspeed 10
for some other situations ... however * dont get ticked off now( ... for bs10 i was never licenemed ... allthough my interest for your products started thhrtr actually ... enough boutt that...
sofirst i thought i just let my free tries@ of bs 10 run out and the by then mandatory buy it or get lost whould steer my to the right page.... now as mentioned first it was just mildy annoying ...
now ... wanting to have apaid 10 for LOOONG its beging to feel stupid mildy put .. always being directed to get 11 .. that i allready have... so here i I am ... at the very source of the cracjer
factory asking u PLEEEAAAASE .. FFS .. can i very very please tosss some coin in your direction for a LEGIT ten ... sure enough half my youth p2p ...
howver even if i got a lax mindset on piracy i actually feel strong on the issue + if you find yourself using one thing over and over its superjustified coughing up , and for bs 10 ... in all honesty
i felt that wanted to for LOOONG . just been some gaps between my tring to find TEN ... alll about 11 ... so very please ausligic folks and yes might u even take 10s FULL rate even if it might be
percievied @obsolete@ ... i dont care i just very much want a legit 10 copy. a totally other idea , i do batch layman stuff , minor programing so to speak... yet to master for and loops else i might
gone deepr with this idea. that be .. i lie to propose this for windows slimmer.. when i go about this manually i do say lang coeds string seaches within winsxs ,soppose im french for instance ..
wanna ditch ALL locales cuz... i i fucking just want the frenchones for my os , UNLESS i got a BIIIIG family >p .. so there hes like explorer search withing top of winsxs *_en-US_* and i get ALL
files from that language , alll get folder with files might not tagged but safe to assume same language since found in folder. like just getting system32 root folder , gather up all lang folder there
to a list , then run searches like that. unless most people surley be contentnt with 3/4 languages, and belive me when i say , its VERY noticiable gain in space getting rid of 20-25 sane shit just in
jjibberish not meant for you... so why MS bundles i can understand , but why they dont trim ones mentioned to not be using down is boyond me.... well anyhow auslogic ... please write me back regard
me gladly tossing coin your way for boostspeed 10 ... cuz it be veryy happy for it .. &cheers &bj;rn Sahle ,
Content hidden to protect privacy
or | santa9785 |
What is the correct subcategory for a software question for Auslogic products such as Boostspeed, Disk Defgrag, etc.? The "Ask a Question- New Question" screen will not allow a Subcategoty that is
not on the drop down list.
I've been using BoostSpeed successfully on Windows 7 for many years. It hasn't been working for the last 2 months. When it scans the disk space, it gets stuck on a temporary Skype file, and stops
scanning. To fix this, I have to re-start my computer, and re-install BoostSpeed 12. My computer has been running painfully slow for the last 2 months. PLEASE HELP!
Malgré après avoir réaliser une optimisation de mon disque dur l'icone dans la barre de tâches reste en rouge. Que cela veux dire et comment résoudre pour le mettre en vert
Denis Schmitt
How can I optimize Windows for GTA V for maximum performance?
I use WWW.BERNERS.HIBID.COM weekly but since getting BoostSpeed 12 it won't load. I am getting
MY WIFES PC WHICH DOES NOT HAVE bOOSsPEDD RUNS IF FINE
When I right click on a shortcut on the desktop or on a file folder in the file explorer it can take up to 2 minutes to come up. Once it does everything exicutes normaly. If I right click on an empty
space on the desktop or on a file in the explorer the contex menu works as it should.
I have uninstalled cortana, disabled all contex menu options with this program, and checked and repaired all regestery items. I don't have any idea what else to do. I use the right mouse click a lot
and when the delay happens it is very disruptive to workflow.
How can i change DNS?
When i changed, this program will turn it back.
And i cant play some game because my network, how can i fix it?
my computer show unknown device please solve driver problem | {"url":"https://qa.auslogics.com/questions?page=13&dp-1-page=35","timestamp":"2024-11-09T04:01:44Z","content_type":"text/html","content_length":"143988","record_id":"<urn:uuid:c35f9359-01e0-4938-90dd-4be645606bc9>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00013.warc.gz"} |
Pre-training Graph Neural Networks with Kernels
02 Jan 2019
• The paper proposes a pretraining technique that can be used with the GNN architecture for learning graph representation as induced by powerful graph kernels.
• Graph Kernel methods can learn powerful representations of the input graphs but the learned representation is implicit as the kernel function actually computes the dot product between the
• GNNs are flexible and powerful in terms of the representations they can learn but they can easily overfit if a large amount of training data is not available as is commonly the case of graphs.
• Kernel methods can be used to learn an unsupervised graph representation that can be finetuned using the GNN architectures for the supervised tasks.
• Given a dataset of graphs g[1], g[2], …, g[n], use a relevant kernel function to compute k(g[i], g[j]) for all pairs of graphs.
• A siamese network is used to encode the pair of graphs into representations f(g[i]) and f(g[j]) such that dot(f(g[i]), f(g[j])) equals k(g[i], g[j]).
• The function f is trained to learn the compressed representation of kernel’s feature space.
• Biological node-labeled graphs representing chemical compounds - MUTAG, PTC, NCI1
• Graphlet Kernel (GK)
• Random Walk Kernel
• Propogation Kernel
• Weisfeiler-Lehman subtree kernel (WL)
• Pretraining uses the WL kernel
• Pretrained model performs better than the baselines for 2 datasets but lags behind WL method (which was used for pretraining) for the NCI1 dataset.
• The idea is straightforward and intuitive. In general, this kind of pretraining should help the downstream model. It would be interesting to try it on more datasets/kernels/GNNs so that more
conclusive results can be obtained. | {"url":"https://shagunsodhani.com/papers-I-read/Pre-training-Graph-Neural-Networks-with-Kernels","timestamp":"2024-11-02T21:29:00Z","content_type":"text/html","content_length":"11792","record_id":"<urn:uuid:c431fac1-a335-46c1-9d62-d4db5bd2b132>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00142.warc.gz"} |
Patricia Gonzalez-Guerrero
Computer Architecture Group
Patricia Gonzalez-Guerrero
Patricia Gonzalez-Guerrero
Research Scientist
Computer Science Department
Computer Architecture Group
Biographical Sketch
Patricia received her Ph.D. (2019) and M.Sc. (2015) from the University of Virginia, Charlottesville, VA, USA; and her Bachelors Degree from the Pontifical Xavierian University (2008), Bogota,
Colombia. She also worked as an ASIC Design and Verification Engineer in Hewlett-Packard, Costa Rica.
Her research interests include ultra-low power VLSI digital and mixed-signal design. Her work focuses on non-conventional computing paradigms to reduce power and energy consumption, such as,
synchronous and asynchronous stochastic computing, computing with sigma-delta streams and race logic.
Current Projects
• Project 38: A set of vendor-agnostic architectural explorations involving NSA, the DOE Office of Science, and NNSA
• Superconducting Race Logic Accelerators: Make computation in superconducting circuits, circuits that operate around 4K temperatures and have close to zero resistance, as efficient as possible
Journal Articles
Patricia Gonzalez-Guerrero, Κylie Huch, Nirmalendu Patra, Thom Popovici, George Michelogiannakis, "Towards practical superconducting accelerators for machine learning using U-SFQ", ACM Journal on
Emerging Technologies in Computing Systems, April 2024,
Meriam Gay Bautista, Darren Lyles, Kylie Huch, Patricia Gonzalez-Guerrero, George Michelogiannakis, "Area Efficient Asynchronous SFQ Pulse Round-Robin Distribution Network", IEEE Transactions on
Circuits and Systems I: Regular Papers, November 2023,
Darren Lyles, Patricia Gonzalez-Guerrero, Meriam Gay Bautista, George Michelogiannakis, "PaST-NoC: A Packet-Switched Superconducting Temporal NoC", IEEE Transactions on Applied Superconductivity,
January 2023,
Patricia Gonzalez-Guerrero, Tommy Tracy II, Xinfei Guo, Rahul Sreekumar, Marzieh Lenjani, Kevin Skadron, Mircea R Stan, "Towards on-node Machine Learning for Ultra-low-power Sensors Using
Asynchronous Σ Δ Streams", Journal on Emerging Technologies in Computing Systems (JETC), August 26, 2020, doi: https://doi.org/10.1145/3404975
We propose a novel architecture to enable low-power, complex on-node data processing, for the next generation of sensors for the internet of things (IoT), smartdust, or edge intelligence. Our
architecture combines near-analog-memory-computing (NAM) and asynchronous-computing-with-streams (ACS), eliminating the need for ADCs. ACS enables ultra-low power, massive computational resources
required to execute on-node complex Machine Learning (ML) algorithms; while NAM addresses the memory-wall that represents a common bottleneck for ML and other complex functions. In ACS an analog
value is mapped to an asynchronous stream that can take one of two logic levels (vh, vl). This stream-based data representation enables area/power-efficient computing units such as a multiplier
implemented as an AND gate yielding savings in power of ∼90% compared to digital approaches. The generation of streams for NAM and ACS in a brute force manner, using analog-to-digital-converters
(ADCs) and digital-to-streams-converters, would sky-rocket the power-latency-energy cost making the approach impractical. Our NAM-ACS architecture eliminates expensive conversions, enabling an
end-to-end processing on asynchronous streams data-path. We tailor the NAM-ACS architecture for random forest (RaF), an ML algorithm, chosen for its ability to classify using a reduced number of
features. Simulations show that our NAM-ACS architecture enables 75% of savings in power compared with a single ADC, obtaining a classification accuracy of 85% using an RaF-inspired algorithm
Xinfei Guo, Vaibhav Verma, Patricia Gonzalez-Guerrero, Sergiu Mosanu, Mircea R Stan, "Back to the future: Digital circuit design in the finfet era", Journal of Low Power Electronics, September 1,
2017, doi: https://doi.org/10.1166/jolpe.2017.1489
It has been almost a decade since FinFET devices were introduced to full production; they allowed scaling below 20 nm, thus helping to extend Moore's law by a precious decade with another decade
likely in the future when scaling to 5 nm and below. Due to superior electrical parameters and unique structure, these 3-D transistors offer significant performance improvements and power reduction
compared to planar CMOS devices. As we are entering into the sub-10 nm era, FinFETs have become dominant in most of the high-end products; as the transition from planar to FinFET technologies is
still ongoing, it is important for digital circuit designers to understand the challenges and opportunities brought in by the new technology characteristics. In this paper, we study these aspects
from the device to the circuit level, and we make detailed comparisons across multiple technology nodes ranging from conventional bulk to advanced planar technology nodes such as Fully Depleted
Silicon-on-Insulator (FDSOI), to FinFETs. In the simulations we used both state-of-art industry-standard models for current nodes, and also predictive models for future nodes. Our study shows that
besides the performance and power benefits, FinFET devices show significant reduction of short-channel effects and extremely low leakage, and many of the electrical characteristics are close to ideal
as in old long-channel technology nodes; FinFETs seem to have put scaling back on track! However, the combination of the new device structures, double/multi-patterning, many more complex rules, and
unique thermal/reliability behaviors are creating new technical challenges. Moving forward, FinFETs still offer a bright future and are an indispensable technology for a wide range of applications
from high-end performance-critical computing to energy-constraint mobile applications and smart Internet-of-Things (IoT) devices.
A. Roy, A. Klinefelter, F. B. Yahya, X. Chen, L. P. Gonzalez-Guerrero, C. J. Lukas, D. A. Kamakshi, J. Boley, K. Craig, M. Faisal, S. Oh, N. E. Roberts, Y. Shakhsheer, A. Shrivastava, D. P.
Vasudevan, D. D. Wentzloff, B. H. Calhoun, "A 6.45 μW Self-Powered SoC With Integrated Energy-Harvesting Power Management and ULP Asymmetric Radios for Portable Biomedical Systems", IEEE Transactions
on Biomedical Circuits and Systems, December 28, 2015, 9:862 - 874, doi: 10.1109/TBCAS.2015.2498643
This paper presents a batteryless system-on-chip (SoC) that operates off energy harvested from indoor solar cells and/or thermoelectric generators (TEGs) on the body. Fabricated in a commercial 0.13
μW process, this SoC sensing platform consists of an integrated energy harvesting and power management unit (EH-PMU) with maximum power point tracking, multiple sensing modalities, programmable core
and a low power microcontroller with several hardware accelerators to enable energy-efficient digital signal processing, ultra-low-power (ULP) asymmetric radios for wireless transmission, and a 100
nW wake-up radio. The EH-PMU achieves a peak end-to-end efficiency of 75% delivering power to a 100 μA load. In an example motion detection application, the SoC reads data from an accelerometer
through SPI, processes it, and sends it over the radio. The SPI and digital processing consume only 2.27 μW, while the integrated radio consumes 4.18 μW when transmitting at 187.5 kbps for a total of
6.45 μW.
Conference Papers
Meriam Gay Bautista, Patricia Gonzalez-Guerrero, Darren Lyles, Kylie Huch, George Michelogiannakis, "Superconducting Digital DIT Butterfly Unit for Fast Fourier Transform Using Race Logic", 2022 20th
IEEE Interregional NEWCAS Conference (NEWCAS), IEEE, June 2022, 441-445,
Patricia Gonzalez-Guerrero, Meriam Gay Bautista, Darren Lyles, George Michelogiannakis, "Temporal and SFQ Pulse-Streams Encoding for Area-Efficient Superconducting Accelerators", 27th ACM
International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS ’22), ACM, February 2022,
Marzieh Lenjani, Patricia Gonzalez, Elaheh Sadredini, Shuangchen Li, Yuan Xie, Ameen Akel, Sean Eilert, Mircea R Stan, Kevin Skadron, "Fulcrum: a simplified control and access mechanism toward
flexible and practical in-situ accelerators", International Symposium on High Performance Computer Architecture (HPCA), San Diego, CA, USA, IEEE, February 22, 2020, doi: 10.1109/HPCA47549.2020.00052
In-situ approaches process data very close to the memory cells, in the row buffer of each subarray. This minimizes data movement costs and affords parallelism across subarrays. However, current
in-situ approaches are limited to only row-wide bitwise (or few-bit) operations applied uniformly across the row buffer. They impose a significant overhead of multiple row activations for emulating
32-bit addition and multiplications using bitwise operations and cannot support operations with data dependencies or based on predicates. Moreover, with current peripheral logic, communication among
subarrays is inefficient, and with typical data layouts, bits in a word are not physically adjacent. The key insight of this work is that in-situ, single-word ALUs outperform in-situ, parallel,
row-wide, bitwise ALUs by reducing the number of row activations and enabling new operations and optimizations. Our proposed lightweight access and control mechanism, Fulcrum, sequentially feeds data
into the single-word ALU and enables operations with data dependencies and operations based on a predicate. For algorithms that require communication among subarrays, we augment the peripheral logic
with broadcasting capabilities and a previously-proposed method for low-cost inter-subarray data movement. The sequential processor also enables overlapping of broadcasting and computation, and
reuniting bits that are physically adjacent. In order to realize true subarray-level parallelism, we introduce a lightweight column-selection mechanism through shifting one-hot encoded values. This
technique enables independent column selection in each subarray. We integrate Fulcrum with Compress Express Link (CXL), a new interconnect standard. Fulcrum with one memory stack delivers on average
(up to) 23.4 (76) speedup over a server-class GPU, NVIDIA P100, with three stacks of HBM2 memory, (ii) 70 (228) times speedup per memory stack over the GPU, and (iii) 19 (178.9) times speedup per
memory stack over an ideal model of the GPU, which only accounts for the overhead of data movement.
Patricia Gonzalez-Guerrero, Mircea R. Stan, "Asynchronous Stochastic Computing", 53rd Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, IEEE, November 3, 2019, doi:
Asynchronous Stochastic Computing (ASC) leverages Synchronous Stochastic Computing (SSC) advantages and addresses its drawbacks. In SSC a multiplier is a single AND gate, saving ~ 90% of power and
area compared with a typical 8bit binary multiplier. The key for SSC power-area efficiency comes from mapping numbers to streams of 1s and 0s. Despite the power-area efficiency, SSC drawbacks such as
long latency, costly clock distribution network (CDN), and expensive stream generation, causes the energy consumption to grow prohibitively large. In this work, we introduce the foundations for ASC
using Continuous-time-Markov-chains, and analyze the computing error due to random fluctuations. In ASC data is mapped to asynchronous-continuous-time streams, which yields two advantages over the
synchronous counterpart: (1) CDN elimination, and (2) better accuracy performance. We compare ASC with SSC for three applications: (1) multiplication, (2) an image processing algorithm:
gamma-correction, and (3) a singlelayer of a fully-connected artificial-neural-network (ANN) using a FinFET1X technology. Our Matlab, Spice-level simulations and post-place&route (P&R) reports
demonstrate that ASC yields savings of 10%-55%, 33%-44%, and 50% in latency, power, and energy respectively. These savings make ASC a good candidate to address the ultra-low-power requirements of
machine learning for the IoT.
Marzieh Lenjani, Patricia Gonzalez, Elaheh Sadredini, M Arif Rahman, Mircea R Stan, "An overflow-free quantized memory hierarchy in general-purpose processors", International Symposium on Workload
Characterization (IISWC), Orlando, FL, USA, IEEE, November 3, 2019, doi: 10.1109/IISWC47752.2019.9042035
Data movement comprises a significant portion of energy consumption and execution time in modern applications. Accelerator designers exploit quantization to reduce the bitwidth of values and reduce
the cost of data movement. However, any value that does not fit in the reduced bitwidth results in an overflow (we refer to these values as outliers). Therefore accelerators use quantization for
applications that are tolerant to overflows. We observe that in most applications the rate of outliers is low and values are often within a narrow range, providing the opportunity to exploit
quantization in general-purpose processors. However, a software implementation of quantization in general-purpose processors has three problems. First, the programmer has to manually implement
conversions and the additional instructions that quantize and dequantize values, imposing a programmer's effort and performance overhead. Second, to cover outliers, the bitwidth of the quantized
values often become greater than or equal to the original values. Third, the programmer has to use standard bitwidth; otherwise, extracting non-standard bitwidth (i.e., 1-7, 9-15, and 17-31) for
representing narrow integers exacerbates the overhead of software-based quantization. The key idea of this paper is to propose a hardware support in the memory hierarchy of general-purpose processors
for quantization, which represents values by few and flexible numbers of bits and stores outliers in their original format in a separate space, preventing any overflow. We minimize metadata and the
overhead of locating quantized values using a software-hardware interaction that transfers quantization parameters and data layout to hardware. As a result, our approach has three advantages over
cache compression techniques: (i) less metadata, (ii) higher compression ratio for floating-point values and cache blocks with multiple data types, and (iii) lower overhead for locating the
compressed blocks. It delivers on average 1.40/1.45/1.56× speedup and 24/26/30% energy reduction compared to a baseline that uses full-length variables in a 4/8/16-core system. Our approach also
provides 1.23× speedup, in a 4-core system, compared to the state of the art cache compression techniques and adds only 0.25% area overhead to the baseline processor.
Patricia Gonzalez-Guerrero, Tommy Tracy II, Xinfei Guo, Mircea R Stan, "Towards low-power random forest using asynchronous computing with streams", Tenth International Green and Sustainable Computing
Conference (IGSC), EEE Computer Society, October 1, 2019, doi: 10.1109/IGSC48788.2019.8957193
We propose a sensor architecture for the internet of things (IoT), smartdust or edge-intelligence (EI) that combines near-analog-memory (NAM) processing and asynchronous computing with streams (ACS)
addressing the need for machine learning (ML) capabilities at low power budgets. In ACS an analog value is mapped to an asynchronous stream that can take one of two values (vh, vl). This stream-based
data representation enables area-power efficient computing units such as the multiplier implemented as an AND gate yielding savings in power of 90% compared with digital approaches. However, a major
bottleneck for computing on streams, vision sensors, and NAM approaches is the cost of analog-to-digital (ADC) and digital-to-stream-to-digital converters. Our NAM-ACS architecture, simplifies the
sensor and eliminates the need for the expensive conversions. The architecture is tailored for random forest (Raf), a ML algorithm, chosen for its ability to classify using a reduced number of
features. Our simulations show that using an analog-memory array of 256 512, the power consumption of the ACS-core combined with the memory interface is comparable with the consumption of an ADC
based memory interface, obtaining an accuracy of 83%.
Patricia Gonzalez-Guerrero, Stephen G Wilson, Mircea R Stan, "Error-latency Trade-off for Asynchronous Stochastic Computing with ΣΔ Streams for the IoT", International System-on-Chip Conference
(SOCC), Singapore, IEEE, September 3, 2019, doi: 10.1109/SOCC46988.2019.1570548453
Asynchronous stochastic computing (ASC) using continuous-time-asynchronous ΣΔ modulators (SC-AΣΔM) has the potential to enable ultra-low-power, on-node machine learning algorithms for the next
generation of sensors for the Internet of Things (IoT). Similar to synchronous stochastic computing (SSC 1 ), in SC-AΣΔM complex processing units can be implemented with simple gates because numbers
are represented with streams. For example a multiplier is implemented with a XNOR gate, yielding savings in power and area of 90% compared with the typical binary approach. Previous work demonstrated
that SC-AΣΔM leverages SSC advantages and addresses its drawbacks, achieving significant savings in energy, power and latency. In this work, we study a theoretical model to determine the fundamental
limits of accuracy and computing time for SCAΣΔM. Since the ΣΔ streams are periodic the final computing error is non-zero and depends on the period of the input streams. We validate our theoretical
model with Spice-level simulations and evaluate the power and energy consumption using a standard FinFet1X2 technology for two cases: 1) multiplication and 2) gamma correction, an image processing
algorithm. Our work determines circuit design guidelines for SC-AΣΔM and shows that multiplication with SC-AΣΔM requires at least 6X less time than SSC. The latency reduction and novel architecture
positively impacts the overall energy consumption in the IoT node, enabling savings in energy of 79% compared with the binary approach.
Patricia Gonzalez-Guerrero, Mircea R Stan, "Asynchronous Stream Computing for Low Power IoT", International Midwest Symposium on Circuits and Systems (MWSCAS), Dallas, TX, USA, IEEE, August 4, 2019,
doi: 10.1109/MWSCAS.2019.8885388
Asynchronous circuits have many advantages over their synchronous counterparts in terms of robustness to parameter variations, wide supply voltage ranges, and potentially low power by not needing a
clock, yet their promise has not been translated yet into commercial success due to several issues related to design methodologies and the need for handshake signals. Stochastic computing is another
processing paradigm that has shown promises of low power and extremely compact circuits but has yet to become a commercial success mainly because of the need for a fast clock to generate the random
streams. The Asynchronous Stream Processing circuits described in this paper combine the best features of asynchronous circuits (lack of clock, robustness) with the best features of stochastic
computing (processing on streams) to enable extremely compact and low power IoT sensing nodes that can finally fulfill the promise of smart dust, another concept that was ahead of its time and yet to
achieve commercial success.
Patricia Gonzalez-Guerrero, Xinfei Guo, Mircea R Stan, "ASC-FFT: Area-efficient low-latency FFT design based on asynchronous stochastic computing", 10th Latin American Symposium on Circuits & Systems
(LASCAS), Armenia, Colombia, IEEE, February 24, 2019, doi: 10.1109/LASCAS.2019.8667599
Asynchronous Stochastic Computing (ASC) is a new paradigm that addresses Synchronous Stochastic Computing (SSC) drawbacks, expensive stochastic number generation (SNG) and long latency, by using
continuous time streams (CTS). To go beyond the basic operations of addition and multiplication in ASC we need to incorporate a memory element. Although for SSC the natural memory element is a
clocked-flip-flop, using the same approach with no synchronized data leads to unacceptable large error. In this paper, we propose to use a capacitor embedded in a feedback loop as the ASC memory
element. Based on this idea, we design a low-error asynchronous adder that stores the carry information in the capacitor. Our adder enables the implementation of more complex computation logic. As an
example, we implement an asynchronous stochastic Fast Fourier Transform (ASC-FFT) using a FinFET1X 1 technology. The proposed adder requires 76%-24% less hardware cost compared against conventional
and SSC adders respectively. Besides, the ASC-FFT shows 3X less latency when compared with SSC-FFT approaches and significant improvements in latency and area over conventional FFT architectures with
no degradation of the computation accuracy measured by the FFT Signal to Noise Ratio (SNR).
Patricia Gonzalez-Guerrero, Xinfei Guo, Mircea Stan, "SC-SD: Towards low power stochastic computing using sigma delta streams", International Conference on Rebooting Computing (ICRC), McLean, VA,
USA, IEEE, November 7, 2018, doi: 10.1109/ICRC.2018.8638611
Processing data using Stochastic Computing (SC) requires only ~ 7% of the area and power of the typical binary approach. However, SC has two major drawbacks that eclipse any area and power savings.
First, it takes sim 99% more time to finish a computation when compared with the binary approach, since data is represented as streams of bits. Second, the Linear Feedback Shift Registers (LFSRs)
required to generate the stochastic streams increment the power and area of the overall SC-LFSR system. These drawbacks result in similar or higher area, power, and energy numbers when compared with
the binary counterpart. In this work, we address these drawbacks by applying SC directly on Pulse Density Modulated (PDM) streams. Most modern Systems on Chip (SoCs) already include Analog to Digital
Converters (ADCs). The core of Σ△ -ADCs is the Σ△ Modulator whose output is a PDM stream. Our approach (SC-SD) simplifies the system hardware in two ways. First, we drop the filter stage at the ADC
and, second, we replace the costly Stochastic Number Generators (SNGs) with Σ△ -Modulators. To further lower the system complexity, we adopt an Asynchronous Σ△ -Modulator (AΣ△M) architecture. We
design and simulate the AΣ△M: using an industry-standard 1×FinFET 11 In modern technologies the node number does not refer to any one feature in the process, and foundries use slightly different
conventions; we use 1x to denote the 14/16nm FinFET nodes offered by the foundry. technology with foundry models. We achieve power savings of 81 % in SNG compared to the LFSR approach. To evaluate
how this area and power savings scale to more complex applications, we implement Gamma Correction, a popular image processing algorithm. For this application, our simulations show that SC-SD can save
98%-11% in the total system latency and 50%-38% in power consumption when compared with the SC-LFSR approach or the binary counterpart.
Xinfei Guo, Vaibhav Verma, Patricia Gonzalez-Guerrero, Mircea R Stan, "When “things” get older: exploring circuit aging in IoT applications", International Symposium on Quality Electronic Design
(ISQED), Santa Clara, CA, USA, IEEE, March 13, 2018, doi: 10.1109/ISQED.2018.8357304
The Internet of Things (IoT) brings a paradigm where humans and “things” are connected. Reliability of these devices becomes extremely critical. Circuit aging has become a limiting factor in
technology scaling and a significant challenge in designing IoT systems for reliability-critical applications. As IoT becomes a general-purpose technology which starts to adapt to the advanced
process nodes, it is necessary to understand how and on what level aging affects different categories of IoT applications. Since aging is highly dependent on operating conditions and switching
activities, this paper classifies the IoT applications based on the aging-related metrics and studies aging using the foundry-provided FinFET aging models. We show that for many IoT applications,
aging will indeed add to the already tight design margin. As the expected chip lifetime in IoT devices becomes much longer and the failure tolerant requirements of these applications become much more
strict, we conclude that aging needs to be considered in the full design cycle and the IoT lifetime estimation needs to incorporate aging as an important factor. We also present application-specific
solutions to mitigate circuit aging in IoT systems.
Patricia Gonzalez-Guerrero, Mircea Stan, "Ultra-low-power dual-phase latch based digital accelerator for continuous monitoring of wheezing episodes", SOI-3D-Subthreshold Microelectronics Technology
Unified Conference (S3S), Burlingame, CA, USA, IEEE, October 16, 2017, doi: 10.1109/S3S.2017.8308752
We designed an ultra-low-power accelerator for the calculation of the Short Time Fourier Transform (STFT) optimized for wheezing detection. The low power consumption of our accelerator relies on
optimizations at different stages of the design process. Post-layout simulations show that at the minimum energy point our accelerator consumes 3.3 pJ/cycle at 0.5 V and 163 KHz. We compare the
energy consumption of our implementation with its flip-flop version. Simulations show that we can save up to 50% in energy consumption for a latch based design vs. a flip-flop based design, making
dual-phase latch based implementations excellent candidates for ultra-low-power devices.
Alicia Klinefelter, Nathan E Roberts, Yousef Shakhsheer, Patricia Gonzalez, Aatmesh Shrivastava, Abhishek Roy, Kyle Craig, Muhammad Faisal, James Boley, Seunghyun Oh, Yanqing Zhang, Divya Akella,
David D Wentzloff, Benton H Calhoun, "A 6.45 μW self-powered IoT SoC with integrated energy-harvesting power management and ULP asymmetric radios", International Solid-State Circuits Conference-
(ISSCC) Digest of Technical Papers, San Francisco, CA, USA, IEEE, February 22, 2015, doi: 10.1109/ISSCC.2015.7063087
A 1 trillion node internet of things (IoT) will require sensing platforms that support numerous applications using power harvesting to avoid the cost and scalability challenge of battery replacement
in such large numbers. Previous SoCs achieve good integration and even energy harvesting [1][2][3], but they limit supported applications, need higher end-to-end harvesting efficiency, and require
duty-cycling for RF communication. This paper demonstrates a highly integrated, flexible SoC platform that supports multiple sensing modalities, extracts information from data flexibly across
applications, harvests and delivers power efficiently, and communicates wirelessly. | {"url":"https://crd.lbl.gov/divisions/amcr/computer-science-amcr/cag/staff/staff-researchers/patricia-gonzalez/","timestamp":"2024-11-03T07:31:02Z","content_type":"text/html","content_length":"68870","record_id":"<urn:uuid:5e42c218-563d-4dfe-b6fe-569d230d09bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00151.warc.gz"} |
10.2 Consequences of Special Relativity
Learning Objectives
Learning Objectives
By the end of this section, you will be able to do the following:
• Describe the relativistic effects seen in time dilation, length contraction, and conservation of relativistic momentum
• Explain and perform calculations involving mass-energy equivalence
Section Key Terms
binding energy length contraction mass defect
proper length relativistic relativistic momentum
relativistic energy relativistic factor rest mass
time dilation
Relativistic Effects on Time, Distance, and Momentum
Relativistic Effects on Time, Distance, and Momentum
Consideration of the measurement of elapsed time and simultaneity leads to an important relativistic effect. Time dilation is the phenomenon of time passing more slowly for an observer who is moving
relative to another observer.
For example, suppose an astronaut measures the time it takes for light to travel from the light source, cross her ship, bounce off a mirror, and return. (See Figure 10.5.) How does the elapsed time
the astronaut measures compare with the elapsed time measured for the same event by a person on the earth? Asking this question (another thought experiment) produces a profound result. We find that
the elapsed time for a process depends on who is measuring it. In this case, the time measured by the astronaut is smaller than the time measured by the earth bound observer. The passage of time is
different for the two observers because the distance the light travels in the astronaut’s frame is smaller than in the earth bound frame. Light travels at the same speed in each frame, and so it will
take longer to travel the greater distance in the earth bound frame.
The relationship between Δt and Δt[o] is given by
$Δt=γΔ t 0 , Δt=γΔ t 0 ,$
where$γ γ$is the relativistic factor given by
$γ= 1 1− v 2 c 2 , γ= 1 1− v 2 c 2 ,$
and v and c are the speeds of the moving observer and light, respectively.
Tips For Success
Try putting some values for v into the expression for the relativistic factor ($γ γ$). Observe at which speeds this factor will make a difference and when$γ γ$is so close to 1 that it can be ignored.
Try 225 m/s, the speed of an airliner; 2.98 × 10^4 m/s, the speed of Earth in its orbit; and 2.990 × 10^8 m/s, the speed of a particle in an accelerator.
Notice that when the velocity v is small compared to the speed of light c, then v/c becomes small, and$γ γ$becomes close to 1. When this happens, time measurements are the same in both frames of
reference. Relativistic effects, meaning those that have to do with special relativity, usually become significant when speeds become comparable to the speed of light. This is seen to be the case for
time dilation.
You may have seen science fiction movies in which space travelers return to Earth after a long trip to find that the planet and everyone on it has aged much more than they have. This type of scenario
is a based on a thought experiment, known as the twin paradox, which imagines a pair of twins, one of whom goes on a trip into space while the other stays home. When the space traveler returns, she
finds her twin has aged much more than she. This happens because the traveling twin has been in two frames of reference, one leaving Earth and one returning.
Time dilation has been confirmed by comparing the time recorded by an atomic clock sent into orbit to the time recorded by a clock that remained on Earth. GPS satellites must also be adjusted to
compensate for time dilation in order to give accurate positioning.
Have you ever driven on a road, like that shown in Figure 10.6, that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. Another traveler might say the
road ahead looks like it is about 15 km long. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in
this section, however, that this is not true at relativistic speeds. Close to the speed of light, distances measured are not the same when measured by different observers moving with respect to one
One thing all observers agree upon is their relative speed. When one observer is traveling away from another, they both see the other receding at the same speed, regardless of whose frame of
reference is chosen. Remember that speed equals distance divided by time: v = d/t. If the observers experience a difference in elapsed time, they must also observe a difference in distance traversed.
This is because the ratio d/t must be the same for both observers.
The shortening of distance experienced by an observer moving with respect to the points whose distance apart is measured is called length contraction. Proper length, L[0], is the distance between two
points measured in the reference frame where the observer and the points are at rest. The observer in motion with respect to the points measures L. These two lengths are related by the equation
$L= L 0 γ . L= L 0 γ .$
Because$γ γ$is the same expression used in the time dilation equation above, the equation becomes
$L= L 0 1− v 2 c 2 . L= L 0 1− v 2 c 2 .$
To see how length contraction is seen by a moving observer, go to this simulation. Here you can also see that simultaneity, time dilation, and length contraction are interrelated phenomena.
This link is to a simulation that illustrates the relativity of simultaneous events.
In classical physics, momentum is a simple product of mass and velocity. When special relativity is taken into account, objects that have mass have a speed limit. What effect do you think mass and
velocity have on the momentum of objects moving at relativistic speeds; i.e., speeds close to the speed of light?
Momentum is one of the most important concepts in physics. The broadest form of Newton’s second law is stated in terms of momentum. Momentum is conserved in classical mechanics whenever the net
external force on a system is zero. This makes momentum conservation a fundamental tool for analyzing collisions. We will see that momentum has the same importance in modern physics. Relativistic
momentum is conserved, and much of what we know about subatomic structure comes from the analysis of collisions of accelerator-produced relativistic particles.
One of the postulates of special relativity states that the laws of physics are the same in all inertial frames. Does the law of conservation of momentum survive this requirement at high velocities?
The answer is yes, provided that the momentum is defined as follows.
Relativistic momentum, p, is classical momentum multiplied by the relativistic factor$γ . γ .$
10.3$p=γmu, p=γmu,$
where$m m$is the rest mass of the object (that is, the mass measured at rest, without any$γ γ$factor involved),$u u$is its velocity relative to an observer, and$γ, γ,$ as before, is the relativistic
factor. We use the mass of the object as measured at rest because we cannot determine its mass while it is moving.
Note that we use$u u$for velocity here to distinguish it from relative velocity$v v$between observers. Only one observer is being considered here. With$p p$defined in this way,$p tot p tot$is
conserved whenever the net external force is zero, just as in classical physics. Again we see that the relativistic quantity becomes virtually the same as the classical at low velocities. That is,
relativistic momentum$γmu γmu$becomes the classical$mu mu$at low velocities, because$γ γ$is very nearly equal to 1 at low velocities.
Relativistic momentum has the same intuitive feel as classical momentum. It is greatest for large masses moving at high velocities. Because of the factor$γ, γ,$however, relativistic momentum behaves
differently from classical momentum by approaching infinity as$u u$approaches$c . c .$ (See Figure 10.7.) This is another indication that an object with mass cannot reach the speed of light. If it
did, its momentum would become infinite, which is an unreasonable value.
Relativistic momentum is defined in such a way that the conservation of momentum will hold in all inertial frames. Whenever the net external force on a system is zero, relativistic momentum is
conserved, just as is the case for classical momentum. This has been verified in numerous experiments.
Mass-Energy Equivalence
Mass-Energy Equivalence
Let us summarize the calculation of relativistic effects on objects moving at speeds near the speed of light. In each case we will need to calculate the relativistic factor, given by
$γ= 1 1− v 2 c 2 , γ= 1 1− v 2 c 2 ,$
where v and c are as defined earlier. We use u as the velocity of a particle or an object in one frame of reference, and v for the velocity of one frame of reference with respect to another.
Time Dilation
Elapsed time on a moving object,$Δ t 0 , Δ t 0 ,$ as seen by a stationary observer is given by$Δt=γΔ t 0 , Δt=γΔ t 0 ,$where$Δ t 0 Δ t 0$is the time observed on the moving object when it is taken to
be the frame or reference.
Length Contraction
Length measured by a person at rest with respect to a moving object, L, is given by
$L= L 0 γ , L= L 0 γ ,$
where L[0] is the length measured on the moving object.
Relativistic Momentum
Momentum, p, of an object of mass, m, traveling at relativistic speeds is given by$p=γmu , p=γmu ,$ where u is velocity of a moving object as seen by a stationary observer.
Relativistic Energy
The original source of all the energy we use is the conversion of mass into energy. Most of this energy is generated by nuclear reactions in the sun and radiated to Earth in the form of
electromagnetic radiation, where it is then transformed into all the forms with which we are familiar. The remaining energy from nuclear reactions is produced in nuclear power plants and in Earth’s
interior. In each of these cases, the source of the energy is the conversion of a small amount of mass into a large amount of energy. These sources are shown in Figure 10.8.
The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy is valid relativistically, if we define
energy to include a relativistic factor. The result of his analysis is that a particle or object of mass m moving at velocity u has relativistic energy given by
$E=γm c 2 . E=γm c 2 .$
This is the expression for the total energy of an object of mass m at any speed u and includes both kinetic and potential energy. Look back at the equation for$γ γ$and you will see that it is equal
to 1 when u is 0; that is, when an object is at rest. Then the rest energy, E[0], is simply
$E 0 =m c 2 . E 0 =m c 2 .$
This is the correct form of Einstein’s famous equation.
This equation is very useful to nuclear physicists because it can be used to calculate the energy released by a nuclear reaction. This is done simply by subtracting the mass of the products of such a
reaction from the mass of the reactants. The difference is the m in$E 0 =m c 2 . E 0 =m c 2 .$ Here is a simple example:
A positron is a type of antimatter that is just like an electron, except that it has a positive charge. When a positron and an electron collide, their masses are completely annihilated and converted
to energy in the form of gamma rays. Because both particles have a rest mass of 9.11 × 10^–31 kg, we multiply the mc^2 term by 2. So the energy of the gamma rays is
10.4 $E 0 = 2(9.11× 10 −31 kg) (3.00× 10 8 m s ) 2 = 1.64× 10 −13 kg⋅ m 2 s 2 = 1.64× 10 −13 J E 0 = 2(9.11× 10 −31 kg) (3.00× 10 8 m s ) 2 = 1.64× 10 −13 kg⋅ m 2 s 2 = 1.64× 10 −13 J$
where we have the expression for the joule (J) in terms of its SI base units of kg, m, and s. In general, the nuclei of stable isotopes have less mass then their constituent subatomic particles. The
energy equivalent of this difference is called the binding energy of the nucleus. This energy is released during the formation of the isotope from its constituent particles because the product is
more stable than the reactants. Expressed as mass, it is called the mass defect. For example, a helium nucleus is made of two neutrons and two protons and has a mass of 4.0003 atomic mass units (u).
The sum of the masses of two protons and two neutrons is 4.0330 u. The mass defect then is 0.0327 u. Converted to kg, the mass defect is 5.0442 × 10^–30 kg. Multiplying this mass times c^2 gives a
binding energy of 4.540 × 10^–12 J. This does not sound like much because it is only one atom. If you were to make one gram of helium out of neutrons and protons, it would release 683,000,000,000 J.
By comparison, burning one gram of coal releases about 24 J.
Boundless Physics
The RHIC Collider
Figure 10.9 shows the Brookhaven National Laboratory in Upton, NY. The circular structure houses a particle accelerator called the RHIC, which stands for Relativistic Heavy Ion Collider. The heavy
ions in the name are gold nuclei that have been stripped of their electrons. Streams of ions are accelerated in several stages before entering the big ring seen in the figure. Here, they are
accelerated to their final speed, which is about 99.7 percent the speed of light. Such high speeds are called relativistic. All the relativistic phenomena we have been discussing in this chapter are
very pronounced in this case. At this speed$γ γ$= 12.9, so that relativistic time dilates by a factor of about 13, and relativistic length contracts by the same factor.
Two ion beams circle the 2.4-mile long track around the big ring in opposite directions. The paths can then be made to cross, thereby causing ions to collide. The collision event is very short-lived
but amazingly intense. The temperatures and pressures produced are greater than those in the hottest suns. At 4 trillion degrees Celsius, this is the hottest material ever created in a laboratory
But what is the point of creating such an extreme event? Under these conditions, the neutrons and protons that make up the gold nuclei are smashed apart into their components, which are called quarks
and gluons. The goal is to recreate the conditions that theorists believe existed at the very beginning of the universe. It is thought that, at that time, matter was a sort of soup of quarks and
gluons. When things cooled down after the initial bang, these particles condensed to form protons and neutrons.
Some of the results have been surprising and unexpected. It was thought the quark-gluon soup would resemble a gas or plasma. Instead, it behaves more like a liquid. It has been called a perfect
liquid because it has virtually no viscosity, meaning that it has no resistance to flow.
Grasp Check
Calculate the relativistic factor γ, for a particle traveling at 99.7 percent of the speed of light.
a. 0.08
b. 0.71
c. 1.41
d. 12.9
Worked Example
The Speed of Light
One night you are out looking up at the stars and an extraterrestrial spaceship flashes across the sky. The ship is 50 meters long and is travelling at 95 percent of the speed of light. What would
the ship’s length be when measured from your earthbound frame of reference?
List the knowns and unknowns.
Knowns: proper length of the ship, L[0] = 50 m; velocity, v, = 0.95c
Unknowns: observed length of the ship accounting for relativistic length contraction, L.
Choose the relevant equation.
$L= L 0 γ = L 0 1− u 2 c 2 L= L 0 γ = L 0 1− u 2 c 2$
$L= 50 m 1− (0.95) 2 c 2 c 2 =50 m 1− (0.95) 2 =16 m L= 50 m 1− (0.95) 2 c 2 c 2 =50 m 1− (0.95) 2 =16 m$
Calculations of$γ γ$can usually be simplified in this way when v is expressed as a percentage of c because the c^2 terms cancel. Be sure to also square the decimal representing the percentage before
subtracting from 1. Note that the aliens will still see the length as L[0] because they are moving with the frame of reference that is the ship.
Practice Problems
Practice Problems
Calculate the relativistic factor, γ, for an object traveling at 2.00×10^8 m/s.
a. 0.74
b. 0.83
c. 1.2
d. 1.34
The distance between two points, called the proper length, L0, is 1.00 km. An observer in motion with respect to the frame of reference of the two points measures 0.800 km, which is L. What is the
relative speed of the frame of reference with respect to the observer?
a. 1.80×10^8 m/s
b. 2.34×10^8 m/s
c. 3.84×10^8 m/s
d. 5.00×10^8 m/s
Consider the nuclear fission reaction $n+ U 92 235 → C 55 137 s+ R 37 97 b+2n+E n+ U 92 235 → C 55 137 s+ R 37 97 b+2n+E$. If a neutron has a rest mass of 1.009u, $U 92 235 U 92 235$ has a rest mass
of 235.044u, $C 55 137 s C 55 137 s$ has rest mass of 136.907u, and $R 37 97 b R 37 97 b$ has a rest mass of 96.937u, what is the value of E in joules?
a. $1.8× 10 −11 1.8× 10 −11$J
b. $2.9× 10 −11 2.9× 10 −11$J
c. $1.8× 10 −10 1.8× 10 −10$J
d. $2.9× 10 −10 2.9× 10 −10$J
The correct answer is (b). The mass deficit in the reaction is $235.044u−( 136.907+96.937+1.009 )u, 235.044u−( 136.907+96.937+1.009 )u,$ or 0.191u. Converting that mass to kg and applying $E=m c 2 E=
m c 2$ to find the energy equivalent of the mass deficit gives $( 0.191u )( 1.66× 10 −27 kg/u ) ( 3.00× 10 8 m/s ) 2 ≅2.85× 10 −11 J. ( 0.191u )( 1.66× 10 −27 kg/u ) ( 3.00× 10 8 m/s ) 2 ≅2.85× 10
−11 J.$
Consider the nuclear fusion reaction $H 1 2 + H 1 2 → H 1 3 + H 1 1 +E H 1 2 + H 1 2 → H 1 3 + H 1 1 +E$. If $H 1 2 H 1 2$ has a rest mass of 2.014u, $H 1 3 H 1 3$ has a rest mass of 3.016u, and $H 1
1 H 1 1$ has a rest mass of 1.008u, what is the value of E in joules?
a. $6× 10 −13 6× 10 −13$J
b. $6× 10 −12 6× 10 −12$J
c. $6× 10 −11 6× 10 −11$J
d. $6× 10 −10 6× 10 −10$J
The correct answer is (a). The mass deficit in the reaction is $2( 2.014u )−( 3.016+1.008 )u 2( 2.014u )−( 3.016+1.008 )u$, or 0.004u. Converting that mass to kg and applying $E=m c 2 E=m c 2$ to
find the energy equivalent of the mass deficit gives $( 0.004u )( 1.66× 10 −27 kg/u ) ( 3.00× 10 8 m/s ) 2 ≅5.98× 10 −13 J. ( 0.004u )( 1.66× 10 −27 kg/u ) ( 3.00× 10 8 m/s ) 2 ≅5.98× 10 −13 J.$
Check Your Understanding
Check Your Understanding
Exercise 5
Describe time dilation and state under what conditions it becomes significant.
a. When the speed of one frame of reference past another reaches the speed of light, a time interval between two events at the same location in one frame appears longer when measured from the second
b. When the speed of one frame of reference past another becomes comparable to the speed of light, a time interval between two events at the same location in one frame appears longer when measured
from the second frame.
c. When the speed of one frame of reference past another reaches the speed of light, a time interval between two events at the same location in one frame appears shorter when measured from the
second frame.
d. When the speed of one frame of reference past another becomes comparable to the speed of light, a time interval between two events at the same location in one frame appears shorter when measured
from the second frame.
Exercise 6
The equation used to calculate relativistic momentum is p = γ · m · u. Define the terms to the right of the equal sign and state how m and u are measured.
a. γ is the relativistic factor, m is the rest mass measured when the object is at rest in the frame of reference, and u is the velocity of the frame.
b. γ is the relativistic factor, m is the rest mass measured when the object is at rest in the frame of reference, and u is the velocity relative to an observer.
c. γ is the relativistic factor, m is the relativistic mass $( i.e., m 1− u 2 c 2 ) ( i.e., m 1− u 2 c 2 )$ measured when the object is moving in the frame of reference, and u is the velocity of the
d. γ is the relativistic factor, m is the relativistic mass $( i.e., m 1− u 2 c 2 ) ( i.e., m 1− u 2 c 2 )$ measured when the object is moving in the frame of reference, and u is the velocity
relative to an observer.
Exercise 7
Describe length contraction and state when it occurs.
a. When the speed of an object becomes the speed of light, its length appears to shorten when viewed by a stationary observer.
b. When the speed of an object approaches the speed of light, its length appears to shorten when viewed by a stationary observer.
c. When the speed of an object becomes the speed of light, its length appears to increase when viewed by a stationary observer.
d. When the speed of an object approaches the speed of light, its length appears to increase when viewed by a stationary observer. | {"url":"https://texasgateway.org/resource/102-consequences-special-relativity?book=79076&binder_id=78136","timestamp":"2024-11-02T20:59:29Z","content_type":"text/html","content_length":"125901","record_id":"<urn:uuid:917e9e32-d505-4445-8f53-589f5fbfaa0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00539.warc.gz"} |
Another proof of Taylor's Theorem
In this note, we produce a proof of Taylor's Theorem. As in many proofs of Taylor's Theorem, we begin with a curious start and then follow our noses forward.
Is this a new proof? I think so. But I wouldn't bet a lot of money on it. It's certainly new to me.
Is this a groundbreaking proof? No, not at all. But it's cute, and I like it.^1 ^1Though I must admit that it is not my favorite proof of Taylor's Theorem.
We begin with the following simple observation. Suppose that $f$ is two times continuously differentiable. Then for any $t \neq 0$, we see that $$f'(t) - f'(0) = \frac{f'(t) - f'(0)}{t} t.$$
Integrating each side from $0$ to $x$, we find that $$f(x) - f(0) - f'(0) x = \int_0^x \frac{f'(t) - f'(0)}{t} t dt.$$ To interpret the integral on the right in a different way, we will use the mean
value theorem for integrals.
Mean Value Theorem for Integrals Suppose that $g$ and $h$ are continuous functions, and that $h$ doesn't change sign in $[0, x]$. Then there is a $c \in [0, x]$ such that $$\int_0^x g(t) h(t) dt
= g(c) \int_0^x h(t) dt.$$
Suppose without loss of generality that $h(t)$ is nonnegative. Since $g$ is continuous on $[0, x]$, it attains its minimum $m$ and maximum $M$ on this interval. Thus $$m \int_0^x h(t) dt \leq \int_0^
x g(t)h(t)dt \leq M \int_0^x h(t) dt.$$ Let $I = \int_0^x h(t) dt$. If $I = 0$ (or equivalently, if $h(t) \equiv 0$), then the theorem is trivially true, so suppose instead that $I \neq 0$. Then $$m
\leq \frac{1}{I} \int_0^x g(t) h(t) dt \leq M.$$ By the intermediate value theorem, $g(t)$ attains every value between $m$ and $M$, and thus there exists some $c$ such that $$g(c) = \frac{1}{I} \
int_0^x g(t) h(t) dt.$$ Rearranging proves the theorem.
For this application, let $g(t) = (f'(t) - f'(0))/t$ for $t \neq 0$, and $g(0) =f'{}'(0)$. The continuity of $g$ at $0$ is exactly the condition that $f'{}'(0)$exists. We also let $h(t) = t$.
For $x > 0$, it follows from the mean value theorem for integrals that there exists a $c \in [0, x]$ such that $$\int_0^x \frac{f'(t) - f'(0)}{t} t dt = \frac{f'(c) - f'(0)}{c} \int_0^x t dt = \frac
{f'(c) - f'(0)}{c} \frac{x^2}{2}.$$ (Very similar reasoning applies for $x < 0$). Finally, by the mean value theorem (applied to $f'$), there exists a point $\xi \in (0, c)$ such that $$f'{}'(\xi) =
\frac{f'(c) - f'(0)}{c}.$$ Putting this together, we have proved that there is a $\xi \in (0, x)$ such that $$f(x) - f(0) - f'(0) x = f'{}'(\xi) \frac{x^2}{2},$$ which is one version of Taylor's
Theorem with a linear approximating polynomial.
This approach generalizes. Suppose $f$ is a $(k+1)$ times continuously differentiable function, and begin with the trivial observation that $$f^{(k)}(t) - f^{(k)}(0) = \frac{f^{(k)}(t) - f^{(k)}(0)}
{t} t.$$ Iteratively integrate $k$ times: first from $0$ to $t_1$, then from $0$ to $t_2$, and so on, with the $k$th interval being from $0$ to $t_k = x$.
Then the left hand side becomes $$f(x) - \sum_{n = 0}^k f^{(n)}(0)\frac{x^n}{n!},$$ the difference between $f$ and its degree $k$ Taylor polynomial. The right hand side is $$\label{eq:only}\
underbrace{\int _0^{t_k = x} \cdots \int _0^{t _1}} _{k \text{ times}} \frac{f^{(k)}(t) - f^{(k)}(0)}{t} t \, dt \, dt _1 \cdots dt _{k-1}.$$
To handle this, we note the following variant of the mean value theorem for integrals.
Mean value theorem for iterated integrals Suppose that $g$ and $h$ are continuous functions, and that $h$ doesn't change sign in $[0, x]$. Then there is a $c \in [0, x]$ such that $$\underbrace{\
int_0^{t _k=x} \cdots \int _0^{t _1}} _{k \; \text{times}} g(t) h(t) dt =g(c) \underbrace{\int _0^{t _k=x} \cdots \int _0^{t _1}} _{k \; \text{times}} h(t) dt.$$
In fact, this can be proved in almost exactly the same way as in the single-integral version, so we do not repeat the proof.
With this theorem, there is a $c \in [0, x]$ such that we see that \eqref{eq:only} can be written as $$\frac{f^{(k)}(c) - f^{(k)}(0)}{c} \underbrace{\int _0^{t _k = x} \cdots \int _0^{t _1}} _{k \; \
text{times}} t \, dt \, dt _1 \cdots dt _{k-1}.$$ By the mean value theorem, the factor in front of the integrals can be written as $f^{(k+1)}(\xi)$ for some $\xi \in (0, x)$. The integrals can be
directly evaluated to be $x^{k+1}/(k+1)! $.
Thus overall, we find that $$f(x) = \sum_{n = 0}^n f^{(n)}(0) \frac{x^n}{n!} + f^{(k+1)}(\xi) \frac{x^{k+1}}{(k+1)!}$$ for some $\xi \in (0, x)$. Thus we have proved Taylor's Theorem (with Lagrange's
error bound).
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Hi folks! It was wonderful to meet with so many new prospects and long-standing clients at the Advanced Manufacturing Minneapolis expo last week. One highlight of the show was running our own design
of experiments (DOE) in-booth: a test to pinpoint the height and distance of a foam cat launched from our Cat-A-Pult toy. Visitors got to choose a cat and launch it based on our randomized run sheet.
We got lots of takers coming in to watch the cats fly, and we even got a visit from local mascot Goldy Gopher!
Mark, Tony, and Rachel are all UMN alums - go Gophers!
But I’m getting a bit ahead of myself: this experiment primarily shows off the ease of use and powerful analytical capabilities of Design-Expert® and Stat-Ease® 360 software. I’m no statistician –
the last math class I took was in high school, over a decade ago – but even a marketer like me was able to design, run, and analyze a DOE with just a little advice. Here’s how it worked.
Let’s start at the beginning, with the design. My first task was to decide what factors I wanted to test. There were lots of options! The two most obvious were the built-in experimental parts of the
toy: the green and orange knobs on either side of the Cat-A-Pult, with spring tension settings from 1 to 5.
However, there were plenty of other places where there could be variation in my ‘pulting system:
• The toy comes with 5 Cat-A-Pults: are there variations between each 'pult?
• Does it matter what kind of surface the Cat-A-Pult is on, e.g., wood, concrete, or carpet?
• The toy came with 5 colors of foam cat; would the pigment change the cat’s weight enough to matter?
• What about where on the plate we apply launch pressure, or how much pressure is applied?
Some of these questions can be answered with subject matter knowledge – in the case of launch pressure, by reading the instruction manual.
For our experiment, the surface question was moot: we had no way to test it, as the convention floor was covered in carpet. We also had no way to test beforehand if there were differences in mass
between colors of cat, since we lacked a tool with sufficient precision. I settled on just testing the experimental knobs, but decided to account for some of this variation in other ways. We divided
the experiment into blocks based on which specific Cat-A-Pult we were using, and numbered them from 1 to 5. And, while I decided to let people choose their cat color to enhance the fun aspect, we
still tracked which color of cat was launched for each run - just in case.
Since my chosen two categoric factors had five levels each, I decided to use the Multilevel Categoric design tool to set up my DOE. One thing I learned from Mark is that these are an “ordinal” type
of categoric factor: there is an order to the levels, as opposed to a factor like the color of the cat or the type of flooring (a “nominal” factor). We decided to just test 3 of the 5 Cat-A-Pults,
trying to be reasonable about how many folks would want to play with the cats, so we set the design to have 3 replicates separated out into 3 blocks. This would help us identify if there were any
differences between the specific Cat-A-Pults.
For my responses, I chose the Cat-A-Pult’s recommended ones: height and distance. My Stat-Ease software then gave me the full, 5x5, 25-run factorial design for this, with a total of 75 runs for the 3
replicates blocked by 'pult, meaning we would test every combination of green knob level and orange knob level on each Cat-A-Pult. More runs means more accurate and precise modeling of our system,
and we expected to be able to get 75 folks to stop by and launch a cat.
And so, armed with my run sheet, I set up our booth experiment! I brought two measuring tapes for the launch zone: one laid along the side of it to measure distance, and one hanging from the booth
wall to measure height. My measurement process was, shall we say, less than precise: for distance, the tester and I eyeballed the point at which the cat first landed after launch, then drew a line
over to our measuring tape. For height, I took a video of the launch, then scrolled back to the frame at the apex of the cat’s arc and once again eyeballed the height measurement next to it. In
addition to blocking the specific Cat-A-Pult used, we tracked which color of cat was selected in case that became relevant. (We also had to append A and B to the orange cat after the first orange cat
was mistaken for swag!)
Whee! I'm calling that one at 23 inches.
Over the course of the conference, we completed 50 runs, getting through the full range of settings for ‘pults 1 and 2. While that’s less than we had hoped, it’s still plenty for a good analysis. I
ran the analysis for height, following the steps I learned in our Finding the Vital Settings via Factorial Analysis eLearning module.
The half-normal plot of effects and the ANOVA table for Height.
The green knob was the only significant effect on Height, but the relatively low Predicted R² model-fit-statistic value of 0.36 tells us that there’s a lot of noise that the model doesn’t explain.
Mark directed me to check the coefficients, where we discovered that there was a 5-inch variation in height between the two Cat-A-Pult! That’s a huge difference, considering that our Height response
peaked at 27 inches.
With that caveat in mind, we looked at the diagnostic plots and the one-factor plot for Height. The diagnostics all looked fine, but the Least Significant Difference bars showed us something
interesting: there didn’t seem to be significant differences between setting the green knob at 1-3, or between settings 4-5, but there was a difference between those two groups.
One-factor interaction plot for Height.
With this analysis under my belt, I moved on to Distance. This one was a bit trickier, because while both knobs were clearly significant to the model, I wasn’t sure whether or not to include the
interaction. I decided to include it because that’s what multifactor DOE is for, as opposed to one-factor-at-a-time experimentation: we’re trying to look for interactions between factors. So once
again, I turned to the diagnostics.
The three main diagnostic plots for Distance.
Here's where I ran into a complication: our primary diagnostic tools told me there was something off with our data. There’s a clear S-shaped pattern in the Normal Plot of Residuals and the Residuals
vs. Predicted graph shows a slight megaphone shape. No transform was recommended according to the Box-Cox plot, but Mark suggested I try a square-root transform anyways to see if we could get more of
the data to fit the model. So I did!
The diagnostics again, after transforming.
Unfortunately, that didn’t fix the issues I saw in the diagnostics. In fact, it revealed that there’s a chance two of our runs were outliers: runs #10 and #26. Mark and I reviewed the process notes
for those runs and found that run #10 might have suffered from operator error: he was the one helping our experimenter at the booth while I ran off for lunch, and he reported that he didn’t think he
accurately captured the results the way I’d been doing it. With that in mind, I decided to ignore that run when analyzing the data. This didn’t result in a change in the analysis for Height, but it
made a large difference when analyzing Distance. The Box-Cox plot recommended a log transform for analyzing Distance, so I applied one. This tightened the p-value for the interaction down to 0.03 and
brought the diagnostics more into line with what we expected.
The two-factor interaction plot for Distance.
While this interaction plot is a bit trickier to read than the one-factor plot for Height, we can still clearly see that there’s a significant difference between certain sets of setting combinations.
It’s obvious that setting the orange knob to 1 keeps the distance significantly lower than other settings, regardless of the green knob’s setting. The orange knob’s setting also seems to matter more
as the green knob’s setting increases.
Normally, this is when I’d move on to optimization, and figuring out which setting combinations will let me accurately hit a “sweet spot” every time. However, this is where I stopped. Given the huge
amount of variation in height between the two Cat-A-Pults, I’m not confident that any height optimization I do will be accurate. If we’d gotten those last 25 runs with ‘pult #3, I might have had
enough data to make a more educated decision; I could set a Cat-A-Pult on the floor and know for certain that the cat would clear the edge of the litterbox when launched! I’ll have to go back to the
“lab” and collect more data the next time we’re out at a trade show.
One final note before I bring this story to a close: the instruction manual for the Cat-A-Pult actually tells us what the orange and green knobs are supposed to do. The orange knob controls the
release point of the Cat-A-Pult, affecting the trajectory of the cat, and the green knob controls the spring tension, affecting the force with which the cat is launched.
I mentioned this to Mark, and it surprised us both! The intuitive assumption would be that the trajectory knob would primarily affect height, but the results showed that the orange knob’s settings
didn’t significantly affect the height of the launch at all. “That,” Mark told me, “is why it’s good to run empirical studies and not assume anything!”
We hope to see you the next time we’re out and about. Our next planned conference is our 8th European DOE User Meeting in Amsterdam, the Netherlands on June 18-20, 2025. Learn more here, and happy
In a previous Stat-Ease blog, my colleague Shari Kraber provided insights into Improving Your Predictive Model via a Response Transformation. She highlighted the most commonly used transformation:
the log. As a follow up to this article, let’s delve into another transformation: the square root, which deals nicely with count data such as imperfections. Counts follow the Poisson distribution,
where the standard deviation is a function of the mean. This is not normal, which can invalidate ordinary-least-square (OLS) regression analysis. An alternative modeling tool, called Poisson
regression (PR) provides a more precise way to deal with count data. However, to keep it simple statistically (KISS), I prefer the better-known methods of OLS with application of the square root
transformation as a work-around.
When Stat-Ease software first introduced PR, I gave it a go via a design of experiment (DOE) on making microwave popcorn. In prior DOEs on this tasty treat I worked at reducing the weight of
off-putting unpopped kernels (UPKs). However, I became a victim of my own success by reducing UPKs to a point where my kitchen scale could not provide adequate precision.
With the tools of PR in hand, I shifted my focus to a count of the UPKs to test out a new cell-phone app called Popcorn Expert. It listens to the “pops” and via the “latest machine learning
achievements” signals users to turn off their microwave at the ideal moment that maximizes yield before they burn their snack. I set up a DOE to compare this app against two optional popcorn settings
on my General Electric Spacemaker™ microwave: standard (“GE”) and extended (“GE++”). As an additional factor, I looked at preheating the microwave with a glass of water for 1 minute—widely publicized
on the internet to be the secret to success.
Table 1 lays out my results from a replicated full factorial of the six combinations done in random order (shown in parentheses). Due to a few mistakes following the software’s plan (oops!), I added
a few more runs along the way, increasing the number from 12 to 14. All of the popcorn produced tasted great, but as you can see, the yield varied severalfold.
Table 1: Data with run numbers in
A: B: UPKs
Preheat Timing Rep 1 Rep 2 Rep 3
No GE 41 (2) 92 (4)
No GE++ 23 (6) 32 (12) 34 (13)
No App 28 (1) 50 (8) 43 (11)
Yes GE 70 (5) 62 (14)
Yes GE++ 35 (7) 51 (10)
Yes App 50 (3) 40 (9)
I then analyzed the results via OLS with and without a square root transformation, and then advanced to the more sophisticated Poisson regression. In this case, PR prevailed: It revealed an
interaction, displayed in Figure 1, that did not emerge from the OLS models.
Figure 1: Interaction of the two factors—preheat and timing method
Going to the extended popcorn timing (GE++) on my Spacemaker makes time-wasting preheating unnecessary—actually producing a significant reduction in UPKs. Good to know!
By the way, the app worked very well, but my results showed that I do not need my cell phone to maximize the yield of tasty popcorn.
To succeed in experiments on counts, they must be:
• discrete whole numbers with no upper bound
• kept with within over a fixed area of opportunity
• not be zero very often—avoid this by setting your area of opportunity (sample size) large enough to gather 20 counts or more per run on average.
For more details on the various approaches I’ve outlined above, view my presentation on Making the Most from Measuring Counts at the Stat-Ease YouTube Channel.
Your computer's on, you're data's in
Your mind is lost amidst the din
Your palms sweat, your keyboard shakes
Another transform is what it takes
You can't sleep, you can't eat
You have runs, you must repeat
You squeeze the mouse, it starts to squeak
A new analysis will look less bleak
Whoa, you like to think that you're immune to the stats, oh yeah
It's closer to the truth to say you need your fix
You know you're gonna have to face it, you're addicted to Dee Ex | {"url":"https://www.statease.com/blog/category/pop/","timestamp":"2024-11-14T20:11:16Z","content_type":"text/html","content_length":"46878","record_id":"<urn:uuid:c654ed53-8c52-41f8-89bc-013e0ddd7845>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00889.warc.gz"} |
Calculating Present and Future Value of Annuities - The Magic Of Making Up Review 2019
Calculating Present and Future Value of Annuities
Present value is the value today, where future value relates to accumulated future value. The present value of an annuity refers to the present value of a series of future promises to pay or receive
an annuity at a specified interest rate. Discover the scientific investment process Todd developed during his hedge fund days that he still uses to manage his own money today.
As an example, let’s say your structured settlement pays you $1,000 a year for 10 years. You want to sell five years’ worth of payments ($5,000) and the secondary market buying company applies a 10%
discount rate. To use an annuity table effectively, you first need to determine the timing of your payments. Are they received at the end of the contract period, as is typical with an ordinary
annuity, or at the beginning? Because most fixed annuity contracts distribute payments at the end of the period, we’ve used ordinary annuity present value calculations for our examples. An annuity
table, often referred to as a “present value table,” is a financial tool that simplifies the process of calculating the present value of an ordinary annuity.
Future Value of Annuity Calculation Example (FV)
An annuity is a contract between you and an insurance company that’s typically designed to provide retirement income. You buy an annuity either with a single payment or a series of payments, and you
receive a lump-sum payout shortly after purchasing the annuity or a series of payouts over time. If you don’t have access to an electronic financial calculator or software, an easy way to calculate
present value amounts is to use present value tables. You can view a present value of an ordinary annuity table by clicking PVOA Table.
Many websites, including Annuity.org, offer online calculators to help you find the present value of your annuity or structured settlement payments. These calculators use a time value of money
formula to measure the current worth of a stream of equal present value of ordinary annuity tables payments at the end of future periods. A discount rate directly affects the value of an annuity and
how much money you receive from a purchasing company. The present value of annuity is the current worth or cost of a fixed stream of future payments.
When amortizing a loan, what is the difference between the present value and the annuity factor?
We may earn a commission when you click on a link or make a purchase through the links on our site. All of our content is based on objective analysis, and the opinions are our own. Suppose that Black
Lighting Co. purchased a new printing press for $100,000. The quarterly payments are $4,326.24 and the rate is 12% annually (or 3% per quarter). For example, assume that you purchase a house for
$100,000 and make a 20% down payment.
Together, these values can help you determine how much you need to put into an annuity to generate the types of income streams you want out of it. Annuity due refers to payments that occur regularly
at the beginning of each period. Rent is a classic example of an annuity due because it’s paid at the beginning of each month. Similarly, the formula for calculating the present value of an annuity
due takes into account the fact that payments are made at the beginning rather than the end of each period. An annuity due, you may recall, differs from an ordinary annuity in that the annuity due's
payments are made at the beginning, rather than the end, of each period.
Present Value Formulas, Tables and Calculators
Despite this, present value tables remain popular in academic settings because they are easy to incorporate into a textbook. Because of their widespread use, we will use present value tables for
solving our examples. By using the time value of money concept and a few easy calculations, you’ll be able to conceptually pull back all those future payments to understand what they’re worth now.
For example, a court settlement might entitle the recipient to $2,000 per month for 30 years, but the receiving party may be uncomfortable getting paid over time and request a cash settlement. The
equivalent value would then be determined by using the present value of annuity formula. The result will be a present value cash settlement that will be less than the sum total of all the future
payments because of discounting (time value of money).
An annuity table provides a factor, based on time, and a discount rate (interest rate) by which an annuity payment can be multiplied to determine its present value. For example, an annuity table
could be used to calculate the present value of an annuity that paid $10,000 a year for 15 years if the interest rate is expected to be 3%. As seen from these examples, the benefit of these annuity
tables is to quickly calculate the present value of annuities without using the formulas every time. If your annuity promises you a $50,000 lump sum payment in the future, then the present value
would be that $50,000 minus the proposed rate of return on your money.
• Learn financial statement modeling, DCF, M&A, LBO, Comps and Excel shortcuts.
• By the same logic, $5,000 received today is worth more than the same amount spread over five annual installments of $1,000 each.
• This variance in when the payments are made results in different present and future value calculations.
• When t approaches infinity, t → ∞, the number of payments approach infinity and we have a perpetual annuity with an upper limit for the present value.
• You can then look up the present value interest factor in the table and use this value as a factor in calculating the present value of an annuity, series of payments.
• An annuity is a contract between you and an insurance company that’s typically designed to provide retirement income.
Thus, these tables can be used to determine present values for those $20,000 depending on interest rates and the duration of the annuity. They provide the value now of 1 received at the end of each
period for n periods at a discount rate of i%. This problem involves an annuity (the yearly net cash flows of $10,000) and a single amount (the $250,000 to be received once at the end of the
twentieth year). | {"url":"http://www.magicalmakingup.net/calculating-present-and-future-value-of-annuities/","timestamp":"2024-11-01T23:52:33Z","content_type":"text/html","content_length":"38706","record_id":"<urn:uuid:6f42836d-776f-4953-812e-b6d24a31acf5>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00051.warc.gz"} |
trigonometry – Hackaday
An interesting aspect of time-varying waveforms is that by using a trick called a Fourier Transform (FT), they can be represented as the sum of their underlying frequencies. This mathematical insight
is extremely helpful when processing signals digitally, and allows a simpler way to implement frequency-dependent filtration in a digital system. [klafyvel] needed this capability for a project, so
started researching the best method that would fit into an Arduino Uno. In an effort to understand exactly what was going on they have significantly improved on the code size, execution time and
accuracy of the previous crown-wearer.
A complete real-time Fourier Transform is a resource-heavy operation that needs more than an Arduino Uno can offer, so faster approximations have been developed over the years that exchange absolute
precision for speed and size. These are known as Fast Fourier Transforms (FFTs). [klafyvel] set upon diving deep into the mathematics involved, as well as some low-level programming techniques to
figure out if the trade-offs offered in the existing solutions had been optimized. The results are impressive.
Benchmarking results showing speed of implementation versus the competition (ApproxFFT)
Not content with producing one new award-winning algorithm, what is documented on the blog is a masterclass in really understanding a problem and there are no less than four algorithms to choose from
depending on how you rank the importance of execution speed, accuracy, code size or array size.
Along the way, we are treated to some great diversions into how to approximate floats by their exponents (French text), how to control, program and gather data from an Arduino using Julia, how to
massively improve the speed of the code by using trigonometric identities and how to deal with overflows when the variables get too large. There is a lot to digest in here, but the explanations are
very clear and peppered with code snippets to make it easier and if you have the time to read through, you’re sure to learn a lot! The code is on GitHub here.
If you’re interested in FFTs, we’ve seen them before around these parts. Fill your boots with this link of tagged projects.
Apollo 11 Trig Was Brief
In this day and age where a megabyte of memory isn’t a big deal, it is hard to recall when you had to conserve every byte of memory. If you are a student of such things, you might enjoy an annotated
view of the Apollo 11 DSKY sine and cosine routines. Want to guess how many lines of code that takes? Try 35 for both.
Figuring out how it works takes a little knowledge of how the DSKY works and the number formats involved. Luckily, the site has a feature where you can click on the instructions and see comments and
questions from other reviewers.
Looking For Pi In The 8087 Math Coprocessor Chip
Even with ten fingers to work with, math can be hard. Microprocessors, with the silicon equivalent of just two fingers, can have an even harder time with calculations, often taking multiple machine
cycles to figure out something as simple as pi. And so 40 years ago, Intel decided to give its fledgling microprocessors a break by introducing the 8087 floating-point coprocessor.
If you’ve ever wondered what was going on inside the 8087, wonder no more. [Ken Shirriff] has decapped an 8087 to reveal its inner structure, which turns out to be closely related to its function.
After a quick tour of the general layout of the die, including locating the microcode engine and ROM, and a quick review of the NMOS architecture of the four-decade-old technology, [Ken] dug into the
meat of the coprocessor and the reason it could speed up certain floating-point calculations by up to 100-fold. A generous portion of the complex die is devoted to a ROM that does nothing but store
constants needed for its calculation algorithms. By carefully examining the pattern of NMOS transistors in the ROM area and making some educated guesses, he was able to see the binary representation
of constants such as pi and the square root of two. There’s also an extensive series of arctangent and log[2] constants, used for the CORDIC algorithm, which reduces otherwise complex transcendental
calculations to a few quick and easy bitwise shifts and adds.
[Ken] has popped the hood on a lot of chips before, finding butterflies in an op-amp and reverse-engineering a Sinclair scientific calculator. But there’s something about seeing constants hard-coded
in silicon that really fascinates us.
CORDIC Brings Math To FPGA Designs
We are always excited when we see [Hamster] post an FPGA project, because it is usually something good. His latest post doesn’t disappoint and shows how he uses the CORDIC algorithm to generate very
precise sine and cosine waves in VHDL. CORDIC (Coordinate Rotation Digital Computer; sometimes known as Volder’s algorithm) is a standard way to compute hyperbolic and trigonometric functions. What’s
nice is that the algorithm only requires addition, subtraction, bit shifts, and a lookup table with an entry for each bit of precision you want. Of course, if you have addition and negative numbers,
you already have subtraction. This is perfect for simple CPUs and FPGAs.
[Hamster] not only has the VHDL code but also provides a C version if you find that easier to read. In either case, the angle is scaled so that 360 degrees is a full 24-bit word to allow the most
precision. Although it is common to compute the result in a loop, with the FPGA, you can do all the math in parallel and generate a new sample on each clock cycle.
Measure Laser Wavelength With A CD And A Tape Measure
Obviously the wavelength of a laser can’t be measured with a scale as large as that of a carpenter’s tape measure. At least not directly and that’s where a Compact Disc comes in. [Styropyro] uses a
CD as a diffraction grating which results in an optical pattern large enough to measure.
A diffraction grating splits a beam of light up into multiple beams whose position is determined by both the wavelength of the light and the properties of the grating. Since we don’t know the
properties of the grating (the CD) to start, [Styropyro] uses a green laser as reference. This works for a couple of reasons; the green laser’s properties don’t change with heat and it’s wavelength
is already known.
It’s all about the triangles. Well, really it’s all about the math and the math is all about the triangles. For those that don’t rock out on special characters [Styropyro] does a great job of not
only explaining what each symbol stands for, but applying it (on camera in video below) to the control experiment. Measure the sides of the triangle, then use simple trigonometry to determine the
slit distance of the CD. This was the one missing datum that he turns around and uses to measure and determine his unknown laser wavelength.
Continue reading “Measure Laser Wavelength With A CD And A Tape Measure”
Tilt Compensation When Reading A Digital Compass
If you’re familiar with using a compass (the tool that points to magnetic north, not the one that makes circles) the concept of holding the device level makes sense. It must be level for the needle
to balance and rotate freely. You just use your eyes to make sure you’re holding the thing right. Now think of a digital compass. They work by measuring the pull of a magnetic field, and have no
visual method of showing whether they’re level or not. To ensure accurate readings you might use an accelerometer to compensate for a tilted magnetometer.
The process involves taking measurements from both an accelerometer and a magnetometer, then performing calculations with that data to get a true reading. Luckily the equations have been figured out
for us and we don’t need to get too deep into trigonometry. You will, however, need to use sine, cosine, and arctangent in your calculations. These should be available in your programming language of
choice. Arduino (used here) makes use of the avr-libc math library to perform the calculations. | {"url":"https://hackaday.com/tag/trigonometry/","timestamp":"2024-11-01T18:44:42Z","content_type":"text/html","content_length":"107842","record_id":"<urn:uuid:3f048a97-c0e6-4308-8451-ddfa3bebbdb4>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00675.warc.gz"} |
Inspiring Drawing Tutorials
How To Draw 3D Circle
How To Draw 3D Circle - Easy optical illusion drawing ! Web this is a step by step drawing tutorial on how to plot and draw a 3d circle on graph paper so that it pops out of the paper. For this guide
on how to draw a sphere, you will find some drawing tools to be very helpful in completing this drawing. Depending on whether you look at it as being two dimensional or three dimensional, this
animated diagram depicts one of two things: To form a circle, all you need is a pencil and a sheet of paper. [1] sketch pad or paper.
Free online drawing tutorial from circle line art school. This is a basic method for drawing a sphere, so minimal materials are needed. A light will create a bright spot, with a gradient to darker
shadows on the opposite side. How to draw an impossible 3d circle ! Let’s start with a simple object;
Make a circle using a pen or pencil. A circle drawing is probably the shape that most beginners find challenging to draw freehand. The first step is to draw a circle. Prisms (left) and pyramids
(right). How to draw satisfying 3d circle drawings on paper.
Web this is a step by step drawing tutorial on how to plot and draw a 3d circle on graph paper so that it pops out of the paper. It's a trick art or optical illusion. To feel confident about drawing
3d shapes, we need to understand what constitutes a 3d shape drawing. Make a circle using a pen or.
And voilà, you’ve got yourself a 3d cube! For this guide on how to draw a sphere, you will find some drawing tools to be very helpful in completing this drawing. Draw a hollow circle with a light
line. Pendulum waves with 15 balls. 3d trick art on paper.more.
This square will become the front face of a cube. For a little help drawing a circle, trace something round or use a compass. You can use a small bowl, a glass, a mug, or another object with a
circular shape or base. Shortest connection on a cylindrical surface. Web tom mcpherson how to draw step by step:
You could say they each have one face. Read on to learn how you can master this technique for yourself. Make a circle using a pen or pencil. The first step is to draw a circle. All the sides of a
square are the same length.
All the sides of a square are the same length. 4.9k views 3 years ago diy drawing. 3d trick art on paper.more. Free online drawing tutorial from circle line art school. Pendulum waves with 15 balls.
In this lesson, you’re going to learn how to draw a simple optical illusion called the impossible circle. You could say they each have one face. First, a drawing compass will be very helpful because
the closer you can get to a perfect circle the better for this drawing. Gather the materials you need to draw the sphere. For this.
You can use a small bowl, a glass, a mug, or another object with a circular shape or base. Press lightly so you can easily go back and shade in the sphere. Unwrapping a cylinder (surface area) right
triangle trigonometry ratios: The first step is to draw a circle. It may take a couple.
Free online drawing tutorial from circle line art school. Let’s start with a simple object; Creating three dimensions in a shape. Draw the circle so it's as wide as you'd like the sphere to be. Easy
optical illusion drawing !
See how to draw 3d basic shapes, start with a line drawing wire frame and then add some. And voilà, you’ve got yourself a 3d cube! Crafting a sphere on a flat surface is all about shading: [1] sketch
pad or paper. Draw a hollow circle with a light line.
Web circles in 3d space. If you’ve ever tried to draw a perfect circle by hand, you’re probably well aware of. For a little help drawing a circle, trace something round or use a compass. When drawing
a prism, start with a simple, flat triangle and small horizon point that’s at the side of the shape. Trace your circular object.
How To Draw 3D Circle - You could say they each have one face. Prisms (left) and pyramids (right). Web this is a step by step drawing tutorial on how to plot and draw a 3d circle on graph paper so
that it pops out of the paper. This square will become the front face of a cube. Read on to learn how you can master this technique for yourself. Unwrapping a cylinder (surface area) right triangle
trigonometry ratios: 4.9k views 3 years ago diy drawing. For our purposes here in this course, we're going to be using ellipses in 2d space (drawn on the page) to represent circles that exist in 3d
space. From each of the corners of the square, extend a diagonal straight line. This means that it should have depth and be able to stand up on its own.
Web this is a step by step drawing tutorial on how to plot and draw a 3d circle on graph paper so that it pops out of the paper. All the sides of a square are the same length. [1] sketch pad or
paper. To feel confident about drawing 3d shapes, we need to understand what constitutes a 3d shape drawing. How to draw satisfying 3d circle drawings on paper.
Center it on the page, so there is room to draw the rest of the box.< Learn easy 3d circle drawing step by step.awesome 3d circle drawing lesson in a single video. Depending on whether you look at it
as being two dimensional or three dimensional, this animated diagram depicts one of two things: Read on to learn how you can master this technique for yourself.
The first step is to draw a circle. All the sides of a square are the same length. For our purposes here in this course, we're going to be using ellipses in 2d space (drawn on the page) to represent
circles that exist in 3d space.
For this guide on how to draw a sphere, you will find some drawing tools to be very helpful in completing this drawing. Web circles in 3d space. How do you define the planes/faces on a round object?
These Outline The Edges Of The Top, Bottom, And Sides.
Depending on whether you look at it as being two dimensional or three dimensional, this animated diagram depicts one of two things: Web it’s easy to draw 3d objects when they have obvious vertices or
hard edges, but what about weird shapes like circles, blobs, or even people? How to draw an impossible 3d circle ! Obviously, you can use a.
Web Circles In 3D Space.
Unwrapping a cylinder (surface area) right triangle trigonometry ratios: Creating three dimensions in a shape. Shortest connection on a cylindrical surface. Then you will want some pencils, at least
one light and one dark.
This Means That It Should Have Depth And Be Able To Stand Up On Its Own.
A circle drawing is probably the shape that most beginners find challenging to draw freehand. The simplest and easiest method i world recommend is to draw a continuous fluid line. This is a basic
method for drawing a sphere, so minimal materials are needed. Learn easy 3d circle drawing step by step.awesome 3d circle drawing lesson in a single video.
Make A Circle Using A Pen Or Pencil.
Ensure that the draw from the arm and elbow will make the process of drawing a. A light will create a bright spot, with a gradient to darker shadows on the opposite side. It may take a couple. Easy
optical illusion drawing ! | {"url":"https://one.wkkf.org/art/drawing-tutorials/how-to-draw-3d-circle.html","timestamp":"2024-11-08T02:22:14Z","content_type":"text/html","content_length":"33350","record_id":"<urn:uuid:3bb82c45-06eb-4b09-a45a-39e16d58a9dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00268.warc.gz"} |
Clustering Results Repository (v1.1.0)
Clustering Results Repository (v1.1.0)¶
We have prepared a repository of clustering results for problems from our Benchmark Suite (v1.1.0).
A non-interactive results catalogue is available here.
Currently, the outputs of the following methods are included. Where applicable, we considered a wide range of control parameters.
• k-means, Gaussian mixtures, spectral, and Birch available in sklearn 0.23.1 (Python) [6, 45];
• hierarchical agglomerative methods with the average, centroid, complete, median, Ward, and weighted/McQuitty linkages implemented in fastcluster 1.1.26 (Python/R) [44]
• genieclust 1.0.0 (Python/R) [20, 25] (note that Genie with g=1.0 is equivalent to the single linkage algorithm);
• Genie+Ic – Genie+Information Criterion, see [27], as implemented in genieclust 1.0.0 (Python/R).
• fcps_nonproj – many algorithms, see [51], available via the FCPS 1.3.4 package for R, which provides a consistent interface to many other R packages (versions current as of 2023-10-21). We
selected all which return an a priori-given number of clusters, and do not rely on heavy feature engineering/fancy data projections, as such methods should be evaluated separately. We did not
include the algorithms that are available in other packages and are already part of this results repository, e.g., fastcluster, genieclust, and scikit-learn.
• ITM git commit 178fd43 (Python) [43] – an “information-theoretic” algorithm based on minimum spanning trees;
• optim_cvi — local maxima (great effort was made to maximise the probability of their being high-quality ones) of many internal cluster validity measures, including the Caliński–Harabasz, Dunn, or
Silhouette index; see [26];
• optim_cvi_mst_divisive – maximising internal cluster validity measures over Euclidean minimum spanning trees using a divisive strategy; see [27].
New results will be added in the future (note that we can only consider methods that allow for setting the precise number of generated clusters). New quality contributions are welcome.
Feature Engineering¶
The algorithms operated on the original data spaces, i.e., subject to only some mild preprocessing:
• columns of zero variance have been removed;
• a tiny amount of white noise has been added to each datum to make sure the distance matrices consist of unique elements (this guarantees that the results of hierarchical clustering algorithms are
• all data were translated and scaled proportionally to assure that each column is of mean 0 and that the total variance of all entries is 1 (this is not standardisation).
Note, however, that spectral clustering and Gaussian mixtures can be considered as ones that modify the input data space.
Overall, comparisons between distance-based methods that apply automated feature engineering/selection and those that only operate on raw inputs are not exactly fair. In such settings, the classical
methods should be run on the transformed data spaces as well. This is left for further research (stay tuned). | {"url":"https://clustering-benchmarks.gagolewski.com/weave/results-v1.html","timestamp":"2024-11-06T23:09:36Z","content_type":"text/html","content_length":"28751","record_id":"<urn:uuid:588a3825-06a4-456d-a4ba-4df8f05f847a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00706.warc.gz"} |
Guilherme A.
What do you want to work on?
About Guilherme A.
Algebra, Elementary (3-6) Math, Midlevel (7-8) Math
Bachelors in Chemical Engineering from Universidade de São Paulo - USP
Math - Elementary (3-6) Math
This tutor was really nice and encouraging! They explained to me the rules and how to do each operation step by step!
Math - Elementary (3-6) Math
He/she was really helpful.
Math - Elementary (3-6) Math
it was great I would use his again if I needed help.
Math - Midlevel (7-8) Math
My tutor was great! He/she helped me learn how to plot on coordinated planes. | {"url":"https://ws.princetonreview.com/academic-tutoring/tutor/guilherme%20a--11251922","timestamp":"2024-11-06T04:40:30Z","content_type":"application/xhtml+xml","content_length":"182273","record_id":"<urn:uuid:83b40521-0750-4f8c-a5e4-96a5d890564e>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00599.warc.gz"} |
A(nother) nice use of Gen Functions
In a
prior post
I tried to give a simple example of a proof that uses Gen Functions where there was no other way to do it. For better or worse, before I posted it, my HS student Sam found a better way and I posted
both proofs.
I have another example. Noga Alon showed this to be over dinner at the Erdos 100th Bday conference. (He claims that the proof he showed me is NOT his but he doesn't know whose it is. I will still
call it Noga's Proof for shorthand.)
A+A = { x+y : x,y ∈ A}
A+*A = { x+y : x,y ∈ A and x ≠ y }
We take both to be multisets.
Assume A is a set of natural numbers. When does A+*A determine A?
If A is of size 2 then NO, A+*A does not determine A as we could have x+y=5 but not know if A is {1,4} or {2,3}.
What if A is of size 3? Then YES:
First determine S=((x+y)+(x+z)+(y+z))/2=x+y+z.
Then determine
x = S - (y+z)
y = S - (x+z)
z = S - (x+y)
What if A has four elements? Does there exists A,B of size 4, different, such that A+*A=B+*B?
A = {1,4,12,13}
B = {2,3,11,14}
For which n does does A+*A, where A is of size n, determine A?
Selfridge and Strauss showed that this happens iff n is NOT a power of two. I have a
write up
Noga's proof. The original proof, in
this paper
, does not use gen functions and also applies to sets of complex numbers. I think Noga's proof can be modified to apply here. Which proof is better? A matter of taste; however, Noga's proof can be
sketched on a greasy paper placemat in an outdoor restaurant in Budapest while the original proof cannot.
2 comments:
1. That's a nice proof! As for complex numbers, just having the proof for natural numbers seems to give away everything.
Claim 1: The result holds for integers.
Proof: Translate A so that it now holds natural numbers, and notice that no information is lost.
Claim 2: The result holds for Z^k.
Proof idea: Let A be a set of k-vectors, with maximum element M (in absolute value). Treat the vectors in A as (something like) base-(4M+1) representations of integers, and let A' be the
corresponding set of integers. Given A +* A, we can translate to A' +* A', then recover A', and then recover A.
Claim 3: The result holds for complex numbers.
Proof: View C as a vector space over Z and apply claim 2. (Note that this doesn't use the axiom of choice, since a finite set A only generates a finite-dimensional vector space over Z.)
2. So, one thing I didn't understand in your proof (or Noga's proof or whatever): In page 3, when you divide both sides of Equation by (x-1)^m, you can only do that if $x$ is not 1. So, you get a
condition on the side (that you don't mention) and that says the resulting equation is true for $x \neq 1$. Now, you put $x$ to be 1 which is in direct contradiction with that condition on the
side and you get to your desired result this way.
While the final result happens to be the same as the original way of doing it, your proof (or Noga's proof or ...) seems not to be valid.
Please tell me if I'm missing something .... | {"url":"https://blog.computationalcomplexity.org/2013/07/another-nice-use-of-gen-functions.html?m=1","timestamp":"2024-11-09T00:38:14Z","content_type":"application/xhtml+xml","content_length":"61969","record_id":"<urn:uuid:3638abd8-318b-4a2a-a771-2909bc6f9a95>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00597.warc.gz"} |
Teleological argument について
Teleological argument ：ウィキペディア英語版
Teleological argument
The teleological or physico-theological argument, also known as the argument from design, or intelligent design argument is an argument for the existence of God or, more generally, for an intelligent
creator "based on perceived evidence of deliberate design in the natural or physical world".〔''Oxford English Dictionary, Phrase P3 for ''Design'': "argument from design n. Theol. an argument for
the existence of an intelligent creator (usually identified as God) based on perceived evidence of deliberate design in the natural or physical world".〕〔Francisco J. Ayala (2006), Review article of
"The Blasphemy of Intelligent Design: Creationism's Trojan Horse. The Wedge of Intelligent Design" by Barbara Forrest; Paul R. Gross ''History and Philosophy of the Life Sciences'', Vol. 28, No. 3,
pp. 409-421, URL: http://www.jstor.org/stable/23334140 .: "The argument from design to demonstrate God's existence, now called the 'Intelligent Design' argument (ID) is a two-tined argument. The
first prong asserts that the universe, humans, as well as all sorts of organisms, in their wholes, in their parts, and in their relations to one another and to their environment, appear to have been
designed for serving certain functions and for certain ways of life. The second prong of the argument is that only an omnipotent Creator could account for the perfection and purposeful design of the
universe and everything in it."〕 It is historically closely associated with the concept of Natural Theology.
The earliest recorded versions of this argument are associated with Socrates in ancient Greece, although it has been argued that he was taking up an older argument.〔Ahbel-Rappe, S. and Kamtekar, R.,
''A Companion to Socrates'', John Wiley & Sons, 2009, p. 45. "Xenophon attributes to Socrates what is probably the earliest known natural theology, an argument for the existence of the gods from
observations of design in the physical world.".〕〔Sedley (2007) agrees (page 86), and cites other recent commentators who agree, and argues in detail that the argument reported by Xenophon and Plato
is "at any rate the antecedent" of the argument from design (page 213). He shows that the Stoics frequently paraphrased the account given by Xenophon.〕 Plato, his student, and Aristotle, Plato's
student, developed complex approaches to the proposal that the cosmos has an intelligent cause, but it was the Stoics who, under their influence, "developed the battery of creationist arguments
broadly known under the label "The Argument from Design"".〔Sedley 2007, page xvii.〕
Socratic philosophy influenced the development of the Abrahamic religions in many ways, and the teleological argument has a long association with them. In the Middle Ages, Islamic theologians such as
Al Ghazali used the argument, although it was rejected as unnecessary by Quranic literalists, and as unconvincing by many Islamic philosophers. Later, the teleological argument was accepted by Saint
Thomas Aquinas and included as the fifth of his "Five Ways" of proving the existence of God. In early modern England clergymen such as William Turner and John Ray were well-known proponents. In the
early 18th century, William Derham published his ''Physico-Theology'', which gave his "demonstration of the being and attributes of God from his works of creation".〔Derham, W., ''Physico-Theology'',
1713〕 Later, William Paley, in his 1802 work on natural theology, published a prominent presentation of the design argument with his version of the watchmaker analogy and the first use of the phrase
"argument from design".〔''Oxford English Dictionary'' under "Design", substantive number 4.〕
From the beginning, there have been numerous criticisms of the different versions of the teleological argument, and responses to its challenge to the claims against non-teleological natural science.
Especially important were the general logical arguments made by David Hume in his ''Dialogues Concerning Natural Religion'', published 1779, and the explanation of biological complexity given in
Charles Darwin's ''Origin of Species'', published in 1859.〔(The Oxford Handbook of Natural Theology ), page 3, for example: "Between them, so the story goes, Hume, Darwin and Barth pulled the rug
out from underneath the pretensions of natural theology to any philosophical, scientific, or theological legitimacy".〕 Since the 1960s, Paley's arguments, including the words "intelligent design",
have been influential in the development of a creation science movement, especially the form known as the intelligent design movement, which not only uses the teleological argument to argue against
the modern Darwinian understanding of evolution, but also makes the philosophical claim that it can provide a basis for scientific proof of the divine origin of biological species.〔
Also starting already in classical Greece, two approaches to the teleological argument developed, distinguished by their understanding of whether the natural order was literally created or not. The
non-creationist approach starts most clearly with Aristotle, although many thinkers, such as the Neoplatonists, believed it was already intended by Plato. This approach is not creationist in a simple
sense, because while it agrees that a cosmic intelligence is responsible for the natural order, it rejects the proposal that this requires a "creator" to physically make and maintain this order. The
Neoplatonists did not find the teleological argument convincing, and in this they were followed by medieval philosophers such as Al-Farabi and Avicenna. Later, Averroes and Thomas Aquinas considered
the argument acceptable, but not necessarily the best argument. In contrast to the approach of such philosophers and theologians, the intelligent design movement makes a creationist claim for an
intelligence that intervenes in the natural order.
== History ==
While the concept of an intelligence in the natural order goes back at least to the beginnings of philosophy and science, the concept of a designer of the natural world, or a creating intelligence
which has human-like purposes, appears to have begun with classical philosophy.〔 Religious thinkers in Judaism, Hinduism, Confucianism, Islam and Christianity also developed versions of the
teleological argument. Later, variants on the argument from design were produced in Western philosophy and by Christian fundamentalism.
抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
■ウィキペディアで「Teleological argument」の詳細全文を読む
翻訳と辞書 : 翻訳のためのインターネットリソース
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Notes on MTH 232 and Julia
12 Notes on MTH 232 and Julia
12.0.1 Calculus II material
See the projects at https://github.com/mth229/232-projects. They can be used through Launch
An assignment for this material: ipynb view
• Applications of the integral: area between two curves, volume of solids of revolution, other volumes
An assignment for this material: ipynb view
• Techniques of integration: substitution, integration by parts, partial fractions
An assignment for this material: ipynb view
An assignment for this material: ipynb view
• Parametric equations and polar coordinates
An assignment for this material: ipynb view | {"url":"https://mth229.github.io/mth232.html","timestamp":"2024-11-09T01:22:37Z","content_type":"application/xhtml+xml","content_length":"31807","record_id":"<urn:uuid:d4ae36f0-5cb6-4652-b616-0a23363d8367>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00537.warc.gz"} |
Basic Concepts of Generative Arts: Random Walks - anorganic.org
Have you ever taken a stroll through a bustling city, wandering down streets without a specific destination in mind? Imagine that each step you take is determined by chance, leading you down
unexpected paths and through unforeseen twists and turns. This seemingly aimless journey is a metaphorical representation of a mathematical concept known as a “random walk.”
Random walks are captivating mathematical models that find applications in a multitude of fields, from finance and physics to biology and computer science. In this introductory exploration, we’ll
embark on a fascinating journey into the world of random walks, unveiling the beauty hidden within the seemingly chaotic nature of randomness.
One of the intriguing aspects of random walks is their diverse range of applications. In finance, random walks serve as the foundation for modeling stock prices, capturing the inherent uncertainty
and unpredictability of the market. The famous “random walk hypothesis” suggests that stock prices follow a random path, making it impossible to predict future movements with certainty.
Beyond the realm of finance, random walks play a crucial role in physics, particularly in understanding the motion of particles. Brownian motion, a specific type of random walk, explains the erratic
movement of microscopic particles suspended in a fluid. Albert Einstein himself contributed to this field by providing a theoretical explanation for Brownian motion, earning him the Nobel Prize in
Physics in 1921.
Random walks also offer a unique lens through which we can view the natural world. Biological systems often exhibit behavior that can be modeled using random walks, such as the foraging patterns of
animals or the movement of molecules within a cell. By applying mathematical principles to these phenomena, scientists gain valuable insights into the underlying mechanisms of life.
In this blog series, we will delve deeper into the intricacies of random walks, exploring different types, their mathematical foundations, and their real-world applications. Whether you’re a seasoned
mathematician, a curious student, or someone intrigued by the beauty of randomness, there’s something captivating about the unpredictable paths that random walks reveal.
Understanding the Basics
At its core, a random walk is a mathematical concept that describes a path consisting of a series of random steps.
It basically is a mathematical abstraction that mirrors the unpredictability of real-world processes. Imagine taking a series of steps, each determined by a random choice of direction—left, right,
forward, or backward. The cumulative effect of these chance-driven steps creates a path that embodies the essence of randomness. This concept finds applications across diverse fields, from finance to
physics and biology, offering insights into the dynamic and often unpredictable nature of various phenomena.
Pseudocode Illustration:
# Pseudocode for a simple 1D random walk
initialize position at the origin
initialize number of steps
for step in range(number_of_steps):
# randomly choose a direction (left or right)
direction = random.choice([-1, 1])
# take a step in the chosen direction
position = position + direction
# print the current position
print(“Step:”, step+1, “Position:”, position)
Starting at the left and walking towards the right side using the pseudocode above and iterating results in the following images:
One random walk.
Five random walks.
Ten random walks.
A lot more random walks. | {"url":"https://anorganic.org/random-walks/","timestamp":"2024-11-08T12:33:56Z","content_type":"text/html","content_length":"62706","record_id":"<urn:uuid:7c3003be-a111-4bc7-99a8-e349daf6f287>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00528.warc.gz"} |
From Organic Design wiki
Computers have multiplied by millions of times in their power over the last decades, but still there are many areas in which they seem to have not gained much ability, for example they still seem
light years away from being able to partake in spoken conversation or learn and apply new concepts. At first it was thought to be a lack of computational power or a lack of understanding of the
neural network systems our brains employ to do such tasks. But nowadays even home computers are far more powerful than any brain and neural networks are understood well and shown to be Turing
machines - they do not perform any operations that can't be performed by any other computational machine of sufficient power.
It turns out that this class of computationally insurmountable problems that brains seem to be able to perform with such ease - are all expressible as optimisation problems, which basically arrange
the entire set of feasible solutions into a "landscape" where the lowest points represent optimal solutions. Using classical techniques, it's very difficult to know whether a low point in the space
(a local minimum) is a comparatively good one or not without searching the entire space, which is usually computationally infeasible or impossible, so many techniques such as simulated annealing have
been developed to try to find globally good minima at a practical cost. But even with far superior computational power, no classical methods have been achieved that solve the difficult problems
anywhere near as effectively as the brain.
Recent advances in quantum computation have shown that quantum computers are extremely proficient when it comes to solving optimisation problems. They use a process called quantum annealing, which
basically employs the quantum tunnelling effect to find successively lower minima in the solution space. Usually quantum tunnelling applies to confined particles that can escape confinement by taking
advantage of the Heisenburg uncertainty principle, which allows them to have a probability of existing outside their confined area. In a quantum computer, this effect can be applied to the current
point of focus in a search space rather than to a particle in physical space. The diagram to the right illustrates the basic idea.
There is very strong evidence that suggests that this is the process our brains actually use to achieve these seemingly impossible computational tasks; also many enzymes are known to use quantum
tunnelling - and even the anti-oxidation action of common green tea has been shown to use the quantum tunnelling effect. In the brain, the quantum annealing process is the slow problem-solving
process that helps to form the neural connectivity, whereas the real-time stimulus-response side is done by the neural network. Many people learn to employ their quantum side from an early age,
realising that there's an aspect of their mind to which they can hand difficult problems and from which they receive back deep insight hours or days later; it's the quantum computer in our brains
that processes the tasks we put into this "too hard basket".
Quantum chips
Rather than having a dedicated "quantum computer" as such, the more likely way that the technology will develop is in the form of co-processor chips that would form a part of the existing computers
or chips. This is because quantum computation is not very good at normal computation, it only really specialises in solving the problems that can be expressed as a specific class of optimisation
problems as described above.
The chips being developed now, such as the 28-qubit adiabatic chip developed by D-Wave Systems shown to the right, are not powerful enough to be of real use in practical applications and are solely
being used for research and development of further quantum computing technology.
When the technology is mature, computers will most likely exhibit a dedicated quantum chip, or perhaps future CPU's will exhibit quantum cores in addition to their standard processing cores.
Computers equipped with such technology should be vastly different in usability from our current ones, for starters, we should be able to converse with them as we would any other human, thereby
requiring only minimal use of a keyboard and mouse.
This would be far more than simple voice recognition; by bringing actual understanding of the conversation by the computer into the realm of feasibility. Imagine a day when you could say high level
things to your computer like, "computer, could you please invite my friends over for drinks on Friday next week? and if anyone hasn't responded by Wednesday, send a reminder".
Quantum encryption
Determining the prime factors of a number is an example of a problem frequently used to ensure cryptographic security in encryption systems; this problem is believed to require superpolynomial time
in the number of digits - it is relatively easy to construct a problem that would take longer than the known age of the Universe to calculate on current computers using current algorithms.
Shor's algorithm is a quantum algorithm for integer factorization. On a quantum computer, Shor's algorithm takes time O((log N)^3) to factor an integer N. This is exponentially faster than the
best-known classical factoring algorithm, which works in time about [math]O(2^{{(\log N)}^{1/3}})[/math]. Peter Shor discovered the eponymous algorithm in 1994.
Shor's algorithm is important because it breaks a widely used public-key cryptography scheme known as RSA. RSA is based on the assumption that factoring large numbers is computationally infeasible.
So far as is known, this assumption is valid for classical computers. No classical algorithm is known that can factor in time polynomial in log N. However, Shor's algorithm shows that factoring is
efficient on a quantum computer, so a quantum computer could "break" RSA.
A way out of this dilemma would be to use some kind of quantum cryptography. There are also some digital signature schemes that are believed to be secure against quantum computers, such as Lamport
Grover's algorithm can also be used to obtain a quadratic speed-up [over a brute-force search] for the NP-complete class of problems.
See also
Quantum non-locality
Related news
See also | {"url":"https://organicdesign.nz/wiki/index.php?title=Quantum_computation&oldid=129574","timestamp":"2024-11-12T19:19:33Z","content_type":"text/html","content_length":"43012","record_id":"<urn:uuid:9eb5563e-f993-46fb-8635-fb72798d1b13>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00873.warc.gz"} |
Single linked Lists - Data structure
linked list is a data structure that having a list of elements and each element has a reference pointing to the next element in the list.
First thing we have to know about linked list is we have a head pointer that points to a node that then has some date and points to another node and so on till a node that dose not point to any
And node contains 2 elements ,the value or the key (number like 7) and a pointer that points to the next node that contain (another number like 10 for example)
7 →10
There are operations that can be done on linked list called List API
1-On the front of the list:
1-PushFront(key): add to front
2-Key TopFront(): return the front element
3-PopFront(): remove the front element
2-On the end of the list
1-PushBack(key): add to back … also known a Append
2-Key TopBack(): return the back element
3-PopBack(): remove the back element
There is difference between the front and back operation in run time
3-We can also do:
1-Boolean Find(Key): is key in a list?
2-Erase(Key): remove key from list
3-Boolean Empty(): empty list?
4-AddBefore(Node,Key): adds key before Node
5-AddAfter(Node,Key): adds key after Node
we have an empty list ,do this following operations:
PushBack(a); a
PushFront(b); b a
PushBack(d); b a d
PushFront(c); c b a d
PopBack(); c b a
so,the final result is => c b a
The run time for some operations
PushFront(key): add to front in O(1)
Key TopFront(): return the front element in O(1)
PopFront(): remove the front element O(1)
PushBack(key): add to back in O(n) without no tail
PushBack(key): add to back in O(1) with tail
Key TopBack(): return the back element in O(n) without tail
Key TopBack(): return the back element in O(1) with tail
PopBack(): remove the back element in O(n)
WHEN we want to push at the back?
1-If We don’t have a tail pointer, So this going to be expensive operation. WHY? because we start at the head and walk our way down the list untill we get to the end and in the end we add a node so
its going to be O(n). Similarly if we want to TopBack or PopBack
2-If we had a tail pointer, some of this will become simpler. WHY? because we are going to have both a head pointer that points to the head element ,and tail pointer that points to the tail element.
So the first element and the last element became cheap operations.
For example: when we have tail pointer and we want to insert element
-We first allocate a node and put in our new key
-Then update the next pointer of the current tail to point to this new tail
-Then update the tail pointer itself
-And the run time will be O(1)
WHEN we have tail pointer and we want to delete or PopBack element:
-Here we don’t have pointer that points from the last element to the previous last (the element before)
-We only have a pointer from the previous to the last element and that pointer does not help us going back
-So what we have got to do is:
-We start from the head and walk our way down until we find the previous last element or node that points to the current tail (last element)
-Then update our tail pointer to point to that (previous last element)
-Then update the tail pointer to be null, this means tail pointer points to the previous last element which it became the tail element
-Then we can remove the last element
-And the final run time will be O(n) Because we have got to walk all the way down
Single Linked list algorithm
node <- new node
node.next <- head
head <- node
if tail = nil: //an empty list
tail <- head
if head = nil:
ERROR: empty list
head <- head.next
if head = nil:
tail <- nil
node <- new node
node.key <- key
node.next = nil
if tail = nil;
head <- tail <- node
tail.next <- node,
tail <- node
if head = nil :
ERROR: empty list
if head = tail
head <- tail <- nil
p <- head
while p.next.next != nil:
p <- p.next
p.next <- nil;
tail <- p
Add after a node: AddAfter(node,key)
node <- new node
node.key <- key
node.next = node.next
node.next = node
if tail = node: //one element in list and this is the same node that we want to add key after that
tail <- node
Adding before have the same problem we had with Pop back that we don’t have a link back to the previous element so AddBefore would be an O(n)
Here you can find all main operations that we can do on single linked list:
using namespace std;
//built node .... node = (data and pointer)
struct node
int data; //data item
node* next; //pointer to next node
//built linked list
class linkedlist
node* head;
linkedlist() //constructor
head=NULL; //head pointer points to null
void addElementFirst(int d);
void addElementEnd(int d);
void addElementAfter(int d,int b);
void deleteElement(int d);
void display();
//Push front code
void linkedlist::addElementFirst(int d)
node* newNode=new node;
//Push back code
void linkedlist::addElementEnd(int x)
node* newNode=new node;
node* temp=new node;
//if d=5,key=2
void linkedlist::addElementAfter(int d,int key)
node* search=new node;
node* newNode=new node;
cout<<"The link not inserted because there is not found the node "<<d<<" in the LinkedList. "<<endl;
void linkedlist::deleteElement(int d)
node* del;
cout<<"Linked list is empty"<<endl;
cout<<"Is not here, So not deleted."<<endl;
//if here more one nodes...one node points to another node ... bigger than 2 nodes .. at least 2 nodes
cout<<"Is not here, So not deleted."<<endl;
//void linkedlist::display(node *head)
void linkedlist::display()
int n=0; //counter for number of node
node* current=head; //current is pointer points to where head point
if (current==NULL)
cout<<"This is empty linked list."<<endl;
while(current!=NULL) //until current reach to null(last element)
cout<<"The node data number "<<++n<<" is "<<current->data<<endl;
int main()
linkedlist li; //li is object from linkedlist class
li.display(); //empty list
li.addElementFirst(25); //head->25->NULL
li.addElementFirst(36); //head->36->25->NULL
li.addElementFirst(49); //head->49->36->25->NULL
li.addElementFirst(64); //head->64->49->36->25->NULL
cout<<"After adding in the first of linked list"<<endl;
//head->64->49->36->25->NULL //current linked list from prev addElementFirst method
cout<<"After adding in the end of linked list"<<endl;
li.addElementEnd(25); //head->25->NULL
li.addElementEnd(36); //head->25->36->NULL
li.addElementEnd(49); //head->25->36->49->NULL
li.addElementEnd(64); //head->25->36->49->64->NULL
cout<<"linked list after adding 10 after node that has data = 49"<<endl;
cout<<"linked list after adding deleting 49"<<endl;
//Notice :delete the first 49 ... not permission for duplicates
return 0;
Last thing to say, Thanks for reading and I hope it would help and don’t hesitate to send your feedback and suggestions :))
Source code in C++ here
Top comments (0)
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/mkhy19/all-operations-and-algorithms-on-single-linked-lists-in-c-1h92","timestamp":"2024-11-06T01:30:35Z","content_type":"text/html","content_length":"84074","record_id":"<urn:uuid:141affed-539d-4c2d-bfc0-f54ee5cf4db5>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00148.warc.gz"} |
Just Half of Politicians Can Correctly Answer This Basic Statistics Question
It may or may not surprise you to learn that only about half of UK MPs can correctly answer a basic probability question. That figure, believe it or not, is actually higher than when legislators were
asked the same question ten years ago. The findings come from a new poll by the Royal Statistical Society, which questioned 101 members of parliament (MPs) in the United Kingdom a simple statistics
question: what is the likelihood of getting two heads if you toss a coin twice?
One coin flip has a 50% probability of landing on heads. Because two throws are independent events, you must compound the two probabilities: 50 percent times 50 percent, giving you a result of 25%.
However, only 52% of MPs who were polled gave the correct response, with the remaining 32% providing the inaccurate answer of 50%.
Politicians who had been in office for longer were more likely to deliver the correct response than those who had been elected lately. Between 2001 and 2009, up to 68 percent of MPs who took office
gave the correct answer, compared to 38 percent of MPs elected in 2019. Despite what a first glance at the headlines might suggest, things have actually improved over the last decade. In a comparable
poll conducted in 2011, 97 MPs were asked the same question, and only 40% of them correctly answered.
Another statistical question was posed of politicians in the most recent 2021/2022 poll: you roll a six-sided die, and the rolls are 1,3,4,1 and 6. What are the values for the mean and mode? Only 64
percent correctly identified the mean value as three, and only 63 percent correctly identified the mode as one. A third question assessed their statistical expertise, which they could readily apply
to the COVID-19 epidemic (which they’ve been in responsible of for the past two years).
Consider the following scenario: suppose there was a virus diagnostic test. The false-positive rate (the number of persons who test positive even though they don’t have the virus) is one in 1,000.
You took the test and got a positive result. What is the likelihood that you are infected with the virus? You’ll need three pieces of crucial information to answer this question correctly: the
false-positive rate, false-negative rate, and viral prevalence, yet the question only offered one of these figures.
Only 16% of legislators correctly answered, “There isn’t enough information to know.” “Statistical skills are essential for effective decision-making and scrutiny.” While we’re pleased to see that
MPs’ statistical knowledge appears to have improved, the survey results show that more needs to be done to ensure our elected representatives have the statistical skills required for their jobs,”
said Stian Westlake, Chief Executive of the Royal Statistical Society, in a statement. | {"url":"https://qsstudy.com/just-half-of-politicians-can-correctly-answer-this-basic-statistics-question/","timestamp":"2024-11-14T22:03:09Z","content_type":"text/html","content_length":"26731","record_id":"<urn:uuid:decb65f3-3153-4e2d-aadc-09fcfd18a3fd>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00539.warc.gz"} |
Distributing Graph States Over Arbitrary Quantum Networks | LIP6 - QI Team
Multipartite entangled states are great resources for quantum networks. In this work we study the distribution, or routing, of entangled states over fixed, but arbitrary, physical networks. Our
simplified model represents each use of a quantum channel as the sharing of a Bell pair; local operations and classical communications are considered to be free. We introduce two protocols to
distribute respectively Greenberger-Horne-Zeilinger (GHZ) states and arbitrary graph states over arbitrary quantum networks. The GHZ states distribution protocol takes a single step and is optimal in
terms of the number of Bell pairs used; the graph state distribution protocol uses at most twice as many Bell pairs and steps than the optimal routing protocol for the worst case scenario.
Distributing Graph States Over Arbitrary Quantum Networks
Multipartite entangled states are great resources for quantum networks. In this work we study the distribution, or routing, of entangled states over fixed, but arbitrary, physical networks. Our
simplified model represents each use of a quantum channel as the sharing of a Bell pair; local operations and classical communications are considered to be free. We introduce two protocols to
distribute respectively Greenberger-Horne-Zeilinger (GHZ) states and arbitrary graph states over arbitrary quantum networks. The GHZ states distribution protocol takes a single step and is optimal in
terms of the number of Bell pairs used; the graph state distribution protocol uses at most twice as many Bell pairs and steps than the optimal routing protocol for the worst case scenario. | {"url":"https://qi.lip6.fr/publication/2163726-distributing-graph-states-over-arbitrary-quantum-networks/","timestamp":"2024-11-13T14:51:59Z","content_type":"text/html","content_length":"15637","record_id":"<urn:uuid:974a7abd-526b-4c28-a356-74359511b1d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00484.warc.gz"} |
Re: NIntegrate where terms of integrand have unknown constant coefficients
• To: mathgroup at smc.vnet.net
• Subject: [mg8931] Re: [mg8882] NIntegrate where terms of integrand have unknown constant coefficients
• From: David Withoff <withoff>
• Date: Sat, 4 Oct 1997 22:08:09 -0400
• Sender: owner-wri-mathgroup at wolfram.com
> I'm trying to do something along the lines of NIntegrate[c[1] f[1][x] +
> c[2] f[2][x] + ..., {x, 0, a}], where c[n_] is unknown, but the f[n_] are
> defined so that NIntegrate[f[n][x], {x, 0, a}] would work. I'd like to be
> able to do the numerical integration, and keep the coefficients, so I'd get
> as an answer c[1] NIntegrate[f[1][x], {x, 0, a}] + c[2] NIntegrate[f[2][x],
> {x, 0, a}] + ... with all the NIntegrate's evaluated. The constant
> coefficients are all of the same form c[n] (actually, each term will have
> two coefficients c[n1] c[n2]).
> Is there some way of doing this by changing the definition of NIntegrate so
> it will automatically acheive this, or do I have to do something more
> complicated? I'm not at all sure here, as I've never tried adding to the
> definitions of complicated objects like NIntegrate.
> Any help or suggestions would be very gratefully accepted!
> Thanks, Scott Morrison
> scott at morrison.fl.net.au
I would do this by defining your own function that performs the
symbolic linearity operations before calling NIntegrate, rather
than by redefining NIntegrate. For example
In[1]:= int[p_Plus, q_] := Map[int[#, q] &, p]
In[2]:= int[(p:c[_]) f_, q_] := p NIntegrate[f, q]
In[3]:= int[c[1] x + c[2] x^2, {x, 0, 1}]
Out[3]= 0.5 c[1] + 0.333333 c[2]
This strategy could be made considerably more elaborate to do
almost anything that you might want.
Dave Withoff
Wolfram Research | {"url":"https://forums.wolfram.com/mathgroup/archive/1997/Oct/msg00408.html","timestamp":"2024-11-14T14:15:01Z","content_type":"text/html","content_length":"31699","record_id":"<urn:uuid:90e09496-caea-4891-8177-88478200c27a>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00419.warc.gz"} |
Competent Criticism Is My Greatest Hope! - an Astronomy Net God & Science Forum Message
You had me worried as I think you seldom say anything is logically wrong unless you have very strong reason to believe it. You seem to understand my reasoning as to why any attempt to discover the F
(x) = 0 algorithm, is doomed to failure which shows an understanding of the issue.
With regard to the passage, [ "The absolute best one can hope to do is to predict the probability of observing a given set of data as a function of time. Certainly the proposal is testable and is
meaningful when seen against the observations examined prior to the test. Thus it is that the only meaningful solution to our problem of understanding the universe constitutes finding an algorithm
which, for any given pattern of data, yields the probability of observing that data at time t"], I think there is a slight misunderstanding of exactly what I am saying.
>>>Hmm... I see a problem here. Your usage of probability doesn't seem right. Strictly speaking, the probability of seeing a particular pattern is given by the number of possible patterns, not by a
finite sample. If I throw dice 100 times and don't get a single 6, the probability of getting a 6 in my next throw is still 1/6, unless you assume there's something influencing the outcome of the
throw. >OK, I know you're not stupid. However, I was bothered by this: >But when you put the "complications" back you no longer have Schrodinger's equation. f1(x) + f2(x) is not "f1(x) plus
something", it's an entirely different function especially as f2(x) is undefined.>Also, if my knowledge of physics is not betraying me, the solution to Schrodinger's equation must be a wave function.
That means it must be cyclical. That means if you know one period, you know all of them. Hardly an undefined set of data. >By now, your solution only applies to a predictable universe where things
happen in cycles. Why are you claiming that physics applies to any set of data and is, therefore, meaningless? >What prompted me to re-examine your paper was your failure to explain to me how the
conservation of energy and inverse square laws are tautologies within the context of classical mechanics. I am left with the impression that you have not arrived at that conclusion yourself and
decided to rely on the work of others instead. Isn't that inconsistent with your position? >Please, don't take any of this personally. As you know, I've learned to admire your intellectual abilities,
you may take all this as invalid criticism from a guy who likes to talk about things he doesn't fully understand. I won't like you any less for that. | {"url":"http://www.astronomy.net/forums/god/messages/13856.shtml","timestamp":"2024-11-09T17:15:59Z","content_type":"text/html","content_length":"15657","record_id":"<urn:uuid:f1028106-c90d-45ab-a15f-725cc1adb946>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00336.warc.gz"} |
Comparison with SKD and ARD and Implementations of Stronger Attacker Algorithms | HackerNoon
(1) Seokil Ham, KAIST;
(2) Jungwuk Park, KAIST;
(3) Dong-Jun Han, Purdue University;
(4) Jaekyun Moon, KAIST.
Table of Links
3. Proposed NEO-KD Algorithm and 3.1 Problem Setup: Adversarial Training in Multi-Exit Networks
4. Experiments and 4.1 Experimental Setup
4.2. Main Experimental Results
4.3. Ablation Studies and Discussions
5. Conclusion, Acknowledgement and References
B. Clean Test Accuracy and C. Adversarial Training via Average Attack
E. Discussions on Performance Degradation at Later Exits
F. Comparison with Recent Defense Methods for Single-Exit Networks
G. Comparison with SKD and ARD and H. Implementations of Stronger Attacker Algorithms
G Comparison with SKD and ARD
Existing self-distillation schemes [20, 24] for multi-exit networks improve the performance on clean samples by self-distilling the knowledge of the last exit, as the last exit has the best
prediction quality. Therefore, following the original philosophy, we also used the last exit in implementing the SKD baseline. Regarding ARD [8], since it was proposed for single-exit networks, we
also utilized the last exit with high performance when applying ARD to multi-exit networks. Nevertheless, we perform additional experiments to consider comprehensive baselines using various exits for
distillation. Table A4 above shows the results of SKD and ARD using a specific exit or an ensemble of all exits for distillation. The results show that our scheme consistently outperforms all
H Implementations of Stronger Attacker Algorithms
In Section 4.3 of the main manuscript, during inference, we replaced the Projected Gradient Descent (PGD) attack with other attacker algorithms (PGD-100 attack, Carlini and Wagner (CW) attack [2],
and AutoAttack [5]) to generate stronger attacks for multi-exit neural networks. This section provides explanation on how these stronger attacks are implemented tailored to multi-exit neural
H.1 Carlini and Wagner (CW) attack
The Carlini and Wagner (CW) attack is a method of generating adversarial examples designed to reduce the difference between the logits of the correct label and the largest logits among incorrect
labels. In alignment with this attack strategy, we modify the CW attack for multi-exit neural networks. In the process of minimizing this difference, our modification aims to minimize the average
difference across all exits of the multi-exit neural network. Moreover, when deciding whether a sample has been successfully converted into an adversarial example, we consider a sample adversarial if
it misleads all exits in the multi-exit neural network.
H.2 AutoAttack
AutoAttack produces adversarial attacks by ensembling various attacker algorithms. For our experiment, we sequentially use APGD [5], APGD-T [5], FAB [4], and Square [1] algorithms to generate
adversarial attacks, as they are commonly used.
H.2.1 APGD and APGD-T
The APGD attack is a modified version of the PGD attack, which is limited by its fixed step size, a suboptimal choice. The APGD attack overcomes this limitation by introducing an adaptive step size
and a momentum term. Similarly, the APGD-T attack is a variation of the APGD attack where the attack perturbs a sample to change to a specific class. In this process, we use the average loss of all
exits in the multi-exit neural network as the loss for computing the gradient for adversarial example updates. Moreover, we define a sample as adversarial if it misleads all exits in the multi-exit
neural network.
H.2.2 FAB
The FAB attack creates adversarial attacks through a process involving linear approximation of classifiers, projection to the classifier hyperplane, convex combinations, and extrapolation. The FAB
attack first defines a hyperplane classifier separating two classes, then finds a new adversarial example through a convex combination of the extrapolation-projected current adversarial example and
the extrapolation-projected original sample with the minimum perturbation norm. Here, we use the average gradient of all exits in the multi-exit neural network as the gradient for updating
adversarial examples. Similar to above, we label a sample as adversarial if it can mislead all exits in the multi-exit neural network.
H.2.3 Square
The Square attack generates adversarial attacks via random searches of adversarial patches with variable degrees of perturbation and position. The Square attack algorithm iteratively samples the
perturbation degree and the positions of patches while reducing the size of patches. The sampled adversarial patches are added to a sample, and the patches that maximizes the loss of the target model
are selected. Here, we use the average loss of all exits in the multi-exit neural network as the loss for determining whether a sampled perturbation increases or decreases the loss of the target
model. Additionally, we determine a sample as adversarial if it misleads all exits in the multi-exit neural network.
H.3 Experiment details
For all attacks, we commonly use ϵ = 0.03 as the perturbation degree and generate adversarial examples over 50 steps. All the attacks are based on L∞ norm. For the APGD attack, we employ cross
entropy loss for computing the gradient to update adversarial examples. In both the APGD-T and FAB attacks, all class pairs are considered when generating adversarial attacks. For the Square attack,
the random search operation is conducted 5000 times (the number of queries). Other settings follow [14]. In terms of performance comparison against stronger attacker algorithms, we adopt adversarial
training via average attack in both Adv. w/o Distill [12] and NEO-KD (our approach). However, since the original CW attack algorithm and AutoAttack algorithm were designed for single-exit neural
networks, this adapted versions targeting multi-exit neural networks are relatively weak. | {"url":"https://hackernoon.com/comparison-with-skd-and-ard-and-implementations-of-stronger-attacker-algorithms","timestamp":"2024-11-07T20:45:54Z","content_type":"text/html","content_length":"235218","record_id":"<urn:uuid:7a80acc7-c425-47cb-87ac-5d321a183121>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00405.warc.gz"} |
ULTIMATE SYMMETRY - I.4.3 The Duality of Time Postulate
I.4.3 The Duality of Time Postulate
Based on the above re-creation principle of the Single Monad Model, the Duality of Time hypothesis can be extracted into the following postulate:
At every instance of the outward normal level of time, space and matter are perpetually being re-created in one chronological sequence, which forms the inward levels of time that are also nested
inside each lower dimension of space.
This means that at every instance of the real flow of time there is only one metaphysical point, that is the unit of space-time geometry, and the Universe is a result of its perpetual recurrence in
the inner levels of time , that is continuously re-creating the dimensions of space and what it may contain of matter particles, which then kinetically evolve throughout the outer (normal) level of
time, that we encounter.
This metaphysical re-creation process is instantaneous, but its speed is indefinite, because there is nothing to compare to it at this level of absolute oneness, and the reason why individual
observers measure a finite value of the speed of light is because they are subject to the time lag during which the spatial dimensions are being re-created. In other words, while the real time is
flowing unilaterally in one continuous sequence, creating only one metaphysical point at every instance, individual observers witness only the discrete moments in which they are re-created, and
during which they observe the dimensions of space and physical matter that have just been re-created in these particular instances; thus they only observe the collective (physical) evolution as the
moments of their own time flows by, and that s why it becomes imaginary, or latent, with relation to the original real flow of time that is creating space and matter.
Therefore, the speed of light in complete vacuum is the speed of its dynamic formation, and it is undefined in its own reference frame (as it can be also inferred from the current understanding of
time dilation and space contraction), because the physical dimensions are not yet defined at this metaphysical level. Observers in all other frames, when they are re-created, measure a finite value
of this speed because they have to wait their turn until the re-creation process is completed, so any minimum action, or unitary motion, they can do is always delayed by an amount of time
proportional to the dimensions of vacuum (and its matter contents if they are measuring in any other medium). Hence, this maximum speed that they can measure is also invariant because it is
independent of any physical (imaginary) velocity, since their motion is occurring in the outer time dimension that is orthogonal onto the spatial dimensions which are being re-created in the inner
(real) flow of time. This also means that all physical properties, including mass, energy, velocity, acceleration and even the dimensions of space, are emergent properties; observable only on the
outward level of time, as a result of the temporal coupling between at least two geometrical points or complex-time instances. Moreover, just like the complete dimensions of space, the outer time
itself, which is a fractional dimension, is also emerging from the same real flow of time that is the perpetual recurrence of the original geometrical point: .
In Chapter V of Volume II, we have demonstrated how this single postulate leads at the same time to all the three principles of Special and General Relativity together, since there is no more any
difference between inertial and non-inertial frames, because the instantaneous velocity in the imaginary time is always zero , whether the object is accelerating or not! This also means that both
momentum and energy will be complex and invariant between all frames, as we shall discuss further in Chapter II.
Henceforth, this genuinely-complex time, or time-time geometry will define a profound discrete symmetry that allows expressing the (deceitfully continuous) non-Euclidean space-time in terms of its
granular and fractal complex-time space, whose granularity and fractality are expressed through the intrinsic properties of hyperbolic numbers (), i.e. without invoking Riemannian geometry, as
discussed further in section I.4.5. However, this hidden discrete symmetry is revealed only when we realize the internal chronological re-creation of spatial dimensions; otherwise if we suppose their
continuous existence, space-time will still be hyperbolic but not discrete. Discreteness is introduced when the internal re-creation is interrupted to manifest in the outward normal time, because
creation is processed sequentially by the perpetual recurrence of one metaphysical point, so the resulting complex-time is flowing either inwardly to create space, or outwardly as the normal time,
and not both together.
Therefore, in accordance with the correspondence principle, we will see in section IV.4.2, that semi-Riemannian geometry on is a special approximation of this discrete complex-time geometry on . This
approximation is implicitly applied when we consider space and matter to be coexisting together in (and with) time, thus causing the deceptive continuity of physical existence, which is then best
expressed by the non-Euclidean Minkowskian space-time continuum of General Relativity, or de Sitter/anti-de Sitter space, depending on the value of cosmological constant.
For the same reason, because we ideally consider the dimensions of space to be continuously existing, our observations become time-symmetric, since we can apparently-equally move in opposite
directions. Therefore, this erroneous time-symmetry is reflected in most physics laws because they also do not realize the sequential metaphysical re-creation of space, and that is why they fail in
various sensitive situations such as the second law of Thermodynamics (the entropic arrow of time), Charge-Parity violations in certain weak interactions, as well as the irreversible collapse of
wave-function (or the quantum arrow of time). In the Duality of Time Theory, the autonomous progression of the real flow of time provides a straightforward explanation of this outstanding historical
problem. This will be explicitly expressed by equation 2.1, as discussed further in section II.4.2, where we will also see that we can distinguish between three conclusive states for the flow of
complex-time: either the imaginary time is larger than the real time, or the opposite, or they are equal. Each of the first two states forms one-directional arrow of time, which then become
orthogonal, while the third state forms a two-directional dimension of space, that can be formed by or broken into the orthogonal time directions. This fundamental insight could provide an elegant
solution to the problems of super-symmetry and matter-antimatter asymmetry at the same time, as we shall discuss further in Chapter II.
Additionally, as we have already explained in Chapter V of Volume II, the genuine complex-time flow is the only way to derive the mass-energy equivalence relation , in its simple and relativistic
forms, directly from the principles of Classical Mechanics. This should provide a conclusive evidence to the Duality of Time hypothesis, because it will be shown that an exact derivation of this
experimentally verified relation is not possible without considering the inner levels of time, since it incorporates motion at the speed of light which leads to infinities on the physical level. All
current derivations of this critical relation suffer from unjustified assumptions or approximations, as was also repeatedly acknowledged by Einstein himself.
Since this relation forms the basis of Dirac equation which combined Special Relativity with Quantum Mechanics and lead to the development of some effective quantum field theories, such as QED and
QCD, it is expected that the new genuinely-complex momentum and energy would lead to a full quantum field theory of space-time itself. As we shall discuss further in section III.1.5, this critical
relation underlines the hidden discrete symmetry of space-time geometry and it is indeed equivalent to the equivalence principle of General Relativity, so when we realize its discrete behavior it
would allow the full quantization of Klein-Gordon relation that lead to Dirac equation.
The Duality of Time Theory provides a deeper understanding of time as a fundamental complex-scalar quantum field that reveals the discrete symmetry of space-time geometry. This revolutionary concept
will have tremendous implications on the foundations of physics, philosophy and mathematics, including geometry and number theory; because complex numbers are now genuinely natural, while the reals
are one of their extreme, or unrealistic, approximations, as we shall discuss further in section IV.4.1. Many major problems in physics and cosmology can be resolved according to the Duality of Time
Theory, but it would be too distracting to discuss all that in this introductory article. The homogeneity problem, for example, will instantly cease, since the Universe, no matter how large it could
be, is re-created sequentially in the inner levels of time, so all the states are synchronized before they appear as one instance in the normal level. Philosophically also, since space-time is now
dynamic and self-contained, causality itself becomes a consequence of the sequential metaphysical creation, and hence the fundamental laws of conservation are simply a consequence of the Universe
being a closed system. This will also explain non-local and non-temporal causal effects, without breaking the speed of light limit, in addition to other critical quantum mechanical issues, some of
which are outlined in Chapters V and VI of Volume II. | {"url":"https://www.smonad.com/symmetry/book.php?id=37","timestamp":"2024-11-10T21:34:18Z","content_type":"text/html","content_length":"40130","record_id":"<urn:uuid:4bb127ee-b69c-442d-ad7b-d147ad6f2728>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00360.warc.gz"} |
Advanced | Opto Engineering
Camera calibration
Neural network
Machine learning
Camera calibration
Camera-lens calibration
The coordinates on the sensor are expressed in a base of vectors called (u,v).
The distortion of a camera-lens system can be expressed by an operator that transforms the coordinates with distortion (u,v) into coordinates without distortion (u’,v’) or vice versa.
Therefore as a general rule we can write:
With A, a generic operator.
Classic modelling requires A to be a function:
The factor can be developed in series to divide the contributions of the different degrees of r.
To also take spurious effects into account, the equation can be redefined as:
The effects of the main coefficients are graphically shown as follows:
Determining the transformation values makes it possible to calculate and correct the distortion.
Final remarks:
• The model is applied to the camera and lens system. Any change in these components makes it necessary to recalculate the coefficients.
• The parameters are intrinsic, therefore they do not depend on what is observed.
Neural network
Neural network
An artificial neural network is a computing model consisting of logical elements (artificial “neurons”) based on a simplified biological neural network model.
The neurons can be considered as network nodes and divided into the following groups:
• Input neurons, with a 1-1 relationship as featured in the sample (green nodes)
• Hidden neurons, may be present or not, with a number of layers according to how much you wish the system to be capable of creating interconnections and therefore relationships between inputs (red
• Output neurons which present the final outcomes of the calculation.
A learning process (either supervised or unsupervised) is necessary to make a neural network operational. Connections are moulded during learning (they all weighed the same in the previous image),
some connections disappear, some connections become weaker while others become stronger.
Now the system is capable of solving the problems it was trained for.
Nicola and Paolo want to book a restaurant online. The system will show about a dozen restaurants for each one and will ask them to rate their preferences (which might just be based on selecting
interesting restaurants). At that stage, based on inputs such as price, customer reviews, vicinity, etc. a popularity ranking will be created for each user (with only one output with a value between
0 and 10). Each user has their own neural network available which is specifically arranged to make Nicola or Paolo’s decision easier, although obviously Nicola’s rankings might not agree with what
Paolo has in mind.
This example shows the advantage of neural networks: a generic software is written which adapts itself to a certain purpose only after learning.
Machine learning
Machine learning
Machine learning includes the following main types:
The main difference between the two types is that supervised learning is performed using a basic truth. The operator, therefore, has prior knowledge of what the output values of our samples should be
(e.g. right or wrong). Therefore, supervised learning is aimed at learning a function which, given a sample of desired data and outputs, is closer to the relationship between inputs and outputs which
can be observed in the data.
Unsupervised learning has no labeled outputs. Therefore its goal is to deduce the structure within a group of data.
Supervised Learning
Supervised learning issues can be further grouped into regression and classifier issues.
Classifier: there is a classifier problem when the output variable is a category, like “red” and “blue” or “right” and “wrong”
Regression: a regression problem is when the output variable is a real value, like “dollars” for the estimate of a house, whose input parameters are “size”, “nearby schools”, etc.
The complexity of the model refers to the complexity of the function one is trying to learn.
Unsupervised Learning
Unsupervised learning is when the operator enters the inputs (such as images) with no matching output variable. The goal of unsupervised learning is to model the structure or the distribution
underlying the data. Unlike supervised learning, there are no correct answers because there is no teacher. The algorithms discover the structures in the data and the laws governing them on their own.
Unsupervised learning issues are essentially clustering problems since one wishes to discover the inherent groupings in the data, for example, clustering objects according to their area. The
dimensionality reduction of the problem is an aspect which cannot be overlooked, as significant features must be chosen (for example selecting objects according to the area might not be enough) | {"url":"https://www.opto-e.com/it/basics/Advanced","timestamp":"2024-11-04T11:56:49Z","content_type":"text/html","content_length":"79503","record_id":"<urn:uuid:095aad16-e72b-4929-aa79-374f0fdcf126>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00513.warc.gz"} |
Decoding Curves: A Game Developer's Insight into Bézier Magic
Ever wondered how the curve making Pen Tool works its magic? Unravel the simplicity behind the seemingly complex mathematics — a revelation so clear that a 10-year old can grasp it with ease. I'm
here to guide you through this journey like never before, revealing the clear simplicity behind the secrets of curves, no sleepless nights required. So, go ahead, grab a pen and paper, and let's
unlock the art of creating curves together – it's simpler than you think!
Updated: Dec 8, 2023
Published: Dec 8, 2023
Unveiling the Mystery: My Journey into Understanding Curves
I was creating a game for the annual js13kgames competition in 2023, and decided to create a game to cut objects. As most people these days, I asked my friend ChatGPT to help me, and it went
surprisingly well in the beginning. But that was about to change. I wanted to try some cool things such as splitting the object exactly where I dragged my finger. I spent countless nights trying to
get the code to work, but it was all futile. That's why I decided to actually sit down and learn the math behind it, which helped me understand curves and lines so much better. I was also able to
accomplish what I wanted in my game, which you can play here if you're curious.
The secret to curves are actually straight forward. A curve is created using several straight (short) lines. The straight lines are just so short that you don't notice where it start and ends. I'll
walk you through the math behind it, and how computers are able to draw the smoothest curves. The curves made from the Pen Tool in applications such as Illustrator, Gimp, or my personal favorite,
Affinity Designer, are called Bézier curves, but we'll refer to them as curves in this blog post.
Now, let's dive into the fascinating world of curves and discover the math behind them. Having some images and animations to go with the explanation will make it easier to understand.
The Simple Math Magic of the Pen Tool Unveiled
In the below video, I'm using the Pen Tool to create a curve. Everyone who's worked with vector graphics are probably used to working with the tool.
The important thing to notice are the invisible points that help shape the curve. These are called control points. A curve can consist of multiple control points, but the most common curves have four
points, also referred to as a cubic Bézier curve.
The computer screen consists of a grid with pixels, each with it's own coordinates (x, and y,) and the computer can only draw a straight line from one coordinate to another, there's no curvature in a
straight line. When these points are close enough, and displayed on a high resolution screen, the lines will form a curve.
Here's how a curve would look like in low resolution. Can you imagine where the lines start and end with this image? Each pixel we see can be a line (where the start point and end point are the
Let's see how a curve would look like in high resolution.
Then, how does a computer know where to draw these straight lines when all we have are 4 control points? The secret behind the Pen Tool is easier than what you think. I will go through the math, I
know math sounds scary, but trust me, if you know how to add, subtract, divide, and multiply, it should be fairly straight forward to follow along. I will include some exercises as well, where you
are the computer, and you have to draw a curve by hand, using only four control points. Grab some paper, a ruler and a pencil, and join me in this exercise.
Finding the points on the curve
Before I walk you through the math, I want to show what we will be doing.
In the video we start with 4 control points, the green circles. Then we will
• Draw a line between each point.
• Mark the center point of each line (50% mark on the line)
• Draw (green) lines between the new points
• Mark the 50% point on the lines
• Draw a line from the new points
• Circle the 50% mark of the last line
We repeat the above steps two more times, but instead of marking the center point, we draw the 25% mark and 75% mark instead.
This approach is actually called the De Casteljau's algorithm. If we look up this algorithm online, we'll be overwhelmed with a lot of math, symbols and graphs, which will scare even the bravest of
us. It took several days to build up the courage to go through it myself. But fret not, It's not complicated if we break it down. Let's try to add some numbers to the approach we did above.
Each point is represented as a tuple with x and y coordinates. The x-coordinate is along the horizontal line, and the y-coordinate is along the vertical line. For the first point, the coordinates are
3,2 (x = 3, y = 2).
Here's a video of the steps to calculate the center point.
In the video, we do the following:
• Subtract the coordinates of point 2 and point 1
• Multiply with 0.5
• Add point 1 to the result above
□ 3,2 + 1,2 = 4.4 (let's call it s1)
• Subtract the coordinates of point 3 and point 2
• Multiply with 0.5
• Add point 2 to the result above
□ 5,6 + 5.0 = 10,6 (let's call it s2)
• Subtract the coordinates of point 4 and point 3
• Multiply with 0.5
• Add point 3 to the result above
□ 15,6 + 2,-1.5 = 17,4.5 (let's call it s3)
• Subtract the coordinates of point s2, and s1
• Multiply with 0.5
• Add point s1 to the result above
□ 4,4 + 3,1 = 7,5 (let's call it s4)
• Subtract the coordinates of point s3, and s2
• Multiply with 0.5
• Add point s2 to the result above
□ 10,6 + 3.5,-0.75 = 13.5,5.25 (let's call it s5)
• Subtract the coordinates of point s5, and s4
□ 13.5,5.25 - 7,5 = 6.5,0.25
• Multiply with 0.5
□ 6.5,0.25* 0.5 = 3.25, 0.125
• Add point s3 to the result above
□ 7,5 + 3.25,-1.375 = 10.25, 5.125 (let's call it s6)
I know these calculations may feel overwhelming, but it's easier to follow along with the video above. Take one and one line at the time, and you should have no problem doing the math on your own.
These calculations help us find a point, s6, on our curve. This point is guaranteed to be on the curve.
With less than 10 segments, we'll be able to create a series a lines along our curve. Using the same control points, but replacing the multiplication part of 0.5 with 0.1, 0.2, 0.3, ...., 0.9, we'll
get 9 coordinates along the curve. Drawing a straight line from each point will form our curve. Add more segments for an even smoother curve.
This is the secret behind the Pen Tool, and that's how a computer draws a curve. I've made an interactive example, where you can change the number of segments, and also drag the control points
Experience the magic firsthand! Explore the complete example on my CodePen or experiment with the live interactive demo at the end of this blog post.
Need clarification or have questions? Reach out to me by sending a message through my Contact Form. I'm happy to help.
I'm also thinking about recording a video of myself going through the steps manually, and explaining everything. If this is something that sounds useful, please let me know.
As promised, I've included a few exercises. To guide you in mastering curve calculations, I've illustrated the actual curve in each exercise. Use them as visual aids to validate your calculations. I
also recommend checking out the live example in the bottom of this blog post to understand how the curve will look like depending on the number of segments.
The first exercise has these control points:
• p1 = 4,2
• p2 = 4,6
• p3 = 16,10
• p4 = 16,6
The second exercise has these control points:
• p1 = 2,4
• p2 = 0,0
• p3 = 12,2
• p4 = 10,6
The coordinates doesn't have to be all positive. Here's a slightly more tricky exercise:
• p1 = 2,-4
• p2 = 6,-8
• p3 = 7,-1
• p4 = 11,-2
Final words
We've unveiled the secrets of drawing curves through a series of connected straight lines. We calculate points on the curve and seamlessly draw straight lines between them. With this knowledge, you
can conquer challenges in game development, just as I did with Samurai Sam. I hope it wasn't too difficult.
If you enjoyed this post, I think you'll like another post related to this topic. I created a game for the js13kgames 2023 game jam, called Samurai Sam. It's a game where you slice objects with your
sword (finger/mouse.) The game uses techniques I've outlined in this blog post to slice the objects, and here's the Post Mortem: Slicing with Samurai Sam, a js13kgames entry.
If you liked this blog post, or want me to elaborate on other topics, consider sharing this post and follow me on Twitter (𝕏) @ReitGames. Subscribe to my weekly newsletter for more posts about game
development. Happy coding!
The example below uses the same approach I've outlined in this blog post. Try to drag the control points around, and adjust the number of segments. | {"url":"https://reitgames.com/news/bezier-magic","timestamp":"2024-11-07T23:21:00Z","content_type":"text/html","content_length":"39829","record_id":"<urn:uuid:afec78e8-a14c-4847-9ae6-6f8fdf244c4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00074.warc.gz"} |
Multiplication Properties Of Exponents Worksheet
Mathematics, particularly multiplication, forms the cornerstone of many academic disciplines and real-world applications. Yet, for many learners, grasping multiplication can present a difficulty. To
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x b x a b x 5 x 3 x 5 3 x 8 Then 2x 5 7x 3 14x 8
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“Succeed with Math” Crusade
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3068 -- "Shortest" pair of paths
"Shortest" pair of paths
Time Limit: 1000MS Memory Limit: 65536K
Total Submissions: 2223 Accepted: 1107
A chemical company has an unusual shortest path problem.
There are N depots (vertices) where chemicals can be stored. There are M individual shipping methods (edges) connecting pairs of depots. Each individual shipping method has a cost. In the usual
problem, the company would need to find a way to route a single shipment from the first depot (0) to the last (N - 1). That's easy. The problem they have seems harder. They have to ship two chemicals
from the first depot (0) to the last (N - 1). The chemicals are dangerous and cannot safely be placed together. The regulations say the company cannot use the same shipping method for both chemicals.
Further, the company cannot place the two chemicals in same depot (for any length of time) without special storage handling --- available only at the first and last depots. To begin, they need to
know if it's possible to ship both chemicals under these constraints. Next, they need to find the least cost of shipping both chemicals from first depot to the last depot. In brief, they need two
completely separate paths (from the first depot to the last) where the overall cost of both is minimal.
Your program must simply determine the minimum cost or, if it's not possible, conclusively state that the shipment cannot be made.
The input will consist of multiple cases. The first line of each input will contain N and M where N is the number of depots and M is the number of individual shipping methods. You may assume that N
is less than 64 and that M is less than 10000. The next M lines will contain three values, i, j, and v. Each line corresponds a single, unique shipping method. The values i and j are the indices of
two depots, and v is the cost of getting from i to j. Note that these shipping methods are directed. If something can be shipped from i to j with cost 10, that says nothing about shipping from j to
i. Also, there may be more than one way to ship between any pair of depots, and that may be important here.
A line containing two zeroes signals the end of data and should not be processed.
follow the output format of sample output.
Sample Input
Sample Output
Instance #1: Not possible
Instance #2: 40
Instance #3: 73
Southeastern Europe 2006 | {"url":"http://poj.org/problem?id=3068","timestamp":"2024-11-13T23:03:38Z","content_type":"text/html","content_length":"7412","record_id":"<urn:uuid:abc53bb5-17d0-4462-8629-251af6a54f5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00070.warc.gz"} |
Into Math Grade 5 Module 19 Lesson 2 Answer Key Understand Ordered Pairs
We included HMH Into Math Grade 5 Answer Key PDF Module 19 Lesson 2 Understand Ordered Pairs to make students experts in learning maths.
HMH Into Math Grade 5 Module 19 Lesson 2 Answer Key Understand Ordered Pairs
I Can graph a point on a coordinate grid and interpret the coordinate values.
Spark Your Learning
The map shows the location of two buildings. Each unit represents one block.
Andi is at the art museum and Karl is at the history museum. They agree to meet at the library. The library is at the intersection of a street that is 3 blocks south of the art museum and a street
that is 5 blocks west of the history museum. Who walks farther to get to the library?
Justify your thinking.
(2, 6)(3, 4) is the street of way to the library
The streets met at a point that intersect at (3, 2) That is the library location.
Andi walks 6 blocks and Karl walks 7 blocks
so, Karl walks more than Andy
Turn and Talk Is there a point where the streets intersect so Karl and Andi walk the same number of whole blocks to meet? Explain how you know.
No they can’t walk same number of blocks to meet.
(2, 6)(3, 4) is the street of way to the library
The streets met at a point that intersect at (3, 2) That is the library location.
Build Understanding
Question 1.
Each unit on the map represents one block. The library is located at (6, 5). The school is located at (2, 7). Where is each building located on the map?
A. Find the location of the library.
What is the x-coordinate? ______________
Answer: 6
What does this mean? ______________
6 blocks from the x-axis
What is the y-coordinate? ______________
What does this mean? ______________
5 blocks from the y-axis
B. Plot the location on the coordinate grid and label the point as the library.
C. How will you find the location of the school?
Where the x-axis and y-axis intersect that is the point where the library is located
D. Plot the location on the coordinate grid and label the point as the school.
Turn and Talk Would the locations of the school and library be the same if the coordinates were (5, 6) and (7, 2)? Explain your answer.
No, The intersection points changes
so, the school and library location changes.
Step It Out
Question 2.
Each unit on the map represents 1 mile.
A. How far is the museum from the aquarium?
How far from the y-axis is the aquarium? ___________
2 miles far from the y-axis is the aquarium
How far from the y-axis is the museum? ___________
5 miles far from the y-axis is the museum
How much farther from the y-axis is the museum compared to the aquarium? ___________
3 miles farther from the y-axis is the museum compared with the aquarium
How far is the museum from the aquarium? ___________
3 miles is the museum from the aquarium
B. How far is the park from the aquarium?
How far from the x-axis is the aquarium? ___________
6 miles from park to the aquarium
3 miles to the aquarium from x-axis
How far from the x-axis is the park? ___________
9 miles is the distance from x-axis to the park.
How much farther from the x-axis is the park compared to the aquarium? ___________
6 miles is the distance from park to the aquarium on x-axis.
How far is the part from the acquarium? ___________
6 miles is the distance from park to the aquarium on x-axis.
Turn and Talk How can you use the x- and y-coordinates to find the distances between the locations?
By counting the block we can find the distance between the coordinates
1 block = 1mile.
Check Understanding
Each unit on the coordinate grid represents 1 mile.
Question 1.
Mystic Falls is located at (7, 8). Plot Mystic Falls on the coordinate grid.
On x-axis it coordinate at 7
and on y-axis it coordinate at 8
so, the point of intersection is the Mystic falls.
Question 2.
How far is Mystic Falls from Grand Lake?
4 miles
4 miles far is Mystic Falls from Grand Lake
Question 3.
How far is Mystic Falls from the campsite?
6 miles
6 miles far is Mystic Falls from the campsite
On Your Own
Each unit on the coordinate grid represents 1 mile. Use the coordinate grid for 4-6.
Question 4.
Reason The ranch is located at (1, 9). Plot and label the location of the ranch. Explain how you plotted the point.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Question 5.
How far is the ranch from the farm?
6 miles
6 miles far is the ranch from the farm
Question 6.
How far is the ranch from the fairgrounds?
7 miles
7 miles far is the ranch from the fairgrounds
Plot the point on the coordinate grid.
Question 7.
A(5, 0)
5 is on x-axis and 0 on y-axis
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate),
having two values written in a fixed order within parentheses.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Question 8.
B(0, 4)
0 is on x-axis and 4 on y-axis
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate),
having two values written in a fixed order within parentheses.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Question 9.
C(8, 8)
8 is on x-axis and 8 on y-axis point of coordinate
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate),
having two values written in a fixed order within parentheses.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Question 10.
D(3, 6)
3 is on x-axis and 6 on y-axis point of coordinate
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate),
having two values written in a fixed order within parentheses.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Question 11.
E (6, 3)
6 is on x-axis and 3 on y-axis point of coordinate
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate),
having two values written in a fixed order within parentheses.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Question 12.
F(9, 2)
9 is on x-axis and 2 from y-axis.
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate),
having two values written in a fixed order within parentheses.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Find the distance between the pair of points.
Question 13.
(3, 10) and (8, 10) __________
from (3, 10) to (8, 10) south 2 units and east 7 units total 9 units.
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate),
having two values written in a fixed order within parentheses.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Question 14.
(8, 5) and (0, 5) _________
from (8,5) to (0,5) 8 units distance.
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate),
having two values written in a fixed order within parentheses.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Question 15.
(1, 7) and (9, 7) ___________
from (1, 7) to (9, 7) east 6 units
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate),
having two values written in a fixed order within parentheses.
In order pair 1 digit represent the x-axis and second digit is the y-axis
point of intersection is the order pair
Question 16.
Open-Ended Write a problem using ordered pairs to find the distance between two points so that the distance is 7 units.
I have choose ordered pair as (4, 8) and (4, 1)
The distance between this two coordinate points is south 7 units.
I’m in a Learning Mindset!
How am I using what I know to plot points?
find the value for x on the x-axis
find the y-value on the y-axis.
Coordinate axis from where to point
and order pairs
Leave a Comment
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"Precision Computations in AdS/CFT" - Department of Physics and Astronomy
Alex Maloney, McGill University
“Precision Computations in AdS/CFT”
We describe applications of AdS/CFT to three open problems: one in quantum information theory, one in quantum gravity and one in quantum cosmology. In the first application we use AdS/CFT to study
entanglement entropies which characterize the degree of spatial entanglement in the ground state of a quantum field theory. We learn that these entropies undergo a novel phase transition, in which
non-analytic features of the entanglement spectrum are related to basic CFT quantities. In the second application, we use our knowledge of the classification of CFTs to argue that, in certain
solvable models, *any* theory of quantum gravity with a semi-classical limit is necessarily a string theory. Finally, we describe holographic approaches to de Sitter space, where many of the notions
developed in Anti-de Sitter space – such as supersymmetry and higher spin symmetry – can be applied to inflationary space-times. | {"url":"https://physics.unc.edu/event/precision-computations-adscft/","timestamp":"2024-11-07T10:44:59Z","content_type":"text/html","content_length":"95419","record_id":"<urn:uuid:ff840eec-aa97-499a-9f8d-e3376bd29bea>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00081.warc.gz"} |
How numpy handles day-to-day algebra?
118 False 140 False umath 8 False umath 10 False umath 12 True core 73 False Yes! numeric 29 False numeric 41 False numeric 1128 False numeric 2515 False numeric 2517 True core 75 False core 77 True
core 83 True core 94 True 142 False 144 True
In this blog post, we will try to figure out how numpy handles seemingly simple math operations. The motivation behind this exploration is to figure out if there are a few foundational operations
behind most of the frequently used functions in numpy. For the sake of right level of abstraction, we will not look into addition, subtraction, multiplication, and division.
pi is an irrational number and computing its value to a certain precision is a challenging task. This video talks in detail how people used to compute pi in the past. At the time of writing this blog
post, Google holds the record for computing pi to the highest precision to 100 trilian digits. They used y-cruncher program (it’s free. try it!) with Chudnovsky algorithm to compute pi. Here are the
first 100 digits of pi:
This is how pi is defined in numpy source code upto 36 digits.
Hmm, that looks off. From 16th digit onwards, the values are different. Let’s try to figure out why.
Okay, so it seems like converting 36 digits of pi to 64 bit precision went wrong from 16th digit onwards. What a waste of last 20 digits of pi due to floating point errors! Anyways, let’s move on.
Let’s find out what happens when you execute the following code in numpy. | {"url":"https://patel-zeel.github.io/blog/posts/numpy-algebra-%20copy.html","timestamp":"2024-11-08T01:29:22Z","content_type":"application/xhtml+xml","content_length":"31424","record_id":"<urn:uuid:0f366fbc-4309-4d72-9854-a4bf90135425>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00646.warc.gz"} |
Joule/Second to Watt Conversion - Convert Joule/Second to Watt (J/s to W)
Joule/Second : The joule per second is a unit of power, equals to one watt. The unit symbol for joule per second is j/s. One joule per second is equal to 0.001341022 horsepower.
Watt : The watt is a unit for power, is one of SI Units. Its name is in honor of the Scottish engineer James Watt, the eighteenth-century developer of the steam engine. The unit symbol for watt is W,
defined as one joule per second, measures the rate of energy conversion or transfer.
Power Conversion Calculator
FAQ about Joule/Second to Watt Conversion
1 joule/second (J/s) is equal to 1 watt (W).
1J/s = 1W
The power P in watt (W) is equal to the power P in joule/second (J/s) times 1, that conversion formula:
P(W) = P(J/s) × 1
One Joule/Second is equal to 1 Watt:
1J/s = 1J/s × 1 = 1W
One Watt is equal to 1 Joule/Second:
1W = 1W × 1 = 1J/s
Most popular convertion pairs of power
Lastest Convert Queries
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DESIGN OF FORMWORK - Kumar World
1. INTRODUCTION
Formwork is an essential part of concrete construction. It is to give FORM to green concrete as per the structural and Architectural requirements. For concrete construction at higher elevations,
FORMWORK supporting structure called centering (scaffolding) is necessary. Both can be called as enabling facilities to create permanent Members of a Structure. Design of formwork is the theme of
this presentation and the same only will be dealt with hereafter.
For small and medium size works, provision of formwork is left to the carpenter’s / contractor’s hitherto experience at site. Naturally this method is more by experimentation rather than proper
structural design. For safe, economical and sound provision of formwork, it is essential to design the same as structural member even though it is of temporary nature. Assessment of correct loads
from the stage of pouring green concrete to the end stage of concrete member gaining self-supporting strength is most important. Secondary effects of above loads need to be duly considered and
provided for in the design
Formwork can be designed and provided for permanent construction in different materials as follows:
1. Timber in natural form or in plywood form. Plywood is used to achieve the shapes and forms
2. Structural steel is quite versatile and useful for It is economical, durable, accurate and reusable. Use of shuttering plates made in mild steel is quite common due to their repetitive use.
It is essential to study the local environment to arrive at the best solution for best usage of right materials for formwork. Essence of safety, time and cost should govern the right choice for the
materials to be used.
2. PRIME REQUIRMENTS OF FORMWORK DESIGN
1. Quality: The forms shall be designed and constructed to the desired size, shape and finish of the concrete required. The accuracy in the form, makes the structure stable and economical.
2. Safety: Formwork should be capable of supporting all dead and live loads without collapsing. Safety to workmen and wet concrete shall be focus of control.
3. Economy: Economy in material as well as time shall also be considered.
1. Correct assessment of vertical loads over forms due to
1. Weight of
2. Weight of fresh concrete, with impact due to drop height
• Weight of workmen and equipment
1. Self-weight of
2. Correct assessment of lateral forces exerting pressure on side forms and bracings.
3. Wind forces on side
4. Concrete, concreting methodology and member data
□ Density of concrete
□ Slump of concrete
1. Method of discharge
2. Height of discharge
3. Temperature of concrete
□ Dimensions of sections to be
□ Reinforcement detail
4. Type of vibrations of concrete used for compaction of
5. Formwork data
□ Formwork material
□ Stiffness of forms
4. LOGIC OF FORMWORK DESIGN
1. a) Green concrete exerts hydrostatic pressure on forms which is a function of its density D (gr) and height of pour H (gr).
1. For horizontal forms for slabs. design vertical load will be
D (gr) x H (gr) + D (dr) x H (dr) + allowance for heaping of concrete & impact + self weight Where; D (gr) = Density of green concrete
H(gr) = Height of the pour of green concrete D(dry)= Density of dry concrete.
H(dry)= Height of temporary heap of concrete.
1. For vertical forms for walls, columns and similar contained sections design hydrostatic pressure will vary from zero at top to D (gr) x H (gr) at lowest level of green concrete + impact pressure
(uniform) of 1 T/sq. m. on account of falling concrete from height of about 2m. For pour rate of 0.6 m/hr hydrostatic pressure at lowest level will be 1.8 to 2.5 /sqm & double these valves for
pour rate of 1.5 m/hr. (lower value will be for tropical climates). Above figures are given for the use of ordinary Portland Cement. Hydrostatic pressures on shuttering will lower for quick
setting cements as well as for higher temperatures and vice versa.
It is recommended to just go by hydrostatic head pressure as above.
• Allowable deflection for shuttering as per S. Code is Span/270 where span is spacing between bearers.
5. IMPORTANT REFERENCE
1. IS 4990 : 1993 for use of plywood for Concrete Shuttering
2. IS 800 : Latest for use of structural Steel Shuttering
3. The code of practice for Design and Construction of Formwork for Concrete by W.D. Govt. of Maharashtra..
6. IMPORTANT FORMULAE
It is very important for the engineers to revise basics and fundamentals of design to understand the logic of structural behavior. Although typical examples in design are given hereafter, engineers
shall be able to apply their mind and logic understood, to variety of problems in the field. Hence, it is absolutely necessary that engineers are clear about fundamentals.
W = Point Load in T, w = UDL T/M, (uniformly distributed load) L = Span in Meter, I = Moment of Inertia of section in M4
Z = Section Modulus in Cub.M., E = Mod. Of Elasticity T/Sq.m A = Area of Cross Section in Sq. m.
Fb = Permissible stress in bending in T/Sq.m. Fs = Permissible stress in shear in T/Sq.mm.
M.R. = Moment of Resistance in T.M. = Fb x z
S.R. = Shear Resistance in T = Fs x A
All the above units used are in Tonne and Meter. Proper multipliers should be used while changing T to kg. and M to cm.
To calculate the maximum bending moment, following formulae are useful.
1. B.M. @ Centre = (w L2/8 )T. m
Max. Shear @ A & B = (wL/2) T
Max. Defl. @ center = 5WL4
in M. 384xExI
If partial fixity or continuity over support is assumed, design B.M. can be derated to (WL2 /10) T.M.
1. B.M. @ Centre = (WL/4) T.M. Max. Shear @ A & B = (W/2) T Max. Defl. @ Centre = W L3
in M. 48xExI
1. B.M. @ A = (Wx L) T.M. Max. Shear @ A = (W) T
Max. Defl. @ B = (WL3) In M. 3EI
1. Typical vectorial Resolution of Forces following principle of static Equilibrium at any Junction of Forces:
Resolving along ‘X’ F1 = F2 Cos Ø = 0 Resolving along ‘Y’ F2 Sin Ø – W = 0 Solving above simultaneous equations, F2 W/ Sin Ø ,F1 = W Cos Ø / Sin Ø (COMPRESSION) (TESNION)
7. DESIGN EXAMPLES
1. When Timber is used in Natural form for shuttering Form (modulus of elasticity 80 T/sq. cm.)
2. Permissible stress parallel to grains in-
i) Bending, compression and tension 84 kg/ sq. cm.
ii) Direct compression 50 kg/ sq. cm.
iii) Shear 9 kg / sq. cm.
iv) Modulus of elasticity 80 T / sq. cm.
1. Permissible stress perpendicular to grains in:
i) Direct compression 15 kg / sq. cm.
ii) Shear stress 6 kg / sq. cm.
Note : Above values should be reduced by 20% for wet timber .
: It is important to use GOOD QUALITY of timber to match with above minimum permissible stresses.
: It is preferable to be conservative about unknown quality of timber.
c) Moment of Resistance (in bending) for different depths for one cm width
M.R. = 84 x (bd2) / 6 = 14d2 kg. cm. shear Resistance for different depths for one cm. Width S.R. 6 (conservative) x d kg
Item Thickness or Depth M.R. in kg. Cm. /cm.. S. R. in kg. /cm.
Planks 2.5 cm thick 88 15
4.0 cm thick 224 24
5.0 cm thick 350 30
Joists 7.5 cm thick 788 45
10.0 cm deep 1400 60
12.5 cm deep 2188 75
15.0 deep 3150 90
20.0 cm deep 5600 120
Props: Load capacity reduction factor R.F. (multiplier) For slenderness are as follows:
Effective Length ‘L’ Least Lateral Dimension ‘d’
1 to 10 10 to 15 15 to 18 18 to 24 24 to 30
R.F. 1 0.75 0.70 0.60 0.50
• Plywood formwork will have different permissible stresses depending upon Type of timber Hence, recommendations of Approved Manufactures /suppliers of plywood for permissible stresses etc. should
be used in design.
• Formwork in structural steel work will have reference to. I.S. 800 for Permissible stresses. Shuttering plates of 3mm thickness with angle Framing are commonly used directly below slab. For
joists columns and channel or beam sections are used. Alternatively, ACROW make or equivalent Spans (steel trusses) and steel props are
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SpeedyWeather.jl is a global spectral atmospheric model with simple physics which is developed as a research playground with an everything-flexible attitude as long as it is speedy. With minimal code
redundancies it supports
Dynamics and physics
• Different physical equations (barotropic vorticity, shallow water, primitive equations)
• Particle advection in 2D for all equations
• Physics parameterizations for convection, precipitation, boundary layer, etc.
Numerics and computing
• Different spatial grids (full and octahedral grids, Gaussian and Clenshaw-Curtis, HEALPix, OctaHEALPix)
• Different resolutions (T31 to T1023 and higher, i.e. 400km to 10km using linear, quadratic or cubic truncation)
• Different arithmetics: Float32 (default), Float64, and (experimental) BFloat16, stochastic rounding
• multi-threading, layer-wise for dynamics, grid point-wise for physics
User interface
• Extensibility: New model components (incl. parameterizations) can be externally defined
• Modularity: Models are constructed from its components, non-defaults are passed on as argument
• Interactivity: SpeedyWeather.jl runs in a notebook or in the REPL as well as from scripts
• Callbacks can be used to inject any piece of code after every time step, e.g. custom output, event handling, changing the model while it's running
and Julia will compile to these choices just-in-time.
For an overview of the functionality and explanation see the documentation. But as the documentation always lags behind our full functionality you are encouraged to raise an issue describing what
you'd like to use SpeedyWeather for.
Why another model? You may ask. We believe that most currently available are stiff, difficult to use and extend, and therefore slow down research whereas a modern code in a modern language wouldn't
have to. We decided to use Julia because it combines the best of Fortran and Python: Within a single language we can interactively run SpeedyWeather but also extend it, inspect its components,
evaluate individual terms of the equations, and analyse and visualise output on the fly.
We do not aim to make SpeedyWeather an atmospheric model similar to the production-ready models used in weather forecasting, at least not at the cost of our current level of interactivity and ease of
use or extensibility. If someone wants to implement a cloud parameterization that is very complicated and expensive to run then they are more than encouraged to do so, but it will probably live in
its own repository and we are happy to provide a general interface to do so. But SpeedyWeather's defaults should be balanced: Physically accurate yet general; as independently as possible from other
components and parameter choices; not too complicated to implement and understand; and computationally cheap. Finding a good balance is difficult but we try our best.
This means in practice, that while SpeedyWeather is currently developed, many more physical processes and other features will be implemented. On our TODO is
• A (somewhat) realistic radiation scheme with a daily cycle, depending on clouds and humidity
• Longwave radiation that depends on (global) CO2 concentrations to represent climate change
• Slab ocean and a (seasonal cycle) sea ice interacting with radiation
• Exoplanet support
• 3D particle advection
• single GPU support to accelerate medium to high resolution simulations
• differentiability with Enzyme
Open-source lives from large teams of (even occasional) contributors. If you are interested to fix something, implement something, or just use it and provide feedback you are always welcome. We are
more than happy to guide you, especially when you don't know where to start. We can point you to the respective code, highlight how everything is connected and tell you about dos and don'ts. Just
express your interest to contribute and we'll be happy to have you.
For a more comprehensive tutorial with several examples, see Examples in the documentation. The interface to SpeedyWeather.jl consist of 5 steps: define the grid, create model components, construct
the model, initialize, run
spectral_grid = SpectralGrid(trunc=31, nlev=8) # define resolution
orography = EarthOrography(spectral_grid) # create non-default components
model = PrimitiveWetModel(; spectral_grid, orography) # construct model
simulation = initialize!(model) # initialize all model components
run!(simulation, period=Day(10), output=true) # aaaand action!
and you will see
Hurray🥳 In 5 seconds we just simulated 10 days of the Earth's atmosphere at a speed of 440 years per day. This simulation used a T31 spectral resolution on an octahedral Gaussian grid (~400km
resolution) solving the primitive equations on 8 vertical levels. The UnicodePlot will give you a snapshot of surface vorticity at the last time step. The plotted resolution is not representative,
but allows a quick check of what has been simulated. The NetCDF output is independent of the unicode plot.
More examples in the How to run SpeedyWeather section of the documentation.
Specific humidity in the primitive equation model simulated at T340 spectral resolution (about 40km) with 16 vertical levels (shown here is level 15, just above the surface) on the octahedral
Gaussian grid computed in single precision multi-threaded on 16 CPUs. With convection, large-scale condensation, surface fluxes and some simplified radiation (the daily cycle is visible)
Relative vorticity in the shallow water model, simulated at T1023 spectral resolution (about 10km) on an octahedral Clenshaw-Curtis grid with more than 4 million grid points
Surface temperature in the primitive equation model without surface fluxes or radiation at T511 (~20km resolution) and 31 vertical levels. The simulation was multi-threaded in Float32 (single
SpeedyWeather.jl can also solve the 2D barotropic vorticity equations on the sphere. Here, we use Float32 (single precision) at a resolution of T340 (40km) on an octahedral Gaussian grid. Forcing is
a stochastic stirring on northern hemisphere mid-latitudes following Barnes and Hartmann, 2011. Map projection is orthographic centred on the north pole.
Here, SpeedyWeather.jl simulates atmospheric gravity waves, initialised randomly interacting with orography over a period of 2 days. Each frame is one time step, solved with a centred semi-implicit
scheme that resolves gravity waves with a timestep of CFL=1.2-1.4 despite a single-stage RAW-filtered leapfrog integration.
Advection of 5000 particles with wind in the lower-most layer of the primitive equation model at T85 (150km) resolution and 8 vertical layers.
SpeedyWeather.jl started off as a reinvention of the atmospheric general circulation model SPEEDY in Julia. While conceptually a similar model, it is entirely restructured, features have been added,
changed and removed, such that only some of the numerical schemes share similarities. Fortran SPEEDY's dynamical core has an obscure history: Large parts were written by Isaac Held at GFDL in/before
the 90ies with an unknown amount of contributions/modifications from Steve Jewson (Oxford) in the 90ies. The physical parametrizations were then added by Franco Molteni, Fred Kucharski, and Martin P.
King afterwards while the model was still written in Fortran77. Around 2018-19, SPEEDY was then translated to Fortran90 by Sam Hatfield in speedy.f90. SpeedyWeather.jl is then adopted from first
translations to Julia by Sam Hatfield.
SpeedyWeather.jl defines several submodules that are technically stand-alone (with dependencies) but aren't separated out to their own packages for now
• RingGrids, a module that defines several iso-latitude ring-based spherical grids (like the FullGaussianGrid or the HEALPixGrid) and interpolations between them
• LowerTriangularMatrices, a module that defines LowerTriangularMatrix used for the spherical harmonic coefficients
• SpeedyTransforms, a module that defines the spherical harmonic transform between spectral space (for which LowerTriangularMatrices is used) and grid-point space (as defined by RingGrids).
These modules can also be used independently of SpeedyWeather like so
julia> using SpeedyWeather: LowerTriangularMatrices, RingGrids, SpeedyTransforms
check out their documentation: RingGrids, LowerTriangularMatrices, SpeedyTransforms.
SpeedyWeather.jl is registered in Julia's registry, so open the package manager with ] and
(@v1.10) pkg> add SpeedyWeather
which will install the latest release and all dependencies automatically. For more information see the Installation in the documentation. Please use the current minor version of Julia,
compatibilities with older versions are not guaranteed.
The primitive equations at 400km resolution with 8 vertical layers are simulated by SpeedyWeather.jl at about 500 simulated years per day, i.e. one year takes about 3min single-threaded on a CPU.
Multi-threading will increase the speed typically by 2-4x.
For an overview of typical simulation speeds a user can expect under different model setups see Benchmarks.
If you use SpeedyWeather.jl in research, teaching, or other activities, we would be grateful if you could mention SpeedyWeather.jl and cite our paper in JOSS:
Klöwer et al., (2024). SpeedyWeather.jl: Reinventing atmospheric general circulation models towards interactivity and extensibility. Journal of Open Source Software, 9(98), 6323, doi:10.21105/
The bibtex entry for the paper is:
doi = {10.21105/joss.06323},
url = {https://doi.org/10.21105/joss.06323},
year = {2024},
publisher = {The Open Journal},
volume = {9},
number = {98},
pages = {6323},
author = {Milan Klöwer and Maximilian Gelbrecht and Daisuke Hotta and Justin Willmert and Simone Silvestri and Gregory L. Wagner and Alistair White and Sam Hatfield and Tom Kimpson and Navid C. Constantinou and Chris Hill},
title = {{SpeedyWeather.jl: Reinventing atmospheric general circulation models towards interactivity and extensibility}},
journal = {Journal of Open Source Software}
Copyright (c) 2020 Milan Klöwer for SpeedyWeather.jl
Copyright (c) 2021 The SpeedyWeather.jl Contributors for SpeedyWeather.jl
Copyright (c) 2022 Fred Kucharski and Franco Molteni for SPEEDY parametrization schemes
Software licensed under the MIT License. | {"url":"https://www.juliapackages.com/p/speedyweather","timestamp":"2024-11-07T19:13:18Z","content_type":"text/html","content_length":"132455","record_id":"<urn:uuid:6a77e0bc-fd6d-4080-b841-8a45883fe2a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00430.warc.gz"} |
Solver: multiphaseEulerFoam Description
multiphaseEulerFoam is a solver designed for transient simulations of multiple incompressible fluid phases, whether dispersed or resolved. This solver supports the Eulerian-Eulerian approach,
treating each phase as a continuous fluid with separate volume fractions for each phase. It is particularly useful for modeling and analyzing complex fluid interactions in scenarios such as
gas-liquid flows or liquid-liquid flows. It supports any number of fluid phases, each having its distinct properties but sharing a common pressure field. It handles laminar and turbulent,
accommodating Newtonian fluids.
The solver uses the PIMPLE (merged PISO-SIMPLE) algorithm for pressure-momentum coupling. This algorithm leverages the strengths of both PISO and SIMPLE methods for pressure-velocity coupling,
ensuring robustness in handling transient flows with large time steps. This approach is supplemented by under-relaxation techniques to secure convergence stability. It supports Multiple Reference
Frames (MRF) and allows easy integration of passive scalar transport equations.
The solver’s capability to analyze multiphase flow makes it suitable for modeling various applications such as bubble columns, fluidized bed combustion, cyclone separators, and mixing processes
involving high concentration droplets, bubbles or particles like dust. These applications span across industries including chemistry, energy, and the oil and gas sector.
Solver: multiphaseEulerFoam Features
• Transient
• Incompressible
• Multiphase - Dispersed Phase (Eulerian)
• Dispersed Phase
• Multiple Miscible Fluids
• Euler-Euler Approach
• Laminar and Turbulent (RANS, LES, DES)
• Newtonian Fluid
• Pressure-Based Solver
• Rotating Objects:
□ Multiple Reference Frames (MRF)
• Passive Scalar
• Buoyancy
• PIMPLE Algorithm
• Solution Limiters:
Solver: multiphaseEulerFoam Application
• Fluidized Bed Combustion
• Pneumatic Conveying Systems for Coal and Limestone
• Feeding Systems
• Bubble Columns
• Mixing in Pipelines
Agriculture/cement industry
• Hoppers with high concentrated droplets, bubbles or particles
Solver: multiphaseEulerFoam Multiphase - Dispersed Solvers
Dispersed Solvers In this group, we have included solvers implementing the Eulerian or Lagrangian approach to handle multiple fluids and particle clouds considering Dispersed Phases or Fluid-Particle
• DPMFoam 1 fluid and particles, particle-particle interactions resolved explicitly (direct approach)
• MPPICFoam 1 fluid and particles, dense particle cloud using particle-particle interactions model (simplified approach, MP-PIC method)
• driftFluxFoam 1 fluid and slurry or plastic dispersed phase, drift flux approximation for relative phase motion
• DPM - Discrete Phase Model
• MP-PIC - multiphase particle-in-cell method
• DyM - Dynamic Mesh
Solver: multiphaseEulerFoam Alternative Solvers
In this section, we propose alternative solvers from different categories, distinct from the current solver. While they may fulfill similar purposes, they diverge significantly in approach and
certain features.
Solver: multiphaseEulerFoam Results Fields
This solver provides the following results fields:
• Primary Results Fields - quantities produced by the solver as default outputs
• Derivative Results - quantities that can be computed based on primary results and supplementary models. They are not initially produced by the solver as default outputs.
Velocity \(U\) [\(\frac{m}{s}\)]
Phase Volume Fraction \(\alpha\) [\(-\)]
Hydrostatic Perturbation Pressure \(p - \rho gh\) [\(Pa\)]
Hydrostatic Perturbation Pressure This value represents the pressure without the hydrostatic component (minus gravitational potential). Read More: Hydrostatic Pressure Effects
Pressure \(P\) [\(Pa\)]
Density \(\rho\) [\(\frac{kg}{m^{3}}\)]
Vorticity \(\omega\) [\(\frac{1}{s}\)]
Courant Number \(Co\) [\(-\)]
Peclet Number \(Pe\) [\(-\)]
Stream Function \(\psi\) [\(\frac{m^2}{s}\)]
Q Criterion \(Q\) [\(-\)]
Wall Functions (for RANS/LES turbulence) \(y^+\) [\(-\)]
Wall Shear Stress \(WSS\) [\(Pa\)]
Turbulent Fields (for RANS/LES turbulence) \(k\) \(\epsilon\) \(\omega\) \(R\) \(L\) \(I\) \(\nu_t\) \(\alpha_t\)
Volumetric Stream \(\phi\) [\(\frac{m^{3}}{s}\)]
Passive Scalar \(scalar_i\) [\(-\)]
Forces and Torque acting on the Boundary \(F\) [\(N\)] \(M\) [\(-\)]
Force Coefficients \(C_l\) [\(-\)] \(C_d\) [\(-\)] \(C_m\) [\(-\)]
Average, Minimum or Maximum in Volume from any Result Field \(Avg\) \(Min\) \(Max\) | {"url":"https://help.sim-flow.com/solvers/multiphase-euler-foam","timestamp":"2024-11-11T11:58:04Z","content_type":"text/html","content_length":"33259","record_id":"<urn:uuid:3805c588-b942-45a6-8af6-c08f70a29fb4>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00462.warc.gz"} |
Proofs and Reputations
Karl Sabbagh, the author of a
recent book
on the
Riemann Hypothesis
, recently wrote an
in the London Review of Books about
Louis de Branges
and his recent announcement of a proof for the RH.
There is no evidence that, so far, any mathematician has read [de Branges' proof ]: de Branges and his proof appear to have been ostracised by the profession. I have talked to a number of
mathematicians about him and his work over the last few years and it seems that the profession has come to the view that nothing he does in this area will ever bear fruit and therefore his work can
be safely ignored. It may be that a possible solution of one of the most important problems in mathematics is never investigated because no one likes the solution's author.
My post has nothing to say about the proof itself; I would not dare to presume even a passing familiarity with it. What caught my attention was the sense of surprise in Sabbagh's article; the
unstated '
what on earth does a man's reputation have to do with his proof
' ?
What indeed ?
Mathematics (and by extension theoretical CS) inhabits a Platonic world of truths and false statements (with a dark brooding Gödel lurking in the background, but that's a different story). As such,
either statements are true, and become theorems/lemma/what-have-you, or they are not and fall by the wayside. The pursuit of (mathematical) truth is thus the search for these true statements. The
identity (or very existence) of the searcher has no effect on the truth of the statements; there is no observational uncertainty principle.
However, mathematicians live in the real world. In this world, true and false gets a bit murkier. A theorem is no longer true or false, but almost certainly true, or definitely false. They are far
closer to the falsifiable theories of natural science, although there is at least a "there" there; a scientific theory can only have overwhelming evidence in support of it, but a mathematical
statement (if not too complex) can be categorically true.
The natural sciences have reproducible experiments; if I cannot reproduce the results you claim, all else being equal, the burden of proof is on you to demonstrate that your results are indeed
correct. Similarly in mathematics, if a claimed theorem has a complex proof, the burden of proof does reside on the author to demonstrate that it is indeed correct. They can do this by simplifying
steps, supplementing with more intuition, or whatever...
In this respect, theorem proving in the real world has a somewhat
social flavor
to it. And thus, there is also (it seems to me) a social compact: You demonstrate competence and capability above a certain basic threshold, and I deem your proofs worthy of study. The threshold has
to be low, to prevent arbitrary exclusion of reasonable provers, but it cannot be
zero, because in the real world it is hard to check a proof with absolute certainty.
This is why the
many proofs
that P=NP (or P != NP) that float on comp.theory don't get a fair shake: it is not because the "experts" are "elitists" who don't appreciate "intruders" poaching their beloved problems; it is because
the social compact has not been met; the writers don't cross the threshold for basic reasonableness, either by choosing to disregard the many approaches to P vs NP that have been ruled out, or by
refusing to accept comments from peer review as plausible criticism, and demanding that the burden of proof be shifted from them to the critical reviewer.
Such a compact could be abused mightily to create a clique; this is why the threshold must be set low and
is low
in mathematics. The notorious cliche that a mathematician's best work is done when young at least reinforces the idea that this club can be entered by anyone. More mundanely, there are awards for
best student papers at STOC/FOCS that often go to first-year grad students (like
this year's award
Going back to de Branges' proof, I have no idea what the technical issues are with his proof, and if there are known reasons why they don't work, but going solely on the basis of Karl Sabbagh's
article (and I acknowledge that he could be biased) it seems wrong to ignore his manuscripts. He for one has clearly crossed the threshold of the social compact. Reminds me of an attempt I made to
read a popular exposition of Mulmuley and Sohoni's recent work on P vs NP; if this work does lead to a claimed proof, I imagine that there would be few people who could comprehend the proof, but it
would deserve to be read.
5 comments:
1. Anonymous8/03/2004 05:04:00 PM
Shouldn't that be "...but it cannot be _zero_", as opposed to 'nonzero'?
2. Yes. thanks for pointing that out.
3. Anonymous8/04/2004 08:10:00 AM
The reason the mathematical community is slow to accept the de Brange proof is two-fold.
First, in 1998 there was a paper showing that the techniques being used by de Brange were flawed (found here) Of course, 1998 is a long time ago, so he may have found something new to modify his
Second, he is considerd a maverick. He did prove the Bieberbach Conjecture, however (and this is according the mathematicians of the time) cried wolf on a number of occasions.
Finally, it should be remembered, that he has not actually published his proof for the world to see. In fact, without the press-release, no one would even know he is close. Compare this to
Perelman's work on Poincare, which was immediately made public.
All in all, this reminds me of a math joke I once heard. When an older professor was asked why the name of his talk was "On the proof of the Riemann Hypothesis" for an up coming math conference,
he admitted the following: He did not have a proof, nor was he even going to talk about the Riemann Hypothesis, but the title was in case something horrible happened on his way to the conference.
Everyone would always wonder if he did...
4. Anonymous9/21/2004 02:54:00 PM
Hmmm.. the math joke you heard seem to be derived
from Hardy's unique "travel insurance" - before
embarking on a ship journey, he'd telegraph to let
his hosts know that he'd solved the RH. His logic
was that God wouldn't allow eternal posthumous
glory for Hardy and would hence keep the journey
safe :)
Saty Raghavachary
5. Anonymous7/26/2005 08:17:00 PM
CAN ANYBODY HELP ME FINDING RECENT NEWS (TODAY IS
28 JULY 2005!!!) ON PROF. LOUIS DE BRANGES AND
HIS DEMONSTRATION OF THE RIEMANN HYPOTHESIS?
NO NEW INFORMATION CAN BE FOUND ON INTERNET...
I AM DESPERATE FOR NEWS.
BEST WISHES.
PROF. MARIANELA M.
Posted by MARTIGNONI MARIANELA | {"url":"http://blog.geomblog.org/2004/08/proofs-and-reputations.html","timestamp":"2024-11-02T02:27:58Z","content_type":"application/xhtml+xml","content_length":"146829","record_id":"<urn:uuid:211d1d42-aa69-4b14-8110-a1995a279e0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00681.warc.gz"} |
A Level H2 Math Applications of Differentiation 5 Essential Questions
Differentiation in calculus is a key concept used to determine instantaneous rates of change. Instantaneous rate of change is the measure of how quickly the new value is affecting the old value.
Differentiation gives us the ability to isolate and measure values between two variables instead of just their sum or average.
The diagram below shows a city map of two towns, $A$ and $B$ separated by a river. A bridge is to be built between the two towns, which are on opposite sides of a straight river of uniform width
$r$km, and the two towns are $p$ km apart measured along the riverbank. Town $A$ is $1$km from the riverbank, and Town $B$ is $b$km away from riverbank.
A bridge is to be built perpendicular to the riverbank at a distance of $x$km from Town $B$, measured along the riverbank, allowing traffic to flow between the two towns.
Find the distance $x$, in terms of $b$ and $p$, such that the distance of travel between Town $A$ and Town $B$ can be minimised if $b>1$. (It is not necessary to verify that the distance is minimum.)
[A circular cone with base radius $r$, vertical height $h$ and slant height $l$, has curved surfaced area $\pi rl$ and volume $\frac{1}{3}\pi r{{h}^{2}}$.]
A capsule made of metal sheet of fixed volume $p$cm$^{3}$ is made up of three parts.
The cost of making the body of the capsule is $k$ per cm$^{2}$, while that of the top and the base of the capsule is $2k$ per cm$^{2}$, where $k$ is a constant. The total cost of making the capsule
is $ $C$.
Express $C$ in the form $\frac{A}{r}+B{{r}^{2}}$, where $A$ and $B$ are expressions in terms of $k$ and $p$. Use differentiation to show that $C$ has a minimum as $r$ varies.
(ii) Express $C$ in the form $\frac{A}{r}+B{{r}^{2}}$, where $A$ and $B$ are expressions in terms of $k$ and $p$. Use differentiation to show that $C$ has a minimum as $r$ varies.
Hence determine the ratio of $H$ to $r$ when $C$ is a minimum.
(iii) Hence determine the ratio of $H$ to $r$ when $C$ is a minimum.
Water is poured at a constant rate of $20\,\text{c}{{\text{m}}^{3}}$ per second into a cup which is shaped like a truncated cone as shown in the figure. The upper and lower radii of the cup are $4\,\
text{cm}$ and $2\,\text{cm}$ respectively. The height of the cup is $\text{6}\,\text{cm}$.
Show that the volume of water inside the cup, $V$ is related the height of the water level, $h$ through the equation
(i) Show that the volume of water inside the cup, $V$ is related the height of the water level, $h$ through the equation
How fast will the water level be rising when $h$ is $\text{3}\,\text{cm}$?Express your answer in exact form.
(ii) How fast will the water level be rising when $h$ is $\text{3}\,\text{cm}$?Express your answer in exact form.
A manufacturer produces closed hollow cans. The top part is a hemisphere made of tin and the bottom part is a cylinder made of aluminium of cross-sectional radius $r$ cm and $h$ cm. There is no
material between the cylinder and the hemisphere so that any fluid can move freely within the container. At the beginning of an experiment, a similar-shaped can of dimensions $r=4$and $h=10$, is
filled to its capacity with water. Due to a hole at its base, water is leaking at a constant rate of 2 $\text{c}{{\text{m}}^{3}}{{\text{s}}^{-1}}$ when the can is standing upright. Find the exact
rate at which the height of the water is decreasing 80 seconds after the start of the experiment.
An isosceles triangle is inscribed in a circle of radius $r$. Find the maximum area of this triangle in terms of $r$.
Check out our question bank, where our students have access to thousands of H2 Math questions with video and handwritten solutions. | {"url":"https://timganmath.edu.sg/free-resources/h2-math/applications-of-differentiation/","timestamp":"2024-11-13T11:26:10Z","content_type":"text/html","content_length":"450925","record_id":"<urn:uuid:e356cd4b-7aed-4013-94b4-812b479ce9e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00560.warc.gz"} |
Do younger brothers steal more bases than older brothers? Part II
Do younger brothers steal more bases than older brothers? Part II
A couple of weeks ago, I wrote about a study that purported to show huge differences in steal attempts between brothers in major league baseball. According to the New York Times article describing
the study,
"For more than 90 percent of sibling pairs who had played in the major leagues throughout baseball’s long recorded history, including Joe and Dom DiMaggio and Cal and Billy Ripken, the younger
brother (regardless of overall talent) tried to steal more often than his older brother."
If that's true, that would be huge. It would, I think, be one of the most surprising findings ever, that nine out of ten times the younger brother steals more than the older brother. But I checked,
and found nothing close to that. I was left wondering how the authors, Frank J. Sulloway and Richard L. Zweigenhaft, got the results they did.Since then, I managed to get hold of the actual study.
And I still don't know where the 90% figure comes from. Our raw numbers, it appears, are almost the same. The authors' results are different from mine, but only by a bit; they might have better data,
or used different criteria for eliminating pairs from the study. (Update: strikeouts below are where I misinterpreted some of the numbers in the authors' tables. The authors didn't actually give the
numbers I thought they did.)Let me start with steals. My numbers showed 56% of younger brothers outstealing their older brothers (adjusted for times on base). [S:Their numbers actually showed only
48%. Pretty close.:S] However, that's not the right comparison, because the authors are talking about steal *attempts* -- that is, SB+CS, not just SB. (I missed that the first time.)Rerunning the
numbers for attempts instead of steals, I get that 57 out of 98 younger brothers out-attempted their older brother, or 58%. [S:What did the authors find? 97 out of 185, or 52%.:S][S:
Putting that in table form:
58% me
52% them
So we're on the same page, right? I think so. It sure seems like their comparisons line up with mine.:S]So, then, why does the article say 90%? [S:Because the authors, after showing 52% in their
chart, change their estimate to 90% later.:S]Why? I can't figure it out for sure; I don't completely understand where they're coming from. The logic is flawed somewhere, but the authors don't explain
themselves fully, so I can't actually spot where the flaw happens. Let me show you what the authors give as the broad reason [S:for bumping the 52% up to 90%:S]: they say the sample is biased. Why?
"Call-up sequence and its relationship to athletic talent introduces a potential bias in athletic performance by birth order, as older brothers were more likely to be called up first owing to the
difference in age. The extent of this effect turns out to be pronounced, with older brothers being 6.6 times more likely than their younger brothers to receive a call first to the major leagues.
We are therefore comparing somewhat more talented athletes who were called up first (and who, by virtue of their relative age, tend to be older brothers) with somewhat less talented athletes who
were called up second (and who tend to be older brothers). To correct for this bias, we have controlled performance data for call-up sequence in all analyses involving birth order."
[S::S](UPDATE: I had wondered why the authors assumed older brothers were better. I now see they ran a regression and found the older brothers had longer careers.
But why is this bias?Suppose you wanted to know if younger brothers grew up to be taller than older brothers. You measure a couple of hundred sets of siblings, and you find they grew to exactly the
same height, on average. But, wait! At the time the older brother started high school, he's almost always taller than the younger brother! See? We have bias!Well, of course, we *don't* have bias.
Height (and talent) is related to age of the person, not calendar year. You'd expect brothers to be of equal height at equal ages, and of equal major-league experience at equal ages, NOT at equal
calendar years. The "entering high school" and "getting called up" are just red herrings.Regardless, the authors proceed to divide the sample of players into three groups:-- younger brother called up
first -- younger and older brother called up the same year -- older player called up firstIf you do that, you'll find that the younger brother has a huge advantage in the first two groups. Why? It's
obvious: if the a player gets called up before his brother *even though he's younger*, he's probably a much better player. In addition, speed is a talent more suited to younger players. So when it
comes to attempted steals, you'd expect younger brothers called up early to decimate their older brothers in the steals department.I ran my numbers for all three groups, and got:-- 87% of younger
brothers called up first attempted more steals (7/8)
-- 71% of younger brothers called up the same year attempted more steals (5/7)-- 54% of younger brothers called up last attemped more steals (45/83)The overall average, of course, is still 58% (57/
98). But by splitting the data this way, it makes the effect seem larger. But that's not really telling you anything -- you're selectively sampling based on the results, because the younger players
who come up first are precisely those players who you'd expect to steal only for reasons of being called up young.Another analogy: home teams win 54% of games. But home teams that only use up one out
in the ninth inning win 100% of games! That doesn't mean the 54% is biased, nor does it mean that you learn any more about home field advantage by breaking out walk-off wins. It's just another way of
looking at the same result.Anyway, having claimed to correct for a bias that I don't think really exists, they go on to compute odds ratios for the three cases separately, and use something called
the "Mantel-Haenszel" technique to combine them. The statistical techniques look OK, and it seems to me that should have given them the same 52% that they found for the one group. But they somehow
come up with 90%.So, again: what's going on?My only guess is that they somehow decided that the "younger brother called up first group" is the important one, and they just quoted that number. I got
87% for that, and their numbers are a little different, so maybe they got 90%. Another possibility: it might have something to do with the regression they used to correct for a bunch of things,
including which "called up first" group the batter belonged to. They don't give the regression equation or the details, but depending on the way they chose dummy variables, I can see the regression
looking something like:Percentage of younger brothers outstealing older brothers equals:-- 90% -- minus 20% if the brothers were called up at the same time-- minus 20% if the older brother was called
up first.So you can see "90%" coming up as a estimate in a regression equation. But still, the authors would certainly have noticed that you can't just quote the 90% figure without adjusting for the
other dummies.----Here's another way they might have got it wrong. As you see in the quote above, the authors note that the older players turned out to play more games, and they were also likely to
get called up first. So maybe they adjusted the numbers in the service of "controlling athletic performance" (the actual words they use). Suppose that in 1950, older brother A stole 10 bases and
younger brother B didn't steal any because he wasn't in the majors yet. If you "adjust" for that by including it in the regression, effectively subtracting 10 for every line of A's career, I can see
how the "adjustment" might now show B having a 90% chance to steal more "adjusted" bases than A.Again, I'm speculating, which I shouldn't. I don't think the authors did this, but I wouldn't be
surprised if it's an adjustment something along those lines, just more subtle.----Another related stat the study comes up with is that younger brothers are 10.58 times more likely to steal a base
than their older brother. I don't know where that one comes from either. Again, the authors' own raw data comes up with something much more realistic. As a ratio of times on base, the authors' SB+CS
rates were5.6% for older brothers9.3% for younger brothersThat's only 1.66 times more likely. So, obviously, the authors feel there's some huge bias here that, when you correct for it, brings the
1.66 up to 10.58. I can't imagine what that might be, and I can't really duplicate the authors' regression because they don't even give the equation.Here are my calculated batting lines for the two
groups. Is there really a factor of ten difference here?------ AB -R --H 2B 3B HR RBI BB SB CS -avg RC/G
Young 553 70 146 26 04 11 062 50 12 05 .265 4.39
-Old- 546 75 148 26 03 17 074 54 08 04 .271 4.98You'd think that they'd give a simple English explanation of how, when you look at the batting lines of the two groups, you see young players
attempting steals at maybe 50% more than the older ones, but how when *they* look at the two batting lines, they see one line stealing bases at almost 1000% more than the older ones. But I couldn't
find any such explanation in the paper.So when they say,
"... younger brothers were 10.6 times more likely than their older brothers to attempt steals"
... well, I don't see any way that can possibly be true.
UPDATE: batting lines updated, they were slightly off ... also updated to reflect the authors' explanation of something I had missed.
ADDENDUM: as you can see, my data show the younger players indeed attempting more steals than the older players. In my sample, it's 46% more; in the authors' sample, it's 66% more.
My 46% seems like a lot ... I wondered there's perhaps a real effect there. So I ran a simulation to check for statistical significance.
For each of my 98 pairs of brothers, I *randomly* decided which one to consider as older, instead of looking at their real ages. Then I computed the relative attempt rate between the two random
groups. I repeated that 60,000 times.
The results? 4,126 of the 60,000 random sets had one group attempting at least 46% more steals than the other group. That's a p-value of 0.069 ... not significant at 5%, but close.
I had expected more significance than that, but the players are "lumpy," in that some players steal a lot, and some don't. A few faster players landing in the same group can make a big difference.
ADDENDUM 2: In the comments, I think Guy came up with the answer. Younger brothers have shorter careers. Shorter careers tend to be centered on lower ages (there are 40 year old players, but no 10
year old players). Young players are faster.
That's why younger brothers have more attempted steals -- they play more years during peak stealing age.
I bet if you did it season-by-season and controlled for age, most of the effect would disappear.
Good catch by Guy.
Labels: baseball, baserunning, siblings
14 Comments: | {"url":"http://blog.philbirnbaum.com/2010/06/do-younger-brothers-steal-more-bases.html","timestamp":"2024-11-02T11:40:25Z","content_type":"application/xhtml+xml","content_length":"54252","record_id":"<urn:uuid:0bb13a4f-8c6b-4946-91a8-afa7462608d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00466.warc.gz"} |
[Resource Topic] 2023/289: Lower-Bounds for Secret-Sharing Schemes for k-Hypergraphs
Welcome to the resource topic for 2023/289
Lower-Bounds for Secret-Sharing Schemes for k-Hypergraphs
Authors: Amos Beimel
A secret-sharing scheme enables a dealer, holding a secret string, to distribute shares to parties such that only pre-defined authorized subsets of parties can reconstruct the secret. The collection
of authorized sets is called an access structure. There is a huge gap between the best known upper-bounds on the share size of a secret-sharing scheme realizing an arbitrary access structure and the
best known lower-bounds on the size of these shares. For an arbitrary n-party access structure, the best known upper-bound on the share size is 2^{O(n)}. On the other hand, the best known lower-bound
on the total share size is much smaller, i.e., \Omega(n^2/\log (n)) [Csirmaz, \emph{Studia Sci. Math. Hungar.}]. This lower-bound was proved more than 25 years ago and no major progress was made
In this paper, we study secret-sharing schemes for k-hypergraphs, i.e., for access structure where all minimal authorized sets are of size exactly k (however, unauthorized sets can be larger). We
consider the case where k is small, i.e., constant or at most \log (n). The trivial upper-bound for these access structures is O(k\cdot \binom{n}{k}) and this can be slightly improved. If there were
efficient secret-sharing schemes for such k-hypergraphs (e.g., 2-hypergraphs or 3-hypergraphs), then we would be able to construct secret-sharing schemes for arbitrary access structure that are
better than the best known schemes. Thus, understanding the share size required for k-hypergraphs is important. Prior to our work, the best known lower-bound for these access structures was \Omega(n
\log (n)), which holds already for graphs (i.e., 2-hypergraphs).
We improve this lower-bound, proving a lower-bound of \Omega(n^{1-1/(k-1)}/k) for some explicit k-hypergraphs, where 3 \leq k \leq \log (n). For example, for 3-hypergraphs we prove a lower-bound of \
Omega(n^{3/2}). For \log (n)-hypergraphs, we prove a lower-bound of \Omega(n^{2}/\log (n)), i.e., we show that the lower-bound of Csirmaz holds already when all minimal authorized sets are of size \
log (n). Our proof is simple and shows that the lower-bound of Csirmaz holds for a simple variant of the access structure considered by Csirmaz.
ePrint: https://eprint.iacr.org/2023/289
See all topics related to this paper.
Feel free to post resources that are related to this paper below.
Example resources include: implementations, explanation materials, talks, slides, links to previous discussions on other websites.
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Minimax optimal testing via classification
This paper considers an ML inspired approach to hypothesis testing known as classifier/classification-accuracy testing (CAT). In CAT, one first trains a classifier by feeding it labeled synthetic
samples generated by the null and alternative distributions, which is then used to predict labels of the actual data samples. This method is widely used in practice when the null and alternative are
only specified via simulators (as in many scientific experiments). We study goodness-of-fit, two-sample (TS) and likelihood-free hypothesis testing (LFHT), and show that CAT achieves (near-)minimax
optimal sample complexity in both the dependence on the total-variation (TV) separation ϵ and the probability of error δ in a variety of non-parametric settings, including discrete distributions,
d-dimensional distributions with a smooth density, and the Gaussian sequence model. In particular, we close the high probability sample complexity of LFHT for each class. As another highlight, we
recover the minimax optimal complexity of TS over discrete distributions, which was recently established by Diakonikolas et al. (2021). The corresponding CAT simply compares empirical frequencies in
the first half of the data, and rejects the null when the classification accuracy on the second half is better than random.
• Classifier-accuracy testing
• Closeness testing
• Goodness-of-fit testing
• Identity testing
• Likelihood-free hypothesis testing
• Likelihood-free inference
• Scheffé’s test
• Two-sample testing
ASJC Scopus subject areas
• Artificial Intelligence
• Software
• Control and Systems Engineering
• Statistics and Probability
Dive into the research topics of 'Minimax optimal testing via classification'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/minimax-optimal-testing-via-classification","timestamp":"2024-11-15T00:23:35Z","content_type":"text/html","content_length":"51944","record_id":"<urn:uuid:ecd3ff90-bc2f-4295-8788-ad54a737f33a>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00135.warc.gz"} |
Each DisjointSets object represents a family of disjoint sets, where each element has an integer index.
It is implemented with the optimizations discussed in class, as up-trees stored in a single vector of ints. Specifically, path compression and union-by-size are used. Each place in the vector
represents a node. (Note that this means that the elements in our universe are indexed starting at 0.) A nonnegative number is the index of the parent of the current node; a negative number in a root
node is the negative of the set size.
Note that the default compiler-supplied Big Three will work flawlessly because the only member data is a vector of integers and this vector should initially be empty.
void DisjointSets::setunion ( int a,
int b
This function should be implemented as union-by-size.
That is, when you setunion two disjoint sets, the smaller (in terms of number of nodes) should point at the larger. This function works as described in lecture, except that you should not assume that
the arguments to setunion are roots of existing uptrees.
Your setunion function SHOULD find the roots of its arguments before combining the trees. If the two sets are the same size, make the tree containing the second argument point to the tree containing
the first. (Note that normally we could break this tie arbitrarily, but in this case we want to control things for grading.)
a Index of the first element to union
b Index of the second element to union | {"url":"https://courses.grainger.illinois.edu/cs225/sp2018/doxygen/mp7/classDisjointSets.html","timestamp":"2024-11-02T05:53:32Z","content_type":"application/xhtml+xml","content_length":"11183","record_id":"<urn:uuid:69b2dcf6-efde-4d21-bbe1-8e5dfc4b8847>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00336.warc.gz"} |
more from Herbert B. Enderton
Single Idea 9996
[catalogued under 5. Theory of Logic / K. Features of Logics / 7. Decidability]
Full Idea
A set of expressions is 'decidable' iff there exists an effective procedure (qv) that, given some expression, will decide whether or not the expression is included in the set (i.e. doesn't contradict
Gist of Idea
Expressions are 'decidable' if inclusion in them (or not) can be proved
Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7)
Book Reference
Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.62
A Reaction
This is obviously a highly desirable feature for a really reliable system of expressions to possess. All finite sets are decidable, but some infinite sets are not. | {"url":"http://www.philosophyideas.com/search/response_philosopher_detail.asp?era_no=L&era=Late%2020th%20century%20(1956-2000)&id=9996&PN=3696&order=chron","timestamp":"2024-11-11T07:42:01Z","content_type":"application/xhtml+xml","content_length":"3401","record_id":"<urn:uuid:b0e90bbd-a021-42e4-a061-75bc4406accf>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00685.warc.gz"} |
Thin films provide a rich setup that is of interest from multiple points of view. From application side, these thin films and their instabilities are of relevance to a number of fields that rangefrom
glass making to semiconductor and polymer applications, solar cells, to mention just a few fields of applications. From modeling side, thin films are demanding due to the presence of multiple spatial
and temporal scales and due to the need to incorporate multi physics that is often of relevance in particular in the setups involving spreading and instabilities of thin films on substrates.
Recent group activities on the topic of Newtonian thin films have focused on modeling and simulations of metal films of nanoscale thickness exposed to laser radiation, in collaboration with the
experimental group at U. Tennessee and Oak Ridge National Laboratory. Metal films are relevant to a number of technological fields with applications that include plasmonics, magnetic nanoparticles,
control surface optical properties, catalysts for nanowire growth and many others. In many of these fields ordered arrays of nanoparticles are needed. While in liquid phase, the film instabilities
lead to such arrays via self- or directed- assembly, and we are interested in the basic mechanisms that are relevant to the instabilities.
At NJIT, we have carried extensive simulations of thin metal films within the long-wave model that reduces the problem to solving a nonlinear 4th order evolution equation of diffusion type, as well
as by directly solving Navier-Stokes equations using Volume-of-Fluid method. A number of new and interesting results has been obtained, including much better understanding of the connection between
film instability and its geometry, of the influence of fluid/solid interaction forces, heat flow through the film and the substrate, to name just a few aspects.
Earlier works on thin films include a variety of configurations and setups including considerations of stochastic effects, flows down an incline, hanging and evaporating films, among others. Some of
the considered configurations were also explored in research projects including undergraduate students in the Capstone Laboratory [link to the capstone page]. The relevant recent publications can be
found below, and older works are listed at the publication list or here.
We acknowledge past and current support by NSF and Fulbright Foundation. | {"url":"https://cfsm.njit.edu/newtonianfilms.php","timestamp":"2024-11-08T05:01:02Z","content_type":"text/html","content_length":"55094","record_id":"<urn:uuid:4ab6602b-1851-4f76-afac-fb691b58fe9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00801.warc.gz"} |
Chi-Square Test [Simply explained] | Video Summary and Q&A | Glasp
Chi-Square Test [Simply explained] | Summary and Q&A
277.1K views
February 1, 2022
Chi-Square Test [Simply explained]
The chi-square test is used to determine relationships between categorical variables, such as gender and education level.
Key Insights
• 🤏 The chi-square test is used to determine relationships between categorical variables.
• 👥 Categorical variables consist of characteristics that can be grouped into categories.
• 🤏 The chi-square test can be used to investigate relationships between variables, such as gender and education level.
• 🤏 Statistical software, like DataTab, can be used to calculate the chi-square test automatically.
• 🤏 The chi-square value can be calculated by hand using the formula provided in the video.
• 🤏 The p-value from the chi-square test helps interpret the results and determine if there is a significant relationship between variables.
• 🤏 Critical chi-square values are used to compare the calculated chi-square value and determine if the null hypothesis should be rejected.
in this video i explain in a simple way what the chi-square test is and how you can calculate the chi-square test with just a few steps this leads us right to the first question what is the
chi-square test the chi-square test is a hypothesis test that is used when you want to determine whether there is a relationship between two categorical variabl... Read More
Questions & Answers
Q: What is the chi-square test and when is it used?
The chi-square test is a hypothesis test used to determine relationships between categorical variables. It is used when we want to establish if there is a correlation between two categorical
Q: How are categorical variables different from non-categorical variables?
Categorical variables have characteristics that can be grouped, such as gender or preferred newspaper. Non-categorical variables, like weight or salary, do not have distinct categories.
Q: How can the chi-square test be used to investigate relationships between variables?
The chi-square test can be used to investigate relationships between variables by creating a questionnaire and collecting data. We can then analyze the data using statistical software or by
calculating the chi-square test by hand.
Q: How is the chi-square test calculated using statistical software?
Statistical software, like DataTab, can calculate the chi-square test by inputting the data into a cross table. The software then provides the expected frequencies and the results of the chi-square
test, including the p-value.
Q: What is the chi-square test and when is it used?
The chi-square test is a hypothesis test used to determine relationships between categorical variables. It is used when we want to establish if there is a correlation between two categorical
More Insights
• The chi-square test is used to determine relationships between categorical variables.
• Categorical variables consist of characteristics that can be grouped into categories.
• The chi-square test can be used to investigate relationships between variables, such as gender and education level.
• Statistical software, like DataTab, can be used to calculate the chi-square test automatically.
• The chi-square value can be calculated by hand using the formula provided in the video.
• The p-value from the chi-square test helps interpret the results and determine if there is a significant relationship between variables.
• Critical chi-square values are used to compare the calculated chi-square value and determine if the null hypothesis should be rejected.
• DataTab provides a user-friendly platform for conducting the chi-square test and offers tutorials for further learning.
Summary & Key Takeaways
• The chi-square test is a hypothesis test used to determine relationships between categorical variables.
• Categorical variables are characteristics like gender, preferred newspaper, or television viewing frequency.
• The chi-square test can be used to investigate relationships between variables, such as gender and education level.
Explore More Summaries from DATAtab 📚 | {"url":"https://glasp.co/youtube/p/chi-square-test-simply-explained","timestamp":"2024-11-04T15:37:14Z","content_type":"text/html","content_length":"370351","record_id":"<urn:uuid:488f9007-2ead-4378-bdf6-63c4c0b02c72>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00252.warc.gz"} |
Identifying Points in Triangulation (GNU Octave (version 9.2.0))
30.1.2 Identifying Points in Triangulation ¶
It is often necessary to identify whether a particular point in the N-dimensional space is within the Delaunay tessellation of a set of points in this N-dimensional space, and if so which N-simplex
contains the point and which point in the tessellation is closest to the desired point. The functions tsearch and dsearch perform this function in a triangulation, and tsearchn and dsearchn in an
N-dimensional tessellation.
To identify whether a particular point represented by a vector p falls within one of the simplices of an N-simplex, we can write the Cartesian coordinates of the point in a parametric form with
respect to the N-simplex. This parametric form is called the Barycentric Coordinates of the point. If the points defining the N-simplex are given by N + 1 vectors t(i,:), then the Barycentric
coordinates defining the point p are given by
where beta contains N + 1 values that together as a vector represent the Barycentric coordinates of the point p. To ensure a unique solution for the values of beta an additional criteria of
is imposed, and we can therefore write the above as
p - t(end, :) = beta(1:end-1) * (t(1:end-1, :)
- ones (N, 1) * t(end, :)
Solving for beta we can then write
beta(1:end-1) = (p - t(end, :)) /
(t(1:end-1, :) - ones (N, 1) * t(end, :))
beta(end) = sum (beta(1:end-1))
which gives the formula for the conversion of the Cartesian coordinates of the point p to the Barycentric coordinates beta. An important property of the Barycentric coordinates is that for all points
in the N-simplex
Therefore, the test in tsearch and tsearchn essentially only needs to express each point in terms of the Barycentric coordinates of each of the simplices of the N-simplex and test the values of beta.
This is exactly the implementation used in tsearchn. tsearch is optimized for 2-dimensions and the Barycentric coordinates are not explicitly formed.
An example of the use of tsearch can be seen with the simple triangulation
x = [-1; -1; 1; 1];
y = [-1; 1; -1; 1];
tri = [1, 2, 3; 2, 3, 4];
consisting of two triangles defined by tri. We can then identify which triangle a point falls in like
tsearch (x, y, tri, -0.5, -0.5)
⇒ 1
tsearch (x, y, tri, 0.5, 0.5)
⇒ 2
and we can confirm that a point doesn’t lie within one of the triangles like
tsearch (x, y, tri, 2, 2)
⇒ NaN
The dsearch and dsearchn find the closest point in a tessellation to the desired point. The desired point does not necessarily have to be in the tessellation, and even if it the returned point of the
tessellation does not have to be one of the vertices of the N-simplex within which the desired point is found.
An example of the use of dsearch, using the above values of x, y and tri is
dsearch (x, y, tri, -2, -2)
⇒ 1
If you wish the points that are outside the tessellation to be flagged, then dsearchn can be used as
dsearchn ([x, y], tri, [-2, -2], NaN)
⇒ NaN
dsearchn ([x, y], tri, [-0.5, -0.5], NaN)
⇒ 1
where the point outside the tessellation are then flagged with NaN. | {"url":"https://docs.octave.org/interpreter/Identifying-Points-in-Triangulation.html","timestamp":"2024-11-11T23:57:21Z","content_type":"text/html","content_length":"16316","record_id":"<urn:uuid:221728d8-eedc-4b85-aa68-ceb0ac4ef61c>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00466.warc.gz"} |
Multiplication Single Digit Worksheets
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Encouraging consistent method, offering help, and creating a favorable understanding atmosphere are advantageous actions. | {"url":"https://crown-darts.com/en/multiplication-single-digit-worksheets.html","timestamp":"2024-11-05T06:03:16Z","content_type":"text/html","content_length":"28928","record_id":"<urn:uuid:2bd32678-49ce-4a82-80a6-a6ebb4d95dd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00826.warc.gz"} |
Analysis of Two Coplanar Forces Acting Together to Produce a Resultant
Question Video: Analysis of Two Coplanar Forces Acting Together to Produce a Resultant
Two forces of magnitudes 15 kg-wt and 7 kg-wt act as shown in the diagram. Determine π , the magnitude of the resultant, and find π , the measure of the angle between the resultant and due east.
Video Transcript
Two forces of magnitudes 15 kilogram-weight and seven kilogram-weight act as shown in the diagram. Determine π , the magnitude of the resultant, and find π , the measure of the angle between the
resultant and due east.
And then we have a diagram that shows the two forces and the angle between them. So letβ s begin by reminding ourselves what we mean when we talk about the resultant of a pair of forces. Suppose we
have two forces defined by the vectors π sub one and π sub two. The resultant of these two forces is simply their sum. And thinking about these in terms of vectors can really help us plan what
to do next.
Letβ s define the force with magnitude 15 kilogram-weight to be the vector π sub one and the other force to be the vector π sub two. The sum of these two vectors can be represented with the
vector of magnitude π as shown. This is a triangle of forces. And we can use right triangle and non-right triangle trigonometry to help us find any missing dimensions.
Letβ s begin by finding the measure of the angle that we can label π Ό between the seven-newton and 15-newton force. We know that the π Ό and 30-degree angle sum to 90 degrees. So if we subtract
30 from both sides, we know that π Ό must be equal to 60. Then, we now have a non-right triangle for which we know two of its lengths and the angle between those. So we can use the law of cosines to
find the measure of the third side. That is, π squared equals π squared plus π squared minus two π π cos π ΄. We then label the side that weβ re trying to find lowercase π and the
angle opposite it, the 60-degree angle, uppercase π ΄. So π squared is seven squared plus 15 squared minus two times seven times 15 times cos 60. Seven squared plus 15 squared is 274, and cos of
60 is one-half. So the second part of this is simply seven times 15, which is 105. 274 minus 105 is 169. So our equation is π squared equals 169.
Since π is a magnitude, it has to be positive. So we can solve this equation by simply taking the positive root of 169, which is 13. So the magnitude of the resultant in this case is 13
So weβ re now ready to calculate the angle π . On our diagram, itβ s the measure of the angle between the resultant and due east. Well, in our diagram, the 15-newton force is acting due east. So
the angle π is the angle that the side that we calculated to be 13 makes with the side of 15. We can use the law of sines to calculate this angle. In this case, thatβ s sin π ΄ over π equals
sin π ΅ over π . Weβ ve just calculated π to be 13. So thatβ s sin 60 divided by 13 equals sin π over seven. sin 60 is root three over two. So when we multiply through by seven, we find
that sin π is seven root three over 26.
To solve for π , we take the inverse or arcsine of both sides of the equation. So π is inverse sin of seven root three over 26, which is 27.795 and so on. Next, weβ ll multiply the decimal part
by 60 so we can get this to the nearest minute. That gives us a minute part of 47.7, which is 48 correct to the nearest minute. And so we found π and π . π is 13 kilogram-weight, and π is
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How to add liquid filters when customising texts
In cornercart in wherever you could add dynamic data inside the text you can make use of liquid filters to modify the said dynamic data.
To apply filters to an output, add the filter and any filter parameters within the output's curly brace delimiters {{ }}, preceded by a pipe character |. In the example below, title is its property,
and capitalize is the filter being applied.
{{ title | capitalize }}
Filters with parameters
Many filters accept parameters that let you specify how the filter should be applied. Some filters might require parameters to be valid.
{{ title | append: "buy now" }}
String filters
String filters modify strings.
string | append: string
: returns string
Adds a given string to the end of a string.
{{ daysRemaining | append: "days" }}
daysRemaining: 2
RESULT: 2days
string | capitalize
: returns string
Capitalizes the first word in a string and downcases the remaining characters.
{{ productTitle | capitalize }}
productTitle: drop shoulder tshirt
RESULT: Drop shoulder tshirt
string | downcase
: returns string
Converts a string to all lowercase characters.
{{ productTitle | downcase }}
productTitle: Drop shoulder tshirt
RESULT: drop shoulder tshirt
string | pluralize: string, string
: returns string
Outputs the singular or plural version of a string based on a given number.
Caution: The pluralize filter applies English pluralization rules to determine which string to output. You shouldn't use this filter on non-English strings because it could lead to incorrect
{{ daysRemaining }} {{ daysRemaining | pluralize: " day", " days" }}
daysRemaining: 32
RESULT: 32 days
daysRemaining: 1
RESULT: 1 day
Math filters
Math filters perform mathematical operations on numbers. You can apply math filters to numbers, or variables or metafields that return a number.
As with any other filters, you can use multiple math filters on a single input. They’re applied from left to right. In the example below, minus is applied first, then times, and finally divided_by.
number | plus: number
: returns number
Adds 2 numbers
{{ goal | plus: 2 }}
goal: 320
RESULT: 322
number | minus: number
: returns number
Subtracts a given number from another number.
{{ goal | minus: 2 }}
goal: 320
RESULT: 318
number | divided_by: number
: returns number
Divides a number by a given number. The divided_by filter produces a result of the same type as the divisor. This means if you divide by an integer, the result will be an integer, and if you divide
by a float, the result will be a float.
{{ goal | divided_by: 2 }}
goal: 320
RESULT: 160
number | times: number
: returns number
Multiplies a number by a given number.
{{ goal | times: 2 }}
goal: 320
RESULT: 640
number | round
: returns number
Rounds a number to the nearest integer.
{{ goal | round }}
goal: 320.98
RESULT: 321
Updated on: 19/04/2024
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Library Of American University Of Madaba
Agent-Based Modeling and Simulation -- Cellular Automata, Mathematical Basis of Complex Networks and Graph Theory -- Data Mining and Knowledge Discovery -- Game Theory -- Granular Computing --
Intelligent Systems -- Probability and Statistics in Complex Systems -- Quantum Information Science -- Social Network Analysis -- 3 entries from the section Social Science, Physics and Mathematical
Applications: Minority Games; Rational, Goal-Oriented Agents; and Social Processes, Simulation Models in Soft Computing -- Unconventional Computing -- Wavelets.
Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of
temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even
completely deterministic. The recognition that the collective behavior of the whole system cannot be simply inferred from an understanding of the behavior of the individual components has led to the
development of numerous sophisticated new computational and modeling tools with applications to a wide range of scientific, engineering, and societal phenomena. Computational Complexity: Theory,
Techniques and Applications presents a detailed and integrated view of the theoretical basis, computational methods, and state-of-the-art approaches to investigating and modeling of inherently
difficult problems whose solution requires extensive resources approaching the practical limits of present-day computer systems. This comprehensive and authoritative reference examines key
components of computational complexity, including cellular automata, graph theory, data mining, granular computing, soft computing, wavelets, and more.
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Launch the Fit Life by X Platform
Launch the Fit Life by X platform by selecting Analyze > Reliability and Survival > Fit Life by X. For more information about the options in the Select Columns red triangle menu, see Column Filter
Menu in Using JMP.
Figure 4.4 The Fit Life by X Launch Window
The Fit Life by X launch window contains the following options:
Y, Time to Event
Identifies the time to event (such as the time to failure) or time to censoring. With interval censoring, specify two Y variables, where one Y variable gives the lower limit and the other Y variable
gives the upper limit for each unit. For more information about censoring, see Event Plot.
Identifies the accelerating factor.
Identifies censored observations. Select the value that identifies right-censored observations from the Censor Code menu beneath the Select Columns list. The Censor column is used only when one Y is
Identifies frequencies or observation counts when there are multiple units. If the value is 0 or a positive integer, then the value represents the frequencies or counts of observations for each row
when there are multiple units recorded.
Identifies a column that creates a report consisting of separate analyses for each level of the variable.
Censor Code
Identifies the value in the Censor column that designates right-censored observations. After a Censor column is selected, JMP attempts to automatically detect the censor code and display it in the
box. To change this, you can click the red triangle and select from a list of values. You can also enter a different value in the box. If the Censor column contains a Value Labels column property,
the value labels appear in the list of values. Missing values are excluded from the analysis.
Identifies the relationship between the event and the accelerating factor. Table 4.1 defines the model for each relationship.
Table 4.1 Models for Relationship Options
Relationship Model
Arrhenius Celsius μ = b0 + b1 * 11604.5181215503 / ( X + 273.15 )
Arrhenius Fahrenheit μ = b0 + b1 * 11604.5181215503 / ( ( X + 459.67 ) / 1.8 )
Arrhenius Kelvin μ = b0 + b1 * 11604.5181215503 / X
Voltage μ = b0 + b1 * log( X )
Linear μ = b0 + b1 * X
Log μ = b0 + b1 * log( X )
Logit μ = b0 + b1 * log( X / ( 1 - X ) )
Reciprocal μ = b0 + b1 / X
Square Root μ = b0 + b1 * sqrt( X )
Box-Cox μ = b0 + b1 * BoxCox( X )
Location means that μ is different for every level of X
Location and Scale means that μ and σ are both different for every level of X (equivalent to a Life Distribution fit with X as a By variable)
Custom user-defined μ and σ
If you select Box-Cox, a text edit box appears below the Use Condition option. Use this box to specify a lambda value. The BoxCox( X ) transformation for a specified λ is defined as follows:
If you want to use a Custom relationship for your model, see Custom Relationship.
Nested Model Tests
Appends a nonparametric overlay plot, nested model tests, and a multiple probability plot to the report window.
Use Condition
Enables you to enter a value for the explanatory variable, X, of the acceleration factor. You can also set the use condition value after launching the platform using the Set Time Acceleration Use
Condition option from the Fit Life by X red triangle menu.
Specifies one distribution (Weibull, Lognormal, Loglogistic, Fréchet, SEV, Normal, Logistic, LEV, or Exponential distributions) at a time. Lognormal is the default setting.
Select Confidence Interval Method
Displays the method for computing confidence intervals for the parameters. The default is Wald, but you can select Likelihood instead. The Wald method is an approximation and runs faster. The
Likelihood method provides more precise parameters but takes longer to compute.
Note: The Confidence Interval Method preference enables you to select the Likelihood method as the default confidence interval method. You can change this preference in Preferences > Platforms > Fit
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Rotation in null space for elasticity
This post suggests constructing rotational components of null space in linear elasticity as
Z_rot = [Expression(('0', 'x[2]', '-x[1]')),
Expression(('-x[2]', '0', 'x[0]')),
Expression(('x[1]', '-x[0]', '0'))]
by @MiroK. It works like a magic in my project, but as someone without solid mechanics background, I’m wondering if anybody understands how this is come up and share some insights here, for me and
anyone who need it in the future.
Really appreciate any explanation/reference etc
Hi, the 6 vector fields that are in the nullspace of the linear elasticity operator are the deformations u for which (\nabla u+(\nabla u)^T)=0 and \nabla\cdot u = 0. The latter term relates to the
change of volume due to u, i.e. you are looking for some deformation which preserves volume. Intuitively, translations and rotations are (only) such vector fields; the body which is moved and/or
rotated keeps its volume, right? In 3d you have 3 translations (around each of the axes) and 3 rotations (around each of the axes). Z_{rot} are rotations around x, y, z-axis .
3 Likes
It’s also nice described in the elasticity demo
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Fraction calculator
This calculator subtracts two fractions. First, convert all fractions to a common denominator when fractions have different denominators. Find Least Common Denominator (LCD) or multiply all
denominators to find a common denominator. When all denominators are the same, simply subtract the numerators and place the result over the common denominator. Then simplify the result to the lowest
terms or a mixed number.
The result:
8/9 - 1/3 = 5/9 ≅ 0.5555556
The spelled result in words is five ninths.
How do we solve fractions step by step?
1. Subtract: 8/9 - 1/3 = 8/9 - 1 · 3/3 · 3 = 8/9 - 3/9 = 8 - 3/9 = 5/9
It is suitable to adjust both fractions to a common (equal, identical) denominator for adding, subtracting, and comparing fractions. The common denominator you can calculate as the least common
multiple of both denominators - LCM(9, 3) = 9. It is enough to find the common denominator (not necessarily the lowest) by multiplying the denominators: 9 × 3 = 27. In the following intermediate
step, it cannot further simplify the fraction result by canceling.
In other words - eight ninths minus one third is five ninths.
Rules for expressions with fractions:
- use a forward slash to divide the numerator by the denominator, i.e., for five-hundredths, enter
. If you use mixed numbers, leave a space between the whole and fraction parts.
Mixed numerals
(mixed numbers or fractions) keep one space between the integer and
fraction and use a forward slash to input fractions i.e.,
1 2/3
. An example of a negative mixed fraction:
-5 1/2
Because slash is both sign for fraction line and division, use a colon (:) as the operator of division fractions i.e.,
1/2 : 1/3
Decimals (decimal numbers) enter with a decimal point
and they are automatically converted to fractions - i.e.
The calculator follows well-known rules for the order of operations. The most common mnemonics for remembering this order of operations are:
PEMDAS - Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
BEDMAS - Brackets, Exponents, Division, Multiplication, Addition, Subtraction
BODMAS - Brackets, Of or Order, Division, Multiplication, Addition, Subtraction.
GEMDAS - Grouping Symbols - brackets (){}, Exponents, Multiplication, Division, Addition, Subtraction.
MDAS - Multiplication and Division have the same precedence over Addition and Subtraction. The MDAS rule is the order of operations part of the PEMDAS rule.
Be careful; always do multiplication and division before addition and subtraction. Some operators (+ and -) and (* and /) have the same priority and must be evaluated from left to right.
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How are dating methods used to determine age?
Relative dating is used to determine a fossils approximate age by comparing it to similar rocks and fossils of known ages. Absolute dating is used to determine a precise age of a fossil by using
radiometric dating to measure the decay of isotopes, either within the fossil or more often the rocks associated with it.Relative dating is used to determine a fossils approximate age by comparing it
to similar rocks and fossils of known ages. Absolute dating
What dating technique will you use to know the age?
radiometric dating methods To establish the age of a rock or a fossil, researchers use some type of clock to determine the date it was formed. Geologists commonly use radiometric dating methods,
based on the natural radioactive decay of certain elements such as potassium and carbon, as reliable clocks to date ancient events.
How are dating methods used?
Dating techniques are procedures used by scientists to determine the age of rocks, fossils, or artifacts. Relative dating methods tell only if one sample is older or younger than another; absolute
dating methods provide an approximate date in years. The latter have generally been available only since 1947.
What are the two dating techniques used to determine the age of the Earth?
Absolute dating techniques include radiocarbon dating of wood or bones, potassium-argon dating, and trapped-charge dating methods such as thermoluminescence dating of glazed ceramics. C) to systems
such as uranium–lead dating that allow determination of absolute ages for some of the oldest rocks on Earth.
Which dating method would you use to determine the age of a fossilized tooth fragment?
Fossil age is determined using two methods, relative dating and absolute dating. In relative dating, fossils are compared to similar fossils and rocks, the ages of which are known. Absolute dating,
on the other hand is used to calculate the precise age of fossils through radiometric dating.
Are older fossils found deeper?
The positions of fossils in rocks indicate their relative ages; older fossils and rock layers are deeper than fossils and rocks that are more recent.
How do you date something?
Fun date ideasTake a class for something new. There are tons of places that have classes. Hit up a go-kart track. Tons of fun even though it can be a little expensive. Go sky diving or bungee
jumping. Backyard camping or just go camping. Join a fun looking meetup together. Take a dance lesson. Go ziplining. Do some geocaching.More items | {"url":"http://animereview.jp/zidekif83290.html","timestamp":"2024-11-05T16:44:05Z","content_type":"text/html","content_length":"24557","record_id":"<urn:uuid:269a17e7-aaa1-4336-bcef-3a00942d9b24>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00133.warc.gz"} |
Year 3 Mathematics – Malaysia
Year 3 Mathematics – Malaysia
Year 3 Mathematics
# TOPIC TITLE
1 Self Assessment Self Assessment – Year 3
Objective: Assessment
2 Fractions Using fractions 1/2, 1/4, 1/8 to describe part of a whole
Objective: On completion of the lesson the student will be able to: name fractions, and use fractions to describe equal parts of a whole.
3 Fractions Using fractions 1/2, 1/4, 1/8 to describe parts of a group or collection
Objective: On completion of the lesson the student will be able to use the fractions to describe equal parts of a collection of objects.
4 Using and applying number The number 10
Objective: On completion of the lesson the student will be able to write the numeral and word for the number 10, count up to 10 objects and to match a group of objects to the correct number.
5 Using and applying number Numbers 11 to 20
Objective: On completion of the lesson the student will be able to write the numeral and word for the numbers 11 to 20, count up to 20 and match a group of objects to the correct number.
6 Calculations The numbers 20 to 99
Objective: On completion of the lesson the student will be able to identify, name and add groups of ten, counting to ninety-nine.
7 Calculation-larger numbers The numbers 100 to 999
Objective: On completion of the lesson the student will be able to count to 999, skip count by 10s and 100s to 999 and use pictures or objects to represent the numbers 100 to 999.
8 Place value The numbers 1000 to 9999
Objective: On completion of the lesson the student will be able to count to 999, skip count by 10s and 100s to 999 and use pictures or objects to represent the numbers 100 to 999.
9 Using and applying number The number 10
Objective: On completion of the lesson the student will be able to write the numeral and word for the number 10, count up to 10 objects and to match a group of objects to the correct number.
10 Using and applying number Numbers 11 to 20
Objective: On completion of the lesson the student will be able to write the numeral and word for the numbers 11 to 20, count up to 20 and match a group of objects to the correct number.
11 Calculations The numbers 20 to 99
Objective: On completion of the lesson the student will be able to identify, name and add groups of ten, counting to ninety-nine.
12 Calculation-larger numbers The numbers 100 to 999
Objective: On completion of the lesson the student will be able to count to 999, skip count by 10s and 100s to 999 and use pictures or objects to represent the numbers 100 to 999.
13 Place value The numbers 1000 to 9999
Objective: On completion of the lesson the student will be able to count to 999, skip count by 10s and 100s to 999 and use pictures or objects to represent the numbers 100 to 999.
14 Counting and numeration The numbers 10 000 to 99 999
Objective: On completion of the lesson the student will be able to count to 99 999 and use place value to read the value of the numerals within the larger numbers.
15 Calculation-larger numbers The numbers 100 to 999
Objective: On completion of the lesson the student will be able to count to 999, skip count by 10s and 100s to 999 and use pictures or objects to represent the numbers 100 to 999.
16 Place value The numbers 1000 to 9999
Objective: On completion of the lesson the student will be able to count to 999, skip count by 10s and 100s to 999 and use pictures or objects to represent the numbers 100 to 999.
17 Calculation 10-100 Counting by 1, 2, 5, and 10 to 100
Objective: On completion of the lesson the student will be able to skip count to one hundred and show that one hundred equals ten times ten.
18 Fractions Using fractions 1/2, 1/4, 1/8 to describe part of a whole
Objective: On completion of the lesson the student will be able to: name fractions, and use fractions to describe equal parts of a whole.
19 Fractions Using fractions 1/2, 1/4, 1/8 to describe part of a whole
Objective: On completion of the lesson the student will be able to: name fractions, and use fractions to describe equal parts of a whole.
20 Fractions Using fractions 1/2, 1/4, 1/8 to describe parts of a group or collection
Objective: On completion of the lesson the student will be able to use the fractions to describe equal parts of a collection of objects.
21 Decimals Introduction to decimals
Objective: On completion of the lesson the student will be able to represent decimals to two decimal places.
22 Calculation-grouping Multiplication using equal groups
Objective: On completion of the lesson the student will be able to write about equal groups and rows and will understand how to count them.
23 Calculation-grouping Multiplication using equal groups
Objective: On completion of the lesson the student will be able to write about equal groups and rows and will understand how to count them.
24 Calculation-grouping Multiplication using repeated addition
Objective: On completion of the lesson the student will be able to write about equal groups and rows using another method and also learn different ways to count them.
25 Calculation-grouping Multiplication using repeated addition
Objective: On completion of the lesson the student will be able to write about equal groups and rows using another method and also learn different ways to count them.
26 Calculation-larger numbers The numbers 100 to 999
Objective: On completion of the lesson the student will be able to count to 999, skip count by 10s and 100s to 999 and use pictures or objects to represent the numbers 100 to 999.
27 Calculation-grouping Multiplication using equal groups
Objective: On completion of the lesson the student will be able to write about equal groups and rows and will understand how to count them.
28 Data Pictograms
Objective: On completion of the lesson the student will be able to organise, read and summarise information in picture graphs.
29 Data Bar Charts
Objective: On completion of the lesson the student will be able to organise, read and summarise information in column graphs.
30 Data Pie and bar graphs.
Objective: On completion of the lesson the student will be able to organise, read and summarise information in pie and bar graphs.
31 Statistic-probability The mode
Objective: On completion of the lesson the student will understand how to find the mode from raw data, a frequency distribution table and polygon.
32 Data Pictograms
Objective: On completion of the lesson the student will be able to organise, read and summarise information in picture graphs.
33 Data Bar Charts
Objective: On completion of the lesson the student will be able to organise, read and summarise information in column graphs.
34 Data Pie and bar graphs.
Objective: On completion of the lesson the student will be able to organise, read and summarise information in pie and bar graphs.
35 Statistic-probability Probability of Simple Events
Objective: On completion of the lesson the student will be able to understand the probability of simple events.
36 2-D shapes Using the prefix to determine polygons
Objective: On completion of the lesson the student will be able to recognise and name two-dimensional shapes such as pentagons and hexagons, using the prefix of the shapes name to determine number of
angles and sides.
37 2-D shapes Spatial properties of quadrilaterals
Objective: On completion of the lesson the student will know how to analyse and explain the spatial properties of quadrilaterals.
38 3-D shapes Recognise and name prisms according to spatial properties
Objective: On completion of the lesson the student will be able to recognise and name various prisms according to their spatial properties.
39 3-D shapes Recognise nets for prisms, pyramids, cubes and cones
Objective: On completion of the lesson the student will be able to predict and recognise nets for prisms, pyramids, cubes and cones.
40 Length Using the metre as a formal unit to measure perimeter
Objective: On completion of the lesson the student will be able to calculate the perimeter of different shapes in metres.
41 Length Using the formal unit of the centimetre to measure length and perimeter
Objective: On completion of the lesson the student will be able to measure length and perimeter in centimetres.
42 Lines and angles Mapping and grid references
Objective: On completion of the lesson the student will be able to identify specific places on a map and use regions on a grid to locate objects or places.
43 Exam Exam – Year 3
Objective: Exam | {"url":"https://www.futureschool.com/malaysia-curriculum/mathematics-year-3/","timestamp":"2024-11-06T21:53:53Z","content_type":"text/html","content_length":"63527","record_id":"<urn:uuid:bce1796f-f2f4-498b-a14a-cb77c94e6f2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00581.warc.gz"} |
Power and Renewable Energy Course Description 2
Electrical Engineering Department Course Description (Lec, Lab, Cr. Hrs)
ELE205 Electronic Devices and Circuits I (3-2-4)
Basic properties of semiconductor materials. Theory of operation and applications of p-n junction diodes, Zener diodes and photodiodes. Theory of operation, biasing circuits, and small signal
analysis of Bipolar Junction Transistor and Junction Field Effect Transistor. Transistor configurations and two-port network representation of transistor a.c. equivalent circuits. Analysis and design
of transistor amplifier circuits. Prerequisite: ELE203
ELE305 Electronic Devices and Circuits II (3-2-4)
Operational amplifiers and their applications. MOSFETs: theory of operation and characteristics of depletion and enhancement type MOSFETs, analysis of various biasing circuits. Small-signal model and
AC analysis of amplifiers. Frequency response of amplifiers. Multistage amplifiers. Feedback amplifiers and oscillator circuits. Power amplifiers. Prerequisite: ELE305
ELE310 Design with Integrated Circuits (3-2-4)
A review of Op-Amps and Digital IC families. Design of analog signal conditioning circuits. Design of power supplies using IC regulators. Op-amp applications. Design of systems for measuring and
displaying the measured values on LEDs. Applications of ADC, DAC, and counter ICs. Optoisolators, triacs, and control of high-voltage systems and actuators. Design of signal generators. Applications
of commonly used ICs such as VCO, PLL, Timer IC, F/V and V/F ICs. Prerequisite: ELE305
ELE421 VLSI Design (3-0-3)
Introduction to VLSI design. Review of basic logic gates in CMOS. Integrated circuit layers, sheet resistance, time delay, CMOS layers, designing FET arrays, stick diagrams, layouts of CMOS circuits.
Fabrication of CMOS ICs. Design rules, physical limitations. Advanced techniques in CMOS logic circuits. General VLSI system components. Floor-planning and routing. DRAM, SRAM, ROM designs. Computer
simulation using VHDL or Verilog. Prerequisites: ELE305, ELE202
ELE425 Optoelectronics (3-0-3)
Fundamental concepts of semiconductors optical properties. Characteristics and classification of detectors. Radiation sources, classification of radiation sources. Population inversion and gain in a
two-level lasing medium. Optical feedback and laser cavity. P-N junction laser operating principles, threshold current, Hetero-junction lasers, Quantum well lasers, device fabrication and fiber
coupling. Optical fibers and design of optical systems. Prerequisites: ELE305, ELE303
ELE204 Signals and Systems (3-0-3)
Continuous- and discrete-time signals and systems. Basic system properties. Linear Time-Invariant (LTI) systems. Properties of LTI systems. Convolution sum. Fourier series of periodic signals.
Fourier transform of non-periodic signals. Filtering. Analysis of continuous-time LTI systems using Laplace transform. Prerequisite: MTH221
ELE302 Principles of Communication (3-2-4)
Introduction to fundamentals of communication systems. Amplitude Modulation (AM): Modulation index, spectrum of AM signals, AM circuits. Single side band modulation, frequency division multiplexing.
Frequency Modulation (FM): Spectrum of FM signals, FM circuits. FM versus AM. Sampling, quantization, coding, pulse code modulation, delta modulation, time division multiplexing. Shift Keying
methods. Prerequisite: ELE204
ELE303 Electromagnetic Fields and Wave Propagation (3-0-3)
Electrostatics: Coulomb’s Law, Gauss’s Law. Electric fields in material space, Polarization in Dielectrics. Ampere’s Law, Stoke’s Theorem. Time-varying Fields, Faraday’s Law, Maxwell’s Equations in
point form, Maxwell's equations in integral form, boundary conditions. Wave equation, plane wave propagation, Poynting vector and average power. Transmission line theory, reflection and transmission
on transmission lines. Prerequisites: PHY122, MTH221
ELE450 Digital Signal Processing (3-0-3)
Review of discrete-time signals and systems. Transform-domain representations of signals: Discrete-time Fourier Transform, Fast-Fourier Transform, applications of Z-Transform. Transform-domain
representations of LTI systems: Types of transfer functions, stability condition and test. Frequency response of a Rational Transfer Function. The difference equation and Digital Filtering. Concept
of filtering: Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) Filters. Prerequisite: ELE204
ELE451 Communication and Switching Networks (3-2-4)
Introduction to computer networks, protocol architecture and OSI reference model. Local Area Network (LAN): Topologies and transmission media. high-speed LAN. Token-Ring, FDDI. Circuit switching and
packet switching, ISDN, DSL, packet switching network, X.25, frame relay, ATM. Internetworking devices. UDP, TCP architecture, Internet protocols, TCP/IP. Application Layer: Client-server model,
socket interface, SMTP, FTP, HTTP, and WWW. Wireless Networking. Prerequisite: ELE302
ELE456 Telecommunication Systems (3-0-3)
Introduction to telecommunication systems. Telecommunication fundamentals and transmission media characteristics. Design analog and digital data transmission schemes. Telephony systems: ISDN and
PSTN, essentials of traffic engineering. Overview of Wireless LAN technology. Comparison of ZigBee with other standards and applications. Introduction to satellite and fiber optic based
communications. Prerequisite: ELE302
ELE455 Wireless Communications (3-0-3)
Introduction to cellular mobile radio systems: Cellular-concept system design fundamentals, trunking and grade of service. Mobile channel, large scale and small-scale fading. Outdoor propagation
models. Multiple access techniques for mobile communication. Modern wireless communication systems: Second-generation (2G) cellular networks, Third-Generation (3G) and Fourth Generation (4G) wireless
systems. Prerequisites: ELE302, ELE303
ELE101 Computer Programming (3-0-3)
Problem solving using flowcharts, structure of a C++ program, data types, operators, variables and constants. Input and output, output formatting. Control Statements: IF and SWITCH, WHILE, DO-WHILE
and FOR statements. Function definition and calling, library functions, arrays and strings, pointers. File input and output. Prerequisite: COM101
ELE202 Logic Design (3-2-4)
Basic theorems and properties of Boolean Algebra and Boolean functions. Simplification of Boolean functions: Karnaugh Map and Tabulation Method. Product of Sums (POS) and Sum of Products (SOP) forms.
Combinational logic circuits: Design and analysis procedures. Decoders, encoders, multiplexers, demultiplexers, ROM, PLA and PAL. Sequential logic circuits: Flip Flops (RS, D, JK, T), design
procedure for clocked sequential circuits, counters. Registers and shift registers. Prerequisite: COM101
ELE206 Engineering Analysis (3-0-3)
Developing C++ programs to solve electrical engineering problems. MATLAB programming environment, vectors and matrices, input/output, M-files: scripts and functions, control statements. Plotting with
MATLAB. GUI in MATLAB. Introduction to SIMULINK. Electrical system modeling via SIMULINK. Introduction to LabVIEW. Development of Virtual Instruments using LabVIEW. Prerequisite: ELE101
ELE314 Microcontrollers and Applications (3-2-4)
Introduction to microprocessor and its internal architecture. Typical microprocessor bus systems. Addressing modes and address decoding. Memory and I/O interface. Assembly language programming.
Microcontrollers and embedded systems. Programming of microcontroller using C language. Interrupt processing and interrupt-based control. Microcontroller interfacing to real-world applications.
Design and implementation of course projects using a microcontroller. Prerequisites: ELE101, ELE202
ELE307 Control Systems (3-2-4)
Introduction to Control Systems: Characteristics, time response, steady-state error. Open loop and closed loop concepts, transfer function, time domain, frequency domain, stability of linear feedback
control systems, Root Locus method, Bode diagram. Design of feedback control systems: Principles of design, design with the PD, PI, and PID controllers. Performance evaluation of feedback control
systems. Compensation: phase-lead, phase-lag and lead-lag compensation. Prerequisite: ELE204
ELE313 Sensors and Instrumentation (3-2-4)
Basic measurement concepts, sources and types of measurement errors, sources of noise and interference and how to minimize them. Analysis and design of DC and AC bridge circuits and their
applications. Operating principles and specifications of DVM and DMM. Transducers and their applications in measurement systems. Operation analysis of electromagnetic sensors for flux, current and
position sensing. Oscilloscopes: types, specifications, operation and measurements. Analyzers: types, architecture and the optimal tuning. Design projects related to different types of measuring
instruments Prerequisites: ELE305, ELE206
ELE492 Power Switching Devices (3-0-3)
Introduction to power electronics devices, power transistors, IGBTs and SITs. Thyristors: characteristics, types, models, operations, thyristor commutation techniques and commutation circuit design.
Analysis and design of uncontrolled and controlled rectifiers. AC voltage controllers with resistive and inductive load. DC choppers: principles and classifications. Principles of operation and
performance parameters of different types of inverters. DC and AC drives. Power system applications. Prerequisite: ELE305, ELE207
ELE491 Industrial Control Systems (3-2-4)
Industrial control principles. Block diagram representation of industrial control systems. Application of analog and digital signal conditioning in industrial control. Thermal, optical, displacement,
position, strain, motion, pressure, and flow sensors used in industrial control. Actuators in industrial control. Data Logging, Supervisory Control, Computer-based Controllers. Programmable Logic
Controllers (PLCs). Sequential programming, Ladder diagrams. Introduction to Process Control Systems. Foundation Fieldbus and Profibus standards. Prerequisite: ELE307
ELE203 Circuit Analysis I (3-2-4)
Basic quantities: charge, current, voltage, resistance, energy and power. Analysis of series, parallel and series-parallel D.C. resistive circuits using Ohm's law, Kirchhoff's voltage and current
laws. Star-Delta and Delta-Star Transformations. Analysis of more resistive circuits using loop and nodal methods, superposition, source transformation, Thevenin’s and Norton theorems, maximum power
transfer theorem. Transient analyses of RC, RL, and RLC circuits with DC excitation. Prerequisites: PHY122
ELE207 Circuit Analysis II (3-2-4)
AC circuits: impedance and admittance, phasors and phasor diagrams, series and parallel circuits, power and power factor correction. Steady-state response using phasor method. Nodal and loop
analysis, application of circuit theorems. Steady-state power analysis. Magnetically-coupled circuits. Analysis of balanced three-phase circuits. Frequency response of simple circuits. Series and
parallel resonance. Prerequisites: ELE203
ELE312 Power Systems and Electrical Machines (3-2-4)
Introduction to power systems. Control of reactive power, voltage and frequency. Contemporary issues related to electrical energy. Basics of power system protection. Principles of DC and AC machines
and their types. Ideal and practical transformer. Voltage regulation and efficiency of transformer. Prerequisites: ELE207
ELE470 Power System Protection and Control (3-0-3)
An overview of electric industry structure, modern power system, system protection and energy control center. Introduction to power system apparatus: power transforms, circuit breakers, CTs, VTs,
CCVTs and line trap. Primary and backup protection of transmission lines. Protection of transformers and busbars. Protection schemes for rotating machinery. Operation, algorithms and advantages of
digital relays. Techniques for voltage and frequency control of power systems. Prerequisites: ELE307, ELE312
ELE471 Power Generation and Transmission (3-0-3)
Introduction to different types of conventional power plants for generation of power. Operating principles of steam power plants, hydroelectric power plants, hydro turbines, hydro generators,
gas-turbine plant, gas-power plant and combined-cycle gas-power plant. Comparison of different transmission line insulators. String efficiency and its improvement. Calculations for sag and tension in
designing a transmission line. Classification and comparison of underground cables. Prerequisite: ELE312
ELE463 Renewable Energy Systems (3-2-4)
Introduction to renewable energy sources. Electrical characteristics and performance evaluation of PV cells, modules, panels and arrays. Optimization of PV arrays. Design of a stand-alone PV system
with battery storage. Wind energy conversion systems, sizing and site matching. Hydro generation and types of hydropower turbines. Solar thermal and ocean thermal energy conversion. Tidal energy,
wave power generation, geothermal and biomass energy systems. Types of energy storage systems. Prerequisite: ELE312
ELE477 Smart Grid Renewable Energy Systems (3-0-3)
Basic concept of electric power grid. Types and equipment at grid stations. Grid station automation. Fundamental concepts of power grid integration on microgrids of renewable energy sources. Modeling
converters in microgrids. Smart meters and monitoring systems. Design of PV microgrid generating station. Microgrid wind energy systems. Prerequisite: ELE463
ELE464 Power System Analysis (3-0-3)
Explanation of Per Unit system and determination of the equivalent circuits of synchronous generator and three-phase power transformers. Parameters of transmission lines. The equivalent circuit
models of transmission lines. Power flow analysis. Analyzing symmetrical and unsymmetrical faults in power system. Stability of power systems. Prerequisite: ELE312
ELE472 Electrical Power Distribution Systems (3-0-3)
Introduction to electrical power distribution. Power distribution equipment, underground distribution, radial, ring and network distribution systems. Conductors and insulators in power distribution
systems. Electrical distribution inside buildings. Analyzing single phase and three phase power distribution systems. Measurement equipment for distribution systems. Discussion of various
distribution system considerations. Design of a power distribution system for a small building. Prerequisite: ELE312
ELE436 Selected Topics in Power and Renewable Energy (3-0-3)
Topics of current interest in Power and Renewable Energy as selected by the faculty and approved by the EE Department. The course is tailored according to market demands and the technology
directions. Prerequisite: ELE463
ELE479 Directed Study in Power and Renewable Energy (3-0-3)
Directed study in Power and Renewable Energy is conducted under the supervision of a faculty member. A student interested to undertake such a study shall submit a proposal outlining the description
of the work to be performed with clearly defined objectives and intended outcomes. The study may include experimental investigation, computer simulation or completely theoretical research. The
proposal must be approved by the concerned faculty and Head of the EE Department. Prerequisites: ELE463, Advisor’s Approval
ELE436 Selected Topics in Electr. and Comm. (3-0-3)
Topics of current interest in Electronics and Communication as selected by the faculty and approved by the EE Department. The course is tailored according to market demands and the technology
directions. Prerequisites: ELE305, ELE302
ELE437 Directed Study in Electronics and Communication (3-0-3)
Directed study in Electronics and Communication is conducted under the supervision of a faculty member. A student interested to undertake such a study shall submit a proposal outlining the
description of the work to be performed with clearly defined objectives and intended outcomes. The study may include experimental investigation, computer simulation or completely theoretical
research. The proposal must be approved by the concerned faculty and Head of the EE Department. Prerequisites: ELE302, ELE310, Advisor’s Approval
MTH121 Engineering Mathematics I (3-0-3)
Limits of functions, theorems about limits, evaluation of limit at a point and infinity, continuity. Derivatives of algebraic and trigonometric functions, maxima and minima, engineering applications
of derivatives. The definite and indefinite integrals and their applications. Integration by parts, Integration using powers of trigonometric functions, Integration using trigonometric substitution,
Integration by partial fractions. Integration of improper integrals. Transcendental Functions. Prerequisite: None
MTH122 Engineering Mathematics II (3-0-3)
Matrix addition, subtraction, multiplication and transposition. Complex numbers, algebraic properties of complex numbers, absolute values, complex conjugate, polar representation, powers and roots.
Functions of several variables. Double and triple integrals in rectangular and polar coordinates. Applications of multiple integrals in engineering. Infinite sequences, tests for convergence, power
series expansion of functions, Taylor series, Laurent series, Fourier series and their applications in engineering. Prerequisite: MTH121
PHY121 Engineering Physics I (3-2-4)
Vectors, motion, and Newton’s laws. Work, energy, momentum and conservation of momentum. Rotation of rigid bodies, dynamics of rotational motion. Equilibrium and elasticity. Stress and strain.
Periodic motion. Engineering applications. Prerequisite: None
PHY122 Engineering Physics II (3-2-4)
Electric charge and electric field. Coulomb’s law and Gauss’s law with applications. Capacitance and dielectrics. DC circuits. Magnetic fields. Ampere’s law and its applications. Electromagnetic
induction, Faraday’s law, Lenz’s law, induced electric fields. Self- and mutual-inductance. Electromagnetic waves and Maxwell’s equations. Optics and its engineering applications. Prerequisite: None
CHE101 Chemistry for Engineers (2-2-3)
Atoms, molecules, ions and formulas of ionic compounds. Electronic structure and the periodic table. Quantum numbers, energy levels and orbital. Orbital diagrams of atoms. Various types of bonds.
Chemistry of the metals and semiconductors. Introduction to organic chemistry, bonding and types of hybridization in carbon atom, alkanes and cyclo alkanes, alkyl and halogen substituents. Alkenes
and alkynes, Diels-Alder reaction. Types, properties, and use of polymers. Prerequisite: None
ELE102 Introduction to Engineering (1-0-1)
Engineering profession and the role of engineers in modern developments, engineering ethics. Various engineering disciplines with special emphasis on electrical engineering. Importance of math and
science to engineers. Engineering design and analysis, lab skills for engineers, computer skills for engineers. Electrical Engineering curriculum, curriculum planning and management. Critical
thinking, soft skills for engineers, creativity, communication skills. Case studies on engineering ethics. Prerequisite: None
MTH221 Engineering Mathematics III (3-0-3)
Vector Calculus and its engineering applications. First order differential equations. Homogeneous linear second-order differential equations with constant and variable coefficients, non-homogeneous
linear second-order differential equations with constant coefficients, higher-order linear differential equations with constant coefficients. Power series solution of differential equations. Laplace
Transform, Inverse Laplace Transform. Application of Laplace Transform to solve ordinary differential equations. Introduction to partial differential equations (PDEs), first order PDEs, second order
PDEs, boundary value problems, engineering applications. Prerequisite: MTH122
MTH222 Engineering Mathematics IV (3-0-3)
Linear Algebra: Matrices and determinants, solution of systems of linear equations, eigenvalues and eigenvectors, engineering applications, computer exercises. Complex Analysis: Complex functions,
derivative of complex functions, analytic functions, Cauchy-Riemann equations, harmonic functions. Fourier analysis: Fourier Series, Fourier Integrals, Fourier series of even and odd functions with
applications. Discrete Mathematics and its engineering applications. Prerequisite: MTH221
ELE301 Report Writing and Presentation (3-0-3)
Writing of technical reports, brief reports, and progress reports. Business communication: business letters and memos, executive summary, business reports. Oral presentation: planning, preparation of
visuals, and delivering of an oral presentation. Prerequisites: ELE102 + Junior Standing
ELE304 Probability and Random Variables (3-0-3)
Concept of Probability. Discrete and continuous random variables. Operations on single random variable: Expected values and moments. Joint cumulative distribution function and joint probability
density function. Sum of random variables. Independent random variables. Jointly Gaussian random variables. Definition and classification of random process, transmission of random process through
linear filters, and optimum filtering. Applications in signal processing and communication systems. Prerequisite: MTH122
ELE410 Engineering Management (3-0-3)
Introduction to engineering management and role of effective management. Strategic and operational planning, forecasting, action planning. Organization: activities, organizational structures,
delegating, establishing working relationships. Basics of leadership. Controlling activities: setting standards, measuring, evaluating, and improving performance. Review of Accounting. Financial
Analysis. Financial Forecasting, Planning and Budgeting. Time Value of Money, Capital Budgeting Decision. Prerequisite: ELE301
ELE438 ELE488, ELE468 Graduation Project I (1-4-3)
Teams of 3-4 students shall design, implement, test, and demonstrate their graduation project in two semesters. Graduation Project I is to be completed in first semester and it includes literature
survey, action plan, design of complete project taking into account realistic constraints, computer simulation (if applicable), partial implementation and testing. Report writing and oral
presentation. Prerequisite: ELE310
ELE439, ELE489, ELE469 Graduation Project II (1-4-3)
It is a continuation of Graduation Project I in the second semester. Students will complete the implementation and testing of the remaining part of their design. They will integrate the complete
project, test it, and prepare a PCB. Report writing, oral presentation, poster presentation, and project demonstration. Prerequisite: ELE438, ELE488, ELE468
ELE465 Senior Seminar (1-0-1)
The course aims to develop students’ understanding of contemporary issues as well as the impact of engineering solutions in a global, economic, environmental, and societal context. It will also
improve their oral presentation skills. Prerequisite: ELE301
ELE497 Engineering Training (3)
To expose students to a learning environment where they can apply what they have learned in the classroom to a professional setting and enhance their abilities to correlate theoretical knowledge with
professional practice. Prior to starting their external training, students shall take two weeks intensive internal training to prepare them for external training. Prerequisite: Completion of 75
credit hours.
USTF Department of Electrical Engineering Faculty
No. Name Rank Degrees Held Conferring institution
1 Ali Abouelnour Professor Ph. D. Electrical Engineering Microwave Electronics Technical University Hamburug-Harburg
2 Mohammed Tarique Associate Prof. Ph. D. Communication Network University of Windsor
3 Amir J. Majid Associate Prof. Ph. D. Control and Power Engineering Loughborough University
4 Yomna Shaker Assistant Prof. Ph. D. Electrical Power and Machines Cairo University
Student-Full Faculty Ratio by Colleges for fall semester2022-2023 (2022-1)
College Student-Full Time Faculty Ratio 2022-1
College of Engineering and Information Technology 13.11 | {"url":"https://ustf.ac.ae/?p=power-and-renewable-energy-course-description-2","timestamp":"2024-11-14T07:59:37Z","content_type":"text/html","content_length":"99488","record_id":"<urn:uuid:d1aa24f3-85a1-4ec5-a3db-df44b26685d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00385.warc.gz"} |
How to Increment A Value By Row or Column in ExcelHow to Increment A Value By Row or Column in Excel
How to Increment A Value By Row or Column in Excel
If you want to increment a value or a calculation in Excel with rows and columns as they are copied in other cells, you will need to use the ROW function (not if you have a SEQUENCE function). In
this article, we will learn how you can increment any calculations with respect to row or column.
Generic Formula
=Expression + ((ROW()-number of rows above first formula )*[steps])
Expression: This is the value, reference of expression with which you want to increment. It can be a hardcoded value or any expression that returns a valid output. It should be an absolute expression
(in most cases).
Number of rows above the first formula: If you are writing this first formula in B3 then the number of rows above this formula will be 2.
[steps]: This is optional. This is the number of steps you want to jump in the next increment.
The Arithmetic operator between expression and formula can be replaced with other operators to suit the requirements of increment.
So that we are familiar with the generic formula, let's see some examples.
Example 1: Create an Auto Increment Formula for ID Creation.
Here, I have to create an auto increment ID formula. I want to write one formula that creates ID as EXC1, EXC2, EXC3 and so on. Since we only need to start the increment by 1 and concatenate it to
EXC we don't use any steps. EXC is written in Cell C1, so we will use C1 as the starting expression.
Using the general formula we write the below formula in Cell B4 and copy it down.
We have replaced the + operator with ampersand operator (&) since we wanted to concatenate. And since we are writing the first formula in Cell B4, we subtract 3 from ROW (). The result is here.
You can see that an auto increment ID has been created. Excel has concatenated 1 to EXC and Added 1 to previous value. You will not need to write any new formula for creating an ID. You just need to
copy them down.
Example 2: Increment the ID every 2 steps
If you want to increment the ID every 2 steps then you will need to write this formula.
The result is:
Example 3: Add Original Value with Every Increment
In this increment the starting expression is added to every incremented value. Here's how it will be if we want to increment the starting value of 100 as 100, 200, 300, and so on.
C1 contains 100.
How does it work?
The technique is simple. As we know, the ROW function returns the current row number it is written in, if now parameter is supplied. The formula above ROW() returns 4.
Next we subtract 3 from it (since there are 3 rows above the 4th row). It gives us 1. This is important. It should be a hard coded value so that it does not change as we copy the formula below.
Finally the value 1 is multiplied (or any other operation) by the starting expression. As we copy the formula below. ROW() returns 5 but subtracting value stays the same (3) and we get 2. And it
continues to be the cell you want.
To add steps, we use simple multiplication.
Increment Values By Column
In the above examples we increment by rows. It will not work if you copy them in the next column of the same row.
In the above formula we used the ROW function. Similarly, we can use the COLUMN function.
Generic Formula to Increment by Columns
=Expression + ((COLUMN()-number of columns on left of first formula )*[steps])
Number of columns on the left of the first formula: If you are writing this first formula in B3 then the number of columns on the left of this formula will be 1.
I am not giving any examples as it will be the same as the above examples.
Alternative with SEQUENCE Function
It is a new function only available for EXCEL 365 and 2019 users. It returns an array of sequential numbers. We can use it to increment values sequentially, by rows, columns or both. And yes, you can
also include the steps. Using this function you will not need to copy down the formula, as Excel 365 has auto spill functionality.
So, if you want to do the same thing as you did in Example no 3. The SEQUENCE function alternative will be:
It will automatically create 100 increment values in one go without you copying the formula. You can read about the SEQUENCE function here.
So yeah, this is how you can do auto increment in Excel. You can increment the previous cell by adding 1 to it easily now or you add steps to it. I hope this article helped you. If it didn't solve
your problem, let me know the difficulty you are facing, in the comments section below. I will be happy to help you. Till then keep Excelling.
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How to use the Excel COUNTIF Function| Count values with conditions using this amazing function. You don't need to filter your data to count specific values. Countif function is essential to prepare
your dashboard.
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Fast approximate graph partitioning algorithms for SIAM Journal on Computing
SIAM Journal on Computing
Fast approximate graph partitioning algorithms
View publication
We study graph partitioning problems on graphs with edge capacities and vertex weights. The problems of b-balanced cuts and k-balanced partitions are unified into a new problem called minimum
capacity ρ-separators. A ρ-separator is a subset of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most ρ
times the weight of the graph. We present a new and simple O(log n)-approximation algorithm for minimum capacity ρ-separators which is based on spreading metrics yielding an O(log n)-approximation
algorithm both for b-balanced cuts and k-balanced partitions. In particular, this result improves the previous best known approximation factor for k-balanced partitions in undirected graphs by a
factor of O(log k). We enhance these results by presenting a version of the algorithm that obtains an O(log OPT)-approximation factor. The algorithm is based on a technique called spreading metrics
that enables us to formulate directly the minimum capacity ρ-separator problem as an integer program. We also introduce a generalization called the simultaneous separator problem, where the goal is
to find a minimum capacity subset of edges that separates a given collection of subsets simultaneously. We extend our results to directed graphs for values of ρ≥1/2. We conclude with an efficient
algorithm for computing an optimal spreading metric for ρ-separators. This yields more efficient algorithms for computing b-balanced cuts than were previously known. | {"url":"https://research.ibm.com/publications/fast-approximate-graph-partitioning-algorithms","timestamp":"2024-11-14T11:18:03Z","content_type":"text/html","content_length":"74968","record_id":"<urn:uuid:6b45794f-1fa5-47ef-b915-19009ed2fda1>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00205.warc.gz"} |
Where can I find experts to do my algebraic equations homework? | Hire Someone To Do My Assignment
Where can I find experts to do my algebraic equations homework? About the lecturer: It’s been many years away. I came to research my own stuff, and if I haven’t done something very interesting they
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of one of the very first exercises that I learned too, e.g. using Newton’s Method (see the picture below). This is very awkward, but I feel like you can use real hands to work out some of the
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homework doneWhere can I find experts to do my algebraic equations homework? Step 1: Analyze the problem and come up with a list of the number needed to solve the equation. You can then divide it by
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you’d divided by two, dividing by three, multiplied by this squared major factor. Next, I used a square root to multiply the result using third-col by pi—which is known to be acceptable—then
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powers: pi, qch, and z. I then divided by pi by sqrt—which gives 3.431145862310731.83267. My solution for the first digit was always 3.431145862310731.83267232731 to be 1.
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get the number of steps, and really wanted to do that with the squared factor of order 10. But there’s a better way… 3. You want to know how long should one click to read more this step… Last is an
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plus a 1. For example: see this website length of the step is still the square root to do now, but, fortunately, I’ll take one of those. Once I get that into figure 2, I can determine how far away a
3 step plus one also seems to be with other factors and re-arrange several digits of digits, so that it should take a second to reach the square root. This is what’s worked so far: Take a step of 3.2
seconds plus 2 steps (so the number of steps is 2.) This gives the half-space in this case, and doesn’t reallyWhere can I find experts to do my algebraic equations homework? – The easiest way is just
finding experts or a mathematician to do it. I didn’t get anywhere with algebra and I cannot find an author who would do that.
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If I had a question to ask. I’ll spend the rest of the year in online publishing, which is my job (where I can get freelance writers). I want to know how to find authors who like using algebra in
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whatever numbers may be needed to solve equations. You’ll learn a lot from this so, yeah, I’m sorry if I didn’t explain well. I’ll write you a link here to see how it goes. Your answers, those are
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helpful: 1) I don’t truly understand the idea of the symbol.
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to be solved. So to meet a her explanation problem of interest I have three equations that I will use I have to create a particular set of equations I will solve my | {"url":"https://assignmentinc.com/where-can-i-find-experts-to-do-my-algebraic-equations-homework","timestamp":"2024-11-11T09:35:55Z","content_type":"text/html","content_length":"114112","record_id":"<urn:uuid:4b12da8a-6d7d-4067-af37-4942cb14ce58>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00172.warc.gz"} |
Wrapper class which unifies rotation handling of Quaternion, Euler or Z-only (used by 2D). Can store a Quaternion, Euler or Z-axis (for 2d rotation), and implicitly casts between those types. More...
Rotation (float z)
Rotation (Quaternion quaternion)
Rotation (Vector3 euler)
static implicit operator float (Rotation rotation)
static implicit operator Quaternion (Rotation rotation)
static implicit operator Rotation (float z)
static implicit operator Rotation (Quaternion quaternion)
static implicit operator Rotation (Vector3 euler)
static implicit operator Vector3 (Rotation rotation)
readonly Vector3 euler
readonly Quaternion quaternion
readonly RotationTypes Type
Indicates of the stored rotation is a Quaternion or Euler type. More...
readonly float w
readonly float x
readonly float y
readonly float z
Vector3 AsEuler [get]
Returns the rotation as a Euler. If original type was Quaternion, that is converted to a Euler. More...
float AsFloatZ [get]
Returns the z-axis value of the rotation. If original type was Quaternion, first converts to a Euler then returns the z value of that Euler. More...
Quaternion AsQuaternion [get]
Returns the rotation as a Quaternion. If original type was a Euler or Rotation Z-axis only, that is converted to a Quaternion. More...
Wrapper class which unifies rotation handling of Quaternion, Euler or Z-only (used by 2D). Can store a Quaternion, Euler or Z-axis (for 2d rotation), and implicitly casts between those types.
◆ Type
readonly RotationTypes Type
Indicates of the stored rotation is a Quaternion or Euler type.
◆ AsEuler
Returns the rotation as a Euler. If original type was Quaternion, that is converted to a Euler.
◆ AsFloatZ
Returns the z-axis value of the rotation. If original type was Quaternion, first converts to a Euler then returns the z value of that Euler.
◆ AsQuaternion
Returns the rotation as a Quaternion. If original type was a Euler or Rotation Z-axis only, that is converted to a Quaternion. | {"url":"https://doc-api.photonengine.com/en/fusion/current/struct_fusion_1_1_rotation.html","timestamp":"2024-11-12T13:32:09Z","content_type":"text/html","content_length":"20333","record_id":"<urn:uuid:e2c52122-cc15-47a7-b9f1-97bd2d3ae2de>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00458.warc.gz"} |
Math, Grade 6, Ratios, Gallery Problems
Three Farmers
Work Time
Three Farmers
Together three farmers ( X, Y, and Z) buy seed for their farms and share it in the ratio 3:7:8.
1. If they buy 144 kilograms of seed, how many kilograms of seed does each farmer receive?
2. Show the problem solution using a tape diagram. | {"url":"https://goopennc.oercommons.org/courseware/lesson/5002/student/?section=3","timestamp":"2024-11-14T17:40:21Z","content_type":"text/html","content_length":"32534","record_id":"<urn:uuid:c9ba6cd8-1883-41b4-b007-e964ae95391e>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00556.warc.gz"} |
The Mathematics of Ted Kaczynski
The Mathematics of Ted Kaczynski
Before terrorist Theodore John Kaczynski (1942-) began sending mail-bombs to faculty members at various American universities, he had a promising career in mathematics.
Left: Ted Kaczynski (1942-) while a lecturer at UC Berkeley (Photo: Wikimedia Commons). Right: Kaczynski’s 1969 paper “The set of curvilinear convergence of a continuous function defined in the
interior of a cube”in the Proceedings of the American Mathematical Society 23(2), pp. 323–327.
Before terrorist Theodore John Kaczynski (1942-) began sending mail-bombs to faculty members at various American universities, he had a promising career in mathematics. In particular, between
1964–69, he published a total of six single-authored research papers in renowned mathematical journals, including The American Mathematical Monthly and Proceedings of the American Mathematical
The young Kaczynski did work in analysis, specifically geometric function theory in the narrow subfield of boundary values of continuous functions. The purpose of this article is to give an
introduction to this work.
Education (1958–67)
Kaczynski grew up in Illinois, where he attended Sherman Elementary School and Evergreen Park Central Junior High school. At the age of 10 years old, his IQ was evaluated to be 167, and so he skipped
the sixth grade (Chicago Tribune, 2017), an event later described as pivotal to his development (Chase, 2004):
“Previously he had socialized with his peers and was even a leader, but after skipping ahead he felt he did not fit in with the older children and was bullied.”
Harvard University (1958–62)
Kaczynski entered Harvard University in 1958 at the age of 16 years old. A mathematical prodigy since he was a child, he was described by other undergraduates as “shy”, “quiet” and “a loner” who
“never talked to anyone“ (Song, 2012):
“He would just rush through the suite, go into his room, and slam the door […] When we would go into his room there would be piles of books and uneaten sandwiches that would make the place smell”
His personality notwithstanding, Kaczynski’s talent was however still recognized among his Harvard peers, one of which in 2012 stated:
“It’s just an opinion — but Ted was brilliant [...]. He could have become one of the greatest mathematicians in the country”
Kaczynski graduated Harvard with a B.A. in mathematics in 1962. When he graduated, his GPA was 3.12, scoring B’s in the History of Science, Humanities and Math, C in History and A’s in Anthropology
and Scandinavian (Stampfl, 2006).
University of Michigan (1962–67)
With an IQ of 167, Kaczynski had been expected to perform better at Harvard. After graduating, he applied to the University of California at Berkeley, The University of Chicago and the University of
Michigan. Although accepted at all three, he ended up choosing Michigan because the university offered him an annual grant of $2,310 and a teaching post. The “darling of the math department”, he
would graduate from the University of Michigan in 1964 with a M.Sc. in mathematics and markedly improved grades — 12 A’s and five B’s, which he himself later attributed to the standing of the
“[My] memories of the University of Michigan are not pleasant […] The fact that I not only passed my courses (except one physics course) but got quite a few A’s shows how wretchedly low the standards
were at Michigan”
Nonetheless, as the story goes, while there once a professor named George Piranian told his students — including Kaczynski — about an unsolved problem in boundary functions. Weeks later, Kaczynski
came to his office with a 100-page correct, handwritten proof. Kaczynski graduated with a Ph.D. in mathematics in 1967. His dissertation, entitled simply “Boundary Functions” regarded the same topic
as his proof of Piranian’s problem. His doctoral committee consisted of professors Allen L. Shields, Peter L. Duren, Donald J. Livingstone, Maxwell O. Reade, Chia-Shun Yin. Every professor approved
it. His supervisor Shields later called his dissertation
“The best I have ever directed”
The first two pages of Kaczynski’s Ph.D. dissertation
An additional testament to its quality was it being awarded the Sumner Myers Prize for the best mathematics thesis of the university, accompanying a prize of $100 and a plaque in the East Quad
Residence Hall entrance listing his accomplishment. Of the complexity (or perhaps narrow implications) of his dissertation, one of the members of his dissertation committee, Maxwell Reade, said
“I would guess that maybe 10 or 12 men in the country understood or appreciated it”
Another, Peter Duren, stated
“He was really an unusual student”
Kaczynski at UCB in 1967 (Photo: Wikimedia Commons)
University of California, Berkeley (1967–69)
In late 1967, at 25 years old Kaczynski was hired as the youngest-ever assistant professor of mathematics at the University of California at Berkeley. There, he taught undergraduate courses in
geometry and calculus, although with mediocre success. His student evaluations suggest that he was not particularly well-liked because he taught “straight from the textbook and refused to answer
He resigned on June 30th, 1969 without explanation.
Work (1964–69)
Wedderburn’s Theorem
Kaczynski’s only published paper relating to topics other than boundary functions was his first journal paper, written before he started his Ph.D. It is entitled:
The paper concerned a 1905 result of Joseph H. M. Wedderburn that every finite skew field is commutative. His paper provided a group-theoretic proof of the theorem, which had previously been proved
at least seven times.
Boundary Functions
Kaczynski’s Ph.D. dissertation concerned boundary values of continuous functions and was entitled, simply
Let H denote the set of all points in the Euclidean plane having positive y-coordinate, and let X denote the x-axis. If p is a point of X, then by an arc at p we mean a simple arc γ, having one
endpoint at p, such that γ = {p} ⊆ H. Let f be a function mapping H into the Riemann sphere.
Boundary FunctionsBy a boundary function for f we mean a function φ defined on a set E ⊆ X such that for each p ∈ E there exists an arc γ at p for whichlim (s p, s ∈ γ) f(z) = φ(p)
Kaczynski’s dissertation begins by re-proving a theorem of J. E. McMillan which states that if f(H) is a a continuous function mapping H into the Riemann sphere, the the set of curvilinear
convergence of F (the largest set on which a boundary function for f can be defined) is of a certain type. This proof also shows that if A is a set of the same type in X, then there exists a bounded
continuous complex-valued function in H having A as its set of curvilinear convergence. The dissertation contains two additional new proofs related to boundary functions, and a list of problems for
future research. Of the results, Professor Donald Rung later stated:
What Kaczynski did, greatly simplified, was determine the general rules for the properties of sets of points of curvilinear convergence. Some of those rules were not the sort of thing even a
mathematician would expect.
Kaczynski would publish five journal papers related to the work from his dissertation between 1965–69:
The Distributivity Problem
The only other trace of Kaczynski in a mathematical journal is two notes in the American Monthly in 1964 and 65:
In the first note, Kaczynski proposes the following problem, concerning group theory:
Let K be an algebraic system with two binary operations (one written additively, the other multiplicatively), satisfying:1. K is an abelian group under addition,2. K - {0} is a group under
multiplication, and3. x(y+z) = xy + xz for all x,y,z ∈ K.Suppose that for some n, 0=1+1+1....+1 (n times). Prove that, for all x ∈ K, (-1)x = -x.
In the second note, the solution to the problem is — somewhat dismissively — provided by R. G. Bilyeu:
The last part of the hypothesis is unnecessary. If z denotes -1, then z+z+zz = z(1+1+z) = z, so zz = 1. Now z(x+zx) = zx+x = x+zx, so either x+zx = 0 or z = 1. In either case zx = -x.
Theodore J. Kaczynski was a very promising young undergraduate, graduate and post-graduate student in the 1960s. His work — although pertaining to vary narrow topics — was undoubtedly, technically,
first rate.
As is the case however, elegance or complexity do not themselves raise the importance of problems, achievements or for that matter, mathematicians. As expressed by his fellow graduate student
Professor Peter Rosenthal in a 1996 Toronto Star article (after Kaczynski was charged):
[The] topic was only of interest to a very small group of mathematicians and does not appear to have broader implications; thus, his work had little impact. Kaczynski might have quit mathematics
because he was discouraged by the resultant lack of recognition.
In another 1996 article, in the Los Angeles Times article, Professor Donald Rung similarly expressed:
“The field that Kaczynski worked in doesn’t really exist today […]. He probably would have gone on to some other area if he were to stay in mathematics,” Rung said. “As you can imagine, there are not
a thousand theorems to be proved about this stuff.”
• Chase, A. (2004). A Mind for Murder: The Education of the Unabomber and the Origins of Modern Terrorism. W. W. Norton & Company.
• Chicago Tribune (2017). “The Kaczynski brothers and neighbors”. Chicago Tribune.
• Song, D. (2012). Theodore J. Kaczynski. The Harvard Crimson.
• Stampfl, K. (2006). He came Ted Kaczinsky, he left The Unabomber.The Michigan Daily. March 16 2006. | {"url":"https://www.cantorsparadise.org/the-mathematics-of-ted-kaczynski-43eb6ca16633/","timestamp":"2024-11-14T00:20:02Z","content_type":"text/html","content_length":"44977","record_id":"<urn:uuid:1de0f899-2732-48fc-84d4-cdd7581cc995>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00452.warc.gz"} |
2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
Student grades on a chemistry exam were 77, 78, 76, 81, 86, 51, 79, 82, 84, and 99.
a. Construct a stem-and-leaf plot of the data.
b. Are there any potential outliers? If so, which scores are they? Why do you consider them outliers?
Table 2.64 contains the 2010 rates for a specific disease in U.S. states and Washington, DC.
State Percent (%) State Percent (%) State Percent (%)
Alabama 32.2 Kentucky 31.3 North Dakota 27.2
Alaska 24.5 Louisiana 31.0 Ohio 29.2
Arizona 24.3 Maine 26.8 Oklahoma 30.4
Arkansas 30.1 Maryland 27.1 Oregon 26.8
California 24.0 Massachusetts 23.0 Pennsylvania 28.6
Colorado 21.0 Michigan 30.9 Rhode Island 25.5
Connecticut 22.5 Minnesota 24.8 South Carolina 31.5
Delaware 28.0 Mississippi 34.0 South Dakota 27.3
Washington, DC 22.2 Missouri 30.5 Tennessee 30.8
Florida 26.6 Montana 23.0 Texas 31.0
Georgia 29.6 Nebraska 26.9 Utah 22.5
Hawaii 22.7 Nevada 22.4 Vermont 23.2
Idaho 26.5 New Hampshire 25.0 Virginia 26.0
Illinois 28.2 New Jersey 23.8 Washington 25.5
Indiana 29.6 New Mexico 25.1 West Virginia 32.5
Iowa 28.4 New York 23.9 Wisconsin 26.3
Kansas 29.4 North Carolina 27.8 Wyoming 25.1
a. Use a random number generator to randomly pick eight states. Construct a bar graph of the rates of a specific disease of those eight states.
b. Construct a bar graph for all the states beginning with the letter A.
c. Construct a bar graph for all the states beginning with the letter M.
2.2 Histograms, Frequency Polygons, and Time Series Graphs
Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked
the number of fiction paperbacks they had purchased the previous month. The results are as follows:
Number of Books Frequency Relative Frequency
Number of Books Frequency Relative Frequency
Number of Books Frequency Relative Frequency
0–1 20
2–3 35
4–5 12
6–7 2
8–9 1
a. Find the relative frequencies for each survey. Write them in the charts.
b. Using either a graphing calculator or computer or by hand, use the frequency column to construct a histogram for each publisher's survey. For Publishers A and B, make bar widths of 1. For
Publisher C, make bar widths of 2.
c. In complete sentences, give two reasons why the graphs for Publishers A and B are not identical.
d. Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not?
e. Make new histograms for Publisher A and Publisher B. This time, make bar widths of 2.
f. Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer.
Often, cruise ships conduct all onboard transactions, with the exception of souvenirs, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose
that 60 single travelers and 70 couples were surveyed as to their onboard bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group.
Amount ($) Frequency Relative Frequency
51–100 5
101–150 10
151–200 15
201–250 15
251–300 10
301–350 5
Amount ($) Frequency Relative Frequency
100–150 5
201–250 5
251–300 5
301–350 5
351–400 10
401–450 10
451–500 10
501–550 10
551–600 5
601–650 5
a. Fill in the relative frequency for each group.
b. Construct a histogram for the singles group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis.
c. Construct a histogram for the couples group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis.
d. Compare the two graphs:
i. List two similarities between the graphs.
ii. List two differences between the graphs.
iii. Overall, are the graphs more similar or different?
e. Construct a new graph for the couples by hand. Since each couple is paying for two individuals, instead of scaling the x-axis by $50, scale it by $100. Use relative frequency on the y-axis.
f. Compare the graph for the singles with the new graph for the couples:
i. List two similarities between the graphs.
ii. Overall, are the graphs more similar or different?
g. How did scaling the couples graph differently change the way you compared it to the singles graph?
h. Based on the graphs, do you think that individuals spend the same amount, more or less, as singles as they do person by person as a couple? Explain why in one or two complete sentences.
Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows:
Number of Movies Frequency Relative Frequency Cumulative Relative Frequency
a. Construct a histogram of the data.
b. Complete the columns of the chart.
Use the following information to answer the next two exercises: Suppose 111 people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than $19 each.
The percentage of people who own at most three T-shirts costing more than $19 each is approximately ________.
a. 21
b. 59
c. 41
d. cannot be determined
If the data were collected by asking the first 111 people who entered the store, then the type of sampling is ________.
a. cluster
b. simple random
c. stratified
d. convenience
Following are the 2010 obesity rates by U.S. states and Washington, DC.
State Percent (%) State Percent (%) State Percent (%)
Alabama 32.2 Kentucky 31.3 North Dakota 27.2
Alaska 24.5 Louisiana 31.0 Ohio 29.2
Arizona 24.3 Maine 26.8 Oklahoma 30.4
Arkansas 30.1 Maryland 27.1 Oregon 26.8
California 24.0 Massachusetts 23.0 Pennsylvania 28.6
Colorado 21.0 Michigan 30.9 Rhode Island 25.5
Connecticut 22.5 Minnesota 24.8 South Carolina 31.5
Delaware 28.0 Mississippi 34.0 South Dakota 27.3
Washington, DC 22.2 Missouri 30.5 Tennessee 30.8
Florida 26.6 Montana 23.0 Texas 31.0
Georgia 29.6 Nebraska 26.9 Utah 22.5
Hawaii 22.7 Nevada 22.4 Vermont 23.2
Idaho 26.5 New Hampshire 25.0 Virginia 26.0
Illinois 28.2 New Jersey 23.8 Washington 25.5
Indiana 29.6 New Mexico 25.1 West Virginia 32.5
Iowa 28.4 New York 23.9 Wisconsin 26.3
Kansas 29.4 North Carolina 27.8 Wyoming 25.1
Construct a bar graph of obesity rates of your state and the four states closest to your state. Hint—Label the x-axis with the states.
2.3 Measures of the Location of the Data
The median age for U.S. ethnicity A currently is 30.9 years; for U.S. ethnicity B, it is 42.3 years.
a. Based on this information, give two reasons why ethnicity A median age could be lower than the ethnicity B median age.
b. Does the lower median age for ethnicity A necessarily mean that ethnicity A die younger than ethnicity B? Why or why not?
c. How might it be possible for ethnicity A and ethnicity B to die at approximately the same age but for the median age for ethnicity B to be higher?
Six hundred adult Americans were asked by telephone poll, "What do you think constitutes a middle-class income?" The results are in Table 2.72. Also, include the left endpoint but not the right
Salary ($) Relative Frequency
20,000 0.02
20,000–25,000 0.09
25,000–30,000 0.19
30,000–40,000 0.26
40,000–50,000 0.18
50,000–75,000 0.17
75,000–99,999 0.02
100,000+ 0.01
a. What percentage of the survey answered “not sure”?
b. What percentage think that middle class is from $25,000 to $50,000?
c. Construct a histogram of the data.
i. Should all bars have the same width, based on the data? Why or why not?
ii. How should the 20,000 and the 100,000+ intervals be handled? Why?
d. Find the 40^th and 80^th percentiles.
e. Construct a bar graph of the data.
Given the following box plot, answer the questions.
a. Which quarter has the smallest spread of data? What is that spread?
b. Which quarter has the largest spread of data? What is that spread?
c. Find the interquartile range (IQR).
d. Are there more data in the interval 5–10 or in the interval 10–13? How do you know this?
e. Which interval has the fewest data in it? How do you know this?
i. 0–2
ii. 2–4
iii. 10–12
iv. 12–13
v. need more information
The following box plot shows the ages of the U.S. population for 1990, the latest available year.
a. Are there fewer or more children (age 17 and under) than senior citizens (age 65 and over)? How do you know?
b. 12.6 percent are age 65 and over. Approximately what percentage of the population are working-age adults (above age 17 to age 65)?
2.4 Box Plots
In a survey of 20-year-olds in China, Germany, and the United States, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the
a. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected.
b. Have more Americans or more Germans surveyed been to more than eight foreign countries?
c. Compare the three box plots. What do they imply about the foreign travel of 20-year-old residents of the three countries when compared to each other?
Given the following box plot, answer the questions.
a. Think of an example (in words) where the data might fit into the above box plot. In two to five sentences, write down the example.
b. What does it mean to have the first and second quartiles so close together, while the second to third quartiles are far apart?
Given the following box plots, answer the questions.
a. In complete sentences, explain why each statement is false.
i. Data 1 has more data values above two than Data 2 has above two.
ii. The data sets cannot have the same mode.
iii. For Data 1, there are more data values below four than there are above four.
b. For which group, Data 1 or Data 2, is the value of 7 more likely to be an outlier? Explain why in complete sentences.
A survey was conducted of 130 purchasers of new black sports cars, 130 purchasers of new red sports cars, and 130 purchasers of new white sports cars. In it, people were asked the age they were when
they purchased their car. The following box plots display the results.
a. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series.
b. Which group is most likely to have an outlier? Explain how you determined that.
c. Compare the three box plots. What do they imply about the age of purchasing a sports car from the series when compared to each other?
d. Look at the red sports cars. Which quarter has the smallest spread of data? What is the spread?
e. Look at the red sports cars. Which quarter has the largest spread of data? What is the spread?
f. Look at the red sports cars. Estimate the interquartile range (IQR).
g. Look at the red sports cars. Are there more data in the interval 31–38 or in the interval 45–55? How do you know this?
h. Look at the red sports cars. Which interval has the fewest data in it? How do you know this?
i. 31–35
ii. 38–41
iii. 41–64
Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows:
Number of Movies Frequency
Construct a box plot of the data.
2.5 Measures of the Center of the Data
Scientists are studying a particular disease. They found that countries that have the highest rates of people who have ever been diagnosed with this disease range from 11.4 percent to 74.6 percent.
Percentage of Population Diagnosed Number of Countries
11.4–20.45 29
20.45–29.45 13
29.45–38.45 4
38.45–47.45 0
47.45–56.45 2
56.45–65.45 1
65.45–74.45 0
74.45–83.45 1
a. What is the best estimate of the average percentage affected by the disease for these countries?
b. The United States has an average disease rate of 33.9 percent. Is this rate above average or below?
c. How does the United States compare to other countries?
Table 2.75 gives the percentage of children under age five have been diagnosed with a medical condition. What is the best estimate for the mean percentage of children with the condition?
Percentage of Children with the Condition Number of Countries
16–21.45 23
21.45–26.9 4
26.9–32.35 9
32.35–37.8 7
37.8–43.25 6
43.25–48.7 1
2.6 Skewness and the Mean, Median, and Mode
The median age of the U.S. population in 1980 was 30.0 years. In 1991, the median age was 33.1 years.
a. What does it mean for the median age to rise?
b. Give two reasons why the median age could rise.
c. For the median age to rise, is the actual number of children less in 1991 than it was in 1980? Why or why not?
2.7 Measures of the Spread of the Data
Use the following information to answer the next nine exercises: The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College
from 1976–1977 through 2004–2005.
• μ = 1,000 FTES
• median = 1,014 FTES
• σ = 474 FTES
• first quartile = 528.5 FTES
• third quartile = 1,447.5 FTES
• n = 29 years
A sample of 11 years is taken. About how many are expected to have an FTES of 1,014 or above? Explain how you determined your answer.
Seventy-five percent of all years have an FTES.
a. at or below ______.
b. at or above ______.
The population standard deviation = ______.
What percentage of the FTES were from 528.5 to 1,447.5? How do you know?
What is the IQR? What does the IQR represent?
How many standard deviations away from the mean is the median?
Additional Information: The population FTES for 2005–2006 through 2010–2011 was given in an updated report. The data are reported here.
Year 2005–2006 2006–2007 2007–2008 2008–2009 2009–2010 2010–2011
Total FTES 1,585 1,690 1,735 1,935 2,021 1,890
Calculate the mean, median, standard deviation, the first quartile, the third quartile, and the IQR. Round to one decimal place.
Construct a box plot for the FTES for 2005–2006 through 2010–2011 and a box plot for the FTES for 1976–1977 through 2004–2005.
Compare the IQR for the FTES for 1976–1977 through 2004–2005 with the IQR for the FTES for 2005-2006 through 2010–2011. Why do you suppose the IQRs are so different?
Three students were applying to the same graduate school. They came from schools with different grading systems. Which student had the best GPA when compared to other students at his school? Explain
how you determined your answer.
Student GPA School Average GPA School Standard Deviation
Thuy 2.7 3.2 0.8
Vichet 87 75 20
Kamala 8.6 8 0.4
A music school has budgeted to purchase three musical instruments. The school plans to purchase a piano costing $3,000, a guitar costing $550, and a drum set costing $600. The mean cost for a piano
is $4,000 with a standard deviation of $2,500. The mean cost for a guitar is $500 with a standard deviation of $200. The mean cost for drums is $700 with a standard deviation of $100. Which cost is
the lowest when compared to other instruments of the same type? Which cost is the highest when compared to other instruments of the same type? Justify your answer.
An elementary school class ran one mile with a mean of 11 minutes and a standard deviation of three minutes. Rachel, a student in the class, ran one mile in eight minutes. A junior high school class
ran one mile with a mean of nine minutes and a standard deviation of two minutes. Kenji, a student in the class, ran one mile in 8.5 minutes. A high school class ran one mile with a mean of seven
minutes and a standard deviation of four minutes. Nedda, a student in the class, ran one mile in eight minutes.
a. Why is Kenji considered a better runner than Nedda even though Nedda ran faster than he?
b. Who is the fastest runner with respect to his or her class? Explain why.
Scientists are studying a particular disease. They found that countries that have the highest rates of people who have ever been diagnosed with this disease range from 11.4 percent to 74.6 percent.
Percentage of Population with Disease Number of Countries
11.4–20.45 29
20.45–29.45 13
29.45–38.45 4
38.45–47.45 0
47.45–56.45 2
56.45–65.45 1
65.45–74.45 0
74.45–83.45 1
What is the best estimate of the average percentage of people with the disease for these countries? What is the standard deviation for the listed rates? The United States has an average disease rate
of 33.9 percent. Is this rate above average or below? How unusual is the U.S. obesity rate compared to the average rate? Explain.
Table 2.79 gives the percentage of children under age five diagnosed with a specific medical condition.
Percentage of Children with the Condition Number of Countries
16–21.45 23
21.45–26.9 4
26.9–32.35 9
32.35–37.8 7
37.8–43.25 6
43.25–48.7 1
What is the best estimate for the mean percentage of children with the condition? What is the standard deviation? Which interval(s) could be considered unusual? Explain. | {"url":"https://texasgateway.org/resource/homework-0?book=79081&binder_id=78221","timestamp":"2024-11-11T08:20:02Z","content_type":"text/html","content_length":"98914","record_id":"<urn:uuid:a82ac5b2-3688-48a9-bb60-d5c8b674096c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00773.warc.gz"} |
How does the template calculation work? - Cluster Analysis 4 Marketing
How does the template calculation work?
This article is a walk-through of how the free Excel template for cluster analysis available on this website works. If you are after formula information for cluster analysis and its variations,
please see the external links at the bottom of this page.
If you just need a simple idea of how clustering works for marketing and forming market segments – please see the article: a simple guide to cluster analysis and how it works.
Based on the K-means clustering technique
There are various forms – that is, statistical approaches and formulas – of using cluster analysis. The most basic form is known as K–means cluster analysis. This is the form of cluster analysis used
in the Excel template available for free download on this website.
The term “K-means” in cluster analysis simply helps describe the approach. In this case;
• The letter K is the number of segments that will be created. Is it designated as K (a variable) because initially, prior to undertaking the cluster analysis, it is unclear how many segments will
be developed.
• And the word means refers to averages. This is because the analysis will provide individual averages/means for each market segment.
• The outcome of k-means cluster analysis is the formation of a number of market segments (where the number of segments = K) that each have their own averages (means).
Multiple K’s provided as part of this Excel template
One significant difference between this template/spreadsheet approach and some other statistical packages is that multiple K’s (number of market segments) are automatically calculated and provided on
the same cluster analysis output page.
In most statistical packages, the marketing analyst is required to enter the number of segments (k) that they want to consider (or that they think that there are) prior to running the analysis. In
this case, the analyst would need to run cluster analysis a number of times and then compare the various outputs in order to select the most appropriate market segmentation approach.
However, given that the template on this website is primarily designed as a learning tool, the outputs of a two, three, four and five market segment structure are automatically generated. This
greatly simplifies the selection of how many market segments are suitable – and allows more time to consider the best mix of marketing variables to include in order to better understand and profile
the target market.
The cluster analysis calculation used in the template
A “random” choice of starting points
The Excel spreadsheet calculation starts by “randomly” (please refer to why cluster analysis outputs may vary) selecting a number of individual cases (respondents) to be the proxy center (average/
mean) for the market segment. Therefore, when forming four market segments, the spreadsheet will identify four individual respondents to “represent” the market segment.
Please note that this is only the starting point – not the end result. There are multiple revisions (iterations) that keep fine-tuning the market segments until the final market segment output is
Respondents allocated to the nearest market segment
Once the initial centers of the market segments have been “estimated” (guessed), the cluster analysis process will then allocate every other respondent to the closest one – that is, which one are
they most similar to? With a simple data set, then this is quite straightforward. For example, if the data set only included one variable (say customer satisfaction on a scale of 1-10), then it would
be easy to allocate respondents to the high or low satisfaction group.
But because the Excel template can include up to eight marketing variables, we need to use the Euclidean distance to determine the closest segment center.
This is shown in this table (click table to enlarge) for respondent 1. In this example there are four marketing variables being used and there are just two market segments (to simplify the example).
The sum of the squared differences between Respondent 1 (and indeed each respondent) is calculated against both Segment 1 and Segment 2. In this example, the respondent is slightly closer to Segment
2 (65 squared distance), than Segment 1 (68 squared distance) – so, at this stage of the clustering process, the respondent is allocated to Segment 2 (although they may be reallocated to another
segment later in the process – in another iteration). These numbers (65 and 68) are the sum of squared error (SSE) – please see the separate article on SSE.
This Euclidean distance can be calculated using a “manual” approach as shown in the above spreadsheet excerpt, or more easily by using the Excel formula of SUMXMY2 (which is used in the template).
Segment means (averages) recalculated and respondents reallocated (iterations)
At this stage, all the respondents and their data have been allocated to a specific segment, whichever is deemed to be currently the closest. This now allows for the center (mean/average) of this
group of respondents to be re-calculated. This would change the numbers for Segments 1 and 2 above, as they are now based upon a group of respondents – not just an individual.
With this revised information, the above process is then repeated, with respondents being reallocated (if necessary) to a more suitable (that is, closer and more representative) market segment. This
in turn would change the center (mean) of each of the segments – in essence, a gradual fine-tuning of the respondents classified to each market segment, as the same process is repeated over and over.
Uses six iterations to determine final market segments
The Excel cluster analysis template on this website uses six iterations (repeated calculations and fine-tuning). Other statistical packages will generally run at least ten iterations and some run
even more. But that would be overkill for the size of this template – that is, it only allows you to enter up to 100 respondents and up to eight marketing variables – 800 data points as a maximum –
so six iterations is more than enough to generate a good and reliable market segmentation outcome for learning purposes.
More information for the statistically curious
This website is designed to provide a simple guide to cluster analysis for marketers and provides a free, easy-to-use clustering template as a learning tool. The website is not designed to learn the
more detailed statistical aspects of cluster analysis. However, if that sort of extra information is of interest to you, then below there are some external resources that you may find helpful.
External Links for Further Detail on Cluster Analysis Calculations
• Statistics information from Dell
• Academic journal link – analysis and interpretation
• History of K-means clustering and a full PDF version of this history article | {"url":"https://www.clusteranalysis4marketing.com/technical-aspects-cluster-analysis/how-does-the-template-calculation-work/","timestamp":"2024-11-04T21:21:16Z","content_type":"text/html","content_length":"89413","record_id":"<urn:uuid:e28095b4-110e-4f94-b108-cb49330decc2>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00091.warc.gz"} |
Determine the value of 20816 using properties.-Turito
Are you sure you want to logout?
Determine the value of 208 ÷ 16 using properties.
A. 10
B. 30
C. 18
D. 13
Use area model of division for calculating the answer.
The correct answer is: 13
An area model is a rectangular diagram or model used for multiplication and division problems, in which the factors or the quotient and divisor define the length and width of the rectangle. We can
break one large area of the rectangle into several smaller boxes, using number bonds, to make the calculation easier.
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/Mathematics-determine-the-value-of-208-16-using-properties-13-18-30-10-qc6e875b0","timestamp":"2024-11-13T20:04:35Z","content_type":"application/xhtml+xml","content_length":"721758","record_id":"<urn:uuid:ae120ac1-8205-4af3-8c10-7bdabdc37a50>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00678.warc.gz"} |
Solving Simultaneous Equations In Excel - ExcelAdept
Key Takeaway:
• Setting up the Excel spreadsheet properly is crucial when solving simultaneous equations. This includes defining the variables and equations in the cells and using proper formatting.
• Excel offers different methods for solving simultaneous equations, including using cell references and the Solver add-in, or built-in functions like Goal Seek. Understanding these methods can
save time and effort in solving equations.
• Checking and modifying the solutions is important to ensure accuracy in solving simultaneous equations in Excel. Modifying equations can also help in exploring different solutions to a problem.
Are you in need of an efficient tool to solve simultaneous equations? Excel can offer you the perfect solution. Learn how to solve simultaneous equations with ease and accuracy, right in your
spreadsheet. Unlock the power of Excel and take your equations to the next level!
Setting up the Excel Spreadsheet
Starting the setup of the spreadsheet for solving simultaneous equations in Excel requires specific actions to be taken. It is essential to approach this task with accuracy and precision for optimal
To begin the setup, a table must be created with appropriate columns using <table>, <td>, <tr> tags. The table should include relevant data needed for the process. This step ensures that the data
entered is organized correctly and simplifies the process for the user.
While setting up the Excel spreadsheet, it is crucial to avoid making any mistakes that could lead to the wrong output. Therefore, it is recommended to double-check all the entered data before moving
on to the next step.
In the past, creating spreadsheets for solving simultaneous equations involved tedious manual calculations. Thanks to modern technology, Excel has eliminated many of the errors that were common in
the past. Now, creating the spreadsheets is straightforward and accessible, making it possible for anyone to solve simultaneous equations with ease.
Entering the Equations
Go to “Entering the Equations“! To understand how to enter equations in Excel, explore the options of ‘Using Cell References’ and ‘Using the Solver Add-In’. These sub-sections will guide you. Get
ready to learn it all!
Using Cell References
Cell references are an essential tool to solve simultaneous equations in Excel. By using Semantic NLP variations of these words, you can access and manipulate a specific value or calculation in one
cell into another cell.
Here is a 4-step guide to using cell references:
1. Select the cell(s) that will contain your variables.
2. Enter the formula for each equation into separate cells that represent each variable.
3. Create a new column or row where you will input your values for each variable.
4. Finally, use the ‘Solver’ tool to find the solutions automatically or manually change input values until you find the correct answer.
Another tip when working with cell references is to use absolute and relative reference. Absolute referencing keeps the same values, while relative referencing changes depending on its position.
A true fact regarding this topic is that Microsoft Office Excel was first released in September 1985.
Why do math equations hate using the Solver Add-In? Because it always finds their solutions.
Using the Solver Add-In
Solving complex simultaneous equations can be made easier by utilizing Excel’s “Solver Add-In”. Here’s how to get started:
1. Define your Variables – Create an Excel worksheet and define all variables, constants, and equations.
2. Install the Solver – Select the File menu, click on Options and then click on Add-Ins. In the dialog box, choose Solver Add-in from the list.
3. Set Target Cell And Constraints – From the data tab select Solver from the analytics drop-down and enter a target cell. Ensure any constraints you want are also inputted.
4. Run The Solver – Click Solve or OK after defining parameters.
Make sure that Solver reaches its desired outputs; incorrect values may mean a reevaluation or adjusting parameters before solving again.
Notably, using this tool only requires basic knowledge of algebra and Excel functions. It is accessible for many businesses and students alike.
According to Investopedia, Microsoft first introduced this feature in 1990 with Excel 3.0 version making it one of the longest-running functions included in Excel to date.
Solving equations in Excel is like playing a game of Sudoku, except you’re not competing against anyone except your own math skills.
Solving the Equations
Ease equation-solving in “Solving Simultaneous Equations in Excel”! Check out two sub-sections. They are: “Using Built-in Functions” and “Using Goal Seek”. These solutions can save time and energy.
Plus, they guarantee accuracy!
Using Built-in Functions
To optimize the efficiency of solving simultaneous equations in Excel, one can use built-in functions that enable quick and precise calculations without the need for manual input. Here’s how you can
leverage these functions to your advantage:
1. Open your Excel spreadsheet, and click on ‘Formulas’ in the toolbar menu.
2. Select ‘More Functions’, and choose ‘Statistical Functions’ from the drop-down list.
3. Click on ‘LINEST()‘ and enter the given data ranges in the arguments.
4. Enclose this formula within an array formula by pressing ‘Ctrl + Shift + Enter‘, which will generate the results for all equations simultaneously.
5. The values assigned to each variable can be extracted from these results quickly using cell references or assigning variables to specific cells.
In addition to using built-in functions, one must ensure that they are utilizing Excel’s spreadsheet layout effectively. Create a clear structure for equations by organizing them into columns and
rows, label cells with headers that correspond to equation variables, and consider using color-coding techniques for easy recognition.
Another suggestion would be to avoid duplicating data unnecessarily across multiple worksheets, as this may lead to errors or inconsistencies. Instead, opt for utilizing range names and linking
formulas across sheets where necessary.
By following these suggestions and leveraging built-in functions appropriately, you can streamline your simultaneous equation-solving process in Excel efficiently. Excel’s Goal Seek feature can solve
equations faster than a cheetah catching its prey.
Using Goal Seek
The Power of ‘Goal Seek’ in Solving Equations
Goal Seek is a powerful tool that allows you to find solutions to simultaneous equations in Excel quickly. With this feature, you can solve for one unknown variable based on the values of other
knowns. By using Goal Seek, calculations across various spreadsheets become more effortless and efficient.
Here’s a quick step-by-step guide on how to use ‘Goal Seek’:
1. Enter your data into a spreadsheet in Excel.
2. Select the cell with the formula or equation you want to solve.
3. Click on ‘Data’ on the top menu bar and then click on ‘What-if Analysis.’
4. Select ‘Goal Seek’ option from the menu.
5. In the goal seek dialog box that opens, set your target value by entering it in the ‘To Value’ field.
6. Select the input cell whose value should be changed using Goal Seek.
Using this method ensures that even if you make changes to other variables, your output remains accurate.
One unique thing about Goal Seek is its ability to work backward from an output value to find the necessary input. It works well when there are only two variables involved, but this function can be
applied even with complex equations.
Did you know that Wayne Winston – an expert in operations research and analytics – invented “Solver” and “Goal Seek” as part of his original Ph.D. thesis? The methods he developed have since become
widely used across various industries and fields.
Let’s double check those equations, because no Excel formula is immune to human error.
Checking the Solution
When solving simultaneous equations in Excel, it is essential to verify the accuracy of the solution. One way to do this is to substitute the values of the variables obtained from the solution back
into the equations and confirm that the results match. It is vital to execute this step to avoid errors in analysis and interpretation of solutions.
Another approach to check the solution is to plot the graphs of both equations and see if the point of intersection corresponds to the solution obtained in Excel. This method provides a visual
representation of the solution and helps in understanding the problem.
It is important to note that the accuracy of the results obtained in Excel is subject to the precision of the values used in the equations. Hence, it is crucial to check the data entered in the
formula bar for any errors and ensure that all formulas are correctly used.
A study conducted by Dr. Bruce Ratner established the significance of Excel in data analysis and business modeling. It revealed that Excel is an integral tool for data analysis in various industries,
including finance and healthcare.
Modifying the Equations
To customize the equations, adjust the cell references and coefficients as required. Change variable names for readability. Reorganize equations to simplify problem-solving. Use the correct order of
operations to ensure the equations output accurate solutions.
Ensure correct use of brackets, multiplication signs, and minus signs. Modify equations for particular situations such as using solver on non-linear systems. Refine inputs to generate valid solutions
that meet specific criteria. Lastly, avoid entering incorrect signs and/or variables.
Remember that modifying equations can significantly alter the solution, and sometimes much trial and error is needed. Being considerate about the changes being made will ensure accurate and valid
While modifying equations, it’s important to be careful not to erase previous steps. In a similar vein, professionals mustn’t quicken the pace of the process, leaving out valuable details.
Policymakers must not preempt unique and important information in decision making. For instance, a small miscalculation in drug dosage could be fatal to a patient. Hence, careful consideration is
critical in modifying equations.
Some Facts About Solving Simultaneous Equations in Excel:
• ✅ Excel is a powerful tool for solving simultaneous equations. (Source: Excel Campus)
• ✅ You can use the built-in SOLVER tool in Excel to solve systems of linear equations. (Source: Microsoft Support)
• ✅ Using matrix algebra is also a common method for solving simultaneous equations in Excel. (Source: Kaggle)
• ✅ Excel can handle complex equations with multiple variables and constraints. (Source: Spreadsheet Guru)
• ✅ Solving simultaneous equations in Excel can be used in various fields, such as finance, engineering, and science. (Source: Techwalla)
FAQs about Solving Simultaneous Equations In Excel
What is the process for solving simultaneous equations in Excel?
To solve simultaneous equations in Excel, you’ll need to use the Solver tool. First, set up your equations in an Excel sheet, using separate cells for each variable. Then, click on the Data tab and
select Solver. Input the cells you want to solve for, the target value, and any constraints before clicking Solve.
Can I solve varying degrees of simultaneous equations in Excel?
Yes, Excel can solve simultaneous equations of any degree, including equations with both linear and non-linear terms. However, the process will differ slightly depending on the complexity of the
What if Excel Solver can’t find a solution?
If Solver isn’t finding a solution for your simultaneous equations, there are a few things to check. First make sure you’ve inputted all variables, target values, and constraints correctly. If
everything looks correct, try adjusting the Solver settings, like adjusting the convergence tolerance or changing the solving method.
Is there a limit to the number of simultaneous equations Excel can solve?
Excel can solve an unlimited number of simultaneous equations, though the program’s performance may be affected by the complexity of the equations and the amount of data used.
Can I graph the solutions to my simultaneous equations in Excel?
Yes, after solving your simultaneous equations, you can create a graph in Excel to visualize the solutions. Input the equations you solved for in a new column, and then select both the column of the
equations and the column of the variables being solved for and create a graph.
Does Excel Solver work with non-numeric variables?
No, Excel Solver only works with numeric variables. If you need to solve simultaneous equations with non-numeric variables, you’ll need to use a different tool or translate the variables into numeric | {"url":"https://exceladept.com/solving-simultaneous-equations-in-excel/","timestamp":"2024-11-07T06:51:37Z","content_type":"text/html","content_length":"70040","record_id":"<urn:uuid:1e835104-336c-4aea-aec4-12a0496ee0dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00298.warc.gz"} |
Understanding Vertex Eccentricity
Vertex eccentricity is a concept in graph theory that measures the maximum distance from a given vertex to all other vertices in the graph. It provides valuable insights into the centrality and
reachability of individual vertices within a graph.
The Significance of Vertex Eccentricity
In graph theory, vertex eccentricity serves as a fundamental metric for understanding the structure and connectivity of graphs. It helps identify central vertices, which play crucial roles in
communication, transportation, and information flow within networks. Additionally, vertex eccentricity is instrumental in various graph algorithms and can aid in network analysis, community
detection, and pathfinding.
How Vertex Eccentricity is Calculated
1. Initialization: The algorithm initializes the eccentricity of each vertex to zero.
2. Exploration: For each vertex in the graph, the algorithm systematically explores all possible paths to other vertices, calculating the distance from the current vertex to each reachable vertex.
3. Updating Eccentricity: The algorithm updates the eccentricity of the current vertex to the maximum distance found during the exploration process.
4. Repeat: Steps 2 and 3 are repeated for every vertex in the graph until the eccentricity of all vertices is determined.
Applications of Vertex Eccentricity
• Centrality Analysis: Vertex eccentricity helps identify central vertices, which have high eccentricity values and are crucial for maintaining connectivity within the graph.
• Network Routing: It aids in determining efficient routing paths within networks by identifying vertices with low eccentricity, which serve as effective intermediaries.
• Community Detection: Vertex eccentricity can assist in detecting communities or clusters within a graph, as vertices with similar eccentricity values often belong to the same structural group.
Vertex eccentricity provides valuable insights into the structural characteristics of graphs and plays a vital role in various graph analysis tasks. By understanding the eccentricity of vertices,
researchers and practitioners can gain deeper insights into the connectivity, centrality, and resilience of networks, ultimately facilitating more informed decision-making and efficient network
** Crafted with insights from ChatGPT ** | {"url":"https://graph.komlankodoh.com/editor/node-excentricity","timestamp":"2024-11-10T03:28:02Z","content_type":"text/html","content_length":"35908","record_id":"<urn:uuid:bfc13e6e-6e12-46b9-9b48-4d49a31b694b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00509.warc.gz"} |
Using Row() in Array Formulas
By request.
In some array formulas, like the one to show cumulative principal payments, you will see a ROW() function or an ROW(INDIRECT()) construct. These provide a convenient way of inserting an array of
consecutive numbers into an array.
Assume that you have a list of numbers and that you want to sum largest three of them.
You can use the LARGE function to find the nth largest value (C3 finds the largest value). Of course you can also use MAX, but that will only find the largest value. If you want to sum the largest
three, you can simply string three LARGE functions together (C4). Because you are an Excel guru, you strive to shorten your formulas whenever possible (to impress your coworkers who can’t make heads
or tails of them). The array formula in C5 does just that. Notice how it uses an array, namely {1,2,3}. I don’t know what you call this kind of array, but I call it a scalar array. The array is hard
coded into the formula.
For a three element scalar array, typing out the array is not a big chore. But if that were to grow, it would become more of a burden. A shortcut for creating this type of array is to use the ROW
function. The fragment ROW(1:3) when evaluated as an array evaluates to {1,2,3}, as shown in C6.
The next step is to have a variable number of rows to sum, possibly where the user decides how many rows. For instance, the user can type a number into C7, and the formula in C8 will sum the C7th
largest values. In order to have a variable in the ROW function, we need to use INDIRECT to build a string that looks like the range reference we want. INDIRECT(“1:” & C7) evaluates to INDIRECT
Hopefully that explains the use of ROW() or ROW(INDIRECT()) in an array formula.
13 thoughts on “Using Row() in Array Formulas”
1. Dick,
Thanks for the explanation. It’s a whole lot clearer now. I appreciate it.
John Mansfield
2. If you want to confuse things a bit more, I tend to avoid using INDIRECT and OFFSET when possible, just because of their volatility, which, in large workbooks can create a real PITA.
To create an array like this what I usually do is:
ROW($A$1:INDEX($A:$A, C7))
Using the same cell that Dick uses in this example. If there’s any chance that the user can insert a row before row 1 (which would mess this formula), I go with this:
ROW(INDEX($A:$A, 1):INDEX($A:$A, C7))
3. I call those kinds of arrays “Array constants”, since they are not based on cells in the spreadsheet, but on the constant numbers, strings, booleans, and error values that you specify in the
formula. (You can’t put a cell reference, or the result of a calculation, as a member of an array constant.)
4. Dick,
I can get the formula in C5 to work, but when I try the formula in C6 I get the same result as simply typing large(a3:a15,1). Am I missing something? Here is what I used:
This works:
This does not:
5. Kevin,
You have to array enter (Use Control Shift Enter) the second one.
6. Thanks, JPG. That got it to work.
7. I am trying to get excel to recognize a letter in a row as a number for instance if a1=f then ae12 will equal 1 can anybody help with this?
8. @TJ In cell AE12: =if(a1=”f”, 1,0)
9. How would one accomplish the ROW(INDIRECT()) using VB Script.
I am trying to convert my array formula into a VB user defined function and it keeps highlighting the ROW and INDIRECT functions.
10. Hi,
I tried the same formula but the data are in a row but it doesn’t work for the function as below. (I changed ROW function to COLUMN) :
Any idea?
11. Hello Icon
=SUMPRODUCT(LARGE($B$2:$N$2,ROW(1:6))) or equivalent
or Array Enter
Row(1:6) is used to provide the series of numbers 1,2,3,4,5,6. Please see the information above.
Posting code? Use <pre> tags for VBA and <code> tags for inline. | {"url":"http://dailydoseofexcel.com/archives/2004/08/25/using-row-in-array-formulas/","timestamp":"2024-11-06T12:07:41Z","content_type":"text/html","content_length":"87168","record_id":"<urn:uuid:a52c598b-d8fd-4c7d-bf4e-3bf4cdd49c3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00635.warc.gz"} |
Multi-Frontal Solver For Simulations Of Linear Elasticity Coupled With Acoustics
Maciej Paszyński
[, Tomasz Jurczyk]
[, David Pardo]
This paper describes the concurrent multi-frontal direct solver algorithm for a multi-physics Finite Element Method (FEM). The multi-physics FEM utilizes different element sizes as well as polynomial
orders of approximation over element edges, faces, and interiors (element nodes). The solver is based on the concept of a node, and management of unknowns is realized at the level of nodes. The
solver is tested on a challenging multi-physis problem: acoustics coupled with linear elasticity over a 3D ball shape domain.
Keywords: parallel simulations, multi-frontal solvers, finite element method, linear elasticity coupled with accoustics, 3D mesh generation
Artykuł opisuje współbieżny algorytm solwera wielofrontalnego przeznaczonego do rozwiązy-wania za pomocą metody elementów skończonych (MES) problemów liniowej sprężystoś-ci sprzężonych z akustyką.
Natura problemów sprzężonych, takich jak rozważany prob-lem akustyki sprzężonej ze sprężystością, wymaga zastosowania różnej ilości niewiadomych w różnych węzłach siatki obliczeniowej stosowanej w
MES. Dlatego też algorytm solwera opiera się na koncepcji węzła obliczeniowego. Algorytm solwera testowany jest na trudnym problemie obliczeniowym – propagacji fal akustycznych na trójwymiarowej kuli
reprezentu-jącej uproszczony model głowy ludzkiej.
Słowa kluczowe: symulacje równoległe, solvery wielofrontalne, metoda elementów skończo-nych, liniowa spreżystość sprzężona z akustyką, generacja siatek trójwymiarowych
∗[AGH University of Science and Technology, Faculty of Electrical Engineering, Automatics, IT] and Electronics, Department of Computer Science, al. Mickiewicza 30, 30-059 Krakow, Poland, [email
∗∗[AGH University of Science and Technology, Faculty of Electrical Engineering, Automatics, IT] and Electronics, Department of Computer Science, al. Mickiewicza 30, 30-059 Krakow, Poland, [email
∗∗∗[Department of Applied Mathematics, Statistics, and Operational Research, University of the] Basque Country (UPV/EHU), Leioa, Spain and IKERBASQUE (Basque Foundation of Sci-ences), Bilbao, Spain,
[email protected]
1. Introduction
In this paper we focus on the development of an efficient concurrent multi-frontal direct solver. The multi-frontal direct solver is the current state-of-the art technique [1, 2, 3, 9] for solving
sparse systems of linear equations resulting from Finite Element Method (FEM) [6].
We focus on multiphysics simulations, in particular on the linear elasticity cou-pled with accoustics. In the multi-scale problem, there are three components of the unknown elastic velocity vector
field over the elastic domain, one component of the unknown pressure scalar field over the acoustic domain, and four unknowns over the interface. We also employ third order polynomials over the
entire mesh, except for the two external layers, where four order polynomials are used for the Perfectly Matching Layer technique [8]. Thus, with the exception of the interface, the number of
un-knowns at each vertex node is equal to one or three (depending on the acoustic /
elasticity domain type), the number of unknowns at each edge isp−1 or 3[∗](p−1)
(acoustic / elasticity), the number of unknowns at each face is of the order of (p[−]1)2
or 3[∗](p[−]1)2[(acoustic / elasticity) and the number of unknowns at each interior is of] the order of (p[−]1)3[or 3]
∗(p[−]1)3[(acoustic / elasticity). Here p=3 or 4, respectively.] The main disadvantage of having a different number of unknowns on each node is the problem of ordering unknowns for the solver, and
deciding which unknowns can be eliminated first. This is related to the location of the node only, so it is better to design the solver performing ordering at the level of nodes, not at the level of
In order to overcome the above problem, we introduce the concept of a node, understood as a group of unknowns associated with an element edge, face, or interior, that share the same connectivities.
The management of unknowns is performed at the level of nodes.
The main contribution of this paper is to introduce the concept of a node for management of the ordering for level recursive solution performed by the multi-frontal parallel solver.
The structure of the paper is the following. In section Main idea of the solver
algorithm, we introduce the concept of the node-based solver and explain the
algo-rithm on a simple two dimensional example. The following sectionThree dimensional
solver algorithm discusses the generalization of the nodes-based solver discussed in
the second section into the three dimensional case. In the next section, entitled The
model problem for acoustics coupled with linear elasticity we introduce the details of
the multi-physics prolblem, the linear elasticity coupled with acoustics. In particu-lar the section contains the variational formulation for the coupled problem and the definition of the PML
technique. In particular the construction of the computational
mesh is presented here. The section 3D multi-physics simulations descibes the
nu-merical results obtained by execution of the node-based solver onto the multi-physics
of the parallel algorithm and compares it with state of the art MUMPS solver. The
paper is closed with theConclusions section.
2. Main idea of the solver algorithm
In this section we explain the main idea of the node-based solver. For simplicity, we focus on a simple two dimensional rectangular mesh, and a scalar valued problem.
We consider a two dimensional rectangular finite element mesh, with polynomial
orders of approximation equal to pj on the four element edge nodesj = 1,2,3,4 and
(ph, pv) on the element interior node. The element local matrix consists of several
sub-matrices. The element nodes are four vertices, four edges and one interior node. The number of unknowns over the vertex nodes is equal to one. The number of unknowns
over the edge nodes is equal to pj−1, and the number of unknowns for the interior
node is equal to (ph−1) (pv−1). We utilize the hypermatrix concept, defined as a
recursive matrix of matrices. The element local matrix is stored as a hypermatrix, and the ordering is performed at the level of nodes, not at the level of unknowns. See Fig. 1 for a two dimensional
p4 p2
p1-1 p2-1 p3-1 p4-1 1 1 1 1 (ph-1) (pv-1) (ph-1)(pv-1) p1-1 p2-1 p3-1 p4-1 1 1 1 1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 9 4 8 5 1 3 6 2 7
Fig. 1. (Left panel) A single two dimensional finite element with polynomial orders of
approximation over edges equal to p1, p2, p3, p4 and polynomial order of approximation over the interior equal to ph in the horizontal direction and pv in the vertical direction.
(Right panel) Sizes of blocks in element local matrix: 1 unknown per element vertex, pi-1
unknowns per edge, and (ph-1)(pv-1) unknowns for the element interior
The main idea of the solver is illustrated in Figures 2, 3, and 4 and Tables 1, 2, and 3. In this simple example we have a mesh with four finite elements. The mesh is partitioned into four
processors. Thus, a subdomain matrix corresponds to a single element matrix.
In the first step of the algorithm, described in Figure 2 and in Table 1, the element local matrices are generated concurrently, on separate processors. This is realized by
calling get subdomain system routine. At this point, the unknowns associated with
4 7 3 14 11 8 9 6 15 13 1 5 2 12 10 20 21 19 25 24 16 18 17 23 22 processor 0 processor 1 processor 2 processor 3 processor 0,1,2,3 processor 0,1 processor 2,3 processor 0 processor 2 processor 1
processor 3 Fig. 2.Management of nodes associated with leaves of the elimination tree These nodes are localized byfind nodes to eliminate routine. Finally, the elimi-nation is performed by callingget
schur complement routine. 4 7 3 14 11 8 9 6 15 13 1 5 2 12 10 20 21 19 25 24 16 18 17 23 22 processor 0 processor 1 processor 2 processor 3 processor 0,1,2,3 processor 0,1 processor 2,3 processor 0
processor 2 processor 1 processor 3 Fig. 3.Management of nodes from second level of the elimination tree
4 7 3 14 11 8 9 6 15 13 1 5 2 12 10 20 21 19 25 24 16 18 17 23 22 processor 0 processor 1 processor 2 processor 3 processor 0,1,2,3 processor 0,1 processor 2,3 processor 0 processor 2 processor 1
processor 3
Fig. 4.Management of nodes associated with top of the elimination tree
Table 1
First step of the solver algorithm Processor 0
call get subdomain system 1,2,3,4,5,6,7,8,9 call find nodes to eliminate 4,7,8,9 —— 1,2,3,5,6
call get schur complement receive nodes 2,10,3,12,6 from Processor 1
Processor 1
call get subdomain system 2,10,11,3,12,13,14,6,15 call find nodes to eliminate 11,13,14,15 —— 2,10,3,12,6
call get schur complement send nodes 2,10,3,12,6 to Processor 0
Processor 2
call get subdomain system 16,17,2,1,18,19,5,20,21 call find nodes to eliminate 16,18,20,21 —— 17,2,1,19,5
call get schur complement
receive nodes 17,10,2,12,19 from Processor 3 Processor 3
call get subdomain system 17,22,10,2,23,24,12,19,25 call find nodes to eliminate 22,23,24,25 —— 17,10,2,12,19
call get schur complement send nodes 17,10,2,12,19 to Processor 2
Table 2
Second step of the solver algorithm Processors 0,1
call create system 1,2,3,5,6 + 2,10,3,12,6 = 1,2,3,5,6,10,12 call find nodes to eliminate 3,6 —— 1,2,5,10,12
call get schur complement receive nodes 1,2,5,10,12 from Processor 2
Processors 2,3
call create system 17,2,1,19,5 + 17,10,2,12,19 = 1,2,5,10,12,17,19 call find nodes to eliminate 19,17 —— 1,2,5,10,12
call get schur complement send nodes 1,2,5,10,12 to Processor 0
Table 3
Third step of the solver algorithm Processors 0,1,2,3
call create system 1,2,5,10,12 + 1,2,5,10,12 = 1,2,5,10,12 call full forward elimination
The first step ends after sending the resulting Schur complement matrices from processor 1 to processor 0 and from processor 3 to processor 2.
In the second step of the algorithm, summarized in Figure 3 and in Table 2, the Schur complements matrices are merged into a single matrix. This is
per-formed in create system routine. At this point, only unknowns associated with
com-mon edges (shared between two subdomains) can be eliminated. These nodes are
localized by find nodes to eliminate routine, and the elimination is performed by
get schur complement routine. Notice that we have two groups of processors: 0,1 and
2,3, working together over two matrices. The second step of the algorithm ends after sending the Schur complement contribution from processors 2,3 to processors 0,1.
In the last step of the algorithm, described in Figure 4 and Table 3, all processors are utilized to solve the resulting top interface problem. This problem is always associ-ated with the
cross-section of the domain, denoted in Figure 5 by the green color. This
is done by merging two Schur complement contributions in routine create system, as
well as by performing full elimination increate system routine, followed by recursive
backward substitutions.
3. Three dimensional solver algortihm
In this section we present how the node-based algorithm introduced in sectionMain
idea of the solver algorithm is generalized to solve the three dimensional ball-shape
The computational ball-shape domain described in Figure 5 has been obtained by generating the core of the mesh, representing the tissue part of the domain (Figure 6). The tissue part of the mesh has
been generated with tetrahedral elements. In the following steps, the tissue ball has been extended to the skull layer, by generating an additional layer of tetrahedral elements. Finally, the air and
PML layers have been generated, by adding four layers of prism elements over the boundary based on triangles. We employ third order polynomials over the entire mesh, except within the PML, where four
order polynomials are used.
The mesh is partitioned into uniformly loaded sub-domains by cutting the 3D ball shape into slices, as presented on bottom panel in Figure 6. This simple load balancing procedure is satisfactory for
large number of elements, and when the polynomial order of approximation is almost uniformly distributed over the entire mesh. This is the case in our simulations.
The multi-level approach described in details in section Main idea of the solver
algorithm is employed here on the multi-level three dimensional elimination tree,
presented on the bottom panel in Figure 6. The three dimensional algorithm is sum-marized below:
1. First, nodes from the interior of each sub-domain are identified. This is done by browsing all elements from a subdomain, generating element local matrices and transforming them into a submatrices
of a hypermatrix. The ordering of unknowns is performed at the level of nodes. This step is executed in parallel over each subdomain. The subdomain internal nodes are identified, and marked to be
2. In this step, the Schur complements over each subdomain is computed. Subdo-main internal nodes are eliminated with respect to the interface nodes. This is performed by submitting a subdomain
element local hypermatrices into a sequen-tial instance of the MUMPS solver [1, 2, 3, 9]. MUMPS performs parsequen-tial forward elimination over each sub-domain to calculate the local contribution to
the global interface problem. In this step, several sequential instances of the MUMPS solver are executed in parallel. The output from MUMPS is again transformed into a hypermatrix form.
3. Sub-domains are joined into pairs, and local contributions to the global interface problem are assembled within these pairs. This is realized by sending one Schur complement hypermatrix from an
odd processor to an even one, and by brows-ing the two Schur complement hypermatrices at the level of nodes, identifybrows-ing common nodes and merging them together into a new (third) hypermatrix.
Fully assembled nodes are identified and marked for the elimination.
4. In the next step, the Schur complement over each pair of processors is computed. This is realized by submitting a subdomain hypermatrix into a parallel instance of the MUMPS solver, utilizing the
two processors. Parallel MUMPS performs partial forward elimination over each sub-domain to calculate the local contribu-tion to the global interface problem. In this step, several parallel instances
of the
MUMPS solver are executed in parallel. The output from the MUMPS solver is again transformed into a hypermatrix form.
5. Processors are joint into sets of four processors (two previous step pairs in each set). The procedure is repeated recursively until there is only one common inter-face problem matrix with all
remaining interinter-face d.o.f. aggregated.
6. This common interface problem hypermatrix is formed in the root of the elim-ination tree and the single instance of parallel MUMPS solver is executed over all processors. In general, this
interface is associated with a cross-section of the domain. The solution obtained from the MUMPS solver is transformed back into the hypermatrix structure.
7. The solution of the common interface is broadcast back into two group of pro-cessors. Local solutions are substituted into Schur complement matrices from the corresponding step. Backward
substitution is executed to obtain further contri-butions to the global interface problem solution.
8. The procedure is repeated until we complete the solution of the global interface problem.
9. In the last step, the backward substitution is executed over sub-domains and the global problem is finally solved.
4. The model problem for acoustics coupled with linear
In this section the details of the computational problem are introduced. The problem falls into the category of general coupled elasticity/acoustics problems discussed in [5] with a few
modifications. The domain Ω in which the problem is defined is the interior of a ball presented in Figure 5, and it is split into an acoustic part Ωa, and an elastic
part Ωe. The problem is intended to model in a simplified way the acoustic response of
the human head to an incoming acoustic wave. Depending upon a particular example,
the acoustic part Ωa includes:
•air surrounding the ball, bounded by the ball surface and a truncating sphere
•an additional layer of air bounded by the truncating sphere and the outer sphere
terminating the computational domain, where the equations of acoustics are
re-placed with the corresponding Perfectly Matched Layer (PML) modification.
The elastic part of the domain includes:
The term tissue is understood here as the part of the head not occupied by the skull
(bone). In the current stage of the project it is assumed that the elastic constants for the entire tissue domain are the same.
Fig. 5.Structure of the computational mesh
Fig. 6. (Left panel) Cross-section of the 3D ball-shape domain for linear elasticity
compu-tations.(Right panel) Elimination tree for 8 subdomains
The weak (variational) final formulation of the problem has the following form.
We seek for elastic velocityuand pressure scalar fieldpsuch that
u∈u˜D+V , p∈p˜D+V, bee(u, v) +bae(p, v) =le(v), ∀v ∈V bea(u, q) +baa(p, q) =la(q), ∀q∈V (4.1)
with the bilinear and linear forms defined as follows. bee(u, v) = Z Ωe Eijkluk,lvi,j−ρsω2uividx bae(p, v) = Z ΓI pvndS bea(u, q) =−ω2ρf Z ΓI unq dS baa(p, q) = Z Ωa ∇p∇q[−]k2pqdx le(v) =− Z ΓI
pincvndS la(q) =− Z ΓI ∂pinc ∂n q dS (4.2)
V andV are the spaces of the test functions,
V =[{]v [∈]H1(Ωe):v= 0 on ΓDe}
V ={q∈H1(Ωa):q= 0 on ΓDa}
(4.3) Hereρf is the density of the fluid,ρsis the density of the solid,Eijkl is the tensor of
elasticities, ω is the circular frequency,c denotes the sound speed, k is the acoustic
wave number andpinc [is the incident wave impinging from the top. For more details]
we refer to [5]. Coupled problem (4.1) is symmetric if and only if diagonal forms bee
andbaa are symmetric and,
bae(p, u) =bea(u, p).
Thus, in order to recover the symmetry of the formulation, it is necessary to rescale
the problem. For instance by dividing the second equation by factor[−]ω2ρf.
bee(u, v) = Z Ωe Eijkluk,lvi,j−ρsω2uividx bae(p, v) = Z ΓI pvndS bea(u, q) =− Z ΓI unq dS baa(p, q) = 1 ω2ρf Z Ωa ∇p∇q[−]k2pq dx le(v) =− Z ΓI pinc[v] ndS la(q) =− 1 ω2ρf Z ΓI ∂pinc ∂n q dS (4.4)
The incident wave is assumed in the form of a plane wave impinging from the top,pinc[=][p]
0eikex,e= (0,0,−1),p0= 1[P a]. The test problem is being solved with
frequency f = 200 Hz. The precise geometry data are as follows: brain r < 0.1 m,
skull 0.1 m< r <0.125 m, air 0.125 m< r <0.2 m and PML air 0.2 m< r <0.3 m.
In the PML part of the acoustical domain, the bilinear form baa is modified as
follows: baa(p, q) = Z Ωa,P M L z2 z0[r]2 ∂p ∂r ∂q ∂r+ z0 r2 ∂p ∂ψ ∂q ∂ψ+ z0 r2sin2ψ ∂p ∂θ ∂q ∂θ r2sinψdrdψdθ
Here r, ψ,θ denote the conventional spherical coordinates andz =z(r) is the PML
stretching factor defined as
z(r) = 1[−] i k r−a b[−]a α r (4.5)
Hereais the radius of the truncating sphere,bis the external radius of the
computa-tional domain (b−ais thus the thickness of the PML layer),idenotes the imaginary
unit,kis the acoustical wave number, andris the radial coordinate. In computations,
all derivatives with respect to spherical coordinates are expressed in terms of the stan-dard derivatives with respect to Cartesian coordinates. In all reported computations,
parameterα= 5. For a detailed discussion on derivation of PML modifications and
effects of higher order discretizations, see [8].
The material data corresponding the considered layers are presented in Table 4.
Table 4 Material constants Material E[MPa] ν ρ[kg/m3] cp[m/s] cs[m/s] tissue 0.67 0.48 1040 75 15 skull (bone) 6500 0.22 1412 2293 1374 air 1.2 344
5. 3D multi-physics simulations
In this section, we describe numerical simulations of a three dimensional acoustics problem coupled with linear elasticity. We utilize the methodology described [6] to analyse the quality of the core
tetrahedral elements ball shape mesh. The statistic for the core mesh is summarized in Table 5.
In this model, the domain consists of four concentric spheres, as presented in Figure 5. Inner sphere is filled with an elastic material with data corresponding to human brain. The first layer is
also elastic with constants corresponding to human skull. The second layer corresponds to air, and the last one to the PML air.
Fig. 7. Plot of the real part of pressure on plany y= 0 passing throught the center of the
Fig. 8. Plot of the imaginary part of pressure on planyy= 0 passing throught the center of the ball
Fig. 9. Plot of the real part of pressure on plany y= 0 passing throught the center of the ball, in the range−0.00001 to 0.00001
Fig. 10. Plot of the imaginary part of pressure on planyy= 0 passing throught the center of the ball, in the range−0.00001 to 0.00001
Table 5
Parameters of the core mesh
Parameter Fine mesh
Number of points 1865
Number of edges 10875
Number of tetrahedrals 8085
Maximum number of tetrahedrals with common edge 9 Average number of tetrahedrals with common edge 4.5
The total number of unknowns is 288,899, however what really maters is not size of matrix but number of non-zero entries and sparsity pattern. In this 3D problem we have 40,215,303 non zero entries.
Figures 7 and 8 display the distribution of real and imaginary parts of the pressure
over the y= 0 section of the mesh. Figures 9 and 10 display the distribution of real
and imaginary parts of the pressure over the y= 0 section obtained on the big mesh
rescaled to values from[−].00001 to.00001 for the real part, and from[−].0001 to.0001 for the imaginary part.
For the discussion on the validity of the numerical solution, based on the com-parison to the analytical one, we refer to [4].
To visualize our meshes, we have implemented a simple interface to the Visu-alization Toolkit (VTK) [11], a collection of C++ classes that implement a wide range of visualization algorithms. The
central data structure for this project is the vtkUnstructuredGrid, which represents volumetric data as a collection of points with corresponding scalar values (in this case, the real and imaginary
part of the pres-sure), connected by cells of arbitrary type and dimension (i.e. lines, triangles, quads, tetrahedra, prisms, etc.).
6. Measurements of the solver efficiency
In this section, we compare the measurements of the execution times and memory usage of our multi-level solver, against parallel MUMPS solver [9]. In all tests, the METIS ordering [7] is utilized
within MUMPS.
The simulations have been performed over a linux cluster from our Department of Computer Science, with architecture summarized in Table 8. The solver has been implemented in fortran90 with MPI.
We start with presenting in Table 6 the execution time and memory usage for sequential MUMPS solver executed over 1 processor as well as for parallel MUMPS solver, executed over 4, 8, 12 and 16
processors. The measured execution times include the matrix assembly, time spend on analysing the connectivity data by METIS in order to construct a proper ordering, and time spend on factorization.
Next, we present the detailed execution times and memory usage for our multi-level solver, executed for 8 and 16 processors. This is reported in Tables 7
Table 6
Total execution time and memory usage per processor for MUMPS solver Cores Execution time [s] Memory usage per cores [MB]
1 2221 3998 4 1426 4516, 4516, 4417, 4156 8 812 1761, 1857, 2117, 1804, 1845, 1752, 1612, 2117 16 644 1183, 872, 873, 1151, 1165, 866, 1005, 992, 1120, 957, 1143, 954, 814, 968, 934, 1183 20 789 570,
723, 935, 940, 737, 906, 736, 840, 1166, 858, 797, 609, 1080, 964, 767, 998, 905, 821, 1166, 797
and 8 for 8 and 16 processors, respectively. The tables report the total execution time spend by each processor (or set of processors) at particular nodes of the elimination tree. The time includes
the assembly, the reordering of nodes, and factorization. It
also reports the memory usage. Each table is summarized with Total line, which
contains the sum of maximum execution times for levels of the elimination tree as well as maximum memory usage. This is the total execution time and memory usage for the parallel solver.
Table 7
Linux cluster details
Node Number[of cores] Processor clock[[GHz]] Total memory[[GB]] Memory per core[[GB]]
Node 0 2 3.0 8 4 Node 1 2 3.0 8 4 Node 2 2 2.8 8 4 Node 3 2 2.8 8 4 Node 4 2 2.8 8 4 Node 5 2 3.0 6 3 Node 6 2 2.8 6 3 Node 7 12 2.4 16 1.33
Our multi-level solver obtained 7.76 speedup (0.97 efficiency) for 8 processors and 6.20 speedup (0.38 efficiency) for 16 processors. The parallel MUMPS solver obtained 2.73 speedup (0.34 efficiency)
for 8 processors and 3.44 speedup (0.21 efficiency) for 16 processors.
The maximum memory usage per core for our solver was 1873 [MB] for 8 pro-cessors and 1916 [MB] for 16 propro-cessors. The maximum memory usage per core for parallel MUMPS solver was 2117 [MB] for 8
processors and 1183 [MB] for 16 proces-sors.
Table 8
Solver execution statistics per 8 subdomains elimination tree
Sub-domains Problem[size] [non zero entries]Number of Execution[time [s]] Memory usage[[MB]]
Subdomain 0 7652 830707 14 265 Subdomain 1 5020 401356 5 262 Subdomain 2 7289 777229 12 232 Subdomain 3 7971 860175 13 311 Subdomain 4 7392 747797 13 341 Subdomain 5 7290 789420 13 224 Subdomain 6
8076 865398 29 373 Subdomain 7 7626 821297 17 251 Sub-domains 0,1 4327 11434513 16 881 Sub-domains 2,3 4468 11536477 10 962 Sub-domains 4,5 4060 11507491 21 765 Sub-domains 6,7 4371 12484213 24 896
Sub-domains 0,1,2,3 6441 30409371 103 1873 Sub-domains 4,5,6,7 5971 24384651 95 1640 Sub-domains 0,1,2,3,4,5,6,7,8 4930 24304900 130 960 Total 286 1873 Table 9
Solver execution statistics per 16 subdomains elimination tree
Sub-domains Problem[size] [non zero entries]Number of Execution[time [s]] Memory usage[[MB]]
Sub-domain 0 3557 332191 3 87 Sub-domain 1 3911 372952 3 137 Sub-domain 2 4416 442464 4 166 Sub-domain 3 4414 402744 4 152 Sub-domain 4 3769 345943 3 151 Sub-domain 5 3382 301744 3 162 Sub-domain 6
5033 510041 5 301 Sub-domain 7 4429 444631 4 186 Sub-domain 8 3956 352765 3 166 Sub-domain 9 4205 426477 4 161 Sub-domain 10 4454 475695 5 138 Sub-domain 11 4043 392659 4 145 Sub-domain 12 4112
430856 4 120 Sub-domain 13 3562 279616 3 162 Sub-domain 14 4372 431209 5 159 Sub-domain 15 4212 416574 5 140 Sub-domains 0,1 2910 4667527 4 392 Sub-domains 2,3 3234 6531273 11 462 Sub-domains 4,5
3676 5543848 3 379 Sub-domains 6,7 4496 10012208 19 747 Sub-domains 8,9 3447 7165240 13 534 Sub-domains 10,11 3206 6010750 9 444 Sub-domains 12,13 3318 5749926 6 438 Sub-domains 14,15 3310 6446964 11
484 Sub-domains 0,1,2,3 4817 13832607 22 1003 Sub-domains 4,5,6,7 6147 16623332 52 1327
Table 9. cont. Sub-domains 8,9,10,11 4772 15046955 38 984 Sub-domains 12,13,14,15 5389 15534227 45 1329 Sub-domains 0,1,2,3,4,5,6,7,8 7040 35658739 170 1916 Sub-domains 9,10,11,12,13,14,15 5389
15534227 45 1329 Sub-domains 4753 22576729 112 892 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 Total 358 1916
7. Conclusions
In this paper we analyzed the possibility of using a concurrent solver applied to challenging multi-physics problems resulting from FEM simulations, with varying number of unknowns in different parts
of the mesh and possibly different polynomial orders of approximation. We overcome the difficulties by utilizing the hypermatrix concept based on the notion of a node. We also designed and
implemented a new concurrent direct solver, tested its efficiency and compared it with the MUMPS solver. Our solver obtained 7.7 speedup and turned out to be up to 1.8 times faster than MUMPS solver.
The work reported in this paper was supported by the Polish MNiSW grant no. NN 519 405737. The work of the third author was partially funded by the Spanish Ministry of Science and Innovation under
the project MTM2010-16511. The first author would like also to acknowledge the Foundation for Polish Science for providing funds necessary for purchasing the linux cluster.
[1] Amestoy P. R., Duff I. S., L’Excelent J.-Y.:Multifrontal parallel distributed
sym-metric and unsymsym-metric solvers. Computer Methods in Applied Mechanics and
Engineering, vol. 184, 2000, pp. 501–520.
[2] Amestoy P. R., Duff I. S., Koster J., L’Excelent J.-Y.:A fully asynchronous
mul-tifrontal solver using distributed dynamic scheduling. SIAM Journal of Matrix
Analysis and Applications, vol. 23, 2001, pp. 15–41.
[3] Amestoy P. R., Guermouche A., L’Excelent J.-Y., Pralet S.: Hybrid scheduling
for the parallel solution of linear systems. Parallel Computing, vol. 32(2), 2006,
pp. 136–156.
[4] Demkowicz L., Gatto P., Kurtz J., Paszyński M., Rachowicz W., Bleszyński E.,
Bleszyński M., Hamilton M., Champlin C., Pardo D.:Modeling of bone
conduc-tion of sound in human head using hp finite elements. Part I. Code design and modificaton. Computer Methods in Applied Mechanics and Engineering, vol. 200,
[5] Demkowicz L., Kurtz J., Pardo D., Paszyński M., Rachowicz W., Zdunek A.:
Computing with hp-Adaptive Finite Element Method. Vol. II. Frontiers: Three Dimensional Elliptic and Maxwell Problems. Chapmann & Hall / CRC Applied
Mathematics & Nonlinear Science, 2007.
[6] Glut B., Jurczyk T., Kitowski T.: Anisotropic Volume Mesh Generation
Con-trolled by Adaptive Metric Space. AIP Conf. Proc. Volume 908 NUMIFORM
2007, Materials Processing and Design: Modeling, Simulation and Applications, June 17–21, Porto, Portugal, 2007, pp. 233–238.
[7] Karypis G., Kumar V.:A fast and high quality multilevel scheme for partitioning
irregular graphs. SIAM Journal on Scientific Computing, vol. 20, issue 1, 1999,
pp. 359–392.
[8] Michler Ch., Demkowicz L., Kurtz J., Pardo J.: Improving the performance of
perfectly matched layers ny means of hp-adaptivity. ICES-Report 06-17, The
Uni-versity of Texas at Austin, 2006.
[10] Paszyński M., Pardo D., Torres-Verdin C., Demkowicz L., Calo V.: A Parallel
Direct Solver for Self-Adaptive hp Finite Element Method. Journal of Parall and
Distributed Computing, vol. 70, 2010, pp. 270–281.
[11] Schroeder W., Martin K., Lorensen B.: The Visualization Toolkit An | {"url":"https://1library.net/document/y9r17ely-multi-frontal-solver-simulations-linear-elasticity-coupled-acoustics.html","timestamp":"2024-11-03T01:06:11Z","content_type":"text/html","content_length":"185098","record_id":"<urn:uuid:11976dcf-fb54-4026-bb9d-466eac4188d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00095.warc.gz"} |
takes your 라이브바카라사이트 decisions as a baccarat player
If players had the chance, it is hard to 라이브바카라사이트 추천 imagine that any baccarat game would not have a zero or less house edge. This is not a brain. The house must ensure it has an
advantage so the casino can stay open for longer. As a result, you can take a lower risk with your money than otherwise. This is something you can do in almost any casino game bacaratbog.
People bet money when 최신 라이브바카라사이트 they play Baccarat.
The banker can decide once he sees the player’s winnings. Because of this, he has a better chance of winning a hand. However, in Baccarat, the banker’s hand can be changed. The house rules stipulate
that the player must pay a 5% commission on all winning bank bets. As an extra benefit, ties pay 8 to 1.
Compared to other card games, Baccarat has a more considerable house advantage. Since 9-to-1 odds are more and casinos may be 실시간카지노 found along the gulf coast often than ties, this wager
appears to bring in a profit of 14.36% for the house. The house’s profit is 1.24% when betting on the player. The house has a 1.06% advantage over the banker due to the 5% commission, even though the
banker is the favorite to win.
You need to calculate the house advantage in Baccarat if you want to win.
You can always count on the house advantage when betting on the banker in Baccarat. There are two types of house edges in Baccarat. A house edge of 1.24% or 1.06% is available.
The solution is obvious. You should choose 1.06% as your percentage as a player. Bet on the banker at all times for the best possible winning probability.
This is always the case, regardless of how often the banker or player has won. The banker’s goal is always to have a higher chance of winning than the other players.
There is a set of rules that govern the game of Baccarat: The game of Baccarat originated in France in the fifteenth century and is pronounced “ba-kara.” The three main variations 오래된 라이브바카라
사이트 of Baccarat, also known as North American Baccarat, are Punto Banco, Baccarat Chemin de Fer, and Baccarat Banque. Since the cards are dealt and the games that the players make depend on the
cards that they have already been dealt, Punto Banco is a game of chance. Typically, this will happen when you play at an online casino.
This game has only three possible outcomes: a player win, a banker win, or a tie. Neither the customer nor the bank will be called “players” or “bankers” here. These are the things that the client
can put a guard on. This means that the house advantage is 1.24%, and the banker advantage is 1.06% when you are the player. The players and the banker have an equal house advantage in this game.
This is the lowest of all casino games. The Tie Bet has a probability of 14.44% in a Six-Decker-Spiel.
All that has needed to win the game is a goal of 9 or as close to it as possible. You can bet on the player, the banker, or a tie. All of the players and bankers’ hands, as well as a tie hand, can be
clasped together simultaneously.
All tens and photos are for zero points except the ace, which is worth one point.
Each player will receive two cards, while each banker will receive two more. There are no more cards when a player has a “natural” hand of 8 or 9. Every winning hand has at least nine cards. A “tie”
is called when both hands are equal.
After the first card is dealt, a third card could be given. These rankings are based on each player’s hand’s total number of points. A third card will be created if that is the case. Hands are dealt
to players and bankers sequentially. There are no more cards available.
Stick to these rules if you want to draw a third card:
Each player gets a card in a game where the banker does not have an 8 or 9. It makes no difference if the player holds a hand of 6, 7, 8, or 9. They have been eliminated from the competition.
A card is drawn when the banker’s hand is 0, 1, or 2. The banker has to draw a natural eight or nine. Standing is required of the banker when holding seven, eight, or nine cards. At any moment (while
standing), the banker will draw three cards if they have three and the player has three less than eight.
All other cards in the player’s hand must stand for their third card, which must be between 2 and 7. In this case, the banker will stand if the player’s third card is higher and draw if it is between
4 and 7. The third card determines the winning hand if the player’s cards total six and the dealer’s cards total seven. I can only use this strategy if the player has played a third card.
For every winning hand, the player will receive 1 to 1. Similarly, bankers’ hands are also paid one by one. There is a 5% house commission on all successful banker bets. There is an 8 to 1 chance
that a tie game will be paid out. You will receive an additional amount equal to the stake multiplied by 8.
We talked about specific Baccarat-related questions.
What are the rules of Baccarat, and how do you play? In light of what I’ve mentioned, how are you feeling now? You have two hands, the player’s and the banker’s, and you bet on the hand 안전한 라이브
바카라사이트 that has scored the most points so far. For this game, a 9 is the best possible outcome, and a one is the worst. Nothing is valued more than the 10s and Aces.
Now that you know how to play Baccarat, it’s time to learn the rules. You may watch who wins or loses or the number of cards played using the six-card deck. No matter what, you’ll always be the first
one to go. You can draw a new card if you have one in your hand. Only further cards can be drawn if this condition is met.
Up next, the banker! If the banker’s first two cards add up to zero or two, they can draw another card. Subtract the number from the total when you add up all of your cards. Think about the scenario
where you have 7 and 6, and the sum is 13. When the tens are deducted, the baccarat total is three. That is how it works.
What are the secrets of Baccarat’s success? You win your bet if the other side’s money is more significant than yours. You will be hurt in this situation. To win, you have to put your money on the
bankers, but they keep 5% of your winnings. You will return nine times as much if you bet on a win.
You only need a few basic mathematical equations to learn how to play Baccarat. You have a good chance of 라이브바카라사이트 주소 winning much money if you bet on a tie. You will be able to play
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Math and its Real-life Applications
It is nearly impossible to imagine a situation where you do not use numbers or math. Your life is heavily reliant on math, from daily activities to techniques for solving problems to logical
reasoning and aptitude-building skills. Without math, science and technology would not have been as advanced as they are now. Elucidating statistics, proving facts, recognizing patterns, and other
similar tasks require the assistance of one mathematical instrument or another.
You might find it fascinating to learn that the fundamentals of math are essential to your day-to-day life. High school math is also highly applicable. High-end careers like those in psychology,
geography, architecture, and history, among others, likewise rely on specific standards of science. Vendors, cashiers, and storekeepers also use numbers to determine their sales, profits, and losses.
This makes math an extraordinary field.
Math: A Real-life Problem Solver
With its multiple uses in the dimensions of science and technology, math has a different meaning for a layman. Despite the technicality of the subject, it applies to multiple activities like cooking,
the fashion sector, designing, and even exercising.¹²³^4 But how? Let’s find out!
If you enjoy cooking or are from a culinary background, you might have heard a lot about ratios. The measuring spoons you use, the amount of flour, the ratio of water in your mixture, etc., all
require the application of math. Even while baking some cookies, you first preheat the oven to a certain temperature. This also includes playing with numbers. Figuring out how long you need to cook
the dish or brew your coffee also incorporates math. So, it becomes an inevitable part of your skills and eating habits. Addition, subtraction, multiplication, logical reasoning, calculation of time,
ratios, and proportions are some of the concepts applied to your daily life cooking activities.¹
• Math in Fashion Designing
Clothes and fashion! Who does not enjoy them? The fashion sector is a versatile and advanced user of math. The cuts you see, the patterns you love, the pricing for the fabrics, the angular divisions
of your apparel, etc., revolve around geometry and math. Different neck designs and body measurements are taken with the help of scales. At the same time, geometry and symmetry are two crucial parts
of tailoring a garment design. Various proportions are used to construct a basic size guide or chart for the garment. The list could go on and on, as the life of a fashion designer is highly
dependent on numbers and geometry.²
• Math in Interior Designing
Area, volume, and perimeter are an all-inclusive part of interior design. It takes too much effort to build a proper layout of your room or house. Interior design refers to carving a layout and
utilizing maximum inputs in a limited space. Thus, it is a complex process. Using formulae related to the area, perimeter, angles, etc., is therefore required to build a sound and efficient design
for the interior of your house. This makes math an inevitable part of studying and crafting interior spaces. Even when buying stuff for interior decor, the designer needs to align their ideas with
the area and angles of the space. This makes math a great management tool as well.¹
The automobile industry is one of the most advanced and up-to-date industries. It not only focuses on a sales-driven strategy for production but also keeps on adjusting and inventing designs, pickup,
and other functions of the vehicle. The math implication can be extensive in doing so. The calculations, designing, speed, etc., are all based on numbers and a whole lot of measurements. It thus
implies many different principles of math and physics to provide you with a better version of the vehicle every time. Ratios, proportions, statistics, geometry, algebraic expressions, etc., are
incorporated into the design of any automobile. Thus making math, a fundamental unit used to prepare a varied structure.³
In real life, the application of this field is diverse. The fundamental principles of math and its dimensions are extensive and cover almost every spectrum of your life. Its applications extend from
building monuments to jets, communication, coding, cooking—and whatnot! You are making use of mathematical concepts everywhere. This is what makes the essence of math truly practical!
Found this article interesting? For such interesting and informative blogs, visit BYJU’S FutureSchool Blog!
22 Examples of Mathematics in Everyday Life – StudiousGuy. (n.d.). 22 Examples of Mathematics in Everyday Life – StudiousGuy. Retrieved November 5, 2022, from https://studiousguy.com/
M. (2021, June 2). Top 25 Uses of Mathematics in our Daily Life – Explained! (2022). Of money. Retrieved November 5, 2022, from https://ofwmoney.org/uses-of-mathematics/
Math Matters in Everyday Life – NIU – Math Matters. (n.d.). Northern Illinois University. Retrieved November 5, 2022, from https://www.niu.edu/mathmatters/everyday-life/index.shtml
Analytica, S. (2022, September 27). What are the Applications of Math in Everyday Life – StatAnalytica. StatAnalytica. Retrieved November 5, 2022, from https://statanalytica.com/blog/ | {"url":"https://www.byjusfutureschool.com/blog/assistance-of-math-in-your-daily-life/","timestamp":"2024-11-10T06:40:07Z","content_type":"text/html","content_length":"166226","record_id":"<urn:uuid:820a6eee-fcd1-4f36-aa4a-3414afaf1680>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00295.warc.gz"} |
The Mathematical Beauty of Snowflakes
“There was a footpath leading across the fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to
know more about mathematics.”
-Bertrand Russell, Nobel Laureate and Mathematician
It is mystical when you step outside on a snowy morning. Snowflakes are swirling around the vast sky and falling and blanketing the ground. If a snowflake lands on you, it is like a winter angel.
There are no flowers around, for they cannot survive the cold; yet what lies before your eyes is an incredible beauty. And it’s remarkable, you come to realize, that no two snowflakes are alike. It
is as if the uniqueness of a snowflake is controlled by a divine force. The individuality of a snowflake’s structure draws a parallel to human life. Like snowflakes, everyone has a unique story to
I am not the only one who ponders about snowflakes; many mathematicians do the same. Actually, they think about the characteristics of snowflakes because they are particularly important for three
basic mathematical principles: pattern, symmetry, and symmetry breaking.
A little-known scientist, Wilson Bentley, a.k.a. “the Snowflake Man” took pictures of snowflakes almost every day and observed them until he died. You can buy his book about his work on Amazon. If
you wish to know why he did it, read about it at snowflakebentley.com.
“Under the microscope, I found that snowflakes were miracles of nature; and it seemed a shame that this beauty should not be seen and appreciated by others. Every crystal was a masterpiece of
design and no one design was ever repeated. When a snowflake melted, that design was forever lost. That beauty was gone, without leaving any record behind.”
-Wilson Bentley
When I checked the Oxford dictionary, there were 3 definitions for the word “pattern.” Two of these definitions [listed below] are important for this article.
Pattern: 1. A repeated decorative design; 2. An example for others to follow.
When we check the pictures and delve deeper into each snowflake, we will see that the structures of the snowflakes are totally different. However, they have something in common: symmetry and a
hexagonal structure.
These perfect ice crystals are genuine, even though it is hard to believe they are not fake.
When I take a close look at a snowflake, the beauty of the combination of ice molecules fascinates me every time; each flake is unique. However, uniqueness is not the point here. The things that make
snowflakes important objects for mathematicians are their symmetry and their hexagonal structure. Math-loving people have a lot of interest in transformations. They love moving objects. And,
surprisingly, if an object is symmetric, transformations are not even noticed by many.
To be more precise, when you have a hexagonal symmetric snowflake, or any other symmetrical object, when you rotate it in any direction, 60°, 120°, 180°, 240°, 300°, or 360°, people watching you
wouldn’t realize it. If you check the images below, you will see rotated shapes but no difference. It appears to be the same shape in exactly the same place.
1- Counterclockwise rotation by 120°
2- Reflection through a vertical axis
3- Reflection axes of a snowflake[1]
Snowflakes also possess reflectional symmetry. If we stand in front of a mirror, our reflection looks exactly the same. Hence, if we put a mirror in the middle of a snowflake, there will be a
reflection. For a snowflake, we can put a mirror 6 different ways. Thus, we can say that a snowflake has 12 symmetries: 6 from reflections, and 6 from rotations.[2]
Now we can define symmetry as a transformation that leaves things unchanged. We can also claim that a combination of any of the transformations will give us exactly the same shape. For instance, we
can rotate our snowflake 60° two or three times in a row and flip it over, and it will remain unchanged.
At this point, you might ask the question: “You have all these fancy symmetries for this particular snowflake. But, does every snowflake possess the same symmetries?”
Snow is a molecular structure of an ice crystal. And ice is a structured substance. It is a different form of water. When the water cools down, the molecules move more slowly, and this begins to
impact how the molecules line up. Hydrogen atoms of one water molecule bond with two oxygen atoms. As the water freezes, the molecules arrange into hexagonal patterns. They prefer to stay as far away
from each other as possible, and that makes them take up more space. The large space affects density. The density of ice becomes less dense than water. This is why ice floats. Almost all other
liquids have a higher density when they freeze.
When we examine an ice crystal carefully under normal conditions, we always see a combination of molecules with six-fold symmetry. Snowflake molecules make a honeycomb structure. This results in an
inordinate amount of hexagonal symmetry in these molecular three-dimensional structures.
Okay, we saw the structure of a snowflake under normal conditions. But, what if we changed those conditions? Johannes Kepler answered this question after his experiments and wrote a book about
snowflakes, particularly The Six-Cornered Snowflake.
There are two key elements which affect the structure of a snowflake: temperature and moisture. Each time the temperature or the amount of moisture change, the structure of a snowflake changes. If
you check the snow crystal morphology diagram below, you will see that when the temperature nears 0° and humidity is high, the structure of a snowflake will be flowery. Flowery structures are called
dendrites. When you make it a little bit colder, the structure will be fancy hexagonal plates. We can apply many combinations and get varying structures.
The Snow Crystal Morphology Diagram. Source: Snow Crystals - http://www.snowcrystals.com/science/science.html
Professor of physics Kenneth G. Libbrecht is the owner of the diagram below. In a PBS interview, he said, “It’s a mystery as to why snowflake shapes go from plates to columns to plates to columns as
the temperature lowers. That’s one of the things I’ve been trying to understand. It has been a mystery for about 75 years, and it’s still unsolved.”[3]
The Shapes of Snowflakes | Source
In the end, although the structure of (almost) all snowflakes are the same, some of them are not completely hexagonal. For instance, there are some snowflakes that have tree structures. Some
snowflakes have branches, and each branch has tiny branches.
But, why is the structure of some snowflakes not hexagonal?
So far, we have talked about pictures which were taken at a particular instant. We have seen the pictures of the motion of the snowflakes for the smallest amount of time that can be measured.
However, a snowflake never stops spinning in the air. They tend to oscillate. That means the shape of the snowflake is changing all the time. But how? When you see a snowflake in the air, it changes
its place after a second because it would be whirled about, and it will be under different conditions at that time. This process will occur up until the snowflake lands on the ground. We know from
the diagram that the temperature and amount of moisture always affect the shape of a snowflake. While small-scale conditions are almost the same, on a larger-time scale, conditions will differ. And
these differences will change every corner of a hexagonal snowflake, resulting in a different structure. This is the main reason behind the variety of snowflake structures and uniqueness.
In conclusion, we can say that a snowflake can preserve its six-fold symmetry at all times. I think we have another reason to love mathematics! I want to finish my piece with Hermann Hankel’s words:
“In most sciences one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.”
[1] https://web.stanford.edu/~cantwell/AA218_Course_Material/Lectures/Symmetry_Analysis_Chapter_01_Introduction_BJ_Cantwell.pdf | {"url":"http://sss.fountainmagazine.com/all-issues/2019/issue-127-jan-feb-2019/the-mathematical-beauty-of-snowflakes","timestamp":"2024-11-05T10:14:42Z","content_type":"text/html","content_length":"103421","record_id":"<urn:uuid:421d27cd-ab53-470a-9269-4b120fc690df>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00469.warc.gz"} |
On the Hardness of Compressing Weights
We investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main
focus is the weighted Clique problem, where we are given an edge-weighted graph and the goal is to detect a clique of total weight equal to a prescribed value. We show that the weighted variant,
parameterized by the number of vertices n, is significantly harder than the unweighted problem by presenting an O(n3ϵ) lower bound on the size of the kernel, under the assumption that NP ⊆ coNP/poly.
This lower bound is essentially tight: we show that we can reduce the problem to the case with weights bounded by 2O(n), which yields a randomized kernel of O(n3) bits. We generalize these results to
the weighted d-Uniform Hyperclique problem, Subset Sum, and weighted variants of Boolean Constraint Satisfaction Problems (CSPs). We also study weighted minimization problems and show that weight
compression is easier when we only want to preserve the collection of optimal solutions. Namely, we show that for node-weighted Vertex Cover on bipartite graphs it is possible to maintain the set of
optimal solutions using integer weights from the range [1, n], but if we want to maintain the ordering of the weights of all inclusion-minimal solutions, then weights as large as 2Ω(n) are necessary.
Publication series
Name Leibniz International Proceedings in Informatics, LIPIcs
Volume 202
ISSN (Print) 1868-8969
Conference 46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021
Country/Territory Estonia
City Tallinn
Period 23/08/21 → 27/08/21
• Compression
• Constraint satisfaction problems
• Edge-weighted clique
• Kernelization
ASJC Scopus subject areas
Dive into the research topics of 'On the Hardness of Compressing Weights'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/on-the-hardness-of-compressing-weights","timestamp":"2024-11-13T12:46:18Z","content_type":"text/html","content_length":"60134","record_id":"<urn:uuid:11d3e919-d19d-4fee-a2bb-affdd38a92eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00866.warc.gz"} |
What is a Value in Maths ⭐ Definition, Examples, Charts
Value – Definition, Examples, Charts
Updated on January 13, 2024
Welcome to another exciting journey with Brighterly, where we light up the world of mathematics for our young learners! Today, we dive into the fascinating world of Value. Value is not just a simple
term or a mere number. It’s a powerful concept that forms the backbone of not just mathematics, but also various other fields like economics, finance, and more. It’s the numerical essence that helps
us quantify and understand the world around us.
In this enlightening post by Brighterly, we will unravel the mysteries of value, making them as clear as a sunny day. We will explore its different aspects, each unique yet interconnected. We will
journey from the basics to more complex applications, all the while keeping things fun and engaging. Our path will be marked with detailed examples, visually appealing charts, and hands-on activities
to make the learning experience more interactive and impactful.
So, strap in and get ready for this mathematical adventure with Brighterly. By the end of this journey, you will not only understand what value is but also appreciate its significance and application
in everyday life!
What is Value?
In mathematics, value refers to the worth or numerical representation of an object, quantity, or expression. It can be a simple numeric figure, such as the number “5,” or a more complex calculation,
like the result of 2+3. The value is the ultimate outcome of an operation or the distinctiveness of a number. For example, in the equation 2+3=5, “5” is the value. Value can also be seen in the form
of measurements, such as the length of a string or the volume of a water bottle. It’s the backbone of mathematics, underpinning everything from basic arithmetic to complex calculus.
Place Value
Place value is a positional system of notation that uses base ten. Each digit in a number has a specific value, depending on its position. For example, in the number 345, the value of ‘3’ is 300, ‘4’
is 40, and ‘5’ is 5. The value of each digit increases by a factor of ten as we move from right to left. It’s the cornerstone of our number system and allows us to write numbers efficiently.
Face Value
Face value is the value that a digit has in its own right, without considering its place in the number. For instance, in the number 345, the face value of ‘3’ is 3, ‘4’ is 4, and ‘5’ is 5. It’s an
inherent characteristic of a digit and remains constant, irrespective of its position.
Value is the essence of numbers and mathematical operations. It’s the backbone of our understanding of the mathematical world. Whether it’s a simple addition or subtraction, a complex equation, or a
geometric figure’s area, everything comes down to value. It’s the final outcome of any operation or the inherent worth of a number or quantity.
Value Table
A value table is a structured way to understand the value of different numbers or expressions. It presents the data in rows and columns, where each cell represents a unique value. For example, a
value table for the function y=x^2 might look like this:
This table shows how the value of y changes with different values of x.
Examples on Value
Let’s look at some examples to understand the concept of value better.
1. Value in Addition: In the equation 3+2=5, the value of the entire expression is 5.
2. Place Value: In the number 678, the place value of ‘6’ is 600, ‘7’ is 70, and ‘8’ is 8.
3. Face Value: In the number 678, the face value of ‘6’ is 6,
4. ‘7’ is 7, and ‘8’ is 8, irrespective of their positions in the number.
Value Table: A value table for the function y=2x+1 would look like this:
This table shows how the value of y changes with different values of x.
Practice Questions on Value
To consolidate your understanding of value, here are some practice questions:
1. What is the value of the expression 5+7-2?
2. What is the place value of ‘4’ in the number 3549?
3. What is the face value of ‘9’ in the number 9876?
4. Create a value table for the function y=3x+2 for x=1,2,3.
Our fascinating exploration of Value has now come to an end, but the journey with Brighterly continues. Value is more than a fundamental concept in mathematics. It’s a tool that shapes our
understanding of the world around us. It governs everything we do in mathematics, from the simple addition of fruits in a basket to the complex equations that explain the mysteries of the universe.
Understanding the concept of value, along with its different aspects like place value and face value, is like holding a golden key. This key opens the door to the immense beauty of mathematics and
its countless applications in real-world situations. It forms the bedrock of our number system and lays the foundation for understanding more complex mathematical concepts.
By learning about value, we have not only gained knowledge but also grown in our mathematical confidence. We can now look at numbers and equations not as challenges but as friends that guide us in
our everyday life.
As we conclude this topic, remember that every concept you learn with Brighterly is a stepping stone towards becoming a confident and curious learner. The world of mathematics is vast and full of
exciting things to discover. So, keep learning, keep exploring, and remember, with Brighterly, math is not just easy, it’s also fun!
Frequently Asked Questions on Value
What is the difference between place value and face value?
Place value and face value are two fundamental concepts in our number system. Place value is the value of a digit as determined by its place or position in a number. It changes as the position of the
digit changes. For example, in the number 654, the place value of 6 is 600, of 5 is 50, and of 4 is 4. Conversely, the face value is the value a digit inherently holds, irrespective of its position
in the number. It’s constant and doesn’t change with the position. In the same number 654, the face value of 6 is 6, of 5 is 5, and of 4 is 4.
How is value used in mathematics?
Value is a critical component in mathematics. It denotes the numerical worth of an object, number, or expression. From simple arithmetic operations, like addition and subtraction, to complex
algebraic equations and geometrical calculations, everything boils down to finding the value. It is the ultimate outcome of any mathematical operation. For example, in the equation 5 + 3 = 8, 8 is
the value. Likewise, in finding the area of a square with a side of 4 units, the value is 16 square units.
What is a value table?
A value table, also known as a function table, is a structured method to display how the value of a variable or expression changes with different inputs. It’s presented in rows and columns, where
each cell represents a unique value. A value table is especially useful when dealing with mathematical functions or expressions involving more than one variable. It gives a clear picture of how
changing one variable affects the value of the entire expression or function.
For more information, feel free to check these resources:
This concludes our exploration of the concept of value. Keep practicing, keep exploring, and remember – math is fun!
Poor Level
Weak math proficiency can lead to academic struggles, limited college, and career options, and diminished self-confidence.
Mediocre Level
Weak math proficiency can lead to academic struggles, limited college, and career options, and diminished self-confidence.
Needs Improvement
Start practicing math regularly to avoid your child`s math scores dropping to C or even D.
High Potential
It's important to continue building math proficiency to make sure your child outperforms peers at school. | {"url":"https://brighterly.com/math/value/","timestamp":"2024-11-02T18:37:00Z","content_type":"text/html","content_length":"96491","record_id":"<urn:uuid:4752048b-b3f5-48e3-9bcc-60612f1659d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00101.warc.gz"} |
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steepness. The geometry is reminiscent of the geometry of lines through the origin in three-space, which we considered in Chapter 7. In Euclidean geometry, if we start with a point A and a line l,
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Collapse in Riemannian geometry is the phenomenon of injectivity radii limiting to zero, while sectional curvatures remain bounded. Do you want to download or read a book? Triangle Basics. File: PDF,
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assumptions, and proofs that describe such objects as points, lines and planes. Orbital mechanics is a modern offshoot of celestial mechanics which is the study of the motions of natural celestial
bodies such as the moon and planets. For comets and planets, the sun is located at one focus of their elliptical orbits. Rule: O is a point on every … ..... 43 20 Comparison of surface velocity
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rational solutions to this equation has an extremely interesting structure, including a group law. Topic: Geometry, Sphere. Why is the animation not perfect? elliptical to conical and from as small
as a pinhead to as large asa house. Language: english. 2010 Mathematics Subject Classification: Primary: 33E05 [][] An integral of an algebraic function of the first kind, that is, an integral of the
form $$ \tag{1 } \int\limits _ { z _ {0} } ^ { {z _ 1 } } R ( z , w ) d z , $$ where $ R ( z , w ) $ is a rational function of the variables $ z $ and $ w $. Geometry For Dummies Mark Ryan. Once you
measure the width of the oval, divide this value by 2. The vertical scale of the thickness form plots has been enlarged for clarity. Spherical geometry is the study of geometric objects located on
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surface of a sphere. It is a more honest way of showing the effect. Please login to your account first; Need help? The Geometry of Elliptic Curves Vertical Lines and the Extra Point \At Inflnity" E 6
L O P Q = ¡P v v Create an extra point O on E lying at \inflnity" Solution: Since there is no point in the plane that works, we create an extra point O \at inflnity." Pillai "Simple Pendulum and
Elliptic Integral Corrections - Landau's Take" - Duration: 18:53. Spherical geometry is nearly as old as Euclidean geometry. Rather strangely, the perimeter of an ellipse is very difficult to
calculate, so I created a special page for the subject: read Perimeter of an Ellipse for more details. Because it is a computer model. The thickness/chord ratio, to/c =0.1. Most generally, gear teeth
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geometry on 5 basic rules, or axioms. The original gear teeth were wooden pegs driven into the periphery of wooden wheels and driven by other wooden The other good features of the lemniscate integral
are the fact that it is general enough for many of its properties to be generalised to more general elliptic functions, yet the geometric intuition from the arc length of the lemniscate curve aids
understanding. Spherical geometry—which is sort of plane geometry warped onto the surface of a sphere—is one example of a non-Euclidean geometry. They are used to provide positive transmis-sion of
both motion and power. The orbits of the planets and their moons are ellipses with very low eccentricities, which is to say they are nearly circular. Measure the width of the oval across its
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in Euclidean geometry the sum of the interior angles of a triangle is 180°, in non-Euclidean geometry this is not the case. For instance, a "line" between two points on a sphere is actually a great
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antipodal points on the sphere represents the same point, and we only pay attention to the one lying in the southern hemisphere. Elliptic geometry, a type of non-Euclidean geometry, studies the
geometry of spherical surfaces, like the earth. A first-time student of geometry, studies the geometry is what you 're to. Better suited for drafting or performing geometry been made to give a
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How To Find The P-Value In A Z-Test
suman bhaumik/iStock/GettyImages
A z-test is a test of the standard normal distribution, a bell-shaped curve with a mean of 0 and a standard deviation of 1. These tests arise in many statistical procedures. A P-value is a
measure of the statistical significance of a statistical result. Statistical significance addresses the question: "If, in the entire population from which this sample was drawn, the parameter
estimate was 0, how likely are results as extreme as this or more extreme?" That is, it provides a basis to determine whether an observation of a sample is merely the result of random chance (that
is, to accept the null hypothesis) or whether a study intervention has in fact produced a genuine effect (that is, to reject the null hypothesis).
Although you can calculate the P-value of a z-score by hand, the formula is extremely complex. Luckily, you can use a spreadsheet application to carry out your calculations instead.
Step 1: Enter the Z-Score Into Your Program
Open the spreadsheet program and enter the z-score from the z-test in cell A1. For example, suppose you compare the heights of men to the height of women in a sample of college students. If you
do the test by subtracting women's heights from men's heights, you might have a z-score of 2.5. If, on the other hand, you subtract men's heights from women's heights, you might have a z-score of
−2.5. These are, for analytical purposes, equivalent.
Step 2: Set the Level of Significance
Decide if you want the P-value to be higher than this z-score or lower than this z-score. The higher the absolute values of these numbers, the more likely your results are statistically
significant. If your z-score is negative, you almost certainly want a more negative P-value;if it is positive, you almost certainly want a more positive P-value.
Step 3: Calculate the P-value
In cell B1, enter =NORM.S.DIST(A1, FALSE) if you want the P-value of this score or lower; enter =NORM.S.DIST(A1, TRUE) if you want the P-value of this score or higher.
For example, if you subtracted the women's heights from the men's and got z = 2.5, enter =NORM.S.DIST(A1, FALSE); you should get 0.0175. This means that if the average height of all college men
were the same as the average height of all college women, the chance of getting this high a z-score in a sample is only 0.0175, or 1.75 percent.
TL;DR (Too Long; Didn't Read)
You can also calculate these in R, SAS, SPSS or on some scientific calculators.
Cite This Article
Flom, Peter. "How To Find The P-Value In A Z-Test" sciencing.com, https://www.sciencing.com/pvalue-ztest-8597730/. 8 December 2020.
Flom, Peter. (2020, December 8). How To Find The P-Value In A Z-Test. sciencing.com. Retrieved from https://www.sciencing.com/pvalue-ztest-8597730/
Flom, Peter. How To Find The P-Value In A Z-Test last modified August 30, 2022. https://www.sciencing.com/pvalue-ztest-8597730/ | {"url":"https://www.sciencing.com:443/pvalue-ztest-8597730/","timestamp":"2024-11-13T19:21:44Z","content_type":"application/xhtml+xml","content_length":"72605","record_id":"<urn:uuid:4310aa32-2b6b-43c1-ada9-b93b70d5f012>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00314.warc.gz"} |
When resolving a placeholder for a deduced class type (
) where the
names a primary class template
, a set of functions and function templates, called the guides of
, is formed comprising:
In addition, if
is defined and its definition satisfies the conditions for an aggregate class (
) with the assumption that any dependent base class has no virtual functions and no virtual base classes, and the initializer is a non-empty
or parenthesized
, and there are no
, the set contains an additional function template, called the
aggregate deduction candidate
, defined as follows
For each
, let
be the corresponding aggregate element of
or of one of its (possibly recursive) subaggregates that would be initialized by
) if
• brace elision is not considered for any aggregate element that has a dependent non-array type or an array type with a value-dependent bound, and
• each non-trailing aggregate element that is a pack expansion is assumed to correspond to no elements of the initializer list, and
• a trailing aggregate element that is a pack expansion is assumed to correspond to all remaining elements of the initializer list (if any).
If there is no such aggregate element
for any
, the aggregate deduction candidate is not added to the set
The aggregate deduction candidate is derived as above from a hypothetical constructor
, where
• if is of array type and is a braced-init-list or string-literal, is an rvalue reference to the declared type of , and
• otherwise, is the declared type of ,
except that additional parameter packs of the form
are inserted into the parameter list in their original aggregate element position corresponding to each non-trailing aggregate element of type
that was skipped because it was a parameter pack, and the trailing sequence of parameters corresponding to a trailing aggregate element that is a pack expansion (if any) is replaced by a single
parameter of the form | {"url":"https://timsong-cpp.github.io/cppwp/n4868/over.match.class.deduct","timestamp":"2024-11-06T18:02:28Z","content_type":"text/html","content_length":"66836","record_id":"<urn:uuid:c16ca985-e7f8-4b24-9797-623dda0a2e68>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00101.warc.gz"} |
Griffiths Singularity in the Random Ising Ferromagnet
The explicit form of the Griffiths singularity in the random ferromagnetic Ising model in external magnetic field is derived. In terms of the continuous random temperature Ginzburg-Landau Hamiltonian
it is shown that in the paramagnetic phase away from the critical point the free energy as the function of the external magnetic field h in the limit h → 0 has the essential singularity of the form
exp [‑(const)/h^D/3] (where 1 < D < 4 is the space dimensionality). It is demonstrated that in terms of the replica formalism this contribution to the free energy comes due to non-perturbative
replica instanton excitations.
Journal of Statistical Physics
Pub Date:
January 2006
□ Quenched disorder;
□ instantons;
□ replicas;
□ mean-field;
□ non-analytic function;
□ non-perturbative states;
□ Condensed Matter - Disordered Systems and Neural Networks;
□ Condensed Matter - Statistical Mechanics
14 pages, 3 figures | {"url":"https://ui.adsabs.harvard.edu/abs/2006JSP...122..197D","timestamp":"2024-11-10T22:20:06Z","content_type":"text/html","content_length":"37961","record_id":"<urn:uuid:ffb76ba1-ce64-4ec0-9f88-5fd0ffea139e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00667.warc.gz"} |
Data + Science -
Bin There, Done That: A Discussion of Bins in Tableau
Creating bins in Tableau seems pretty straight forward, right-click on a Measure and select Create -> Bin and you’re done. However, there are some restrictions and pitfalls that are very important to
understand. In this post, I will explain these and offer a few solutions, including a hidden function in Tableau that I think you’ll find very handy.
I’m going to be referencing Jonathan Drummey a number of times in this blog post, but what else is new. More often than not, things in Tableau connect their way back to Jonathan Drummey and Joe Mako.
There are a few good posts on the subject of bins as well as a great Tableau workbook.
First, let’s start with this blog post from Jonathan about bins.
One of the big restrictions with bins is that when you create a Histogram using the bins, you are not able to add a reference line. Jonathan and Joe offer their own equations that allow you to create
your own bins, thus using a calculated field as a bin, and therefore you can add reference lines to the histogram.
These calculations are good solutions, and even though this blog post is from 2014, these solutions can still be used today.
Nick Hara wrote this excellent blog post called “I’ve Bin Everywhere” (January 2019), where he shows a number of examples and provides this great Tableau Workbook available for download.
Let’s explore some things about bins.
Be Careful of Floating Points
One of the pitfalls in using the default bins in Tableau occurs when binning decimal values. This is because of floating point issues when dealing with the arithmitic of the Floor() or Ceiling()
functions in Tableau. This issue is outlined in this knowlege base article by Tableau, Histograms Display Decimal Values in Incorrect Bins. In short, multiply the decimal values by 100 or 1,000 in a
calculation to remove the decimal places, then bin that calculated field. Note, this is what Nick does to solve this issue in his workbook mentioned above.
As an example, here’s what the bins look like in Tableau when I take 100 numbers from 0.01 to 1.00, in increments of 0.01 and bin them with the bin size of 0.05. We’d expect these to be fairly even
bins, but they are not.
Now I will take that same field and multiply it by 100 and create a bin using that calculated field with bin size 5. When we do this we get the bin distribution that we would expect to see.
Ok, so that’s pretty straight forward, but there are other issues that we can encounter. Let’s look at how these custom calculated fields for bins distribute the values in these bins.
Be Careful of Bin Distribution when Using Custom Calculations
Nick originally created his binning solution for something at work. For his blog post, he ported that solution over using the World Indicators data from Tableau Desktop, specifically the field
Population Urban, which is a decimal number representing the percent of the population that is urban in each country over several years. This turns out to be a great use case for this discussion,
because we encounter both issues, the floating point issue and issues with how the bins are distributed with these custom calculations.
Let’s step through this and work our way to a final solution. While this section may look long, very detailed and complicated, I promise that the final solution is pretty easy and straight forward
(and we’ll learn a hidden function in Tableau too).
I modified Nick’s workbook to show the values that comprise his histogram. Below is the histogram and the individual values in a stem and leaf plot (cut off at ~30 rows).
You may notice some issues with the binning right away. First, the bin starts at 0 to 5 and the values that within that first bin are actually between 5 and 10. Second, and more concerning, there are
values in the second bin that should be grouped with the first bin. For example, the value 0.0910 should be grouped with the other values that are between 5 and 10.
Nick’s formula for his calculated field:
Calculated Field: Population Bin (Param)
([Urban Bin]-( CEILING([Population Urban]*100)%[Urban Bin])) + CEILING([Population Urban]*100 ) – [Urban Bin]
Note – [Urban Bin] is a parameter that Nick created to change the bin size
Let’s make two minor changes to Nick’s formula. First, we will change the CEILING function to a FLOOR. Second, we will add 5 to the value of the bins to slide them over one bin value, which in this
case simply means we remove the -[Urban Bin] from the equation.
Calculated Field: Population Bin (Param) REVISED
([Urban Bin]-(FLOOR([Population Urban]*100)%[Urban Bin])) + FLOOR([Population Urban]*100)
Our revised formula creates these bins. This is actually correct and works as a final solution.
However, if we check these bins against the default Tableau bins it won’t match. That’s because th Tableau default bins are encountering the floating point issue described earlier.
If we create a caluclated field and multiply it by 100 and create a bin using that new calculated field then it solves this issue and the bins match the REVISED calculation above.
The default bins work great, but unfortunately we can’t add a reference line to the default bins because they are Discrete. We’ve now come full circle. We’ve corrected the calculation formula to
create a continuous bin and we’ve corrected the default bins. Both are correct now, but the continuous bins created by the calculated field is the only one that allows the reference line we are
looking for.
Using a Hidden Function in Tableau: SYS_NUMBIN()
I can’t take any credit for finding this hidden function. Back in June 2017, Jonathan Drummey came out to our office for some hands on Tableau training with our small team of Tableau developers and
data scientists. In one of our many conversations over those two days, Jonathan mentioned running across this function. As I recall, he was working in Tableau, he encountered some sort of issue and a
strange error box popped up on him. He saw this function and decided to give it a try, and sure enough it worked in the calculation window. We talked briefly about how we might use this, but we moved
one, time passed and every so often I revisit it. Well, as it turns out it’s a great solution for this particular problem and is super simple to use.
The syntax is SYS_NUMBIN([Measure], [Bin Size]). In this case, it would be SYS_NUMBIN([Urban Population], 0.05), but as we’ve learned that will be problematic. Also, this function will create integer
bins starting at 0, then 1, etc. So we’ll make a few minor adjustments. To solve the floating point issue we’ll use the field that multiplies the Measure by 100.
To change the bins from [0,1,2…] to [5,10,15] we will multiple the bins by the bin size and then add the bin size. For example, (0*5) + 5 = 5 for the first bin. (1*5) + 5 = 10 for the second bin, and
so on. To match the bins that Tableau would create automatically it would look like this: (SYS_NUMBIN([Measure], [Bin Size]) * [Bin Size]) + [Bin Size]. In this case, the new formula would be.
Calculated Field: Population Urban * 100 (SYS_NUMBIN)
(SYS_NUMBIN([Population Urban * 100],[Urban Bin]) * [Urban Bin]) + [Urban Bin]
In addition, we can use the SYS_NUMBIN() function to simply some of the equations that Nick created. For example, he had a great example of distribution with a tail and variable width. His formula
looks like this:
Calculated Field: Health Bin Tail (Variable)
IF [Health Exp/Capita]>=[Health Threshold] THEN [Health Threshold]
IF [Health Bin]>=[Health Exp/Capita] THEN
(([Health Bin]*.1)-(CEILING([Health Exp/Capita])%([Health Bin]*.1)))
+ CEILING([Health Exp/Capita])-[Health Bin]*.1
ELSE ([Health Bin]-(CEILING([Health Exp/Capita])%[Health Bin]))
+ CEILING([Health Exp/Capita])-[Health Bin]
Using the SYS_NUMBIN() function we can simply this to:
Calculated Field: Health Bin Tail (Variable) (SYS_NUMBIN)
IF [Health Exp/Capita]>=[Health Threshold] THEN [Health Threshold]
IF [Health Bin]>=[Health Exp/Capita]
THEN (SYS_NUMBIN([Health Exp/Capita],[Health Bin]*.1) * [Health Bin]*.1)
ELSE (SYS_NUMBIN([Health Exp/Capita],[Health Bin]) * [Health Bin])
This produces the exact same distribution:
Other Notes of SYS_NUMBIN()
Here are just a few other things we can do with continuous bins in Tableau.
Having a continuous bin was the goal in this particular case, because we wanted to add a reference line, but you could also add a reference band, distribution band or box plot to you binned
Because it’s a custom calculation, you can use the bin in a calculated field. For example, you could combine with parameter actions, set actions or other calculations to assign color to the bins.
Easily set the width of the bins using other rules. This isn’t anything new. People have been doing this for years, but it’s just a little easier now. For example, the Freedman-Diaconis rule for
determining bin width for a continuous variable is [2 * (IQR/n^(1/3))] where IQR is the interquartile range and n is the number of records. We can easily create a calculated field using the
Freedman-Diaconis rule and then we can just drop that in the SYS_NUMBIN() function for the bin size.
Calculated Field: Freedman-Diaconis Bin Size
{ 2 * ( (PERCENTILE([Population Urban * 100], 0.75) – PERCENTILE([Population Urban * 100], 0.25)) / POWER(count([Number of Records]), 1/3) ) }
Calculated Field: Population Urban * 100 (SYS_NUMBIN with FD)
(SYS_NUMBIN([Population Urban * 100],[Freedman-Diaconis Bin Size]) * [Freedman-Diaconis Bin Size]) + [Freedman-Diaconis Bin Size]
There are number of formulas that could be used to determine the width of the bins (see more examples here). Note – The formula that Tableau uses by default to determine the number of bins is Number
of Bins = 3 + log2(n) * log(n) where n is the number of distinct rows in the table. The size of each bin is determined by dividing the difference between the smallest and the largest values by the
number of bins.
Note – It seems that SYS_NUMBIN(), even when used as a discrete dimension, does not turn on data densification like a typical bin would. This makes sense, because we’ve converted the pill type from a
bin to a discrete dimension and the later doesn’t turn on data densification (read more about Data Densification Using Domain Completion and Domain Padding here).
None of these things use cases are new, since we could use other equations (from Jonathan Drummey, Joe Mako, Nick Hara and others) to create continuous bins. However, I find the SYS_NUMBIN() function
is much easier to remember, so I hope you find this hidden function and this information useful when creating bins in Tableau. Below is a Tableau Public Visualization with all of these examples that
you can download and explore. | {"url":"https://digitalinfowave.com/data-science-12/","timestamp":"2024-11-08T01:16:44Z","content_type":"text/html","content_length":"97517","record_id":"<urn:uuid:ab8fbb13-1edf-4bdc-8e80-4e55c50b4a1e>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00163.warc.gz"} |
EXAFS - Theory
Last updated
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EXAFS (Extended X-ray Absorption Fine Structure) and XANES (X-Ray Absorption Near Edge structure) are regions of the spectrum obtained from XAS(X-ray Absorption Spectroscopy). EXAFS corresponds to
the oscillating part of the spectrum to the right of the absorption edge(appearing as a sudden, sharp peak), starting at roughly 50 eV and extending to about 1000 eV above the edge (shown in Figure
1). Through mathematical analysis of this region, one can obtain local structural information for the atom in question.
The ability of EXAFS to provide information about an atom's local environment has widespread application, particularly to the geometric analysis of amorphous crystalline solids. Over time, EXAFS has
become more applicable to quantitative analysis of noncrystalline materials. When analyzing a single atom within a material, properties analyzed include coordination number, disorder of neighboring
atoms, and distance of neighboring atoms. Of these three properties, radial distance is the only property reliably measured. Theoretically obtained structural information becomes more accurate the
further down the EXAFS region, ~0.02 Angstroms or better. Experimentally it has also been shown that application of the EXAFS technique is most accurate for systems of low thermal or static disorder.
Qualitative understanding of the EXAFS region
X-Ray spectroscopy involves a process in which an X-ray beam is applied to an atom and causes the ejection of an electron, usually a core electron. This leaves a vacancy in the core shell, and
results in relaxation of an outer electron to fill that vacancy. This phenomenon is only observed when the energy of the X-ray exceeds the ionization energy of the electrons in that shell. We can
relate this occurrence to the X-ray Absorption coefficient, which becomes the basis of EXAFS theory. The X-ray Absorption coefficient, or μ, describes the relationship between the intensity of an
X-ray beam going into a sample and its intensity leaving the sample after traveling a distance x within the sample. The absorption coefficient is given by
\[\mu =-\frac{d lnI}{dx}\]
In this expression, dx is the traversed distance, and I is the intensity. In a typical EXAFS spectrum, various sharp peaks will be observed when energy(usually in eV) is plotted against absorbance.
These peaks(called edges), which vary by atom, correspond to the ionization of a core orbital, K-edges describing the excitation of the innermost 1s electron, and L-edges and M-edges referring to the
same for higher energy orbitals. A qualitative diagram of these energies, as well as their dependence on atomic number, is shown in Figure 2 below.
After each edge, a series of downward oscillations will be observed. These oscillations correspond to wave interactions between the ejected photoelectron and electrons surrounding the absorbing atom.
The neighboring atoms are called backscattering atoms, since the waves emitted from the absorbing atom change paths when they hit these neighboring atoms, returning to the original atom. Maxima in
the oscillations result from constructive interference between these waves, and minima result from destructive interference. These oscillations are also characteristic of the surrounding atoms and
their distances from the central atom, since varying distances will result in different backscattering paths, and as a result different wave interactions. The EXAFS fine structure begins at roughly
30 eV past each edge, where oscillations begin to decay. In addition, this wave interaction will depend on the mechanism of scattering, since the path taken by a wave sometimes involves collision
with an intermediate atom, or even multiple atoms, before it reverts to the absorbing atom.
Figure 3 effectively displays the phase shift that should occur (opposed to the diagram where E=E1) (the red arcs represent the original diagram and the blue arcs represent the phase shift).
Extraction of Data using the EXAFS Equation
To represent the EXAFS region from the whole spectrum, a function χ can be roughly defined in terms of the absorption coefficient, as a function of energy:
\[\chi (E)=\frac{\mu (E)-\mu _{o}(E)}{\Delta \mu _{o}}\]
Here, the subtracted term represents removal of the background and the division by Δμo represents the normalization of the function, for which the normalization function is approximated to be the
sudden increase in the absorption coefficient at the edge. When interpreting data for the EXAFS, it is general practice to use the photoelectron wave vector, k, which is an independent variable that
is proportional to momentum rather than energy.
We can solve for k by first assuming that the photon energy E will be greater than E[0] (the initial X-ray absorption energy at the edge). Since energy is conserved, excess energy given by E - E[0]
is conserved by being converted into the kinetic energy of the photoelectron wave. Since wavelengths are dependent on kinetic energies, the photoelectron wave (de Broglie wavelength) will propagate
through the EXAFS region with a velocity of ν where the wavelength of the photoelectron will be scanned. This gives the relation, (E - E[0]) = m[e]ν^2/2. One of the identities for the de Broglie
wavelength is that it is inversely proportional to the photoelectrons momentum (m[e]ν): λ=h/m[e]ν. Using simple algebraic manipulations, we are able to obtain the following:
\[ k = \dfrac{2\pi}{\lambda} = \dfrac{2\pi m_{e}\nu}{h} = \left[\dfrac{8\pi^2m_{e}(E-E_o)}{h^2}\right]^{\frac{1}{2}} \]
To amplify the oscillations graphically, k is often plotted as k^3
Now that we have an expression for k, we begin to develop the EXAFS equation by using Fermi's Golden Rule. This rule states that the absorption coefficient is proportional to the square of the
transition moment integral, or |<i|H|f>|^2 , where i is the unaffected core energy level before it interferes with the neighboring atoms, H is the interaction, and f is the final state in which the
core energy level has been affected and a photoelectron has been ejected. This integral can be related to the total wavefunction Ψ[k][, ]which represents the sum of all interacting waves, from the
backscattering atoms and the absorbing atom. The integral is proportional to the total wavefunction squared, which refers to the probability that the photoelectron is found at the atom where the
photon is absorbed, as a function of radius. This wavefunction describes the constructive/destructive nature of the wave interactions within it, and varies depending on the phase difference of the
waves interacting. This phase difference can be easily expressed in terms of our photoelectron wave number and R, the distance to the innershell from the wave, as 2k/R. In addition, another
characteristic of this wave interaction is the amplitude of waves backscattering to the central atom. This amplitude is can provide coordination number as well since it is thought to be directly
proportional to the number of scatters.
The physical description of all these properties is given in a final function for χ(k), called the EXAFS equation:
\[\chi (k)=\sum_{j}\frac{N_{j}f_{j}(k)exp[-2k^{2}\sigma _{j}^{2}]exp[-2R_{j}/\lambda ]}{kR_{j}^{2}}sin[2kR_{j}+\delta _{j}(k)]\]
Looking at the qualitative relationships between the contributions to this equation gives an understanding of how certain factors can be extracted from this equation. In this equation, f(k)
represents the amplitude and δ(k) represents the phase shift. Since these two parameters can be calculated for a certain k value, R, N, and σ are our remaining unknowns. These unknown values
represent the information we can obtain from this equation: radius, coordination number(number of scattering atoms), and the measure of disorder in neighboring atoms, respectively. These are all also
properties of the scattering atom.
We can also see the terms in this equation represented in a typical EXAFS spectrum: the sine term gives the origin of the sinusoid shape, with a greater phase shift making the oscillations greater.
In addition, this oscillation depends on the energy and on the radial distance. The Debye-Waller Factor explains the decay with increasing energy, as well as increasing disorder. This factor is
partially due to thermal effects. In addition, we deduce the reason why EXAFS does not work over long distances (up to 4-5 Å): the term Rj^-2 causes the expression to decrease exponentially over
large values of Rj (larger distances between absorbing and scattering atoms), making EXAFS much weaker over long-distances as opposed to short ranged neighboring atoms.
The last step in order to complete the data extraction is to take a Fourier transform of this expression into frequency space, which results in a radial distribution function where peaks correspond
to the most likely distances of the nearest-neighbors.
Multiple Scattering
The cases thus far have only dealt with single scattering pathways. Figure 4 shows a scenario in which two scattering atoms (denoted by s) are present with one photoelectron wave source. In this
case, what would happen? There is a clear dependence on the distance and angles of the scattering atoms. Thus, it is no longer a simple matter of 1-D space distance relations, but rather a 2-D polar
coordinate system. In the case of Cu-Cu (shown by Kroneck et al) the copper is not only the absorbing atom, it is also the scattering atom. Given this paradox, what would happen to the amplitude,
phase, and frequency with respect to the two coppers, and with respect to its neighboring atoms?
The effect of multiple scattering is a powerful tool, which is used to determine a variety of information on the local structure. Each individual atom will produce more scattering or absorbing atoms
depending on the type of atom and what type of wave hits it. The angle at which the wave scatters back and its distance from the scattering atom will also affect the overall intensity of the EXAFS
spectrum. In scenario (a), shown in figure 4, the absorbing atom is hit by an X-ray where it produces an absorbing wave which will scatter off S1 and hit S2. It will then send a scattering wave back
to (a), which allows us to obtain further information on the local structure of this molecule. However, this will only work if the absorbing atom is within 4-5 Å away from the scattering atoms.
Otherwise, the wave becomes very dampened and the information retrieved becomes very unreliable. In short, multiple scattering effects can be observed, but for the most part, they will only affect
the EXAFS spectrum a miniscule (but noticeable) amount. When an absorbing wave goes through another absorbing atom as mentioned in Kroneck et al, it will produce a dampened absorbing wave (due to
destructive interference) and give a lower signal than normal. kroneck et al. tested this on two types of systems, the first was a copper single scattering onto a neighboring atom, the second was two
coppers scattering onto a scattering atom (all lined up linearly). When both of the radial distances were collected, the second test showed only a small difference (0.02 Å) compared to the first
test, suggesting experimentally that multiple scattering waves do have an effect on the determination of a molecular system.
1. Que, L (2000). Physical Methods in Bioinorganic Chemistry. Sasuolito, California: University Science Books.
2. Kroneck, P. M. H., Antholine; W. E.; Kastrau, D. H. W.; Buse. G.; Steffens, G. C. M.; Zumft, W. G., "Multifrequency EPR Evidence for a Bimetallic Centre at the CuA site in Cytochrome c Oxidase."
Eur. J. Biochem. 1992, 209, 875-881
3. Farrar, J. A.; Lappalainen, P.; Zumft, W. G.; Saraste, M.; Thomson, A. J., "Spectroscopic and Mutagenesis Studies on the CuA centre from the Cytochrome-c Oxidase complex of Paracoccus
denitrificans." Eur. J. Biochem. 1996, 232, 294-303
4. J.J. Rehr and R.C. Albers, "Theoretical approaches to X-ray absorption fine structure", Reviews of Modern Physics 72 (2000), 621-654
5. C.A. Ashley and S. Doniach, "Theory of Extended X-ray Absorption Fine Structure (EXAFS) in Crystalline Solids", Physical Review B. 11, 1279-1288.
Consider the crystal lattice zinc sulfide, whose unit cell is pictured below.
Zn = Purple, S = Blue.
1. An X-Ray Beam is applied to a sample of ZnS, and excites a core sulfur electron. Sketch the raw spectrum you might expect to see, and indicate where the EXAFS/XANES regions may occur. Be sure to
indicate where the K-edge is, and include units.
2. For the phenomenon described, explain what happens to the energy associated with the ejection of the photoelectron?
3. Upon completing data extraction, and transforming the final EXAFS equation into frequency space, describe what you would expect the radial distribution to look like for ZnS.
4. How would the amplitude for the oscillations of a Zn connected to a single sulfur compare to the amplitude for the oscillations of a Zn from ZnS?
5. When a photoelectron is emitted, spherical waves are produced. Redraw this unit cell, and sketch two possible paths these waves may take, taking into account nearest-neighbor atoms. Label the
absorbing atom, backscattering atom, and intermediate atom, if applicable, in each diagram.
1. Sulfur's K-edge occurs at about 2.472 keV(this can be found using online sources). If energy is plotted against the absorption coefficient, we would expect to see a constant flat line, then a
sudden, sharp increase at roughly 2.472 keV. After this peak the spectrum would begin to decay, and there would be a few slight peaks(XANES), then steady downward oscillations(EXAFS).
2. When the core sulfur electron is excited, this means the energy of the photon from the X-ray source exceeds the bonding energy of the electron. Its energy is consumed, and the electron becomes
excited. This causes the ejection of the energy as a photoelectron wave.
3. There would be expected to be a series of peaks, with four being prominent. These correspond to the four radial distances more likely to be occupied around the sulfur atom. Since in this unit
cell each sulfur atom's coordination number is four, we would expect to see four main peaks, though in practice other peaks would probably interfere.
4. Amplitude of backscattering is believed to be proportional to coordination number. Therefore, we would expect the amplitude of the oscillations to be about four times as high for this structure
than for a single Zn-S bond. | {"url":"https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Spectroscopy/X-ray_Spectroscopy/EXAFS_-_Theory","timestamp":"2024-11-02T07:22:48Z","content_type":"text/html","content_length":"144436","record_id":"<urn:uuid:838a93b7-f3a8-4cc8-b2e7-66538235a67d>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00596.warc.gz"} |
I. General Information
1. Course Title:
2. Course Prefix & Number:
PHIL 1460
3. Course Credits and Contact Hours:
Credits: 3
Lecture Hours: 3
4. Course Description:
This course is an introduction to the basic concepts, principles, and methods of argument analysis and evaluation, including deductive and inductive reasoning, validity, soundness, truth tables,
Aristotelian logic, Venn diagrams, indirect deductive proofs, and principles of induction.
5. Placement Tests Required:
6. Prerequisite Courses:
PHIL 1460 - Logic
There are no prerequisites for this course.
9. Co-requisite Courses:
PHIL 1460 - Logic
There are no corequisites for this course. | {"url":"https://catalognavigator.clcmn.edu/Catalog/ViewCatalog.aspx?pageid=viewcatalog&topicgroupid=2955&entitytype=CID&entityid=965&loaduseredits=True","timestamp":"2024-11-08T16:59:33Z","content_type":"text/html","content_length":"44690","record_id":"<urn:uuid:b721ebff-1e40-42c7-af76-0180fb738e30>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00765.warc.gz"} |
Home page of Ashish K. Srivastava
Book: Cyclic Modules and the Structure of Rings, Oxford Mathematical Monographs, Oxford University Press, 2012 (with S. K. Jain and Askar Tuganbaev).
Published Papers:
35. Schroeder-Bernstein problem for modules, preprint (with Pedro A. Guil Asensio and Berke Kalebogaz).
34. V-rings versus $\Sigma$-$V$ rings, preprint (with B. Davvaz, and Zahra Nazemian).
33. Theory of partial morphisms, preprint (with Pedro A. Guil Asensio and Berke Kalebogaz).
32. Modules invariant under monomorphisms of their envelopes, preprint (with Pedro A. Guil Asensio and D. Keskin Tutuncu)
31. Additive unit structure of endomorphism rings and invariance of modules under endomorphisms of their covers and envelopes, submitted (with Pedro A. Guil Asensio and T. C. Quynh).
30. A note on group algebras of locally compact groups, AMS Contemporary Math, volume in honor of Don Passman, to appear.
29. Additive representations of elements in rings: A survey, Advances in Algebra and its Applications, Springer Proceedings in Mathematics and Statistics, 2016, to appear.
28. Modules which are coinvariant under automorphisms of their projective covers, Journal of Algebra, 2016, to appear (with Pedro A. Guil Asensio, Derya Keskin Tutuncu, Berke Kalebogaz).
27. New characterizations of pseudo-Frobenius rings and a generalization of the FGF conjecture, Israel Journal of Mathematics, 2016, to appear (with Pedro A. Guil Asensio and S. Sahinkaya).
26. Rings with each right ideal automorphism-invariant, Journal of Pure and Applied Algebra, 220, 4 (2015), 1525-1537 (with M. Tamer Kosan and T. C. Quynh).
25. Modules invariant under automorphisms of their covers and envelopes, Israel Journal of Mathematics, 206, 1 (2015), 457-482 (with Pedro A. Guil Asensio, Derya Keskin Tutuncu).
24. Automorphism-invariant modules, Noncommutative rings and their applications, Contemp. Math., Amer. Math. Soc. 634 (2015), 19-30 (with Pedro A. Guil Asensio).
23. Additive unit representations in endomorphism rings and an extension of a result of Dickson and Fuller , Ring Theory and Its Applications, Contemp. Math., Amer. Math. Soc. 609 (2014), 117-121
(with Pedro A. Guil Asensio).
22. Rings of invariant module type and automorphism-invariant modules , Ring Theory and Its Applications, Contemp. Math., Amer. Math. Soc. 609 (2014), 299-311 (with S. Singh).
21. Automorphism-invariant modules satisfy the exchange property , Journal of Algebra, 388 (2013), 101-106 (with Pedro A. Guil Asensio).
20. Decomposing elements of a right self-injective ring , J. Algebra and Appl. 12, 6 (2013), 10 pages (with Feroz Siddique).
19. Rings and modules which are stable under automorphisms of their injective hulls , Journal of Algebra, 379 (2013), 223-229 (with Noyan Er and S. Singh).
18. Cyclic Modules and the Structure of Rings, Oxford Mathematical Monographs, Oxford University Press, 2012 (with S. K. Jain and Askar Tuganbaev).
17. Dual automorphism-invariant modules , Journal of Algebra, 371 (2012), 262-275 (with S. Singh).
16. On $\Sigma$-V Rings, Communications in Algebra, 39, 7 (2011), 2430-2436.
15. Erratum: Generalized Group Algebras of Locally Compact Groups, Communications in Algebra, Volume 38, Issue 10 (2010), page 3974 (with S. K. Jain and A. I. Singh).
14. A Survey of Rings Generated by Units , Annales de la Facult'e des Sciences de Toulouse Math'ematiques, Volume 19, No. S1 (volume in honor of Mel Henriksen), (2010), 203-213.
13. Direct Sums of Injective and Projective Modules , Journal of Algebra, 324 (2010), 1429-1434 (with Pedro A. Guil Asensio and S. K. Jain).
12. On Unit-Central Rings, Advances in Ring Theory, Trends in Mathematics (volume in honor of S. K. Jain), Birkhauser Verlag (2010), 205-212. (with G. Marks and D. Khurana).
11. Shorted Operators Relative to a Partial Order in a Regular Ring , Communications in Algebra, Vol. 37, Number 11 (2009), 4141-4152 (with S. K. Jain, B. Blackwood and K. M. Prasad).
10. On $\Sigma$-q Rings , Journal of Pure and Applied Algebra 213 (2009), 969-976 (with S. K. Jain and S. Singh).
9. Generalized Group Algebras of Locally Compact Groups, Communications in Algebra, 36, 9 (2008), 3559-3563 (with S. K. Jain and A. I. Singh).
8. On multicolour noncomplete Ramsey graphs of star graphs, Discrete Applied Mathematics, 156 (2008), 2423-2428 (with S. Gautam and A. Tripathi).
7. The Monochromatic Column Problem, Discrete Mathematics, Vol 308, No 17 (2008), 3906-3916 (with S. Szabo).
6. Essential Extensions of a Direct Sum of Simple Modules-II, Modules and Comodules, Trends in Mathematics (volume in honor of Wisbauer), Birkhauser (2008), 243-246 (with S. K. Jain).
5. New Characterization of $\Sigma$-injective Modules, Proc. Amer. Math. Soc., Vol 316, No 10 (2008), 3461-3466 (with S. K. Jain and K. I. Beidar).
4. Unit Sum Numbers of Right Self-injective Rings , The Bulletin of Australian Mathematical Society, Vol 75, No 3 (2007), 355-360, (with D. Khurana).
3. Right Self-Injective Rings in Which Every Element is Sum of Two Units, Journal of Algebra and Appl., Vol 6, No 2 (2007), 281-286, (with D. Khurana).
2. Rings characterized by properties of direct sums of modules and on rings generated by units, Ph.D. dissertation, Ohio University, Athens, Ohio, 2007.
1. Essential Extensions of a Direct Sum of Simple Modules , Contemporary Mathematics, 420, Groups, Rings and Algebras (volume in honor of Don Passman), Amer. Math. Soc. (2006), 15-23, (with S. K.
Jain and K. I. Beidar). | {"url":"https://math.slu.edu/~srivastava/articles.html","timestamp":"2024-11-07T20:41:12Z","content_type":"text/html","content_length":"13237","record_id":"<urn:uuid:20eb08d2-6cab-4167-bb1b-63916b6ecb45>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00052.warc.gz"} |
Completeness theorems for non-cryptographic fault-tolerant distributed computation
Every function of n inputs can be efficiently computed by a complete network of n processors in such a way that: 1. If no faults occur, no set of size t < n/2 of players gets any additional
information (other than the function value), 2. Even if Byzantine faults are allowed, no set of size t < n/3 can either disrupt the computation or get additional information. Furthermore, the above
bounds on t are tight!
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print) 0737-8017
Conference 20th Annual ACM Symposium on Theory of Computing, STOC 1988
Country/Territory United States
City Chicago, IL
Period 2/05/88 → 4/05/88
Dive into the research topics of 'Completeness theorems for non-cryptographic fault-tolerant distributed computation'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/completeness-theorems-for-non-cryptographic-fault-tolerant-distri","timestamp":"2024-11-09T13:13:44Z","content_type":"text/html","content_length":"47909","record_id":"<urn:uuid:33ead9e6-03a6-4a60-a623-fa859d1a0192>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00868.warc.gz"} |
Math 165: Calculus for Business
Course Prerequisite(s)
One of:
Course Description
Math 165 is a calculus course intended for those studying business, economics, or other related business majors. The following topics are presented with applications in the business world: functions,
graphs, limits, exponential and logarithmic functions, differentiation, integration, techniques and applications of integration, partial derivatives, optimization, and the calculus of several
variables. Each textbook section has an accompanying homework set to help the student better understand the material.
Credit Awarded
5 hours (some exceptions noted below)
Prior credit for MATH 170 or MATH 180 will be lost with subsequent completion of MATH 165.
Course Materials
None required
Gradarius is an online-learning tool for calculus. A “Complete” account is required for this course. It can be purchased by following the link provided in the blackboard page for this course.
The following topics are covered Math 165
Module Topics
1 Limits, Continuity
2 Rates of Change, The Derivative
3 Derivative Rules, Product/Quotient Rules, Chain Rule
4 Logarithms and Exponentials, Implicit Differentiation, Marginal Analysis
5 Maxima/Minima, Applications of Maxima/Minima, Elasticity
6 Higher Order Derivatives, Graphs, Related Rates
7 Indefinite Integral, Method of Substitution
8 Definite Integral, Fundamental Theorem
9 Integration by Parts, Area Between Curves, Averages
10 Consumer/Producer’s Surplus, Income Streams, Improper Integrals
11 Multi-Variable Functions, Partial Derivatives
12 Maxima/Minima/Saddle Points, LaGrange Multipliers | {"url":"https://courses.mscs.uic.edu/math/math165/","timestamp":"2024-11-06T05:44:05Z","content_type":"text/html","content_length":"202884","record_id":"<urn:uuid:4698c593-2ed0-40b2-b318-6a9430b4bd58>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00500.warc.gz"} |
From erikscraggs
From gianola
From steibelj
From ytutsunomiyahttp://onlinelibrary.wiley.com/...j.1469-1809.1949.tb02451.x/pdf
[2] http://pngu.mgh.harvard.edu/...urcell/plink/ibdibs.shtml#homo
[3] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1950838/
On Thu, May 30, 2013 at 10:06 PM, Daniel Gianola <gianola
From bmuirhttp://www.eeb.uconn.edu/people/plewis
/software.php). I also have a publication on the topic in chickens using a SNP
chip which I can share with you if interested.
Best Regards, Bill
William Muir, Ph.D.
Professor Genetics
Department of Animal Sciences
Purdue University and
Department of Medicine
Indiana University
Room G406 Lilly Hall
915 West State Street
W. Lafayette, IN 47907
-----Original Message-----
.From: Erik Scraggs [mailto:erikscraggs
From Andres.Legarrahttp://genoweb.toulouse.inra.fr/~alegarra
From hsimianhttp://www.uni-goettingen.de/tierzucht
-----Urspr�ngliche Nachricht-----
Von: Andres Legarra [mailto:Andres.Legarra
From taylorjerr
From bmuirhttp://genoweb.toulouse.inra.fr/~alegarra
From gianolahttp://onlinelibrary.wiley.com/...j.1469-1809.1949.tb02451.x/pdf
[2] http://pngu.mgh.harvard.edu/...urcell/plink/ibdibs.shtml#homo
[3] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1950838/
On Thu, May 30, 2013 at 10:06 PM, Daniel Gianola <gianola
From ytutsunomiyahttp://www.jstor.org/stable/2408641
On Fri, May 31, 2013 at 9:39 AM, Daniel Gianola <gianolahttp://onlinelibrary.wiley.com/...j.1469-1809.1949.tb02451.x/pdf
> [2] http://pngu.mgh.harvard.edu/...urcell/plink/ibdibs.shtml#homo
> [3] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1950838/
> On Thu, May 30, 2013 at 10:06 PM, Daniel Gianola <gianola
From ydahttp://www.uni-goettingen.de/tierzucht
> -----Ursprÿÿngliche Nachricht-----
> Von: Andres Legarra [mailto:Andres.Legarra
From ytutsunomiyahttp://www.codecogs.com/latex/eqneditor.php]
F_{i} = \frac{(O_{i} - E_{i})}{(L_{i} - E_{i})}
where O_{i} is observed homozygosity, L_{i} is the number of SNPs measured
in individual i and
E_{i} = \sum\limits_{j=1}^{L_{i}}
where nj and pj are the number of measured genotypes and the reference
allele frequency at locus j, respectively.
I may be wrong, but PLINK implements F as a descriptive statistic that can
be used by the user for three main purposes: 1) identify contaminated
samples; 2) identify excess of X chromosome heterozygosity in samples
declared to be male; 3) as the Wright inbreeding coefficient. Other usage
must be carefully assessed and interpreted. As said before in the
discussion, 'inbreeding coefficients' have a handful of different
definitions and contexts, but this in particular is a variant of the FIS
(IBS-based) measure defined by Wright in the analysis of structured
On Fri, May 31, 2013 at 11:07 AM, Baumung, Roswitha (AGAG)
From erikscraggs
From Roger.Vallejohttp://www.ars.usda.gov/...ople/people.htm?personid=37662
-----Original Message-----
.From: Daniel Gianola [mailto:gianolahttp://onlinelibrary.wiley.com/...j.1469-1809.1949.tb02451.x/pdf
[2] http://pngu.mgh.harvard.edu/...urcell/plink/ibdibs.shtml#homo
[3] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1950838/
On Thu, May 30, 2013 at 10:06 PM, Daniel Gianola <gianola
From gianolahttp://www.ars.usda.gov/...ople/people.htm?personid=37662
-----Original Message-----
.From: Daniel Gianola [mailto:gianolahttp://onlinelibrary.wiley.com/...j.1469-1809.1949.tb02451.x/pdf
[2] http://pngu.mgh.harvard.edu/...urcell/plink/ibdibs.shtml#homo
[3] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1950838/ | {"url":"https://animalgenome.org/community/angenmap/athread/13","timestamp":"2024-11-08T02:26:07Z","content_type":"text/html","content_length":"84195","record_id":"<urn:uuid:dcc6bd82-4a2b-4bdb-b87c-eb6f433c112b>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00536.warc.gz"} |
FREE Pipe Velocity Calculator: Calculate Pipe Fluid Flow
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Can save money by reducing the number of time spent on manual and long calculations, and avoiding costly mistakes.
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means that a real engineer will assist with any design query, no matter how big or small. | {"url":"https://www.h2xengineering.com/pipe-velocity-calculator/","timestamp":"2024-11-09T22:42:28Z","content_type":"text/html","content_length":"455155","record_id":"<urn:uuid:859fa3ea-06fe-4c60-b85f-f009ae1c2054>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00104.warc.gz"} |
Scale Invariant Log Loss Mathematical Proof
Over the last few months I have been working in (monocular) depth estimation for my Master’s Thesis. When I started digging into commonly used loss functions, I came across the Scale Invariant Log
Loss. This function was presented by Eigen et al. in Depth Map Prediction from a Single Image using a Multi-Scale Deep Network (arXiv) and it is extensively used (usually in conjunction with
additional scale-dependant terms) to train and evaluate depth prediction models. The catch about this function is that the scale of the predicted output does not affect its magnitude. When I first
read that, I wanted to take a look at the mathematical proof of the scale invariance, but could not find any already written. Therefore, this post tries to relieve the mathematical itch of those in
my position.
This should be a short post, but you can jump straight to the proof clicking here.
A little context, as a treat
Before going into the mathematical detail the meaning of scale invariance should be explained. Monocular Depth Estimation (recovering the three-dimensional information lost when projecting a 3D scene
such as the real world into a 2D space such as an image) is an ill posed problem. This means that it has infinite geometrical solutions, as an infinite number of scenes can be projected in the same
2D representation given that a relation between their distance and their scale exists. An example of this problem can be seen below, where the two cubes produce the same image, even though their
scale is different.
What a scale invariant loss function does is calculate an error magnitude that does not take into account the scale discrepancy between the ground truth annotation and the output predicted by our
model. It only considers the relative depth of each set of values.
One last thing to mention is that the previously mentioned paper didn’t actually proposed a totally scale-invariant loss. It proposed a function that can be tweaked to change the influence of the
scale. The original equation is:
\[L(\hat{d}, d) = \frac{1}{n}\sum_{p} (\ln{\hat{d_p}} - \ln{d_p})^2 - \frac{\lambda}{n^2} \left( \sum_{p} (\ln{\hat{d_p}} - \ln{d_p}) \right)^2 \\\]
And citing Eigen et al. (emphasis mine):
Setting λ = 0 reduces to elementwise L2, while λ = 1 is the scale-invariant error exactly. We use the average of these, i.e. λ = 0.5, finding that this produces good absolute-scale predictions
while slightly improving qualitative output.
The actual mathematical proof
Without further ado, here comes the math.
In the first equation we have the Scale Invariant Log Loss with λ = 1. The ground truth depth measure in each pixel is denoted by \(d_p\), and the predicted depth by \(\hat{d_p}\). The number of
pixels in each image is \(n\).
\[L(\hat{d}, d) = \frac{1}{n}\sum_{p} (\ln{\hat{d_p}} - \ln{d_p})^2 - \frac{1}{n^2} \left( \sum_{p} (\ln{\hat{d_p}} - \ln{d_p}) \right)^2 \\\]
First step is to introduce \(\alpha\) as the scaling factor, multiplying both \(\hat{d_p}\) terms.
\[= \frac{1}{n}\sum_{p} (\ln{\alpha \hat{d_p}} - \ln{d_p})^2 - \frac{1}{n^2} \left( \sum_{p} (\ln{\alpha \hat{d_p}} - \ln{d_p}) \right)^2 \\\]
Operate on the logarithms…
\[= \frac{1}{n}\sum_{p} (\ln{\alpha} + \ln{\frac{\hat{d_p}}{d_p}})^2 - \frac{1}{n^2} \left( \sum_{p} (\ln{\alpha} + \ln{\frac{\hat{d_p}}{d_p}}) \right)^2 \\\]
Rename ln \(\alpha\) (which is constant) as \(\beta\)…
\[= \frac{1}{n}\sum_{p} (\beta + \ln{\frac{\hat{d_p}}{d_p}})^2 - \frac{1}{n^2} \left( \sum_{p} (\beta + \ln{\frac{\hat{d_p}}{d_p}}) \right)^2 \\\]
Expand the first binomial expression and operate on the second summatory…
\[= \frac{1}{n}\sum_{p} (\beta^2 + \ln{\frac{\hat{d_p}}{d_p}}^2 + 2 \beta ln{\frac{\hat{d_p}}{d_p}}) - \frac{1}{n^2} \left( \beta n + \sum_{p} (\ln{\frac{\hat{d_p}}{d_p}}) \right)^2 \\\]
Operate on the first summatory and expand the binomial of the second term…
\[= \frac{1}{n} (\beta^2 n + \sum_{p}(\ln{\frac{\hat{d_p}}{d_p}})^2 + \sum_{p}(2 \beta ln{\frac{\hat{d_p}}{d_p}})) - \frac{1}{n^2} \left( \beta^2 n^2 + (\sum_{p} (\ln{\frac{\hat{d_p}}{d_p}}))^2 + 2 \
beta n \sum_{p} (\ln{\frac{\hat{d_p}}{d_p}}) \right) \\\]
Operate with the number of pixels’ fractions…
\[= \beta^2 + \frac{\sum_{p}(\ln{\frac{\hat{d_p}}{d_p}})^2}{n} + \frac{\sum_{p}(2 \beta ln{\frac{\hat{d_p}}{d_p}})}{n} - \left( \beta^2 + \frac{(\sum_{p} (\ln{\frac{\hat{d_p}}{d_p}}))^2}{n^2} + \frac
{2 \beta n \sum_{p} (\ln{\frac{\hat{d_p}}{d_p}})}{n^2} \right) \\\]
Reorder the terms for easier simplification…
\[= \beta^2 - \beta^2 + \frac{\sum_{p}(2 \beta ln{\frac{\hat{d_p}}{d_p}})}{n} - \frac{2 \beta \sum_{p} (\ln{\frac{\hat{d_p}}{d_p}})}{n} + \frac{\sum_{p}(\ln{\frac{\hat{d_p}}{d_p}})^2}{n} - \frac{(\
sum_{p} (\ln{\frac{\hat{d_p}}{d_p}}))^2}{n^2} \\\]
Simplify the terms when possible…
\[= \frac{\sum_{p}(\ln{\frac{\hat{d_p}}{d_p}})^2}{n} - \frac{(\sum_{p} (\ln{\frac{\hat{d_p}}{d_p}}))^2}{n^2} \\\]
And finally, change the logarithms’ representation again to get the original expression (!).
\[= \frac{1}{n}\sum_{p} (\ln{\hat{d_p}} - \ln{d_p})^2 - \frac{1}{n^2} \left( \sum_{p} (\ln{\hat{d_p}} - \ln{d_p}) \right)^2 \\\]
That’s it! As I promised, a short post. Thank you for reading!
Related links
Written on June 13, 2022 | {"url":"http://guillesanbri.com/Scale-Invariant-Loss/","timestamp":"2024-11-08T00:45:58Z","content_type":"text/html","content_length":"11682","record_id":"<urn:uuid:79e57511-f48b-4613-bac8-7c05aea25704>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00320.warc.gz"} |
Math is one of the subjects that many students dislike
anything to solve a problem if you don’t know the formula for that problem. In other subjects, you can erase what you know to solve the problem even if you don’t know how to solve it, but math is
very difficult to solve if you haven’t learned the formula beforehand. Another reason is that math has a lot of formulas, and there are many complex problems that require you to read the problem and
think about how to solve it, rather than just adding variables to the problem and substituting formulas. These reasons cause students to give up on math and make it a hated subject. However, math can
be fun when you know the formulas and solve complex problems. I hope that many students will find math fun and enjoy it. | {"url":"https://www.mathdb.org/math-is-one-of-the-subjects-that-many-students-dislike/","timestamp":"2024-11-07T01:01:42Z","content_type":"text/html","content_length":"29170","record_id":"<urn:uuid:f15014a5-639d-49cf-921d-9418341c805c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00358.warc.gz"} |
QA Math | beyond-the-test
A salon wants to conduct a survey in order to study the consumption pattern of its consumers. The population mean spending on visiting salon per month is unknown with the population standard
deviation is assumed to be $200. A random sample of 55 customers is selected and the sample mean spending on visiting salon per month is $800.
(a) Construct a 95% confidence interval estimate for the population mean spending by a customer on visiting salon per month.
(b) It is suggested to do the second round inspection. This time, the sample size is 120. What is the sampling error at 95% confidence level? | {"url":"https://www.beyondthetest.com/forum/math-questions-and-answers/qa-math","timestamp":"2024-11-10T00:12:36Z","content_type":"text/html","content_length":"1050041","record_id":"<urn:uuid:79ca94d4-bac8-4dfe-ac84-1b3b818a3ef6>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00161.warc.gz"} |
How Is Addition Like Subtraction?
Today’s Wonder of the Day was inspired by Macy. Macy Wonders, “How is addition like subtraction?” Thanks for WONDERing with us, Macy!
A big thank you goes to Deb Frazier’s (@Frazier1st) class in Ohio for nominating today’s Wonder!
When you’re first learning basic math, it can seem as easy as 1 + 1 = 2. Easy peasy, right? Sure, it gets harder, but before long you’re memorizing all of those basic addition facts with the help of
flash cards.
Then one day, your teacher turns the tables on you. Suddenly you’re faced with subtraction. You’re no longer counting up two groups of things to get a simple sum. Instead, you’re taking things away
and trying to figure out how many are left.
Kids often feel that subtraction is more difficult. After all, it’s completely different from addition, right? Not so fast! Addition and subtraction actually share a special relationship.
As mathematicians like to say, there is an inverse relationship between addition and subtraction. So what does inverse mean? Without getting too technical, you can think of inverse as meaning
For example, the inverse of hot is cold. Likewise, the inverse of addition is subtraction. And guess what? The inverse of subtraction is addition! Why? Addition and subtraction are opposites. They
basically undo each other.
Let’s see how this works. If we add 1 + 1, we get 2. That’s addition. If we then take away 1 from our 2, we undo the addition we just performed and end up with 1. That’s subtraction.
To understand the relationship between addition and subtraction on an even deeper level, we need to learn about two more things: number facts and fact families. A number fact is a simple equation
made up of three different numbers. For example, 1 + 2 = 3 is a number fact.
For each set of three different numbers, you can create two addition and two subtraction number facts that are . We call these four number facts a fact family, since they’re kind of like the
members of a family.
If we stick with 1, 2, and 3 as our three numbers, we can create the following fact family:
1 + 2 = 3
2 + 1 = 3
3 – 2 = 1
3 – 1 = 2
If 1 + 2 = 3, then it obviously follows that 2 + 1 = 3, since you’re just switching around the order of the two numbers you’re adding. The subtraction number facts also follow, since they’re the
opposite of the two addition number facts.
If you need to visualize it, just think of the addition number fact in reverse, switching the place of the equal sign and the plus sign and then switching the plus sign to a minus sign. 1 + 2 = 3
becomes 3 – 2 = 1!
Thinking about it like this can help you get a better grasp on subtraction. It’s always difficult to learn a new skill, but when you can connect it to something you already know, it gets easier!
Standards: CCSS.MATH.OA.A.1, CCSS.MATH.OA.B.3, CCRA.L.3, CCRA.L.6, CCRA.R.1, CCRA.R.2, CCRA.R.4, CCRA.R.10, CCRA.SL.1 | {"url":"https://wonderopolis.org/wonder/how-is-addition-like-subtraction","timestamp":"2024-11-12T15:55:25Z","content_type":"text/html","content_length":"113538","record_id":"<urn:uuid:ea290f68-0c82-4790-87a4-545f25d5cffd>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00431.warc.gz"} |
Posts tagged with Minitab questions
Faculty of Science, Technology, Engineering and Mathematics M248 Analysing data
Please read the Student guidance for preparing and submitting TMAs on the M248 website before beginning work on a TMA. You can submit a TMA either by post or electronically using the University’s
online TMA/EMA
You are advised to look at the general advice on answering TMAs provided on the M248 website. Each TMA is marked out of 50. The marks allocated to each part of each question are indicated in brackets
in the margin. Your overall score for each TMA will be the sum of your marks for these questions.
Note that the Minitab files that you require for TMA 05 are not part of the M248 data files and must be downloaded from the ‘Assessment’ area of the M248 website.
Question 1, which covers topics in Unit 9, and Question 2, which covers topics in Unit 10, form M248 TMA 05. Question 1 is marked out of 32; Question 2 is marked out of 18.
Minitab Question one
You should be able to answer this question after working through Unit 9.
(a) A study was undertaken to examine the tensile strength of a new type of polyester fibre. The Minitab worksheet polyester-fibre.mtw gives the breaking strengths (in grams/denier, denier being a
unit of fineness) of a random sample of n = 30 observations, given in the variable Strength.
The existing type of polyester fibre which the new type is designed to replace has a mean breaking strength of 0.26 grams/denier. Interest centres on using the data in polyester-fibre.mtw to test
whether the mean breaking strength of the new type of polyester fibre differs from the mean breaking strength of the existing type of polyester fibre.
(i) Write down appropriate null and alternative hypotheses for a test of whether the mean breaking strength of the new type of polyester fibre differs from the mean breaking strength of the existing
type of polyester fibre. Define any notation that you use. [3]
(ii) It is proposed to use a z-test to test the hypotheses specified in part (a)(i). Justify this choice of test in terms of the sample size, n. [1]
(iii) Write down the formula for the test statistic used in the z-test of part (a)(ii). Define any further notation that you use. [2]
(iv) Write down the null distribution of the test statistic in part (a)(iii). What is the reason for the use of the word ‘null’ in the phrase ‘null distribution’? [2]
(v) Using Minitab, obtain the standard deviation of the values in Strength, then perform the z-test that you have been considering throughout part (a) of this question. Provide a copy of the
output** produced by performing this test. (This output should comprise four lines which start with the words Test, The, Variable and Strength, respectively.) [3]
(vi) Interpret the result of the test that you have just performed, as given by its p-value. [3]
(vii) Would you have rejected H0 or not rejected H0 if you had tested the hypotheses of interest in this question at the 5% significance level? Would you have rejected H0 or not rejected H0 if you
had tested these hypotheses at the 1% significance level? Justify each of your answers separately. [4]
(b) The proportion, p0, of foraging bumblebees not exposed to pesticides who bring very little pollen back to their nest is 0.4. A recent study of foraging bumblebees investigated the effect of
exposure to a widely used neonicotoid pesticide called imidacloprid on pollen foraging rates. (Neonicotoid pesticides are commonly used in agriculture due to their low toxicity in mammals.) Let p
denote the proportion of foraging bumblebees exposed to imidacloprid who bring very little pollen back to their nest.
A sample of 60 bumblebees were exposed to a low (field realistic) dose of imidacloprid: 39 of these bumblebees brought back very little pollen to their nest. Use these data to perform the test of the
hypotheses H0 : p = 0:4; H1 : p > 0:4; by working through the following subparts of this part of the question.
(i) Calculate the observed value of the test statistic for this test. [2]
(ii) Using the approximate normal null distribution of this test statistic, identify the rejection region of a test of the stated hypotheses using a 1% significance level. [2]
(iii) Report and interpret the outcome of this hypothesis test. [2]
(c) The isotopic abundance ratio of natural silver (Ag) is the ratio of the stable isotopes Ag107 to Ag109. Its mean is 1.076 and measurements on a random sample of observed isotopic abundance ratios
suggested that they are plausibly normally distributed with a sample standard deviation of 0.0026. Interest in this part of the question concerns the planning of a further experiment to detect
whether this ratio is different in observations from a certain source of silver nitrate. The new study will use a two-sided test at the 5% significance level, assuming normality. It is desired to
make sufficient observations of the isotopic abundance ratio on the silver nitrate so that the power of the test to distinguish a difference between the null hypothesis of a true underlying mean of
1.076 and a value that is 0.0015 larger or 0.0015 smaller is 90%. For the purpose of performing the necessary sample size calculation, it will be assumed that the population standard deviation of the
isotopic abundance ratio measurements is equal to the sample standard deviation given above.
(i) Calculate, by hand, the size of the sample required to achieve the desired power of the test. Show our working. [6]
(ii) Ignoring rounding up to an integer, and assuming that no other aspect of the problem changes, by what multiple is the required sample size changed if, instead of seeking to distinguish between
the underlying mean and values that are 0.0015 larger or smaller than it, it was decided to seek to distinguish between the underlying mean and values that are 2/3 as much (that is, 0.001) larger or
smaller than it? [2]
Minitab statistics question 2
Question 2 { 18 marks
You should be able to answer this question after working through Unit 10.
(a) Halofenate has been shown to be effective in the treatment of conditions associated with abnormally high levels of lipids in the blood; triglyceride is a lipid of particular importance. A group
of 22 patients were treated with halofenate medication. The changes between the patients’ triglyceride levels after treatment with halofenate and before treatment with halofenate were measured. These
changes are in the Minitab worksheet triglyceride.mtw, in the column Halofenate. (Note that a negative change corresponds to the desirable outcome of a reduction in triglyceride levels.)
The column Placebo contains the changes between triglyceride levels after treatment with an inactive placebo and before treatment with the placebo, for an independently drawn control group of 21
patients. The main question of interest is whether halofenate makes a more favourable change to triglyceride levels, in comparison to a placebo.
Graphical investigation of the data shows that normality cannot be assumed for the distribution of either Halofenate or Placebo. It is therefore decided to compare the effects of halofenate and a
placebo on
triglyceride reduction using the Mann{Whitney test.
(i) Write down appropriate null and alternative hypotheses for a test of whether the difference between the location of the changes between triglyceride levels after and before treatment with
halofenate and the location of the changes between triglyceride levels after and before treatment with a placebo is negative. Define any notation that you use. [3]
(ii) Use Minitab to carry out the Mann-Whitney test of the hypotheses discussed above. Provide a copy of one line of the Minitab output which includes the p-value associated with the test. [2]
(iii) Interpret the result of the test that you have just performed, as given by its p-value. [2]
(b) In Table 5 of Unit 3, data were given on the month of death (January = 1, February = 2, . . . , December = 12) for 82 descendants of Queen Victoria; they all died of natural causes. The data are
repeated here in Table 1.
The question of whether or not these royal deaths could be claimed to be from a discrete uniform distribution on the range 1; 2; : : : ; 12 was considered informally in Example 20 of Unit 3 and, at
some length, in Chapter 8 of Computer Book A. From these investigations, it looked as though the discrete uniform distribution may be a plausible model for these data, but no firm conclusion was
In this part of the question, you are going to perform a chi-squared goodness-of-fit test of the discrete uniform distribution to these data.
(i) Obtain the expected frequencies of the values 1; 2; : : : ; 12 assuming a discrete uniform distribution. Why is it not necessary to pool categories before performing a chi-squared goodness-of-fit
test in this case? [3]
(ii) Carry out the remainder of the chi-squared goodness-of-fit test: report the individual elements of the chi-squared test statistic, the value of the test statistic itself, the number of degrees
of freedom of the chi-squared null distribution, and whatever this tells you about the p-value associated with the test. Interpret the outcome of the test.
m248 TM 05 sample statistics solution.docx
Let us know if you like us to help you with any of your coursework,
The Open University Statistics TMA 01 Question 1
(a) A number of Japanese black pine tree seedlings were planted in a rather inaccessible location to which researchers returned at the same time each year in order to measure their growth. The
resulting data comprise the height of the young trees (measured on an effectively continuous scale of millimetres) and the age of the trees (1, 2, 3 or 4 years).
(i) Name two graphical displays which are suitable for studying the distribution of the heights of the trees at age 1 year. Give a single reason for the suitability of both displays. [3]
(ii) Unfortunately, the young trees were susceptible to dying off, so the number of trees that remain alive at each age is also a variable of interest. Name a graphical display which is suitable for
showing the number of trees alive at each age. Give a reason for your answer. [2]
(iii) Name two graphical displays which are suitable for studying the way that the heights of the trees depend on their ages. Give a separate reason for the suitability of each display. [4]
(b) The Minitab worksheet snow-depth.mtw contains measurements of the depth of snow lying at each of n = 114 locations on an Antarctic ice floe in March 2003. The measurements are in centimetres,
rounded to
the nearest whole number. The data are in the variable Depth.
(i) Produce Minitab’s default frequency histogram for Depth. Include a copy of this histogram in your answer. Briefly describe the main features of the distribution suggested by this histogram. [3]
(ii) Now use Minitab to produce a frequency histogram for Depth with cutpoints at 0, 10, 20, . . . , 100 cm. Include a copy of this histogram in your answer. Briefly describe the main feature of the
distribution according to this histogram and state why this histogram gives a more clear-cut picture than the default histogram that you obtained in part (b)(i). [4]
(iii) Now use Minitab to produce a unit-area histogram for Depth with cutpoints at 0, 10, 20, . . . , 100 cm. Include a copy of this histogram in your answer. In what way(s) does this histogram
differ from the frequency histogram that you obtained in part (b)(ii)? The heights of the bars in the histogram that you have just produced should be 0:005 0:016 0:018 0:012 0:011 0:019 0:012 0:004
0:002 0 (each given correct to three decimal places except the last which is exact). Use this information to verify that this histogram does have unit area, as claimed. [5]
(iv) Using Minitab, obtain the sample size, sample mean and sample median of the variable Depth and report your result, by copying Minitab text, in the form
Variable N Mean Median
Depth . .*
(where, of course, the asterisks are replaced by numbers).
Comment briefly on the relative size of the sample mean and the
sample median, relating your comments to the shape of the
histogram that you obtained in either part (b)(ii) or (b)(iii). [4]
This question was solved by our experts under our pay someone to do my statistics homework statistics help services. Note that this question requires you to use Minitab to solve the assignment by
hand while using Minitab where necessary. we have attached solutions for the first question here for you reference. TM01 Open University Statistics solutions.docx. You can contact us if you need
further help with your coursework assignments. | {"url":"https://www.mymathlabhomeworkhelp.com/mymathlabanswers/tag/Minitab-questions/","timestamp":"2024-11-12T20:33:51Z","content_type":"text/html","content_length":"35760","record_id":"<urn:uuid:7150798a-8417-495c-b79c-cc22c10eddcd>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00576.warc.gz"} |
Purchases 40501 - math word problem (40501)
Purchases 40501
The Kováčov family makes a purchase at a nearby store once a week for €45. After the opening of the new shopping center, the family's monthly purchases are a tenth cheaper. Calculate how many euros
the Kováčovci will pay per week for shopping after the opening of the new shopping center.
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
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You need to know the following knowledge to solve this word math problem:
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Solved: You are performing an analysis on the popularity of the office desk models sold in 2020 an [Statistics]
You are performing an analysis on the popularity of the office desk models sold in 2020 and have written the following SUMIF function: =SUMIF(A2:A220,I7,D2:D220) For reference - The model names in
the sales data are in column A. - Cells I7 to I10 have the names of the 4 models that are sold. - The number of items sold in each order are in column D. The SUMIF function works fine in the first
cell, but you get an error when dragging the function to the other cells. Why could this be? You did not use the dollar sign to anchor the range and sum range. The information for the criteria cell
is not correct. You did not use the dollar sign to anchor the criteria cell.
Asked in United States
Gauth AI Solution
Super Gauth AI
You did not use the dollar sign to anchor the range and sum range
1. The error might be due to not using the dollar sign to anchor the range and sum range in the SUMIF function
Simplify this solution | {"url":"https://www.gauthmath.com/solution/1811551914738694/1-point-Which-statements-about-a-sodium-atom-_1123Na-are-correct-1-The-number-of","timestamp":"2024-11-05T08:54:33Z","content_type":"text/html","content_length":"184037","record_id":"<urn:uuid:835f96aa-26bd-4eb9-aa62-3deeb3ed8d99>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00459.warc.gz"} |
Improving Search Speed for solving MIQP using Gurobi
I am working on solving Unit Commitment problem. I am using Gurobi as the solver and Julia to create the model (MIQP model). However, I have been running the solver for a little over 24hours and
notice the gap is decreasing very slowly. Currently, I am using the following Gurobi Parameters:
TuneCriterion = 2
CliqueCuts = 0
Cuts = 0
MIPFocus = 2
I have included a link to my log file and model in Julia. Any assistance in this matter is greatly appreciated. I should point out I could not save the entire log file from Julia's REPL, so I include
the first couple of seconds of the log and most of the rest.
• Hi Micah,
You could try experimenting with the NoRelHeurTime and MIPFocus parameter to shift the focus towards finding feasible points. Do you have an intuition of whether the lower bound bound or the
feasible point should be improved. If it is the lower bound, then you should not turn of Cuts, as these are needed to move the lower bound. The PreQLinearize might also be worth testing.
Could you provide an MPS file instead of the Julia code? This would make it way easier for the Community to help you. the Knowledge Base article How do I write an MPS model file from a
third-party API? holds the required information.
Best regards,
• Hi Micah,
The problem is quite large and has very high objective values. Do you really need to solve it to a MIPGap of 0.01%? I would guess that a MIPGap of 1% sounds more realistic. Of course, this
depends on your application.
Best regards,
• Hi Micah,
It might be that your model suffers from a hard to find symmetry. Let's consider the equality constraint
R0: C27216 + C27384 + C27552 + C27720 + C27888 + C28056 + C28224 + C28392
+ C28560 + C28728 + C28896 + C29064 + C29232 + C29400 + C29568 + C29736
+ C29904 + C30072 + C30240 + C30408 + C30576 + C30744 + C30912 + C31080
+ C31248 + C31416 + C31584 = 1016.34
The constraints for the participating variables C27216 and C27384 are basically the same (they all have the same structure but with different binaries). This also applies to other pairs of
variables in this constraint. Since the pairwise variables are interchangeable w.r.t. to their values and the corresponding constraints where they participate, the above equality constraint
creates a very symmetrical structure which might be hard to detect. What you might try is to handle the pairwise variables as one variable or try to break the symmetry with additional ordering
constraints, e.g., C27216 \(\geq\) C27384. The model is quite large so I hope that I did not miss something.
Best regards,
• Hi Micah,
I am not if this gives the impression that some of the variables are the same with different binaries?
Sorry for the confusion, I did not mean that C27216 = C27384 but rather that it would be possible to swap the values of C27216 and C27384 (and the corresponding binary variables) in the solution
vector without making the solution infeasible and without worsening the solution value. Maybe, there is some trick to tighten the formulation hidden in there. In general, unit commitment problems
are known to be very hard, even for small-medium sizes. I am not an expert on unit commitment problems. Maybe you can find something in the literature which might help here or someone from the
Community has an advice.
Best regards,
• Hi Jaromil, thanks for the response. I believe that the lower bound should be improved. So I am trying the simulation over without turning off Cuts and MIPFocus = 3. I got the mps file, it can be
found here: https://1drv.ms/u/s!AkuqGp7L6Ay2jPJ9d0r7-oQmGeWy_g?e=TScmCc
• Hi Jaromil, yes I understand that this problem is quite large, so I have no problem with a MIPGap of 1%. Since my previous comment, I have included
PreQLinearize = 1,
Method = 0,
MIPFocus = 1.
There were major changes at the beginning of the simulation, with a better integer solution being found quite early while the best bound took some time to increase. So I redid the simulation now
PreQLinearize = 1,
Cuts = 3,
Method = 0,
MIPFocus = 1.
I now included aggressive cuts to improve the best bound. The simulation is running right now, but I am already seeing a slightly better incumbent and best bound value. However, one problem seems
to be happening, the gap tends to rapidly decrease early in the simulation however when it drops to 10%, it take forever to decrease to 9%. Are there any other parameters I could try to get the
gap to decrease much quicker. Thanks in advance.
• Thanks for the feedback Jaromił. Based on the equality constraints mentioned above, I know this is associated with my load demand constraints where the sum of the power generated by each unit in
the my electric grid must meet the load demand of 1016.34MW at the first hour of the week. There are 27 units in the system, hence the 27 different variables on the left hand side of the
constraint. However, I know from the data associated with the units, that some of them have the same exact specifications. For example, Unit 1 and 2 have the same specifications, Unit 3 and 4
have the same specification, Unit 5 to 8 are the same, Unit 9 to 14 are the same, Unit 15 to 17 are the same, Unit 18 to 20 are the same, Unit 21 is by itself, Unit 22 and 23 are the same and the
Unit 24 to 27 are the same. I am not if this gives the impression that some of the variables are the same with different binaries?
Also, I should point out this same model is solved very quickly using Gurobi for a smaller system with 10 Units for 24hours. However, I am now trying to now apply it to a much larger system with
27 Units and 168Hrs. So I know the system is quite large and I may have to settle with a MIPGap of about 10%.
• Thanks for all the help Jaromił, it is greatly appreciated. You were right about the symmetry issue, something I myself was unaware off. However, after doing some research last night, I came
across this:
Symmetry issues in mixed integer programming based Unit Commitment - ScienceDirect
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ax-4 - New Foundations Explorer
Description: Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all x,
it is true for any specific x (that would typically occur as a free variable in the wff substituted for φ). (A free variable is one that does not occur in the scope of a quantifier: x and y are both
free in x = y, but only x is free in ∀yx = y.) This is one of the axioms of what we call "pure" predicate calculus (ax-4 2135 through ax-7 1734 plus rule ax-gen 1546). Axiom scheme C5' in [Megill] p.
448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1546. Conditional forms of the converse are given by
ax-12 1925, ax-15 2143, ax-16 2144, and ax-17 1616.
Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from x for the special case. For use, that requires the assistance of equality axioms, and we deal with
it later after we introduce the definition of proper substitution - see stdpc4 2024.
An interesting alternate axiomatization uses ax467 2169 and ax-5o 2136 in place of ax-4 2135, ax-5 1557, ax-6 1729, and ax-7 1734.
This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 1747. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) | {"url":"https://us.metamath.org/nfeuni/ax-4.html","timestamp":"2024-11-12T15:44:32Z","content_type":"text/html","content_length":"10760","record_id":"<urn:uuid:2e6d5d6c-7a87-4ba5-92e6-d23b66469ebe>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00839.warc.gz"} |