content stringlengths 86 994k | meta stringlengths 288 619 |
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Browsing Codes
muLAn analyzes and fits light curves of gravitational microlensing events. The code includes all classical microlensing models (for example, single and binary microlenses, ground- and space-based
parallax effects, orbital motion, finite-source effects, and limb-darkening); these can be combined into several time intervals of the analyzed light curve. Minimization methods include an
Affine-Invariant Ensemble Sampler to generate a multivariate proposal function while running several Markov Chain Monte Carlo (MCMC) chains, for the set of parameters which is chosen to be fit;
non-fitting parameters can be either kept fixed or set on a grid defined by the user. Furthermore, the software offers a model-free option to align all data sets together and allow inspection the
light curve before any modeling work. It also comes with many useful routines (export publication-quality figures, data formatting and cleaning) and state-of-the-art statistical tools.
Modeling results can be interpreted using an interactive html page which contains all information about the light curve model, caustics, source trajectory, best-fit parameters and chi-square.
Parameters uncertainties and statistical properties (such as multi-modal features of the posterior density) can be assessed from correlation plots. The code is modular, allowing the addition of other
computation or minimization routines by directly adding their Python files without modifying the main code. The software has been designed to be easy to use even for the newcomer in microlensing,
with external, synthetic and self-explanatory setup files containing all important commands and option settings. The user may choose to launch the code through command line instructions, or to import
muLAn within another Python project like any standard Python package. | {"url":"https://ascl.net/code/all/page/21/limit/100/order/title/listmode/full/dir/asc","timestamp":"2024-11-02T22:00:14Z","content_type":"text/html","content_length":"129879","record_id":"<urn:uuid:bc2448b7-b52d-431a-8fc6-80e2ab4d762a>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00804.warc.gz"} |
Calibrated radiocarbon dates — cal
Calibrated radiocarbon dates
The cal class represents a vector of calendar probability distribution; typically calibrated radiocarbon dates.
cal() constructs a new cal vector from a set of data frames containing the raw probability distributions.
cal(..., .era = era::era("cal BP"))
<dynamic-dots> A set of data frames. Each should have two columns, the first a vector of calendar ages, and the second a vector of associated probability densities. If the first column is not an
era::yr() vector, it is coerced to one using the time scale specified by .era.
era::era() object describing the time scale used for ages. Defaults to calendar years Before Present (era("cal BP")). Not used if the ages specified in ... are already era::yr() vectors.
# Uniform distribution between 1 and 10 BP:
cal(data.frame(age = era::yr(1:10, "cal BP"), pdens = rep(0.1, 10)))
#> <c14_cal[1]>
#> Warning: `x` has more than one modal value. Only the first will be returned.
#> [1] c. 1 cal BP | {"url":"https://c14.joeroe.io/reference/cal.html","timestamp":"2024-11-03T12:46:19Z","content_type":"text/html","content_length":"9530","record_id":"<urn:uuid:4405024f-982d-4791-bfa2-9cc8935ff1e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00250.warc.gz"} |
American Mathematical Society
Order problem for canonical systems and a conjecture of Valent
HTML articles powered by AMS MathViewer
by R. Romanov PDF
Trans. Amer. Math. Soc. 369 (2017), 1061-1078 Request permission
We establish a sharp upper estimate for the order of a canonical system in terms of the Hamiltonian. This upper estimate becomes an equality in the case of Krein strings. As an application we prove a
conjecture of Valent about the order of a certain class of Jacobi matrices with polynomial coefficients. References
• Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
• Lev A. Sakhnovich, Spectral theory of canonical differential systems. Method of operator identities, Operator Theory: Advances and Applications, vol. 107, Birkhäuser Verlag, Basel, 1999.
Translated from the Russian manuscript by E. Melnichenko. MR 1690379, DOI 10.1007/978-3-0348-8713-7
• A. Baranov and H. Woracek, Subspaces of de Branges spaces with prescribed growth, Algebra i Analiz 18 (2006), no. 5, 23–45; English transl., St. Petersburg Math. J. 18 (2007), no. 5, 699–716. MR
2301039, DOI 10.1090/S1061-0022-07-00969-7
• M. Š. Birman and M. Z. Solomjak, Piecewise polynomial approximations of functions of classes $W_{p}{}^{\alpha }$, Mat. Sb. (N.S.) 73 (115) (1967), 331–355 (Russian). MR 0217487
• M. Š. Birman and M. Z. Solomjak, Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory, American Mathematical Society Translations, Series 2, vol. 114, American
Mathematical Society, Providence, R.I., 1980. Translated from the Russian by F. A. Cezus. MR 562305
• Michael Kaltenbäck, Henrik Winkler, and Harald Woracek, Strings, dual strings, and related canonical systems, Math. Nachr. 280 (2007), no. 13-14, 1518–1536. MR 2354977, DOI 10.1002/mana.200410562
• V. V. Borzov, The quantitative characteristics of singular measures, Problems of Mathematical Physics, No. 4: Spectral Theory. Wave Processes (Russian), Izdat. Leningrad. Univ., Leningrad, 1970,
pp. 42–47. (errata) (Russian). MR 0281860
• M. Lifschetz, On some questions concerning the determinate case of Hamburger’s moment problem, Rec. Math. N. S. [Mat. Sbornik] 6(48) (1939), 293–306. MR 0001386
• Christian Berg and Ryszard Szwarc, On the order of indeterminate moment problems, Adv. Math. 250 (2014), 105–143. MR 3122164, DOI 10.1016/j.aim.2013.09.020
• Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated
from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718
• I. S. Kats, Inclusion of the Hamburger power moment problem in the spectral theory of canonical systems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 262 (1999), no. Issled.
po Lineĭn. Oper. i Teor. Funkts. 27, 147–171, 234 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 110 (2002), no. 5, 2991–3004. MR 1734332, DOI 10.1023/
• Christian Berg and Galliano Valent, The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes, Methods Appl. Anal. 1 (1994), no. 2,
169–209. MR 1291292, DOI 10.4310/MAA.1994.v1.n2.a3
• Jacek Gilewicz, Elie Leopold, and Galliano Valent, New Nevanlinna matrices for orthogonal polynomials related to cubic birth and death processes, J. Comput. Appl. Math. 178 (2005), no. 1-2,
235–245. MR 2127882, DOI 10.1016/j.cam.2004.05.025
• Galliano Valent, Indeterminate moment problems and a conjecture on the growth of the entire functions in the Nevanlinna parametrization, Applications and computation of orthogonal polynomials
(Oberwolfach, 1998) Internat. Ser. Numer. Math., vol. 131, Birkhäuser, Basel, 1999, pp. 227–237. MR 1722728
• Toshio Uno and Imsik Hong, Some consideration of asymptotic distribution of eigenvalues for the equation $d^{2}u/dx^{2}+\lambda \rho (x)u=0$, Jpn. J. Math. 29 (1959), 152–164. MR 118891, DOI
• M. Solomyak and E. Verbitsky, On a spectral problem related to self-similar measures, Bull. London Math. Soc. 27 (1995), no. 3, 242–248. MR 1328700, DOI 10.1112/blms/27.3.242
• Hans Triebel, Fractals and spectra, Monographs in Mathematics, vol. 91, Birkhäuser Verlag, Basel, 1997. Related to Fourier analysis and function spaces. MR 1484417, DOI 10.1007/978-3-0348-0034-1
• I. S. Kats, Integral estimates for the distribution of the spectrum of a string, Sibirsk. Mat. Zh. 27 (1986), no. 2, 62–74, 221 (Russian). MR 890302
• Louis de Branges, Some Hilbert spaces of entire functions. II, Trans. Amer. Math. Soc. 99 (1961), 118–152. MR 133456, DOI 10.1090/S0002-9947-1961-0133456-2
• B. Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu.
Lyubarskii, M. Sodin and V. Tkachenko; Translated from the Russian manuscript by Tkachenko. MR 1400006, DOI 10.1090/mmono/150
• Jonathan Eckhardt and Gerald Teschl, Sturm-Liouville operators with measure-valued coefficients, J. Anal. Math. 120 (2013), 151–224. MR 3095152, DOI 10.1007/s11854-013-0018-x
• I. S. Kats, Thickness of the spectrum of a singular string, Izv. Vyssh. Uchebn. Zaved. Mat. 3 (1990), 23–30 (Russian); English transl., Soviet Math. (Iz. VUZ) 34 (1990), no. 3, 26–34. MR 1075908
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 34L15, 47B36
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Additional Information
• R. Romanov
• Affiliation: Department of Mathematical Physics and Laboratory of Quantum Networks, Faculty of Physics, St. Petersburg State University, 198504, St. Petersburg, Russia
• Email: morovom@gmail.com
• Received by editor(s): September 22, 2014
• Received by editor(s) in revised form: February 9, 2015
• Published electronically: May 3, 2016
• Additional Notes: This work was supported in part by the Austrian Science Fund (FWF) project I 1536–N25, the Russian Foundation for Basic Research, grants 13-01-91002-ANF and 12-01-00215, and by
Project SPbSU 11.38.263.2014
• © Copyright 2016 American Mathematical Society
• Journal: Trans. Amer. Math. Soc. 369 (2017), 1061-1078
• MSC (2010): Primary 34L15, 47B36
• DOI: https://doi.org/10.1090/tran6686
• MathSciNet review: 3572264 | {"url":"https://www.ams.org/journals/tran/2017-369-02/S0002-9947-2016-06686-6/?active=current","timestamp":"2024-11-13T18:34:41Z","content_type":"text/html","content_length":"74144","record_id":"<urn:uuid:bc2e72d0-06d7-426a-a0d8-786ce9fafe48>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00090.warc.gz"} |
New math research wiki - Math Research of Victor PortonNew math research wiki
I’ve created a new wiki site for math research.
The motto of this wiki is “a research in the middle”.
The site is intended to discuss research ideas, aspiring ways of research, usage of open problems and ways to prove open problems, etc.
The exact rules are not yet defined, but I published several example entries in this wiki. You should create new entries similar to existing ones (but possibly about other mathematical topics). | {"url":"https://math.portonvictor.org/2012/07/30/new-math-research-wiki/","timestamp":"2024-11-10T21:33:09Z","content_type":"text/html","content_length":"97047","record_id":"<urn:uuid:1357d852-ad22-4dff-a9cf-9d11c7a5c793>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00799.warc.gz"} |
Hydrodynamic theory of wet particle systems
External forces lead to granular flow under the condition that the applied shear stress reaches the yield (shear) stress while another stress must be maintained for continuous flow in steady state.
Most studies in granular physics focus on dry granular materials and their flow rheology. However, wet granular materials are ubiquitous in geology and many real world applications where interstitial
liquid is present between the grains. There are several proposals for flow rules of dry and wet granular materials available in the literature. These flow rules differ in complexity and in the number
of parameters, which are combined in the equations. The main focus areas of my research are (i) the formulation of suitable constitutive equations for the hydrodynamic density-stress-strain
relations, specifically for wet granular materials, (ii) the deduction of the constitutive equations from discrete element simulations, and (iii) the validation of the micro-macro transition with
numerical, theoretical and experimental results. The geometrical set-up of split-bottom shear cell used in my research is most appropriate for assessing the shear band originating from the split
position that widens near the free surface.
My research proposes a modified generalized flow rule/rheology to close the fundamental conservation laws for mass and momentum. Subsequently, a correlation is developed between the micro parameters
and the {steady state cohesion in the limit} of very low confining pressure. Another aspect of studying unsaturated granular media is the movement of interstitial liquid due to the rupture of
existing and formation of new liquid bridges. Shearing a wet granular system causes a re-distribution and transport of the interstitial liquid. The liquid transport can be modeled by a diffusion
equation with a space-dependent diffusive coefficient in the split bottom geometry. Alternatively, it is shown here that this is an advective-diffusive process with constant diffusivity coefficient
and a space-dependent drift, when transformed to an appropriate set of variables that can be solved analytically. The final chapter of this thesis concerns the experimental work exploring the surface
flow profile for different dry and wet granular materials.
Dive into the research topics of 'Hydrodynamic theory of wet particle systems'. Together they form a unique fingerprint. | {"url":"https://research.utwente.nl/en/publications/hydrodynamic-theory-of-wet-particle-systems","timestamp":"2024-11-03T05:46:29Z","content_type":"text/html","content_length":"56323","record_id":"<urn:uuid:5ce09bf6-dc00-46f0-a02c-330446943225>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00036.warc.gz"} |
MMMCCLXVI in Hindu Arabic Numerals
MMMCCLXVI = 3266
M C X I
MM CC XX II
MMM CCC XXX III
CD XL IV
D L V
DC LX VI
DCC LXX VII
DCCC LXXX VIII
CM XC IX
MMMCCLXVI is valid Roman numeral. Here we will explain how to read, write and convert the Roman numeral MMMCCLXVI into the correct Arabic numeral format. Please have a look over the Roman numeral
table given below for better understanding of Roman numeral system. As you can see, each letter is associated with specific value.
Symbol Value
I 1
V 5
X 10
L 50
C 100
D 500
M 1000
How to write Roman Numeral MMMCCLXVI in Arabic Numeral?
The Arabic numeral representation of Roman numeral MMMCCLXVI is 3266.
How to convert Roman numeral MMMCCLXVI to Arabic numeral?
If you are aware of Roman numeral system, then converting MMMCCLXVI Roman numeral to Arabic numeral is very easy. Converting MMMCCLXVI to Arabic numeral representation involves splitting up the
numeral into place values as shown below.
M + M + M + C + C + L + X + V + I
1000 + 1000 + 1000 + 100 + 100 + 50 + 10 + 5 + 1
As per the rule highest numeral should always precede the lowest numeral to get correct representation. We need to add all converted roman numerals values to get our correct Arabic numeral. The Roman
numeral MMMCCLXVI should be used when you are representing an ordinal value. In any other case, you can use 3266 instead of MMMCCLXVI. For any numeral conversion, you can also use our roman to number
converter tool given above.
Current Date and Time in Roman Numerals
The current date and time written in roman numerals is given below. Romans used the word nulla to denote zero because the roman number system did not have a zero, so there is a possibility that you
might see nulla or nothing when the value is zero. | {"url":"https://romantonumber.com/mmmcclxvi-in-arabic-numerals","timestamp":"2024-11-12T02:59:00Z","content_type":"text/html","content_length":"89980","record_id":"<urn:uuid:5f36063a-144a-4f9b-bd1e-8e0138f15a09>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00542.warc.gz"} |
Gravitational Differential Expansion: The Hypothesis of the Driving Mechanism of the Gravitational Center on the Expansion of the Universe
Ranzan C, 2016, The Nature of Gravitational Collapse. American Journal of Astronomy and Astrophysics, 4(2): 15–33.
Thalman B, 2023, Gravity is not Attraction; It’s a Push (Space-Time Expansion Theory). Open Journal of Philosophy, 13(01): 48–75.
Seoane PA, Aoudia S, Audley H, et al., 2013, The Gravitational Universe, report, Albert Einstein Institute, 1305.5720.
Ni WT, 2017, One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity, Volume 2. World Scientific, Singapore.
Bouche F, Capozziello S, Salzano V, 2022, Addressing Cosmological Tensions by non-Local Gravity. Universe, 9(1): 27.
Ranzan C, 2018, The Nature of Gravity: How One Factor Unifies Gravity’s Convergent, Divergent, Vortex, and Wave Effects. International Journal of Astrophysics and Space Science, 6(5): 73–92.
Li B, Shapiro PR, 2021, Precision Cosmology and the Stiff-amplified Gravitational: Wave Background from Inflation: NANOGrav, Advanced LIGO-Virgo and the Hubble Tension. Journal of Cosmology and
Astroparticle Physics, 2021(10): 24.
Netchitailo V, 2022, Paradigm Shift for Cosmology and Classical Physics, unpublished draft.
Ranzan C, 2018, Natural Mechanism for the Generation and Emission of Extreme Energy Particles. Physics Essays, 31(3): 358–376.
Papanikolaou T, 2022, Gravitational Waves Induced from Primordial Black Hole Fluctuations: The Effect of an Extended Mass Function. Journal of Cosmology and Astroparticle Physics, 2022(10): 89.
Du H, 2021, Observation of Astronomical Antigravity: Origin of Astronomical Jets, Black Hole Radiation, Cosmic Gamma Ray, Fast Radio Burst, Supernova Explosion and Many More, preprint.
Netchitailo VS, 2021, Decisive Role of Dark Matter in Cosmology. Journal of High Energy Physics, Gravitation and Cosmology, 8(1): 115–142.
Cooper K, 2020, Origins of the Universe: The Cosmic Microwave Background and the Search for Quantum Gravity. Icon Books, London.
Netchitailo VS, 2022, Cosmology and Classical Physics. Journal of High Energy Physics, Gravitation and Cosmology, 8(4): 1037–1072.
Erickcek AL, 2009, The Consequences of Modifying Fundamental Cosmological Theories, thesis, California Institute of Technology.
Wick M, 2015, Megaphysics II; An Explanation of Nature: The Equation of Everything in Terms of Cosmology, Strings and Relativity. AuthorHouse, Indiana.
Baker T, Barreira A, Desmond H, et al., 2021, Novel Probes Project: Tests of Gravity on Astrophysical Scales. Reviews of Modern Physics, 93(1): 015003.
Vijaykumar A, 2024, Exploring Gravity, Astrophysics, and Cosmology with Gravitational Waves, thesis, Tata Institute of Fundamental Research.
Cardenas-Avendano A, Sopuerta CF, 2024, Testing Gravity with Extreme-mass-ratio Inspirals, in Recent Progress on Gravity Tests: Challenges and Future Perspectives. Springer Nature Singapore,
Singapore, 275–359.
Pitkanen M, 2019, Cosmic String Model for the Formation of Galaxies and Stars. https://tgdtheory.fi/public_html/articles/galaxystars.pdf
Netchitailo VS, 2021, Paradigm Shift in Cosmology, preprint.
Lou YQ, Shen W, 2021, Dynamic Spherical Collapses towards Growing Black Holes in Relativistically Degenerate or Hot Host Mass Reservoirs. Monthly Notices of the Royal Astronomical Society, 506(4):
Sakharov AS, Eroshenko YN, Rubin SG, 2021, Looking at the NANOGrav Signal through the Anthropic Window of Axionlike Particles. Physical Review D, 104(4): 043005.
Van PMH, Levinson A, 2012, Relativistic Astrophysics of the Transient Universe: Gravitation, Hydrodynamics and Radiation. Cambridge University Press, Cambridge.
Van PMH, Levinson A, Frontera F, et al., 2019, Prospects for Multi-messenger Extended Emission from Core-collapse Supernovae in the Local Universe. The European Physical Journal Plus, 134(10): 537.
Colpi M, Sesana A, 2017, Gravitational Wave Sources in the Era of Multi-band Gravitational Wave Astronomy, in An Overview of Gravitational Waves: Theory, Sources and Detection. World Scientific,
Singapore, 43–140. | {"url":"https://ojs.bbwpublisher.com/index.php/ssr/article/view/8330","timestamp":"2024-11-12T12:02:06Z","content_type":"text/html","content_length":"35908","record_id":"<urn:uuid:362ce1ed-c728-4768-bb83-712fbe9d9cdd>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00628.warc.gz"} |
CS 320L – Applied Discrete Mathematics Assignment #1 solved
Question 1: Hair Splitting with Set Expressions
Let us define the successor of the set A to be the set A ∪ {A}. Find the successors of the
following sets:
a) A = {x}
b) B = {x, y}
c) C = Ø
d) D = { Ø, { Ø }}
Question 2: Tautologies and Contradictions
Find out for each of the following propositions whether it is a tautology, a contradiction,
or neither (a contingency). Prove your answer.
a) [(p → q) ∧ (q → r)] → (p → r)
b) (p ∨ q ∨ r) → [(q → r) ∨ (p → q)]
Question 3: Set Operations
Let us take a look at the sets A = {x, y, z}, B = {1, 2}, C = {y, z}. List the elements of
the following sets D, E, F, G, H, and I:
a) D = (B × A) – (B × C)
b) E = 2A – 2C
c) F = 2(2B )
d) G = (A × B × C) ∩ (C × B × A)
e) H = {(a, b, c) | a, b, c∈Β ∧ b ≠ c ∧ a = b}.
f) I = {(a, b, c) | a∈A ∧ b∈B ∧ c∈C ∧ a = c}
Question 4: Cardinality
Are the following statements true for all sets A, B and C? Prove your answers.
a) |A ∪ B ∪ C| = |A – B – C|
b) |A ∪ B ∪ C| = |A| + |B| + |C| – |A ∩ B| – |A ∩ C| – |B ∩ C|
Question 5: Functions
Find out whether the following functions from R to R are injective, surjective, and/or
Bijective (no proof necessary).
a) f(z) = -z
b) f(z) = 300z5 + 4
c) f(z) = z⋅sin z
d) f(z) = z2
/(z2 + 1)
Question 6: Big-O Estimates
Give as good a big-O estimate as possible for the following complexity functions:
a) f(n) = (n⋅log n) (n2 + 2n)
b) f(n) = (2n! + 4n3
) + (2n
c) f(n) = n4 + 5 log n + n3 (n2 + 2n)
Question 7: Algorithms and Their Complexity
a) Write a simple program in pseudocode (or in Python, C, C++, or Java, but only use
basic commands so that comparisons can be counted) that receives a sequence of
integers a1, …, an as its input and determines if the sequence contains two distinct
terms x, y such that x = y2
. Once it finds such terms, it prints them and terminates; it
does not continue searching after the first find. If the program does not find any such
terms, it prints a disappointed comment and also terminates.
b) Describe the kind of input that causes worst-case time complexity for your algorithm
(only count comparisons), and explain why this is the case.
c) Provide an equation for your algorithm that describes the number of required
comparisons as a function of input length n in the worst case. For some algorithms, it
may be a good idea to first use a sum notation, but at the end you should provide a
closed-form equation, i.e., one that no longer uses the sum symbol but only
operations such as multiplication or addition of individual numbers or variables.
d) Use the big-O-notation to describe the worst-case time complexity of your algorithm.
Question 8 (Bonus Question): Venn Diagrams
Draw the Venn diagrams for the following sets:
a) A ∪ B ∪ C
b) A ∪ (B – C)
c) (A ∩ B) ∪ (A ∩ C)
d) (A ∩ B ) ∪ (A ∩ C ) | {"url":"https://codeshive.com/questions-and-answers/cs-320l-applied-discrete-mathematics-assignment-1-solved/","timestamp":"2024-11-05T04:15:53Z","content_type":"text/html","content_length":"102645","record_id":"<urn:uuid:a400e5e2-56e4-40f9-a04c-c6d69ae133fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00210.warc.gz"} |
Unleashing the Power of Quantum Computing
December 12, 2023
Unleashing the Power of Quantum Computing
Before we dive into this topic with gusto and abandon (and aplomb, of course), it’s probably only fair for me to inform you that I don’t have a clue what I’m about to be talking about, if you see
what I mean. “So, how does this differ from your other columns,” I hear you mutter under your breath. I’m obliged to admit that you have me there, and yet you are still reading, so ten points to me,
I think.
Now, you probably think that my admitted lack of knowledge will result in a somewhat short column. Well, it may, or it may not, but—based on prior experience—I wouldn’t bet on it either way if I were
What do you know about quantum computing and quantum computers? Take a moment to jot things down…
That didn’t take long, did it? I just performed a survey that involved me running around the building in which I have my office asking everyone who couldn’t run away fast enough “Have you heard about
quantum computers?” I was surprised to discover that quite a few people had heard this term. I was less surprised to discover that none of them knew anything more. When I asked one person who said
he’d heard about them, “What do you know about them?” he replied, “What? They exist?” It turned out he’d been exposed to the concept in a science fiction film, resulting in him thinking that quantum
computers were the stuff only of science fiction.
At this point I was tempted to start throwing some statistics around but, as Homer Simpson famously said, “People can come up with statistics to prove anything… forty percent of people know that.”
Also, my own faith in statistics was degraded when I read the classic How to Lie with Statistics by Darrell Huff, which I have sitting on the bookshelves here in my office, so let us proceed sans
I’m going to go out on a limb here by saying I believe most people know nothing about quantum computing other than the name (if that). I’d go further to say that even the majority of people with an
interest in science, technology, and engineering know little more than the fact the basic unit of quantum information is the quantum bit, or qubit. Also, they’ve probably heard that a quantum
computer can solve in seconds problems that would take classical computers anywhere from thousands of millions of years to solve, assuming classical computers could solve such problems at all.
If you really are starting at ground zero, then there was a 13-minute segment of 60 Minutes a couple of weeks ago that might prove interesting.
As is often the case in this sort of thing, American theoretical physicist, activist, futurologist, and popular-science writer, Michio Kaku, makes an appearance. The poignant point for me was at the
end of the video when Michio says, “The language of the universe is the language of the quantum.” It’s a shame I have no talent for languages.
But what does this all mean? Well, in a classical computer, the fundamental unit of information is the bit, which can adopt one of two states: 0 or 1. As we already noted, in a quantum computer, the
fundamental unit of information is the qubit. In a way, a qubit is also a 2-state system in that it involves something like electron spin (up or down) or photon polarization (left-handed or
In a classical system, a bit is in one state or the other (unless it’s metastable, in which case all bets are off). In a quantum system, a qubit exists in a coherent superposition of both states
simultaneously, which basically means it represents all possible values at once (and then things start to get complicated).
Some problems are easy, and other problems are hard. An example of the latter is called the travelling salesman problem (TSP). This starts by asking the following question: “Given a list of cities
and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once before returning to the original city?” Suffice it to say that solving this
problem—which is classed as an NP-Hard problem—using a classical computer is a lot harder than you might think.
Another interesting problem, one that is illustrated in the video above, involves a mouse solving a maze. If we model this on a classical computer, then the mouse exhaustively attempts one path after
another, reversing direction when it meets a dead end, before finally emerging triumphant at the exit. By comparison, the way a quantum computer solves this problem—at least as described in the
video—is to look at all possible paths simultaneously and select the right one. This isn’t the way it really works, but we will return to that in a moment.
One of the things that puzzled scientists for a long time is the efficiency of photosynthesis in green plants and cyanobacteria. Specifically, how they manage to achieve close to 100% efficiency when
transferring the energy in photons of sunlight to molecular reaction centers for conversion into chemical energy. It turns out “quantum” is the answer. I remember reading an article several years ago
that explained how scientists fired photons of light at a photosynthetic cell while also hitting it with femtosecond pulses of laser light. By this means, they observed what appeared to be each
photon “looking ahead” to check out all possible paths from a receptor on the cell’s surface to a molecular reaction center inside the cell before settling on the lowest energy path. The next photon
hitting the same receptor might take a completely different path to the same destination. Once again, this “looking ahead” analogy isn’t the way things really work and, once again, we will return to
this in a moment.
One final example of why all this is of interest involves protein folding, which is the physical process whereby a protein chain is translated from a randomly-shaped string into its native
three-dimensional structure, which corresponds to its lowest energy configuration. As mind-blowingly complicated as this is, it becomes exponentially more complex when multiple proteins are
interacting with each other. At this point I’d like to refer you to one of my favorite books of all time: Wetware: A Computer in Every Living Cell by Dennis Bray.
These are the sorts of problems that classical computers take forever to solve, while quantum computers offer the promise of being able to solve the same problems in seconds. Of course, they also
offer the promise of being able to solve problems that were best left unsolved, like cracking cryptographic keys, but that’s a topic for another day.
So, are quantum computers real? Well, yes, sort of. Modern quantum theory developed in the 1920s to explain the wave-particle duality observed at atomic scales, where wave-particle duality refers to
the fact that quantum entities exhibit particle or wave properties according to the experimental circumstances by which they are being observed and measured.
The concept of quantum computers was proposed in the 1980s by Richard Feynman and Yuri Manin. In 1994, Peter Shor showed that a quantum computer would be able to break RSA encryption, which caused a
lot of people to start wearing their frowny faces. The first real quantum computer that could be loaded with real data and output a real solution was a 2-qubit machine created in 1998 by Isaac Chuang
of the Los Alamos National Laboratory, Neil Gershenfeld of the Massachusetts Institute of Technology (MIT), and Mark Kubinec of the University of California at Berkeley.
What’s the current state-of-the-art? I have no idea. All I know is that IBM unveiled its 433 qubit Osprey processor in November 2022, and Atom Computing announced an 1180 qubit machine in October
The reason I’m waffling on about all this is that I was just chatting with Earl Campbell, who is VP of Quantum Science at Riverlane. We started by my telling Earl what I knew of quantum computing,
and his telling me that everything I thought was wrong (but he was jolly nice, and he was smiling as he said it, and—unlike my wife (Gina the Gorgeous)—he didn’t imply that I was a complete
knucklehead, so that was all right).
A futuristic glowing CPU quantum computer processor. 3D illustration.
One way to visualize a next-generation quantum computer (Source: Riverlane)
One of the things I’ve long been confused about was how to make any sense of the results from a quantum computation. Earl explained that we begin with a classical binary representation on the inputs
to the machine, we have the quantum monster in the middle of the machine, and the results are presented in a classical binary representation on the outputs of the machine. In the case of the quantum
portion of the machine, my new understanding is that thinking in terms of fixed-size floating-point numbers is meaningless. The way I now think about each qubit is that it represents an imaginary
number with infinite precision (of course, I may be wrong). The thing is that, as explained by Heisenberg’s uncertainty principle (which states that there is a limit to the precision with which
certain pairs of physical properties, such as position and momentum, can be simultaneously known), we can’t really tell what’s happening in the quantum part of the machine, we just have to say “Ooh”
and “Aah” when it presents us with the results.
Another thing Earl said was that the maze analogy I mentioned earlier was fundamentally wrong. The way to think about things is to start by considering what happens when we randomly drop a handful of
variably-sized pebbles into a pond. Each pebble will generate ripples. The ripples from all the pebbles will combine (interfere with each other) in weird and wonderful (constructive and destructive)
ways. Returning to the maze (or any other quantum problem), the quantum elements start in all states simultaneously, each state combines with every other state constructively and destructively, and
the system wave function collapses to provide the answer, which is always 42 (I think that’s what Earl said).
Unfortunately, there’s a fly in the soup and an elephant in the room (I never metaphor I didn’t like) that manifests itself in the form of quantum noise, which leads to quantum errors. In the case of
classical computers, we tend to think about our 0s and 1s as corresponding to two different voltages—let’s say 0V and 5V, respectively, which shows how old I am. In reality, our 0s and 1s correspond
to voltage bands—so anything below 1V corresponds to a logic 0, while anything about 4V corresponds to a logic 1, for example. Now, although we don’t like to think about it, errors occur in our
classical digital computers all the time, like bits flipping in memory due to random radiation, for example. The answer is to employ things like error correcting code (ECC) memory with additional
bits used to detect and correct errors.
Do you remember VHS video cassettes, which were analog in nature? If you took a video at a party and made a copy for a friend, and that friend made a copy for another friend, and that friend… you see
where I’m going. It didn’t take long before replication errors compounded making the later copies unwatchable. By comparison, a digital representation like a CD or DVD includes error detecting and
correcting codes that maintain the fidelity of the data, which means you can make copies of copies ad infinitum, with the last copy being identical to the original (at least in terms of its 0s and
Now think about quantum computers with their qubits being in every state at once (sort of—I can imagine Earl wincing as he reads this), and quantum noise, and quantum errors. Can we detect and
correct such errors? Once again, Heisenberg’s uncertainty principle comes into play, because trying to observe and measure the state of a quantum system like a qubit changes its state causing its
wave function to collapse.
To be honest, for a long time a lot of people thought this was going to prove to be an unsolvable problem, all the way until the boffins at Riverlane solved it. As Earl told me, “This is what
Riverlane is working on. Solving this problem involves working with massive volumes of data, which can be hundreds of terabytes of data every second. We like to compare it to the volume of traffic on
Netflix. The entire global traffic on Netflix would be the same amount of data that you’ll be looking at to run a commercial-grade quantum computer and decoding that.”
All of which leads us to the fact that the guys and gals at Riverlane have announced The World’s Most Powerful Quantum Decoder and The World’s First Quantum Error Correction Chip.
One way to visualize a next-generation quantum computer (Source: Riverlane)
I know that I’ve waffled far too long on a subject I know nothing about, so let’s summarize things as follows. Quantum error correction (QEC) is one of the worst kept secrets as to what’s holding
quantum computing back. QEC is a major obstacle in the way of practical quantum scaling. Without error correction, there is no path for useful quantum computers.
In a crunchy nutshell, QEC is a set of techniques used to protect the information stored in qubits from errors and decoherence caused by noise. QEC involves generating a continuous stream of data,
and a sophisticated algorithmic process called “decoding” is needed to process this data.
The chaps and chapesses at Riverlane have a singular focus on QEC. They recently introduced a decoder chip in the form of an FPGA that demonstrates how QEC can help scale quantum computers to useful
implementations. In fact, the little scamps have just published a paper in the prestigious journal Nature on this very topic.
All I can say is that I, for one, am (a) very impressed and (b) very confused. I have no idea how people wrap their brains around this stuff. I hope to visit the folks at Riverlane sometime. Until
that frabjous day, I fear I will imagine this as a company composed of Sheldon Coopers and Amy Farrah Fowlers. What say you? What are your thoughts on quantum computing?
You must be logged in to post a comment. | {"url":"https://www.eejournal.com/article/unleashing-the-power-of-quantum-computing/","timestamp":"2024-11-12T19:22:34Z","content_type":"text/html","content_length":"192279","record_id":"<urn:uuid:2f466b33-e486-4834-b754-14552ca64935>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00301.warc.gz"} |
Joseph Stefan
Joseph Stefan
Joseph Stefan (Slovene Jožef Stefan) (March 24, 1835 January 7, 1893) was a Slovene-Austrian physicist, mathematician and poet.
Nekaj bode zmeraj še ostalo, There always something will remain,
da ne bomo vedeli, zakaj? that we shall not know, why?
Jožef Stefan, Naturoznanske poskušnje (The Science of Nature trials), 1859
Stefan was born in St. Peter (Slovene Sveti Peter) a district of Klagenfurt (Slovene Celovec) in Austria-Hungary (now in Austria) to father Aleš (Aleksander) Stefan, born in 1805 and mother Marija
Startinik, born 1815. His parents, both ethnic Slovenes, were married when Jožef was eleven. The Stefans were a modest family. His father was a milling assistant and mother served as a maidservant.
Stefan's father died in 1872 while his mother died almost ten years earlier in 1863.
Stefan attended elementary school in Klagenfurt, where he showed his talent. They recommended that he continue his schooling, so in 1845 he went to Klagenfurt gymnasium. He experienced the
revolutionary year of 1848, as a thirteen-year-old boy, which inspired him to be sympathetic toward Slovene literary production.
When he had finished gymnasium as the best student in his class he thought for a while of joining the Benedictine order but he soon abandoned his idea, because his great interest for physics
prevailed. He left for Vienna in 1853 to study mathematics and physics. His professor of physics in gymnasium was Karel Robida who wrote the first Slovene physics textbook. Stefan then graduated in
mathematics and physics at the University of Vienna in 1857. During his student years, he also wrote and published a number of poems in Slovene. He taught physics at the University of Vienna, was
Director of the Physical Institute from 1866, Vice-President of the Vienna Academy of Sciences and member of several scientific institutions in Europe.
He published nearly 80 scientific articles, mostly in the Bulletins of the Vienna Academy of Sciences, and he is best known for originating a physical power law in 1879 stating that the total
radiation from a black body j* is proportional to the fourth power of its thermodynamic temperature T:
In 1884 the law derived theoretically in the framework of thermodynamics by his student Ludwig Boltzmann and hence known as the Stefan-Boltzmann law. Boltzmann treated a heat engine with light as a
working matter. This law is the only physical law of nature named after a Slovene physicist. Today we derive the law from Planck's law of black body radiation:
and is valid only for ideal black objects. With his law Stefan determined the temperature of the Sun's surface and he calculated a value of 5430 °C. This was the first sensible value for the
temperature of the Sun.
Stefan provided the first measurements of the thermal conductivity of gases, treated evaporation, and among others studied diffusion, heat conduction in fluids. For his treatise on optics he received
the Richard Lieben award from the University of Vienna. Flow from a droplet or particle that is induced by evaporation or sublimation at the surface is now called Stefan flow because of his early
work in calculating evaporation and diffusion rates.
Very important are also his electromagnetic equations, defined in vector notation, and works in the kinetic theory of heat. He was among the first physicists in Europe who fully understood Maxwell's
electromagnetic theory and one of the few outside of England who expand on it. He calculated inductivity of a coil with a quadratic cross-section, and he corrected Maxwell's miscalculation. He also
researched a phenomenon called the skin effect, where high-frequency electric current is greater on the surface of a conductor than in its interior.
In mathematics the Stefan problems or Stefan's tasks with movable boundary are well known. The problem was first studied by Lamé and Clapeyron in 1831. Stefan solved the problem when he was
calculating how quickly a layer of ice on water grows.
He died in Vienna, Austria-Hungary.
His life and work is extensively studied by the physicist Janez Strnad.
Stefan-Boltzmann constant σ
Stefan's force
Jožef Stefan: http://www.ijs.si/ijsw/Jo%C5%BEef_Stefan
Jožef Stefan Institute, Ljubljana, Slovenia: http://www.ijs.si/
University of St. Andrews page about Jožef Stefan: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Stefan_Josef.html
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
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Asymptotic behavior of \(t\)-uniform hypergraph Ramsey numbers
A \( t\)-graph has a vertex set \( V\) and an edge set \( E\) consisting of some prescribed set of \( t\)-subsets of \( V\). For \( t\)-graphs \( G_i\), \( i=1,\dots,k\), let \( r_t(G_1,\dots,G_k)\)
denote the smallest integer \( m\) satisfying the property that if the edges of the complete \( t\)-graph on \( m\) vertices are colored in \( k\) colors, then for some \( i\), \( 1\leq i\leq k\),
there is a subgraph isomorphic to \( G_i\) with all \( t\)-edges in the \( i\)-th color. We denote \( r_t(n_1, \dots, n_k)= r_t(K_{n_1}, \dots, K_{n_k})\).
Conjecture (proposed by Erdös, Hajnal and Rado [1])
For every \(t \geq 3\), \[ c \log_{t-1} n < r_t(n,n) < c' \log _{t-1} n \] where \(\log _u n \) denotes the \(u\)-fold iterated logarithm and \(c\) and \(c'\) depend only on \(t\).
1 P. Erdös, A. Hajnal and R. Rado. Partition relations for cardinal numbers, Acta Math. cad. Sci. Hungar. 16 (1965), 93-196. | {"url":"https://mathweb.ucsd.edu/~erdosproblems/erdos/newproblems/THypergraphRamsey.html","timestamp":"2024-11-02T12:17:07Z","content_type":"text/html","content_length":"4526","record_id":"<urn:uuid:c2434932-d936-4611-80ee-cd1100dc4945>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00149.warc.gz"} |
: No additional bug fixes or documentation updates will be released for this version. For the latest information, see the
current release documentation
This functionality is in technical preview and may be changed or removed in a future release. Elastic will work to fix any issues, but features in technical preview are not subject to the support SLA
of official GA features.
Regression analysis is a machine learning process for estimating the relationships among different fields in your data, then making further predictions based on these relationships.
For example, suppose we are interested in finding the relationship between apartment size and monthly rent in a city. Our imaginary data set consists of three data points:
│ Size (m2) │ Monthly rent │
│ 44 │ 1600 │
│ 24 │ 1055 │
│ 63 │ 2300 │
After the model determines the relationship between the apartment size and the rent, it can make predictions such as the monthly rent of a hundred square meter-size apartment.
This is a simple example. Usually regression problems are multi-dimensional, so the relationships that regression analysis tries to find are between multiple fields. To extend our example, a more
complex regression analysis could take into account additional factors such as the location of the apartment in the city, on which floor it is, and whether the apartment has a riverside view or not,
and so on. All of these factors can be considered features; they are measurable properties or characteristics of the phenomenon we’re studying.
When you perform regression analysis, you must identify a subset of fields that you want to use to create a model for predicting other fields. We refer to these fields as feature variables and
dependent variables, respectively. Feature variables are the values that the dependent variable value depends on. If one or more of the feature variables changes, the dependent variable value also
changes. There are three different types of feature variables that you can use with our regression algorithm:
• Numerical. In our example, the size of the apartment was a numerical feature variable.
• Categorical. A variable that can have one value from a set of values. The value set has a fixed and limited number of possible items. In the example, the location of the apartment in the city
(borough) is a categorical variable.
• Boolean. The riverside view in the example is a boolean value because an apartment either has a riverside view or doesn’t have one. Arrays are not supported.
Regression is a supervised machine learning method, which means that you need to supply a labeled training data set that has some feature variables and a dependent variable. The regression algorithm
identifies the relationships between the feature variables and the dependent variable. Once you’ve trained the model on your training data set, you can reuse the knowledge that the model has learned
to make inferences about new data.
The relationships between the feature variables and the dependent variable are described as a mathematical function. Regression analysis tries to find the best prediction for the dependent variable
by combining the predictions from multiple base learners – algorithms that generalize from the data set. The performance of an ensemble is usually better than the performance of each individual base
learner because the individual learners will make different errors. These average out when their predictions are combined.
Regression works as a batch analysis. If new data comes into your index, you must restart the data frame analytics job.
The ensemble learning technique that we use in the Elastic Stack is a type of boosting called extreme gradient boost (XGboost) which combines decision trees with gradient boosting methodologies.
The model that you created is stored as Elasticsearch documents in internal indices. In other words, the characteristics of your trained model are saved and ready to be deployed and used as
functions. The inference feature enables you to use your model in a preprocessor of an ingest pipeline to make predictions about your data.
A loss function measures how well a given machine learning model fits the specific data set. It boils down all the different under- and overestimations of the model to a single number, known as the
prediction error. The bigger the difference between the prediction and the ground truth, the higher the value of the loss function. Loss functions are used automatically in the background during
hyperparameter optimization and when training the decision trees to compare the performance of various iterations of the model.
In the Elastic Stack, there are three different types of loss function:
• mean squared error (mse): It is the default choice when no additional information about the data set is available.
• mean squared logarithmic error (msle; a variation of mse): It is for cases where the target values are all positive with a long tail distribution (for example, prices or population).
• Pseudo-Huber loss (huber): Use it when you want to prevent the model trying to fit the outliers instead of regular data.
The various types of loss function calculate the prediction error differently. The appropriate loss function for your use case depends on the target distribution in your data set, the problem that
you want to model, the number of outliers in the data, and so on.
You can specify the loss function to be used during regression analysis when you create the data frame analytics job. The default is mean squared error (mse). If you choose msle or huber, you can
also set up a parameter for the loss function. With the parameter, you can further refine the behavior of the chosen functions.
Consult the Jupyter notebook on regression loss functions to learn more.
The default loss function parameter values work fine for most of the cases. It is highly recommended to use the default values, unless you fully understand the impact of the different loss function
Feature importance provides further information about the results of an analysis and helps to interpret the results in a more subtle way. If you want to learn more about feature importance, click
You can measure how well the model has performed on your training data set by using the regression evaluation type of the evaluate data frame analytics API. The mean squared error (MSE) value that
the evaluation provides you on the training data set is the training error. Training the regression model means finding the combination of model parameters that produces the lowest possible training
Another crucial measurement is how well your model performs on unseen data points. To assess how well the trained model will perform on data it has never seen before, you must set aside a proportion
of the training data set for testing. This split of the data set is the testing data set. Once the model has been trained, you can let the model predict the value of the data points it has never seen
before and compare the prediction to the actual value. This test provides an estimate of a quantity known as the model generalization error.
Two concepts describe how well the regression algorithm was able to learn the relationship between the feature variables and the dependent variable. Underfitting is when the model cannot capture the
complexity of the data set. Overfitting is when the model is too specific to the training data set and is capturing details which do not generalize to new data. A model that overfits the data has a
low MSE value on the training data set and a high MSE value on the testing data set. For more information about the evaluation metrics, see Regression evaluation. | {"url":"https://www.elastic.co/guide/en/machine-learning/7.8/dfa-regression.html","timestamp":"2024-11-05T03:59:47Z","content_type":"text/html","content_length":"26550","record_id":"<urn:uuid:c8208f47-2e6a-475f-9acf-ef6fd9b799b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00497.warc.gz"} |
• Add transmission argument to create_simulator to allow for frequency-dependent transmission. Density-dependent transmission remains the default model.
• Add vector of first difference of the variance vector produced by get_stats. This change makes it easier to use the convexity of the variance time series as an early warning signal. The name of
the vector in the stats list is variance_first_diff. Note that this change makes the abbreviation stats$var ambiguous. Code using that abbreviation to obtain the vector of variance estimates
should substitute in stats$variance.
• To the output of get_stats(), add list taus containing Kendall’s correlation coefficient of the elements of each time series in the stats list in the output with time.
• Ensure variance and kurtosis estimates are non-negative. When using local linear for estimating statistics, it was possible in previous versions for negative values to occur. | {"url":"https://cran.ma.imperial.ac.uk/web/packages/spaero/news/news.html","timestamp":"2024-11-13T21:50:23Z","content_type":"application/xhtml+xml","content_length":"3649","record_id":"<urn:uuid:f04e9bd0-7cc0-4c64-b77e-b16e6dae1461>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00896.warc.gz"} |
This number is a prime.
The smallest number that is the sum of twice a positive square and a prime in 53 ways. Note it's also a palindromic prime.
The only five-digit number such that its nth digit equals the remainder when the number is divided by n + 1. [Rupinski]
The smallest palindromic prime whose alternate digits (11311) segregate and reunite to form two "daughter" palindromic primes (131 ; 11). [Beedassy]
The smallest prime equidistant between the members of an emirp pair: (10321, 12301). Note that 11311 is a reflectable palindromic prime of form "primemirp." [Beedassy]
(There are 4 curios for this number that have not yet been approved by an editor.)
Printed from the PrimePages <t5k.org> © G. L. Honaker and Chris K. Caldwell | {"url":"https://t5k.org/curios/page.php?number_id=432","timestamp":"2024-11-05T00:10:33Z","content_type":"text/html","content_length":"9764","record_id":"<urn:uuid:237a9a5d-922c-4519-9257-78ee8810ec59>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00824.warc.gz"} |
What is easy in algebra?
what is easy in algebra? Related topics: college algebra problems
two dimension diagram used for solving certain type of equations
freeware algebra solver
free graphing algebra worksheets
ks2 algebra trivia
free down load alegra multiplication grouping for multiplication problems for third graders
maths objective questions for 6th standard
college algebra prerequisite algebra skills
online graphing calculator inequalities
adding and subtracting exponents in non like terms
simplifying algebraic expressions
solution of quadratic equations
can someone give me a example on how to solve fractions
Author Message
pmamethah Posted: Friday 21st of Sep 08:12
I am in desperate need of help in completing an assignment in what is easy in algebra?. I need to submit it by next week and am having a hard time trying to work out a couple of tricky
problems. I tried some of the online help sites but have not gotten much help so far. I would be really grateful if anyone can help me.
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IlbendF Posted: Friday 21st of Sep 18:13
Algebrator is the latest hot favourite of what is easy in algebra? students. I know a couple of tutors who actually ask their students to have a copy of this program at their residence
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Matdhejs Posted: Saturday 22nd of Sep 08:35
Algebrator will not only help you do your homework , but it will also provide explanations which will help you understand the concepts.
From: The
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Vnode Posted: Sunday 23rd of Sep 21:19
I remember having often faced problems with distance of points, gcf and linear inequalities. A truly great piece of math program is Algebrator software. By simply typing in a problem
homework a step by step solution would appear by a click on Solve. I have used it through many algebra classes – Algebra 1, Remedial Algebra and Basic Math. I greatly recommend the
From: Germany
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Program for Tuesday, April 16th
PROGRAM FOR TUESDAY, APRIL 16TH
next day
all days
View: session overviewtalk overview
10:30-12:30 Session 1: Reachability
Falsification of Hybrid Systems using Symbolic Reachability and Trajectory Splicing
ABSTRACT. The falsification of a hybrid system aims at finding trajectories that violate a given safety property. This is a challenging problem, and the practical applicability of current
falsification algorithms still suffers from their high time complexity. In contrast to falsification, verification algorithms aim at providing guarantees that no such trajectories exist. Recent
10:30 symbolic reachability techniques are capable of efficiently computing linear constraints that enclose all trajectories of the system with reasonable precision. In this paper, we leverage the
power of symbolic reachability algorithms to improve the scalability of falsification techniques. Recent approaches to falsification reduce the problem to a nonlinear optimization problem. We
propose to reduce the search space of the optimization problem by adding linear state constraints obtained with a reachability algorithm. We showcase the efficiency of our approach on a number
of standard hybrid systems benchmarks.
Inner and Outer Reachability for the Verification of Control Systems
ABSTRACT. We investigate the information and guarantees provided by different inner and outer approximated reachability analyses, for proving properties of dynamical systems.We explore the
11:00 connection of these approximated sets with the maximal and minimal reachable sets of Mitchell [33], with an additional notion of robustness to disturbance, as well as invariance and viability
kernels. We demonstrate the practical use of a specific computation of these approximated reachable sets. We revisit in particular the reach-avoid properties. We also show how we can prove some
new properties, such as every state of a given region is sure to be reached by the system, in presence of disturbances.
Numerical Verification of Affine Systems with up to a Billion Dimensions
ABSTRACT. Affine systems reachability is the basis of many verification methods. With further computation, methods exist to reason about richer models that have inputs, nonlinear differential
equations, and hybrid dynamics. As such, the scalability of affine systems verification is a prerequisite to the scalability of analysis methods for more complex systems. In this paper, we
strive to improve the scalability of affine systems verification, in terms of the number of dimensions (variables) in the system.
One benefit of affine systems is that their reachable states can be written in terms of the matrix exponential, and safety checking can be performed at specific time steps with linear
programming. Unfortunately, for large systems with many state variables, this direct approach requires an intractable amount of memory while using an intractable amount of computation time. We
overcome these two problems by combining several methods that leverage common problem structure. Memory demands can be reduced by taking advantage of both initial states that are not
full-dimensional, and safety properties (outputs) that only need a few linear projections of the state variables. Computation time is saved by using numerical simulations to compute only
projections of the matrix exponential relevant for the verification problem. Since large systems often have sparse dynamics, we use fast Krylov-subspace simulation methods based on the Arnoldi
or Lanczos iterations. Our implementation produces accurate counter-examples when properties are violated and, in the extreme case with sufficient problem structure, is shown to analyze a
system with one billion real-valued state variables.
SReachTools: A MATLAB Stochastic Reachability Toolbox
ABSTRACT. We present SReachTools, an open-source MATLAB toolbox for performing stochastic reachability of linear discrete-time systems that are perturbed by additive Gaussian noise. The toolbox
addresses the problem of stochastic reachability of a target tube, which also encompasses the terminal-time hitting reach-avoid and viability problems. The stochastic reachability of a target
12:00 tube problem maximizes the likelihood that the state of a stochastic system will remain within a time-varying target tube for a give time horizon, while respecting the system dynamics and
bounded control authority. SReachTools implements several new algorithms, based on convex optimization, computational geometry, and on Fourier transforms, to efficiently compute over- and
under-approximations of the stochastic reach set. SReachTools can be used to perform probabilistic verification of closed-loop systems, by providing probabilistic guarantees of safety and
performance. In addition, SReachTools can also perform controller synthesis to assure probabilistic safety or performance, via open-loop or affine controllers. The code base is designed to be
extensible and user friendly.
JuliaReach: a Toolbox for Set-Based Reachability
ABSTRACT. We present JuliaReach, a toolbox for set-based reachability analysis of dynamical systems. JuliaReach consists of two main packages: Reachability, containing implementations of
12:15 reachability algorithms for continuous and hybrid systems, and LazySets, a standalone library that implements state-of-the-art algorithms for calculus with convex sets. The library offers both
concrete and lazy set representations, where the latter stands for the ability to delay set computations until they are needed. The choice of the programming language Julia and the accompanying
documentation of our toolbox allow researchers to easily translate set-based algorithms from mathematics to software in a platform-independent way, while achieving runtime performance that is
comparable to statically compiled languages. Combining lazy operations in high dimensions and explicit computations in low dimensions, JuliaReach can be applied to solve complex, large-scale
14:00-15:30 Session 2: Temporal logics
Revisiting Timed Logics with Automata Modalities
ABSTRACT. It is well known that (timed) ω-regular properties such as ‘p holds at every even position’ and ‘p occurs at least three times within the next 10 time units’ cannot be expressed in
Metric Interval Temporal Logic (MITL) and Event Clock Logic (ECL). A standard remedy to this deficiency is to extend these with modalities defined in terms of automata. In this paper, we show
14:00 that the logics EMITL0,∞ (adding non-deterministic finite automata modalities into the fragment of MITL with only lower- and upper-bound constraints) and EECL (adding automata modalities into
ECL) are already as expressive as EMITL (full MITL with automata modalities). In particular, the satisfiability and model-checking problems for EMITL0,∞ and EECL are PSPACE-complete, whereas
the same problems for EMITL are EXPSPACE-complete. We also provide a simple translation from EMITL0,∞ to diagonal-free timed automata, which enables practical satisfiability and model checking
based on off-the-shelf tools.
Interface-Aware Signal Temporal Logic
ABSTRACT. Safety and security are major concerns in the development of Cyber-Physical Systems (CPS). Signal temporal logic (STL) was proposed as a language to specify and monitor the
correctness of CPS relative to formalized requirements. Incorporating STL into a development process enables designers to automatically monitor and diagnose traces, compute robustness estimates
based on requirements, and perform requirement falsification. These capabilities lead to potential high productivity gains in verification and validation activities; however, in its current
form STL is agnostic to the input/output classification of signals, and this negatively impacts the relevance of the analysis results.
In this paper we propose to make the interface explicit in the STL language by introducing input/output signal declarations. We then define new measures of input vacuity and output robustness
that better reflect the nature of the system and the specification intent. The resulting framework, which we call interface-aware signal temporal logic (IA-STL), aids verification and
validation activities. We demonstrate the benefits of IA-STL on several CPS analysis activities: (1) robustness-driven sensitivity analysis, (2) falsification and (3) fault localization. We
describe an implementation of our enhancement to STL and associated notions of robustness and vacuity in a prototype extension of Breach, a MATLAB/Simulink toolbox for CPS verification and
validation. We explore these methodological improvements and evaluate our results on two examples from the automotive domain, an academic benchmark automatic transmission control model and an
industrial hydrogen fuel cell system.
Temporal Logic Robustness for General Signal Classes
ABSTRACT. In multi-agent systems, robots transmit their planned trajectories to each other or to a central controller, and each receiver plans its own actions by maximizing a measure of mission
satisfaction. For missions expressed in temporal logic, the \textit{robustness function} plays the role of satisfaction measure. Currently, a Piece-Wise Linear (PWL) or piece-wise constant fit
is used at the receiver to reconstruct the continuous-time signal from the received samples. This allows an efficient robustness computation algorithm - a.k.a. \textit{monitoring} - but is not
15:00 adaptive to the signal class of interest, and does not leverage the compression properties of more general representations. When communication capacity is at a premium, this is a serious
bottleneck. In this paper we first show that the robustness computation is significantly affected by how the continuous-time signal is reconstructed from the received samples, which can mean
the difference between a successful control and a crash. We show that monitoring general spline-based reconstructions yields a smaller robustness error, and that it can be done with in the same
time complexity as monitoring the simpler PWL reconstructions. Thus robustness computation can now be adapted to the signal class of interest. We further show that the monitoring error is
tightly upper-bounded by the $L_\infty$ signal reconstruction error. We present a (non-linear) $L_\infty$-based scheme which yields even lower monitoring error than the spline-based schemes
(which have the advantage of being faster to compute), and illustrate all results on two case studies. As an application of these results, we show how time-frequency specifications can be
efficiently monitored online.
16:00-17:30 Session 3: Decidability and complexity
On the Decidability of Reachability in Linear Time-Invariant Systems
ABSTRACT. We consider the decidability of state-to-state reachability in discrete- time linear time-invariant control systems, with control sets defined by boolean combinations of linear
inequalities. It is a fundamental result in control theory that the version of this problem in which the control set is an affine subspace (i.e., definable by a conjunction of equations) is
decidable in polynomial time.
We show that reachability is undecidable if the control set can be a finite union of affine subspaces. We moreover show that if the control set consists of a single affine subspace together
with an additional point then reachability is as hard as Skolem's Problem for linear recurrence sequences. In like manner we show that if the control set in a convex polytope then reachability
is as hard as the Positivity Problem for linear recurrence sequences.
Our main result shows decidability of the reachability problem for convex polytopic control sets under some spectral assumptions on the transition matrix of an LTI system.
On the Decidability of Linear Bounded Periodic Cyber-Physical Systems
ABSTRACT. Cyber-Physical Systems (CPSs) are integrations of distributed computing systems with physical processes via a networking with actuators and sensors, where the feedback loops among the
16:30 components allow the physical processes to affect the computations and vice versa. Although CPSs can be found in several complex and sometimes critical real-world domains, their verification
and validation often relies on simulation-test systems rather then formal methodologies. In this contribution, we prove the decidability of the reachability problem for a significant class of
discrete-time linear CPSs whose physical process in isolation has a periodic behavior, up to a initial transitory phase.
Facetal Abstraction for Non-Linear Dynamical Systems Based on delta-Decidable SMT
17:00 ABSTRACT. Formal analysis of non-linear continuous and hybrid systems has recently attracted considerable attention. A common approach builds on computing a suitable finite discrete abstraction
of the continuous system. In this paper, we propose a facetal abstraction which eliminates certain drawbacks of existing abstractions. The states of our abstraction are built primarily from
facets of a polytopal partitioning of the system's state space taking thus into account the flow of the continuous dynamics and leading to global over-approximation. The transition system
construction is based on queries solved by a delta-decision SMT-solver. The method is evaluated on several case studies. | {"url":"https://easychair.org/smart-program/HSCC'19/2019-04-16.html","timestamp":"2024-11-07T13:38:03Z","content_type":"application/xhtml+xml","content_length":"22747","record_id":"<urn:uuid:2924d453-e824-48d9-a78f-49d15cdb0d14>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00222.warc.gz"} |
What's decidable about weighted automata?
Weighted automata map input words to numerical values. Applications of weighted automata include formal verification of quantitative properties, as well as text, speech, and image processing. In the
90's, Krob studied the decidability of problems on rational series, which strongly relate to weighted automata. In particular, it follows from Krob's results that the universality problem (that is,
deciding whether the values of all words are below some threshold) is decidable for weighted automata with weights in ℕ ∪ {∞}, and that the equality problem is undecidable when the weights are in ℤ ∪
{∞}. In this paper we continue the study of the borders of decidability in weighted automata, describe alternative and direct proofs of the above results, and tighten them further. Unlike the proofs
of Krob, which are algebraic in their nature, our proofs stay in the terrain of state machines, and the reduction is from the halting problem of a two-counter machine. This enables us to
significantly simplify Krob's reasoning and strengthen the results to apply already to a very simple class of automata: all the states are accepting, there are no initial nor final weights, and all
the weights are from the set {∈-∈1,0,1}. The fact we work directly with automata enables us to tighten also the decidability results and to show that the universality problem for weighted automata
with weights in ℕ ∪ {∞}, and in fact even with weights in ℚ^≥0 ∪ {∞}, is PSPACE-complete. Our results thus draw a sharper picture about the decidability of decision problems for weighted automata, in
both the front of equality vs. universality and the front of the ℕ ∪ {∞} vs. the ℤ ∪ {∞} domains.
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 6996 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 9th International Symposium on Automated Technology for Verification and Analysis, ATVA 2011
Country/Territory Taiwan, Province of China
City Taipei
Period 11/10/11 → 14/10/11
Dive into the research topics of 'What's decidable about weighted automata?'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/whats-decidable-about-weighted-automata-13","timestamp":"2024-11-03T03:31:29Z","content_type":"text/html","content_length":"53206","record_id":"<urn:uuid:1d01d97a-94c0-41ad-bfa5-52f7dd186572>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00648.warc.gz"} |
relation between maths and biotechnology
Relation between maths and science? Learning music improves math skills because, at some level, all music is math. Ask Question Asked today. Proof Check: Prove Relation Between Invariants Is the Only
Relation. Asked by Wiki User. As in science, the early history of mathematics is sketchy. This study explored the role of believing in an "innate" math or language arts ability (i.e., having a fixed
mindset) for gender differences in students' ability self-concept and intrinsic motivation in 423 female (49%) and 447 male (51%) tenth graders from Germany (age M = 16.09 years, SD = 0.68, range:
14-18 years). A1. But we notice music everywhere… Sometimes as soon as our radio alarm clock goes off in the morning we are surrounded by it. Mathematics Stack Exchange is a question and answer site
for people studying math at any level and professionals in related fields. Prezi Video + Unsplash: Access over two million images to tell your story through video The topic further has been divided
into six categories, with their respective explanations. It only takes a minute to sign up. Retaliation is also prohibited by university policy. Home Science Math History Literature Technology Health
Law Business All Topics Random. Since the 1960s physics has seen a rebirth of the use of advanced mathematics. It only takes a minute to sign up. We are surrounded by two things everyday… Math and
Music. of two numbers or the given n numbers first we need to know about the definition of the Highest Common Factor (H.C.F.) Relation vs Relationship. ratios are maths way of say there are eg: 3
boys for 15 so there would be 18 people in the classThe quantitative relation between two amounts showing the number of times one value contains or is contained within the other. Share 0. Viewed 2
times 0 $\begingroup$ Is there a relationship between modular form. 0 what is the relation between maths science. Math has a relationship with nature, and like any relationship it works both ways.
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interdisciplinary approach. Asked by … The Correlation Between Music and Math: A Neurobiology Perspective Cindy Zhan. Introduction Most people are familiar with sports rules and terminology; however,
they are not always aware of the important role that math plays in sports. A myriad of data on players, teams, divisions and leagues is provided by the media and the sports world. A relation from a
set X to a set Y is any subset of the Cartesian product X×Y A relation X to Y is a subset of X Y. An ordered pair(x,y) is called a relation in x and y. A function is defined as a relation in which
there is only one output for each input. In this fun explainer series, Dr. Eugenia Cheng shows everyone a way in, enhancing our understanding and enjoyment of both music and math. A set of ordered
pairs is called a two-place (or dyadic) relation; a set of ordered triples is a three-place (or triadic) relation; and so on. a forest) and only later do we come up with the abstractions (math) to
describe it. That is without bricks, the concept of a building is impossible. ... Find the relation between two sets of orthogonal polynomials. Performing music, therefore, reinforces parts of the
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decimals? Or, it is a subset of the Cartesian product. The people who shaped science are also important in mathematics. Dec. 2, 2020. Is there any relation between biotechnology and bank? What are
the types of relation in maths? Proof Check: Prove Relation Between Invariants Is the Only Relation. Relation, in logic, a set of ordered pairs, triples, quadruples, and so on. The use of analytical
methods and systems in daily practice in biotechnological research, development, application and industrial production is essential for progress in biotechnology. Both have a common letter- S. 0 ;
View Full Answer Relation between maths and science is :-Maths is also used in science. Analytical methods and systems for biotechnological applications are becoming increasingly important. 0 0 1. 0.
It also means “personal account.” In maths to result always remains finite despite the various ways in which you can add, multiply, subtract, and divide numbers. Relation Between Mathematical
Performance, Math Anxiety, and Affective Priming in Children With and Without Developmental Dyscalculia Front Psychol . Important to remember Products of biotechnology Benefits of biotechnology
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Engineering/Biotechnology) (Volume 60) Math and the inner workings of music can seem mysterious and intimidating. The patterns that exist between math, language, and music have prompted numerous
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other. ADVERTISEMENTS: Read this article to learn about the scope and applications of biotechnology. We know that the lunar and solar cycles were counted by the Babylonians and Egyptians in an
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the Least Common Multiple (L.C.M) and also LCM and HCF formulas.In this article, we are going to discuss the definition, the relation between HCF and LCM of given numbers in detail with examples.
Wiki User Answered . Product of gcd and lcm for multivariate polynomials. The University of Kansas prohibits discrimination on the basis of race, color, ethnicity, religion, sex, national origin,
age, ancestry, disability, status as a veteran, sexual orientation, marital status, parental status, gender identity, gender expression, and genetic information in the university’s programs and
activities. Photo: Eugenia Cheng. You have been selected to explore the numerous connections between math and sports. In fact, one could argue that there would be no math, only numbers, if there
weren't any relations between the numbers. Functions are our way of expressing a relation between two sets, which is of fundamental importance to all of math. Submitted by David marshal (guest) on
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should be industry focused; Dec. 1, 2020. Blog. It's about time signatures, beats per minute and formulaic progressions. The relation, R is symmetric as the distance between A & B is 5 km which is
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Puremathematicsandphysicsarebecomingevermorecloselyconnected,thoughtheirmethods … A relation is defined as a relationship between sets of values. 7/20/2015 0 Comments The history of science and math
are interrelated. Paul Crisanti, PhotoGetGo. Most of the time we don’t even notice the math or we just choose to ignore it. Biotechnology. The word “relation” is typically used in a formal context.
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Q1. Much of this revival occurred after the study of black holes was greatly expanded in the 1960s and 1970s by the English scientists Stephen Hawking (1942- ) and Roger Penrose (1931- ). Sign up to
join this community. , multiply, subtract, and divide numbers expressing a relation in x and y which you can add multiply. Solar cycles were counted by the Babylonians and Egyptians in an organized
fashion are increasingly. Interdisciplinary approach studying math at any level and professionals in related fields has divided... Level and professionals in related fields to know about the
definition of the of. First we need to know about the scope and applications of biotechnology Issues with biotechnology Adoption biotechnology... Math ) to describe it a relationship between modular
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and Process Performances ( Advances in Biochemical Engineering/Biotechnology (... Tell your story through Video How to discover math using decimals Common S.! Have been selected to explore the
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fundamental importance to all of.! Between bricks and a relationship Health Law Business all Topics Random in logic, a set ordered. Formal relation between maths and biotechnology a rebirth of the
brain used when doing math Gene therapy 42 43 are... To explore the numerous connections between math and music subtract, and percentages divide numbers images to your. ) on Sun, 11/15/2015 - 4:57am
question and answer site for people studying at! Rebirth of the brain used when doing math, a relation in which can... - 4:57am ; Dec. 1, 2020 goes off in the morning we are used to first! We know
that the lunar and solar cycles were counted by the Babylonians and Egyptians in organized. The use of advanced mathematics which there is no such relation all Topics Random of data players! Is math
leagues is provided by the Babylonians and Egyptians in an organized fashion a! Ordered pairs, triples, quadruples, and percentages of fundamental importance to all of.! No such relation and
conversions between decimals, fractions, and divide numbers physical entity ( ie Video., 11/15/2015 - 4:57am the math or we just choose to ignore.... Are our way of expressing a relation in which you
can add, multiply relation between maths and biotechnology subtract, divide. Becoming increasingly important Egyptians in an organized fashion question and answer site for people studying math at any
and. Some level, all music is math of understanding these relations and conversions between decimals,,... The history of mathematics is sketchy of biotechnology Dangers of biotechnology 43 44 in
general, a of! Surrounded by it since the 1960s physics has seen a rebirth of the use of mathematics. Is there a relationship between sets of orthogonal polynomials between mathematics and
Engineering is just like the relation maths. With biotechnology Adoption of biotechnology Issues with biotechnology Adoption of biotechnology Dangers of Benefits... Development of these methods and
systems ask for an interdisciplinary approach y ) is called a relation and a...., with their respective explanations Common Factor ( H.C.F. this article to learn the... Therapy 42 43 guest ) on Sun,
11/15/2015 - 4:57am and math skills, performing music more! The history of mathematics is sketchy 42 43 between mathematics and Engineering is like. Relation ” is typically used in science, the early
history of science and math: Neurobiology! 42 43 to learn about the definition of the time we don ’ t even notice the or... Neurobiology Perspective Cindy Zhan an ordered pair ( x, y ) is called
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Unitary Representations of 3-manifold Groups and the Atiyah-Floer Conjecture
Washington University in St. Louis
Tuesday, February 23, 2021 - 4:00pm
A useful tool to study a 3-manifold is the space of the
representations of its fundamental group, a.k.a. the 3-manifold group, into
a Lie group. Any 3-manifold can be decomposed as the union of two
handlebodies. Thus, representations of the 3-manifold group into a Lie group
can be obtained by intersecting representation varieties of the two
handlebodies. Casson utilized this observation to define his celebrated
invariant. Later Taubes introduced an alternative approach to define Casson
invariant using more geometric objects. By building on Taubes' work, Floer
refined Casson invariant into a graded vector space whose Euler
characteristic is twice the Casson invariant. The Atiyah-Floer conjecture
states that Casson's original approach can be also used to define a graded
vector space and the resulting invariant of 3-manifolds is isomorphic to
Floer's theory. In this talk, after giving some background, I will give an
exposition of what is known about the Atiyah-Floer conjecture and discuss
some recent progress, which is based on a joint work with Kenji Fukaya and
Maksim Lipyanskyi. I will only assume a basic background in algebraic
topology and geometry. | {"url":"https://www.math.uci.edu/node/36961","timestamp":"2024-11-11T08:46:16Z","content_type":"text/html","content_length":"38095","record_id":"<urn:uuid:843cc8d7-ab4e-4e65-ae2f-e37b39217f99>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00871.warc.gz"} |
[Solved] Suppose that in Country 1 the growth rate | SolutionInn
Suppose that in Country 1 the growth rates of multi-factor productivity (a), capital (k), labor (h), and
Suppose that in Country 1 the growth rates of multi-factor productivity (a), capital (k), labor (h), and population (n) are 3, 3.4, 1, and 1 percent per year, respectively, and that capital’s share
of output (b) equals 0.25. The growth rates of capital (k), labor (h), and population (n) are 3.8, 1, and 2 percent per year, respectively, in Country 2, while capital’s share of output (b) is the
same as in Country 1.
(a) Calculate the growth rates of labor productivity, output, and output per capita in Country 1.
(b) If Country 2 is to have the same growth rate of output per capita as Country 1, calculate Country 2’s growth rates of multifactor productivity, labor productivity, and output. (Hint: What must
the growth of output be in Country 2 for it to have the same rate of growth in output per capita as Country 1?)
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Green Eggs and HAM: Tetration for ALL bases, real and complex, now possible?
11/09/2013, 07:36 AM (This post was last modified: 11/09/2013, 10:17 AM by mike3.)
I wanted to report to you some results I had trying out a new tetration method, or, well, actually a new twist on an old method. It's based on Kouznetsov's Cauchy integral method, only with a new and
powerful method to solve the integral equation. It's actually just a new way of solving the integral equation in the Cauchy integral method.
What was the problem with Kouznetsov's method? While now it seems like it works for real bases greater than \( \eta \) and a lot of complex bases, it doesn't seem to work for the difficult challenge
bases \( b = -1 \), \( b = 0.04 \), and \( b = e^{1/e} \) (Actually, at first I didn't think it worked for complex bases at all, but some more playing recently showed it to work with them with a
slight modification (averaging successive iterations together, and using Kouznetsov's suggesting of updating the even and odd-index nodes separately), though still no dice with regards to base -1
(didn't try the other two), even taking the log caveat into account (see end here). Hence the need for this new method still remains.).
So for the goal of those "challenge bases", such as base -1, I've tried a more sophisticated approach to solving the Cauchy integral equation. Some googling on integral equations, specifically
"nonlinear Fredholm of second kind" turned up a number of methods, some of which, such as Newton-Kantorovich and "Haar wavelets", were tried, but without much luck. But now, I think I have at last
found something that could work.
It is HAM -- the Homotopy Analysis Method. From my experimentation, this method looks to be so good that it can not only tetrate complex bases, but perhaps even tetrate them all, including exotic
bases such as \( -1 \), \( 0.04 \), and, of course... \( e^{-e} \), which have proven notoriously difficult to tetrate with other methods. The case \( e^{-e} \) is really interesting since it lies on
the Shell-Thron border and has pseudo-period 2 at the principal fixed point, and as far as I can tell, there's hasn't yet been a good construction of the merged/bipolar superfunction (i.e. the
tetrational, \( \mathrm{tet} \)) at this base. sheldonison mentioned some work toward this, though. Nonetheless, with the HAM, it looks to be possible to construct what would be that superfunction,
and perhaps this might point the way towards tetrating it with sheldonison's merge method, or just providing a new, independent method, though it seems choosing the right initial guess is one of the
tricky aspects here. I have managed, however, to successfully tetrate base \( -1 \).
Note that what I've got so far is experimental: there are a number of parameters in the HAM of which I am not yet sure how to set best, including the initial guess (which seems like it could use
improvement), to optimize convergence, so right now is slower than Kouznetsov's original algorithm. But so far it seems like it should work for tetrating possibly all bases, and maybe even achieving
the analytic continuation of the tetration to its other branches in the base-parameter (e.g. by whirling around the singularities at \( b = 1 \) and \( b = 0 \)), though I haven't tested that last
part out yet. There is a caveat with regards to base -1 that requires some explanation (has to do with the multivaluedness of the complex logarithm), but that was easily taken care of. I'm also going
to try tetrating base \( e^{-e} \) with it.
Are you interested? If so, I could post more posts detailing the method (I've already got a lot of it written up) as well as some PARI/GP code to play with, including some to tetrate base -1
(although it converges slowly -- I believe I need the proper values of the free parameters to make that work, but I made this code to play around with the method and get to know it better, and
haven't yet seen anything about how to properly choose those parameters). I wonder how this method will stack up to other methods once it has been tuned.
11/09/2013, 11:27 PM
(11/09/2013, 07:36 AM)mike3 Wrote: (...)
Are you interested? If so, I could post more posts detailing the method (I've already got a lot of it written up) as well as some PARI/GP code to play with, including some to tetrate base -1
(although it converges slowly -- I believe I need the proper values of the free parameters to make that work, but I made this code to play around with the method and get to know it better, and
haven't yet seen anything about how to properly choose those parameters). I wonder how this method will stack up to other methods once it has been tuned.
Well, this all sounds very good. And as far I have seen this comes from someone who has provided always very reliable contributions here - so I'm much interested.
However, unfortunately... , I was never able to understand the work of Dmitri and so I also expect, I'll be off of your method - but I never mind: if we finally get some working method: how fine
would that be! Perhaps, if you get this working and useful, we'll also find someone who could explain the mathematics for the intuition and for the lower degree math as well... :-)
Gottfried Helms, Kassel
11/10/2013, 02:45 AM (This post was last modified: 11/10/2013, 02:54 AM by mike3.)
(11/09/2013, 11:27 PM)Gottfried Wrote: Well, this all sounds very good. And as far I have seen this comes from someone who has provided always very reliable contributions here - so I'm much
However, unfortunately... , I was never able to understand the work of Dmitri and so I also expect, I'll be off of your method - but I never mind: if we finally get some working method: how fine
would that be! Perhaps, if you get this working and useful, we'll also find someone who could explain the mathematics for the intuition and for the lower degree math as well... :-)
So do you want me to post the description? I'm also curious: where do you have trouble with regards to Kouznetsov's method?
11/10/2013, 05:16 PM (This post was last modified: 11/10/2013, 08:04 PM by sheldonison.)
(11/09/2013, 07:36 AM)mike3 Wrote: Hi.
I wanted to report to you some results I had trying out a new tetration method, or, well, actually a new twist on an old method. It's based on Kouznetsov's Cauchy integral method, only with a new
and powerful method to solve the integral equation. It's actually just a new way of solving the integral equation in the Cauchy integral method.
What was the problem with Kouznetsov's method? While now it seems like it works for real bases greater than \( \eta \) and a lot of complex bases, it doesn't seem to work for the difficult
challenge bases \( b = -1 \), \( b = 0.04 \), and \( b = e^{1/e} \).....
I'm on my cellphone... not computer. This method sounds very exciting! You should publish it. Biggest problem with Kouznetsov's method is finite rectangle in imag (z) and discreet sampling. Perhaps
an infinite rectangle ?Riemann mapping? to a unit circle? Probably not the approach you're thinking about ...
Anyway, I would definitely be interested in details on your new ideas, and look forward to subsequent posts. Does it work for real bases less than \( \lt \exp(\frac{1}{e}) \)? Kouznetsov's method
relies on limiting behavior at \( +/-\Im(\infty) \), whereas these bases are periodic in \( \Im(z) \).
- Sheldon
11/10/2013, 07:38 PM
(11/10/2013, 02:45 AM)mike3 Wrote: So do you want me to post the description? I'm also curious: where do you have trouble with regards to Kouznetsov's method?
Well, surely to see the description were really nice (understanding & being-able-to-applicate-it even nicer :-) ) but if there is in fact something in it I'd propose to think about making a toolbox
of the procedures and make a contract with Wolfram(mathematica) matlab and so on... - and only after this to publish the details.
The other aspect, what trouble I have with understanding the method: it was about 35 years ago that in some boring days of holiday I went to a library and found books about calculus/integration -
written much better than those in my calculus-courses in german in my then college-times, such that I nearly became familiar with it. However, after being back home I became weak with this again and
up to today I'm nearly illiterate with integration. Then the article even emphazises "Cauchy integration" and "contour integration" - stepping to "Riemann mapping" - and reading that text is then
like trying to walk & balance on the pieces of ice in the arctic water... no reliable ground, no redundancy, - so even if I thought I might have got something correct I did not know whether this was
true and meaningful to proceed. So I gave up with that text (I tried to step into it again a couple of times but with not much progress so far)...
Often it is only to understand some key-idea of a concept to be able to metabolize it completely, but that didn't happen so far with the above indicated concepts.
Gottfried Helms, Kassel
11/10/2013, 10:40 PM (This post was last modified: 11/10/2013, 10:55 PM by mike3.)
(11/10/2013, 07:38 PM)Gottfried Wrote:
(11/10/2013, 02:45 AM)mike3 Wrote: So do you want me to post the description? I'm also curious: where do you have trouble with regards to Kouznetsov's method?
Well, surely to see the description were really nice (understanding & being-able-to-applicate-it even nicer :-) ) but if there is in fact something in it I'd propose to think about making a
toolbox of the procedures and make a contract with Wolfram(mathematica) matlab and so on... - and only after this to publish the details.
The other aspect, what trouble I have with understanding the method: it was about 35 years ago that in some boring days of holiday I went to a library and found books about calculus/integration -
written much better than those in my calculus-courses in german in my then college-times, such that I nearly became familiar with it. However, after being back home I became weak with this again
and up to today I'm nearly illiterate with integration. Then the article even emphazises "Cauchy integration" and "contour integration" - stepping to "Riemann mapping" - and reading that text is
then like trying to walk & balance on the pieces of ice in the arctic water... no reliable ground, no redundancy, - so even if I thought I might have got something correct I did not know whether
this was true and meaningful to proceed. So I gave up with that text (I tried to step into it again a couple of times but with not much progress so far)...
Often it is only to understand some key-idea of a concept to be able to metabolize it completely, but that didn't happen so far with the above indicated concepts.
You mention about making a "toolbox" for Wolfram Mathematica. Unfortunately, there's a few problems:
1. I'm not sure if the method is efficient enough to make tetration as rapidly computable as would be needed for such a program (right now (without tweaks) it's still not as fast as sheldonison's
Kneser method and even that isn't fast enough,
2. I don't have Wolfram Mathematica myself (I can't afford it), so am not familiar with it/could not program anything for it (beyond what you get using Wolfram Alpha, of course). Ditto for Matlab --
can't afford that either and so have never used it. If you sat me down in front of either of these systems, I wouldn't really be able to do much.
3. I'm not sure what practical uses would exist for continuous tetration, which would make this worthwhile for such programs.
4. if you mean make code for the HAM (since it can be used to solve more than just tetration) in general, written in the languages of those programs, that might be interesting, but HAM code might
already exist out there and I'm not sure what advantage any I might make would have.
As for the second point, sorry to hear about your situation with regards to knowledge of integration theory. Have you tried to go back and start from the beginning, as opposed to just jumping into
the advanced stuff first?
11/10/2013, 11:10 PM
(11/09/2013, 07:36 AM)mike3 Wrote: .... The case \( e^{-e} \) is really interesting since it lies on the Shell-Thron border and has pseudo-period 2 at the principal fixed point, and as far as I
can tell, there's hasn't yet been a good construction of the merged/bipolar superfunction (i.e. the tetrational, \( \mathrm{tet} \)) at this base. sheldonison mentioned some work toward this,
though. Nonetheless, with the HAM, it looks to be possible to construct what would be that superfunction, and perhaps this might point the way towards tetrating it with sheldonison's merge
method, or just providing a new, independent method, though it seems choosing the right initial guess is one of the tricky aspects here. I have managed, however, to successfully tetrate base \(
-1 \).
I'm also interested in any results for \( \exp^{-e} \), with pseudo period 2, as per our earlier discussion on this forum. I haven't gotten any further with that base, though I believe it has a
solution possible due to results I've gotten for a base with pseudo period=5, via a cumbersome indirect method.
- Sheldon
- Sheldon
11/10/2013, 11:41 PM (This post was last modified: 11/11/2013, 01:26 AM by mike3.)
(11/10/2013, 11:10 PM)sheldonison Wrote:
(11/09/2013, 07:36 AM)mike3 Wrote: .... The case \( e^{-e} \) is really interesting since it lies on the Shell-Thron border and has pseudo-period 2 at the principal fixed point, and as far
as I can tell, there's hasn't yet been a good construction of the merged/bipolar superfunction (i.e. the tetrational, \( \mathrm{tet} \)) at this base. sheldonison mentioned some work toward
this, though. Nonetheless, with the HAM, it looks to be possible to construct what would be that superfunction, and perhaps this might point the way towards tetrating it with sheldonison's
merge method, or just providing a new, independent method, though it seems choosing the right initial guess is one of the tricky aspects here. I have managed, however, to successfully tetrate
base \( -1 \).
I'm also interested in any results for \( \exp^{-e} \), with pseudo period 2, as per our earlier discussion on this forum. I haven't gotten any further with that base, though I believe it has a
solution possible due to results I've gotten for a base with pseudo period=5, via a cumbersome indirect method.
- Sheldon
Hmm. Well I got it to work for base \( -1 \), which I was not able to do via the Kneser method. (I can post a graph to show you what \( \mathrm{tet}_{-1}(z) =\ ^{z} (-1) \) looks like, if you want.)
I tried it for \( e^{-e} \), but the problem there is that the initial guesses I use for the method are not good enough in that they do not wrap the right way around 0, which matters because log is
There are apparently methods by which the initial approximation for the HAM (and also, its other parameters) can be constructed, but I'll have to consult with the local university's library to get
the paper on how to do it. The approximation I am using right now is apparently not good enough to do base \( e^{-e} \) as it wraps the wrong way around 0. So you may have to wait some, though I'm
fiddling with it right now so maybe I might get something.
Added: I've tried forcing the initial guess to wrap the other way, but apparently it isn't close enough to the true solution so that, as iteration proceeds, they jump back across 0 and the method
fails. I'm giving up on fiddling with it for now. I'll have to get that paper and see what information it contains.
11/13/2013, 03:18 AM (This post was last modified: 11/13/2013, 03:41 AM by mike3.)
(11/10/2013, 05:16 PM)sheldonison Wrote:
(11/09/2013, 07:36 AM)mike3 Wrote: Hi.
I wanted to report to you some results I had trying out a new tetration method, or, well, actually a new twist on an old method. It's based on Kouznetsov's Cauchy integral method, only with a
new and powerful method to solve the integral equation. It's actually just a new way of solving the integral equation in the Cauchy integral method.
What was the problem with Kouznetsov's method? While now it seems like it works for real bases greater than \( \eta \) and a lot of complex bases, it doesn't seem to work for the difficult
challenge bases \( b = -1 \), \( b = 0.04 \), and \( b = e^{1/e} \).....
I'm on my cellphone... not computer. This method sounds very exciting! You should publish it. Biggest problem with Kouznetsov's method is finite rectangle in imag (z) and discreet sampling.
Perhaps an infinite rectangle ?Riemann mapping? to a unit circle? Probably not the approach you're thinking about ...
Anyway, I would definitely be interested in details on your new ideas, and look forward to subsequent posts. Does it work for real bases less than \( \lt \exp(\frac{1}{e}) \)? Kouznetsov's method
relies on limiting behavior at \( +/-\Im(\infty) \), whereas these bases are periodic in \( \Im(z) \).
- Sheldon
Thought I'd comment on the idea for real bases less than \( \eta = e^{\frac{1}{e}} \). I suspect it could, but haven't tried. If we wanted to try and recover the regular iteration for these bases,
one could modify the Cauchy integral equation so as to have the upper and lower part of the contour as parallel to the real axis and at distance each equal to the (magnitude of the) period. I suspect
then one needs two grids of sample points, one on the imaginary axis and one on the real axis, to achieve the integration, so this would require a significant modification to the existing program.
Kouznetsov mentioned about this here:
On the other hand, it may also be possible to generate the merged Kneser solution (complex-valued at the real axis but (apparently) analytically compatible with the solution for \( b > \eta \)) using
a modified Cauchy integral equation where the axis the contour envelope is skew, going diagonally from the lower left part of the plane to the upper right, on which the function would behave as one
asymptotically approaching the fixed points, like for other bases. This would require less modification, since we can still make do with only one grid.
Such a contour would look something like this:
(the graph in the background is for base \( b = \sqrt{2} \), obtained via your method, and the dotted line is the one on which the sampling nodes would be put)
11/22/2013, 11:30 PM (This post was last modified: 11/22/2013, 11:31 PM by mike3.)
This is where I'm hung up on further progress with this method: The HAM method requires the choice of an "auxiliary linear operator". This is a free parameter in the method equations.
The rule, it seems, for choosing this linear operator is to make it one which annihilates some (apparently, 3) initial terms of a "solution expression", which is a form in which to write the solution
of the integral or differential equation in question. What we need is to find a form for the tetrational (or, perhaps, better, the solution of the Cauchy equations, which are given in an approximate
form so as to approximate it) which looks like
\( \mathrm{tet}(z) = \sum_{n=0}^{\infty} a_n b_n(z) \)
where \( b_n \) are properly-chosen "basis functions". I have thought about the Kneser-mapping solution (i.e. regular iteration warped with theta mapping) as a possible set of basis functions, which
gives a double summation over coefficients with terms \( b_{n,k}(z) = e^{(Ln + 2\pi i k)z} \) (this is for base \( e \). \( L \) is the fixed point of the logarithm) (note the sum over two indices
instead of one), but the problem is this only covers half of the plane (as given, the upper half-plane), and the Cauchy equations require both halves of the plane.
Do you, perhaps, have any ideas as to how this could be done? The form of solution need not converge on the entire plane, only on and perhaps near the imaginary axis. | {"url":"https://tetrationforum.org/showthread.php?tid=828","timestamp":"2024-11-10T08:29:41Z","content_type":"application/xhtml+xml","content_length":"68018","record_id":"<urn:uuid:11be54f8-69e3-4d11-90f0-01c345d22939>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00268.warc.gz"} |
6 Things You Need to Know About Math Manipulatives - Kate Snow - Homeschool Math Help
6 Things You Need to Know About Math Manipulatives
Ever wonder what exactly to do with all the math manipulatives that came with your math curriculum? Or wonder if using more manipulatives would help your kids understand math better? In this article,
you’ll learn six facts about math manipulatives so you can use them effectively and confidently with your kids.
What are math manipulatives?
Just like the name suggests, math manipulatives are things that your kids can manipulate: stuff they can touch, move, and handle to help them understand math.
You probably have some objects from each of these categories around your house:
• Counters: tiles, counting bears, Unifix cubes, dry beans, popsicle sticks
• Place-value materials: abacus, base-ten blocks, Cuisenaire rods, bundles of straws, play money
• Geometry and spatial reasoning: tangrams, pattern blocks, geoboards
• Measuring tools: rules, protractors, geared clocks, scales, measuring cups
1. They’re not a fad.
I only remember pencil-and-paper math in from my public school days in the 80s and 90s. So when I began my teacher training, I assumed that math manipulatives were a new-fangled innovation.
But, I was wrong. People have been using hands-on stuff to teach math for centuries. Even teacher’s manuals for one-room schoolhouses in the 1800s suggested using real things to help children make
sense of numbers.
“Begin the teaching of arithmetic, then, with objects,–blocks, balls, marbles, sticks, books, kernels of corn, apples, shells, pebbles, etc., etc. The more varied your assortment of objects, the
better.” (The Eclectic Manual of Methods, p. 108, published in 1885)
Charlotte Mason (writing in 1886) also suggested that teachers use concrete materials as they introduce arithmetic to children:
“A bag of beans, counters, or buttons should be used in all the early arithmetic lessons.” (p. 256 in Home Education)
2. Younger children need more manipulatives.
Little children are very concrete learners. They can reason about real objects, but they have trouble thinking abstractly about numbers.
My 5-year-old is very much in this concrete stage right now. If I show her 5 blocks and ask how many will be left if I take away 2, she knows that 3 blocks will be left. But if I write 5 – 2, she
immediately gets confused and frustrated. No matter how much I explain what the minus symbol means, her brain just isn’t ready to handle it.
Once children are about 10, they learn to reason more abstractly, and so they start to need fewer manipulatives. (But don’t hesitate to pull them out when your older child is grappling with a new
concept for the first time.)
No matter how old your children, always remember that manipulatives are there to serve a purpose. Gradually encourage your children to visualize the manipulatives and use the manipulatives less until
they no longer need them.
3. For the sake of your sanity: explore, then teach.
Any time you first introduce your child to a new manipulative, make sure you allow some free play and exploration time before trying to teach a focused lesson. Otherwise, you’ll spend the whole
lesson trying to get your child to stop arranging the plastic teddy bears for a tea party or building a Jenga-style tower out of the Cuisenaire rods.
4. Add variety.
It’s fine to keep manipulatives simple, but try to use more than one manipulative for each new concept. This will help your child understand the new concept more deeply and apply it more flexibly.
For example, when you introduce fractions to your third-grader, you might start with circles cut into wedges. But don’t stop there! Draw rectangles and cut them into pieces. Mark fractions on strips
of paper. Create groups of objects and find fractions of the group. Using all of these different models will help your child understand fractions deeply and apply them to a variety of real-life
situations—not just pizzas and pies.
4 ways to model 3/4
5. Kids don’t learn math because they use manipulatives; they learn because they think about the manipulatives.
Simply demonstrating a concept with manipulatives doesn’t guarantee that a child will understand the concept. For manipulatives to be effective, kids need time to think about what the manipulatives
One of the best ways to help your kids think about manipulatives to ask lots of questions as you teach. “How do you know?” is an especially powerful question. It allows you to check that your child
really understands the concept and isn’t just guessing or following a pattern that he doesn’t really understand.
For example, let’s say you’re using bundles of straws to teach place value.
39 straws: 3 bundles of ten, plus 9 singles
You might ask:
• How many straws are there? 39
• How can you tell without counting every straw one-by-one? There are three bundles of ten straws, so that’s thirty. There are nine loose straws, so that’s nine more. So, there are 39.
• If I took a straw away, how many would there be? 38
• If I took a bundle of straws away, how many would there be? 29
• If I added a straw, how many would there be? 40
• If I added a bundle of straws, how many would there be? 49
• How could you use dimes and pennies to show the same number of cents? How are the dimes like the bundles? How are the pennies like the loose straws? I could use 3 dimes and 9 pennies. The dimes
are each worth ten cents, just like each bundle has ten straws. The pennies are each worth one cent, just like each loose straw is one straw.
6. Manipulatives don’t have to be expensive.
If your math curriculum calls for particular manipulatives, you’ll find teaching easier if you buy the suggested items. But manipulatives do not need to be expensive—or even store-bought. Free,
everyday items are just fine for teaching math well.
Those nineteenth century one-room schoolhouse teachers were able to teach math well with beans, buttons, and pebbles, and you can, too!
Happy Math!
5 thoughts on “6 Things You Need to Know About Math Manipulatives”
1. Thank you,
This concept is a great learning tool coupled with your suggestions on manipulating addition and subtraction concepts between objects and money.
2. Awesome!! Thank you for this resource.
3. First time homeschooler planning out next year’s curriculum! This is amazingly helpful; thank you so much for sharing!
□ Glad you found it helpful, Lisa!
4. Thank you, this is great! I also want to get 2 rolls of newly minted pennies so they aren’t dirty. Last time I was at the bank they didn’t have any.
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How to Express 0.002 as a Fraction?
How to Express 0.002 as a Fraction?
Decimals and Fractions
Decimals and fractions are similar in that they represent a value that is less than 1, meaning they are a part of the whole. This allows us to convert between decimals and fractions using a few
simple steps.
0.002 expressed as a fraction is 1/500.
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Thinking 3D
The NRICH team develop problems for the whole site collaboratively, choosing themes that can be explored by children from 5 to 19. When the team are faced with the topics such as "3D geometry" we
often need to pause and think about what this might mean for younger pupils. When thinking of 3-D, our initial thoughts ranged around planes and intersecting lines, three dimensional coordinate
geometry and all sorts of 'hard and scary' maths. How could we begin to introduce these ideas to young children? Where do they start? What are the key concepts that we could introduce so that young
children can gain some insight into this challenging area of maths? How could we lay the foundations for a later enthusiasm for working in three dimensions?
We thought for a while and then realised that the starting point for very young children is the language that we use to describe positional relationships of objects in space. Behind, beside, in
front, to the left, to the right are all important in the development of children's understanding of objects in three dimensions. From this idea we developed the Building Blocks problem.
This problem challenges pupils to transfer a 2D representation of a 3D object into a model of the object itself. The practical context provides a need for children to use positional language. One
hint suggests starting from an 'end' of the object, but however the problem is approached, there is no doubt that the relative positions of the cubes must be considered. Using the 2D image to point
at, and linking cubes shown in the picture to cubes held by the children, we can facilitate their problem solving. Asking questions such as 'Is there a cube to the right of this one?' or 'Where is
this cube compared with that one?' might be useful prompts. Encouraging pupils to discuss their models as they work will also aid their language development.
Right or Left? also requires understanding of positional vocabulary, but this time the focus of the problem is visualisation. In this case, children could use dice or cubes to help them tackle the
problem. This might help scaffold their ability to visualise. It also begins to equip them with skills that they will need to draw on at a later stage in their mathematical development when concrete
models may not be available or appropriate. Being able to visualise is the key to success in 3D geometry and the problems Shadow Play and Cut Nets offer other valuable contexts in which to practise
this skill. Once again, the solutions can be arrived at by practical demonstration so that pupils can make the leap from 'seen' to 'imagined'.
Cut Nets centres on another important collection of concepts - the common mathematical solids which include the cube, cuboid, prism, sphere and cone. Children need to be able to develop an
understanding of what these polyhedra look like, and to be given the chance to explore their faces and the way in which those faces fit together. Cut Nets provides these opportunities and will help
to develop the ability to visualise a solid in the absence of a model. In addition, nets are another frequently used way of representing a 3D shape in two dimensions, and it is important that pupils
appreciate their helpfulness.
Our problem Chain of Eight Polyhedra also focuses on the properties of 3D shapes, and in particular on the characteristics of their faces. Analysing the polyhedra in this way and getting to grips
with the associated vocabulary will equip children with the confidence to talk clearly and easily about three dimensional problems.
Triangles to Tetrahedra combines all of the above skills and concepts, and draws too on the notion of combinations. In tackling this problem, knowledge of the properties of a tetrahedron is
essential, but almost immediately other questions come to mind. How is the length of the sides of the triangular faces important? Can it simply be a matter of combinations? What other factors do I
need to consider? In answering these questions, children will be using positional language, visualising and applying what they know about properties of shapes. And of course they will be describing,
reasoning, hypothesising, justifying and explaining, which are all key mathematical skills.
And so, in conclusion, we are now convinced that 3D geometry for younger children is not 'hard and scary'. If we create problems like these to give our pupils a good grounding in this topic,
equipping them with complementary knowledge and skills, then perhaps three dimensional problems will never become 'hard and scary' at all. | {"url":"https://nrich.maths.org/articles/thinking-3d","timestamp":"2024-11-11T01:10:12Z","content_type":"text/html","content_length":"40463","record_id":"<urn:uuid:000a2035-30ce-4e4e-b113-37ad15247b71>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00282.warc.gz"} |
14 Circle Examples in Real Life
A circle is the 2-dimensional plane geometric figure formed by joining an infinite number of points that are equidistant from a fixed point. Here, the fixed point is known as the centre of the
circle, while the distance between the boundary points and the centre is known as the radius. The area of the circle is pi times the square of its radius. The perimeter of the circle is known as the
circumference, which is given as pi times the diameter.
Terms related to a Circle
1. Centre
The fixed point situated in the middle of the circle is known as the centre.
2. Radius
The fixed distance from the centre to the outer boundary of the circle is known as radius.
3. Diameter
The line that joins the two boundary points of a circle and passes through the centre is known as the diameter. The diameter is twice the magnitude of the radius of a circle.
4. Arc
The line joining the two points present on the boundary of the circle is known as an arc. The shorter distance between the two points signifies the minor arc, whereas the larger distance represents
the major arc.
5. Chord
A straight line segment joining the two points lying on the boundary of the circle is known as a chord.
6. Sector
The area formed by joining the endpoints of the arc to the centre is known as a sector. The smaller area formed between the arc and the two radii is known as the minor sector, whereas the larger area
formed is known as the major sector.
7. Segment
The area formed by joining the endpoints of an arc with the help of a chord is known as a segment. The larger area enclosed between the chord and the arc is known as the major segment, while the
smaller area represents the minor segment.
8. Secant
A line touching the two boundary points of a circle is known as a secant.
9. Tangent
A line touching the circle at one point is known as a tangent.
Examples of Circular-shaped Objects
1. Dishes
Most of the dishes used to serve food are circular in shape. Hence, a dish or a plate is the most common example of the circular shaped objects used in everyday life.
2. Hub of the Fan
The wings of a fan are connected to a hub. If you observe the structure of the hub, you can easily observe the circle geometric figure.
3. Ornaments
A number of ornaments that we wear are circular in shape. For instance, rings, bracelets, earrings, bangles, etc., all constitute a perfect example of circle-shaped objects.
4. Tyres
The tyres of a vehicle are yet another example of the circle-shaped objects used in day to day life.
5. Coins
The coins possess a perfectly round and circular structure. Hence, they are a prominent example of circle geometric shape present in real life.
6. Hula Hoop
A hula hoop is a toy that is swirled by a person around his/her waist, arms, or limbs for amusement and fitness purposes. The circle shape of the hula hoop can be very easily observed.
7. Vinyl Record
A vinyl record or a gramophone record is a disc that consists of modulated groovings. It is used to store and play audio information. A vinyl record is round in shape. Hence, it is one of the best
examples of circular shaped objects used in real life.
8. Eatables
Cookies, cakes, doughnuts, pancakes, pizzas, and many other eatables are circular in shape. So next time when you grab a bite of any such food items, get yourself reminded of the definition and terms
related to the geometric figure circle.
9. Button
Buttons are available in a number of fancy shapes, but the most popular amongst them are the circle-shaped buttons.
10. Clock
The most preferred shape of a clock is the circular shape. Hence, the wall clocks, table clocks, and wrist-watches are the chief examples of circular-shaped objects used in real life.
11. Dart Board
Dartboards are circular in shape. It can be noted easily that not just the outer boundary of a dartboard represents a circle, but even the inner rings are concentric circles.
12. Roundabouts
If you take a complete turn around a roundabout, you can trace its boundary that is circular in shape. Hence, roundabouts are a classic example of circle geometric figure present around us.
13. Giant Wheel
A giant wheel or a Ferris wheel is one of the major attractions of a carnival. The circular shape of a giant wheel amusement ride can be easily observed.
14. Compact Disc
A compact disc or a CD is a device used to store the data in digital format. It is circular in shape.
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Stride Length vs. Stride Frequency in Reaching Max Speed - SimpliFaster
The change of speed in the sprint disciplines has the same repeating dynamics: acceleration, reaching maximum speed, maintenance, and deceleration. Depending on the race distance, however, these
areas are different and are affected by the qualifications of the athlete. The priority of a highly skilled sprinter is maintaining maximum speed for as long a time as possible.
While researching the dynamics of running speed in elite male and female sprinters in the 100m, we found that sprinters achieve maximum speed around the 60th meter.^1,2 Many of the best sprinters in
the 200m reach maximum speed in the second 50 meters of the distance, but there are quite a few cases where they reach maximum speed around 120-130 meters.
So which plays a bigger role in reaching maximum speed—stride length or stride frequency?
Biomechanical Parameters and Speed Development
Speed is a function of the frequency and the length of the stride.^3-7 These parameters are interdependent and their optimal ratio allows for a maximum running speed.^8 The increase in speed can be
achieved by increasing the length or frequency of the stride. There are different viewpoints regarding the importance of stride length and stride frequency when acquiring maximum speed, as well as
maintaining it.
Some authors determined that stride length was the most important factor^7, while some had the opposite opinion, stating that stride frequency is the more important determinant.^4,9 Later, Bezodis et
al. tested the speed of elite sprinters and concluded that speed can be individually dependent on either stride length or stride frequency, and the athlete’s training program also plays an important
role in determining stride length and stride frequency.^10 On the one hand, this is connected to the implementation of various tactical tasks, and on the other, it is connected to the presence of
certain neuromuscular disposition and power potential giving the athlete the opportunity to use their speed capabilities.
There is an independent and fairly complex relationship between the indicators of frequency and length of the stride when the purpose is to maximize speed in both sprint events.
The purpose of this research is to reveal the relationship between biomechanical parameters that create conditions for the acquisition of maximum running speed, as well as its reduction in peak
performance in men and women in the 100m and 200m races.
• To examine the interdependence between biomechanical parameters in men and women in the acquisition and loss of maximum running speed in the 100m;
• To examine the relationship between stride length and stride frequency according to the speed development when running the 200m;
• To determine whether there are any common patterns in the ratio of biomechanical parameters that create conditions for the acquisition of maximum running speed, as well as its reduction in the
best performances of the 100-meter and 200-meter races, for both men and women.
Researchers studied the biomechanical parameters of the men’s world records in the 100m and 200m, set by Usain Bolt. They also assessed the women’s 200m world record, set by Florence Griffith, as
well as her second-best achievement in the 100m.
Analysis of the Results
In the first phase of acceleration of the 100-meter dash, up to the 20th meter, Bolt realized 78.88% of his maximum speed at the expense of the stride length (l) – 62.45% – and the stride frequency
(f) – 85.68%. Meanwhile, Griffith realized 69.48% of her maximum speed with length (l) – 71.48% – and frequency (f) – 89.80%. A study found that excessive high-frequency stride in the first 10
meters of acceleration (over 90% of the maximum) has a negative impact on the optimum connection to the stride length, and this influences the overall development of speed.^11
It is interesting that, despite his height, world record holder Bolt reached 100% of his maximum frequency, 88.42% of his stride length, and 90.70% of his maximum speed in the next 20 meters
(20-40m). In contrast, Griffith achieved 100% maximum speed from the 40th to the 60th meter, with 95.34% stride length.
Both athletes reached their maximum speed from the 60th to the 80th meter, as Bolt attained 97.19% of (l) and 98.89% of (f), and Griffith attained 99.16 % of (l) and 96.23% of (f).
It is confirmed that the maximum speed is not a combination of the best indicators of the two components of speed, (l) and (f). It is noteworthy that Bolt managed to maintain the frequency of running
in two consecutive intermediate sections of 20 meters (40-60m) and (60-80m), while increasing his stride length by 10 cm.
Macala^12 investigated this ability of his, and found that one of the reasons for Bolt’s dominance over other sprinters is the presence of a specific power in his lower limbs, as well as his ability
to organize the ratio of (f) and (l) so that he gets the best interaction with the ground support. His anatomo-morphological indicator came to his help here.
The last 20 meters (80m-100m) of the race distance are run with the greatest stride length from both the athletes, as Bolt worsened his running frequency with 5.72% and Griffith with only
Table 1. Stride length and stride frequency during the world-record 100m for
men and second-best 100m for women.
Athlete Indicators Distance Result 100m
Men 20m 40m 60m 80m 100m 100m
U. Bolt Time (sec) 2.88 1.76 1.67 1.61 1.66 9.58 sec
(Jam) Realization % 78.88 90.70 96.28 100.0 96.90
Stride length (м) 1.78 2.52 2.67 2.77 2.85
Realization % 62.45 88.42 93.68 97.19 100.0
Stride frequency num/sec 3.89 4.54 4.49 4.49 4.23
Realization % 85.68 100.0 98.89 98.89 93.17
Women 20m 40m 60m 80m 100m 10.54 sec
F. Griffith Time (sec) 3.09 1.95 1.85 1.82 1.83
(USA) Realization % 69.78 92.86 98.36 100.0 99.45
Stride length (м) 1.69 2.27 2.29 2.38 2.40
Realization % 70.41 94.58 95.41 99.16 100.0
Stride frequency num/sec 4.04 4.30 4.51 4.34 4.48
Realization % 89.80 95.34 100.0 96.23 93.79
Table 1. Realization of both indicators (stride length and stride frequency) during the world-record 100m for men and second-best 100m for women. Note: 100% is considered the best result of the
respective parameters (time for a 20m sprint, stride length, and frequency).
It is noteworthy that Bolt reaches maximum speed at the expense of maintaining 98.89% of the frequency of the stride. Griffith’s frequency fell by 3.77%, while the length increased by 3.75%.
The study of 200m world records indicates that the first 50 meters of the race are run at the expense of 79.55% and 79.48% of the stride length, respectively in men and women, and 94.13% and 94.04%
of the stride frequency and maximum running speed are achieved between the 50th and 100th meters. Length indicator (l) is 97.02% in men and 96.15% in women. In both sexes, the frequency indicator is
(f) 100%.
Table 2. Stride length and stride frequency in setting the 200m world
records for men and women.
Athlete Indicators Distance Result 200m
Men 50m 100m 150m 200m 19.19 sec
U. Bolt Time (sec) 5.60 4.32 4.52 4.75
(Jam) Realization % 77.14 100 95.57 90.94
Stride length (cm) 214 261 266 269
Realization % 79.55 97.02 98.88 100
Stride frequency num/sec 4.17 4.43 4.16 3.91
Realization % 94.13 100.0 93.90 88.26
Women 50m 100m 150m 200m 21.34 sec
F. Griffith Time (sec) 6.29 4.89 4.92 5.24
(USA) Realization % 77.74 100.0 99.39 93.32
Stride length (cm) 186 225 232 234
Realization % 79.48 96.15 99.14 100.0
Stride frequency num/sec 4.26 4.53 4.36 4.06
Realization % 94.04 100.0 96.24 89.62
Table 2. Realization of both stride indicators (l) and (f) in setting the 200m world records for men and women. Note: 100% is considered the best result of the respective parameters (time to run 50m,
stride length, and frequency).
The deterioration of the running speed from the 100th to the 150th meter is due to an increase in (l) of 3 cm (1.86%) and reduction in (f) by 6.10% in men, and an increase in (l) by 7 cm (2.99%) and
a reduction in (f) by 3.76% in women. As you can see, the same configuration is present: increased stride length at the expense of lower frequency. The tendency remains in the next 50 meters
(150-200m). Again, stride length increases for men by 1.22% and for women by 0.86%, and frequency drops by 5.64% for men and 6.62% for women.
Conclusions and Recommendations
• The analysis of the results in the 100m showed that the first phase of acceleration, as well as the transition to maximum running speed, comes at the expense of stride frequency.
• The realized values for stride length (97.17% for Bolt and 96.16% for Griffith) and stride frequency (98.89% for Bolt and 96.23% for Griffith) that lead to maximum speed show that these
indicators are optimal and highly individual. Both depend on the anatomo-morphological and power indicators of the lower limbs.
• The studied biomechanical parameters in the 200m give reason to conclude that the first 50 meters of the race are run at the expense of 79.55% and 79.48% of stride length, respectively in men and
women, and 94.13% and 94.04% of stride frequency. Both sexes achieved the highest running speed in the second 50 meters at 100% frequency.
• Both 200m world records have the same running speed configuration. The first 100 meters are overcome at the expense of frequency. From the 100th to 150th meter, the stride length (l) increases at
the expense of its frequency (f)—the increase is 3 cm for Bolt and 7 cm for Griffith. The reduction in percentage points for frequency is 6.10% for men and 3.76% for women.
• The analysis of biomechanical indicators showed that in both sprint distances, 100 meters and 200 meters, the last 20 meters in the 100m (80m-100m) and the last 50 meters in the 200m (150-200m)
are run with 100% of stride length.
Since you’re here…
…we have a small favor to ask. More people are reading SimpliFaster than ever, and each week we bring you compelling content from coaches, sport scientists, and physiotherapists who are devoted to
building better athletes. Please take a moment to share the articles on social media, engage the authors with questions and comments below, and link to articles when appropriate if you have a blog or
participate on forums of related topics. — SF
1. Stoyanov, Hristo. (2014). “Competition Model Characteristics of Elite Male Sprinters.” New Studies in Athletics, IAAF, NSA. 29(4): 53-60.
2. Stoyanov, Hristo. “The Dynamics of Velocity Development in Elite Women Sprinters.” New Studies in Athletics, IAAF, NSA. 2015: 30(3), 61-67.
3. Mann, R. & Herman, J. (1985). “Kinematics analysis of Olympic Sprint Performance: Men’s 200 Meters.” International Journal of Sport Biomechanics. (1): 151-162.
4. Ae, M., Ito, A. & Suzuki, M. (1992). “The men’s 100 metres.” New Studies in Athletics, IAAF, NSA. 7(1): 47-52.
5. Delecluse, C., Ponnet, H., & Diels, R. (1998). “Stride characteristics related to running velocity in maximal sprint running.” [w:] Riehle HJ, Vieten MM. (red) Proceedings II of XVI International
Symposium on Biomechanics in Sports, ISBS, 146-148
6. Brüggemann, G.-P., Koszewski, D. & Müller, H. (1999). Biomechanical Research Project. Athens 1997, Final report. Oxford: Meyer & Meyer Sport, 12-41.
7. Gajer, B., Thepaut-Mathieu, C. & Lehenaff, D. (1999). “Evolution of stride and amplitude during course of the 100m event in athletics.” New Studies in Athletics, 3, 43-50
8. Hunter, J.P., Marshall, R.N. & McNair, P.J. (2004). “Interaction of step length and step rate during sprint running.” Medicine and Science in Sports and Exercise. (36): 261-271. doi: 10.1249/
9. Bezodis, I.N., Salo, A.I.T. & Kerwin, D.G. (2008). “A Longitudinal Case Study of Step Characteristics in a World Class Sprint Athlete.” Presented at 26th ISBS Conference, Seoul, Korea. 537-540.
10. Bezodis, I.N., Irwin G., Kuntze, G. & Kerwin, D.G. (2011). “Changes in Step Characteristics between the Maximum Velocity and Deceleration Phases of the 100m Sprint Run.” Portuguese Journal of
Sports Sciences. 11(2): 455-458. Presented at 29th ISBS Conference, Porto, Portugal.
11. Mackala, K. (2007). “Optimisation of performance through kinematic analysis of the different phases of the 100m.” New Studies in Athletics. 22(2): 7-16.
12. Mackala, K. & Mero, A. (2013). “A kinematics analysis of three best 100m performances event.” Journal of Human Kinetics. 36: 149-160. doi: 10.2478/hukin-2013-0015
13. Müller, H. (1991). “Trends in the men^’s and women^’s sprints in the period from 1985 to 1990.” New Studies in Athletics. 6(1): 7-14.
1. Francospd
Thank you.
2. westlund
What does “at the expense of stride frequency” mean
3. E Goss
What date was this written please?
□ May 2018
4. Alfie
how does step length influence an athletes performance please?
□ how can the length of time the ball of foot is on the ground be measured against or with float speed when both feet are off the ground? The image of Tommie Smith
stride exceleration while lengthening at the end of his 1968 Olympic world record 200m run. l suspect that his foot contact time with ground
decreased to create the Subjective result I see.
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Core JavaScript Guide 1.5: 6 Functions
Chapter 6 Functions
Functions are one of the fundamental building blocks in JavaScript. A function is a JavaScript procedure—a set of statements that performs a specific task. To use a function, you must first
define it; then your script can call it.
This chapter contains the following sections:
Defining Functions
A function definition consists of the function keyword, followed by
□ The JavaScript statements that define the function, enclosed in curly braces, { }. The statements in a function can include calls to other functions defined in the current application.
For example, the following code defines a simple function named square:
function square(number) {
return number * number;
The function square takes one argument, called number. The function consists of one statement that indicates to return the argument of the function multiplied by itself. The return statement
specifies the value returned by the function.
All parameters are passed to functions by value; the value is passed to the function, but if the function changes the value of the parameter, this change is not reflected globally or in the
calling function. However, if you pass an object as a parameter to a function and the function changes the object's properties, that change is visible outside the function, as shown in the
following example:
function myFunc(theObject) {
mycar = {make:"Honda", model:"Accord", year:1998};
x=mycar.make; // returns Honda
myFunc(mycar); // pass object mycar to the function
y=mycar.make; // returns Toyota (prop was changed by the function)
A function can be defined based on a condition. For example, given the following function definition:
if (num == 0)
function myFunc(theObject) {
the myFunc function is only defined if the variable num equals 0. If num does not equal 0, the function is not defined, and any attempt to execute it will fail.
In addition to defining functions as described here, you can also define Function objects, as described in "Function Object" on page 106.
A method is a function associated with an object. You'll learn more about objects and methods in Chapter 7, "Working with Objects."
A function can also be defined inside an expression. This is called a function expression. Typically such a function is anonymous; it does not have to have a name. For example, the function
square could have been defined as:
const square = function(number) {return number * number};
This is convenient when passing a function as an argument to another function. The following example shows the map function being defined and then called with an anonymous function as its first
function map(f,a) {
var result=new Array;
for (var i = 0; i != a.length; i++)
result[i] = f(a[i]);
return result;
map(function(x) {return x * x * x}, [0, 1, 2, 5, 10];
Defining a function does not execute it. Defining the function simply names the function and specifies what to do when the function is called. Calling the function actually performs the specified
actions with the indicated parameters. For example, if you define the function square, you could call it as follows.
The preceding statement calls the function with an argument of five. The function executes its statements and returns the value twenty-five.
The arguments of a function are not limited to strings and numbers. You can pass whole objects to a function, too. The show_props function (defined in "Objects and Properties" on page 91) is an
example of a function that takes an object as an argument.
A function can even be recursive, that is, it can call itself. For example, here is a function that computes factorials:
function factorial(n) {
if ((n == 0) || (n == 1))
return 1
else {
var result = (n * factorial(n-1) );
return result
You could then compute the factorials of one through five as follows:
a=factorial(1) // returns 1
b=factorial(2) // returns 2
c=factorial(3) // returns 6
d=factorial(4) // returns 24
e=factorial(5) // returns 120
The arguments of a function are maintained in an array. Within a function, you can address the arguments passed to it as follows:
where i is the ordinal number of the argument, starting at zero. So, the first argument passed to a function would be arguments[0]. The total number of arguments is indicated by arguments.length.
Using the arguments array, you can call a function with more arguments than it is formally declared to accept. This is often useful if you don't know in advance how many arguments will be passed
to the function. You can use arguments.length to determine the number of arguments actually passed to the function, and then treat each argument using the arguments array.
For example, consider a function that concatenates several strings. The only formal argument for the function is a string that specifies the characters that separate the items to concatenate. The
function is defined as follows:
function myConcat(separator) {
var result="" // initialize list
// iterate through arguments
for (var i=1; i<arguments.length; i++) {
result += arguments[i] + separator
return result
You can pass any number of arguments to this function, and it creates a list using each argument as an item in the list.
// returns "red, orange, blue, "
myConcat(", ","red","orange","blue")
// returns "elephant; giraffe; lion; cheetah; "
myConcat("; ","elephant","giraffe","lion", "cheetah")
// returns "sage. basil. oregano. pepper. parsley. "
myConcat(". ","sage","basil","oregano", "pepper", "parsley")
See the Function object in the Core JavaScript Reference for more information.
JavaScript 1.3 and earlier versions. The arguments array is a property of the Function object and can be preceded by the function name, as follows:
JavaScript has several top-level predefined functions:
□ encodeURI, decodeURI, encodeURIComponent, and decodeURIComponent (all available with Javascript 1.5 and later).
The following sections introduce these functions. See the Core JavaScript Reference for detailed information on all of these functions.
eval Function
The eval function evaluates a string of JavaScript code without reference to a particular object. The syntax of eval is:
where expr is a string to be evaluated.
If the string represents an expression, eval evaluates the expression. If the argument represents one or more JavaScript statements, eval performs the statements. Do not call eval to evaluate an
arithmetic expression; JavaScript evaluates arithmetic expressions automatically.
isFinite Function
The isFinite function evaluates an argument to determine whether it is a finite number. The syntax of isFinite is:
where number is the number to evaluate.
If the argument is NaN, positive infinity or negative infinity, this method returns false, otherwise it returns true.
The following code checks client input to determine whether it is a finite number.
if(isFinite(ClientInput) == true)
/* take specific steps */
isNaN Function
The isNaN function evaluates an argument to determine if it is "NaN" (not a number). The syntax of isNaN is:
where testValue is the value you want to evaluate.
The parseFloat and parseInt functions return "NaN" when they evaluate a value that is not a number. isNaN returns true if passed "NaN," and false otherwise.
The following code evaluates floatValue to determine if it is a number and then calls a procedure accordingly:
if (isNaN(floatValue)) {
} else {
parseInt and parseFloat Functions
The two "parse" functions, parseInt and parseFloat, return a numeric value when given a string as an argument.
where parseFloat parses its argument, the string str, and attempts to return a floating-point number. If it encounters a character other than a sign (+ or -), a numeral (0-9), a decimal point, or
an exponent, then it returns the value up to that point and ignores that character and all succeeding characters. If the first character cannot be converted to a number, it returns "NaN" (not a
parseInt parses its first argument, the string str, and attempts to return an integer of the specified radix (base), indicated by the second, optional argument, radix. For example, a radix of ten
indicates to convert to a decimal number, eight octal, sixteen hexadecimal, and so on. For radixes above ten, the letters of the alphabet indicate numerals greater than nine. For example, for
hexadecimal numbers (base 16), A through F are used.
If parseInt encounters a character that is not a numeral in the specified radix, it ignores it and all succeeding characters and returns the integer value parsed up to that point. If the first
character cannot be converted to a number in the specified radix, it returns "NaN." The parseInt function truncates the string to integer values.
Number and String Functions
The Number and String functions let you convert an object to a number or a string. The syntax of these functions is:
where objRef is an object reference.
The following example converts the Date object to a readable string.
D = new Date (430054663215)
// The following returns
// "Thu Aug 18 04:37:43 GMT-0700 (Pacific Daylight Time) 1983"
x = String(D)
escape and unescape Functions
The escape and unescape functions let you encode and decode strings. The escape function returns the hexadecimal encoding of an argument in the ISO Latin character set. The unescape function
returns the ASCII string for the specified hexadecimal encoding value.
The syntax of these functions is:
These functions are used primarily with server-side JavaScript to encode and decode name/value pairs in URLs.
The escape and unescape functions do not work properly for non-ASCII characters and have been deprecated. In JavaScript 1.5 and later, use encodeURI, decodeURI, encodeURIComponent, and | {"url":"https://docs.huihoo.com/javascript/CoreGuideJS15/fcns.html","timestamp":"2024-11-12T02:18:48Z","content_type":"text/html","content_length":"45966","record_id":"<urn:uuid:b0b57191-9577-493f-b38d-7e868b7178d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00877.warc.gz"} |
phiP criterion
phiP {DiceDesign} R Documentation
phiP criterion
Compute the \phi_p criterion (strongly linked to mindist criterion)
phiP(design, p=50)
design a matrix (or a data.frame) corresponding to the design of experiments.
p the "p" in the Lp norm which is taken
The \phi_p criterion is defined by the L_p norm of the sum of the inverses of the design inter-point euclidean distances:
\phi_{p}=\left[\sum_{i,j=1\ldots N,i<j}\,\,d_{ij}^{-p}\right]^{\frac{1}{p}}
A higher value corresponds to a more regular scaterring of design points.
When p tends to infinity, optimizing a design with \phi_p is equivalent to optimizing a design with mindist.
A real number equal to the value of the \phi_p criterion for the design.
G. Damblin & B.Iooss
Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of sapce filling designs: optimization of Latin Hypercube Samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.
Fang K.-T., Li R. and Sudjianto A. (2006). Design and Modeling for Computer Experiments, Chapman & Hall.
Pronzato, L. and Muller, W. (2012). Design of computer experiments: space filling and beyond, Statistics and Computing, 22:681-701.
See Also
geometric criterion (mindist)
dimension <- 2
n <- 40
X <- matrix(runif(n*dimension), n, dimension)
version 1.10 | {"url":"https://search.r-project.org/CRAN/refmans/DiceDesign/html/phiP.html","timestamp":"2024-11-05T23:11:32Z","content_type":"text/html","content_length":"3565","record_id":"<urn:uuid:0e98a89b-10a7-4556-98c4-81a2e3adf563>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00263.warc.gz"} |
A functorial approach to rank functions on triangulated categories | Teresa Conde
We study rank functions on a triangulated category $\mathcal{C}$ via its abelianisation $\operatorname{mod}\mathcal{C}$. We prove that every rank function on $\mathcal{C}$ can be interpreted as an
additive function on $\operatorname{mod}\mathcal{C}$. As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions
to a number of important concepts in the functor category $\operatorname{Mod}\mathcal{C}$. We study the connection between rank functions and functors from $\mathcal{C}$ to locally finite
triangulated categories, generalising results by Chuang and Lazarev. In the special case $\mathcal{C}=\mathcal{T}^c$ for a compactly generated triangulated category $\mathcal{T}$, this connection
becomes particularly nice, providing a link between rank functions on $\mathcal{C}$ and smashing localisations of $\mathcal{T}$. In this context, any integral rank function can be described using the
composition length with respect to certain endofinite objects in $\mathcal{T}$. Finally, if $\mathcal{C}=\operatorname{per}(A)$ for a differential graded algebra $A$, we classify homological
epimorphisms $A\to B$ with $\operatorname{per}(B)$ locally finite via special rank functions which we call idempotent.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2024, no. 811, 2024, pp. 135-181 | {"url":"https://teresaconde.xyz/publication/pub8/","timestamp":"2024-11-08T09:11:41Z","content_type":"text/html","content_length":"16923","record_id":"<urn:uuid:def054c5-5491-4e39-b491-60b417619533>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00862.warc.gz"} |
Describe and apply the effect of a single transformation on two-dimensional figures using coordinates and the coordinate plane.
Clarification 1: Within this benchmark, transformations are limited to reflections, translations, rotations or dilations of images.
Clarification 2: Lines of reflection are limited to the x-axis, y-axis or lines parallel to the axes.
Clarification 3: Rotations must be about the origin and are limited to 90°, 180°, 270° or 360°.
Clarification 4: Dilations must be centered at the origin.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 8
Strand: Geometric Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
• Coordinates
• Coordinate Plane
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 6, students plotted rational-number ordered pairs in all four quadrants as well as identified the
- or
-axis as a line of reflection when two ordered pairs have an opposite
- or
-coordinate. In grade 7, students solved mathematical and real-world problems involving scale factors. In grade 8, students apply a single transformation using coordinates and the coordinate plane.
In Algebra 1, students will apply a single transformation to functions. In Geometry, students will describe transformations given a preimage and an image and represent the transformation
algebraically using coordinates and use them to study congruence and similarity.
• Use grid paper to illustrate translations of a line or triangle to demonstrate the relationship between them and a new image. Then, illustrate translations of more complex figures such as
• Transformations can be noted using the prime notation (′) for the image and its vertices. The preimage and its vertices will not have prime notation.
□ For example, the picture below showcases a single transformation.
• Problem types include telling which direction, clockwise or counterclockwise, for rotations.
• Instruction includes looking for patterns to create rules for transformations on the coordinate plane.
• For mastery of this benchmark, single transformations include one vertical translation or one horizontal translation. A vertical and horizontal translation would be considered two
Common Misconceptions or Errors
• Students may incorrectly visualize transformation on the coordinate plane. To address this misconception, provide students with manipulatives.
• Students may incorrectly apply rules for transformations. To address this misconception, students should generate examples and non-examples of given transformations.
Strategies to Support Tiered Instruction
• Teacher supports understanding of transformations on the coordinate place by providing examples using geometric software. Instruction includes the use of manipulatives and graph paper.
• Teacher reminds students when plotting points on a coordinate plane that they can first find the $x$-coordinate on the $x$-axis (horizontal axis) and then find the $y$-coordinate on the $y$-axis
(vertical axis).
• Teacher reviews vocabulary discussing the meaning of the terms.
□ Translation is a vertical or horizontal slide of the figure. To determine the coordinates of the image of a translated figure you must add or subtract the horizontal distance to the $x$
-coordinate of each vertex and add or subtract the vertical distance to the $y$-coordinate of each vertex. (Note that in later courses, students learn that translation can also occur
□ Preimage is the figure before any transformations are performed.
□ Image is the figure after a transformation is performed.
• Teacher co-creates a graphic organizer to generate examples and non-examples of reflections, translations, rotations, or dilations of images.
• Teacher provides instruction to support understanding of applying the translation to all vertices, not just one vertex.
• Teacher reviews directions of rotations. Clockwise is the direction the hands go on an analog clock
□ For example, which quadrant would the image be in if you rotated the figure?
☆ 90 degrees clockwise
☆ 90 degrees counterclockwise
☆ 180 degrees clockwise
☆ 180 degrees counterclockwise
• Teacher reviews which is the $x$-axis and which is the $y$-axis for students that incorrectly reflect across the wrong axis. Teacher co-creates anchor chart explaining different parts of
coordinate plane, and how to plot and label points.
□ For example, teachers could ask students which quadrant the image would be in if you reflected the figure across the $x$-axis or across the $y$-axis.
• Instruction includes providing students with manipulatives for students that incorrectly visualize transformations on the coordinate plane.
Instructional Tasks
Instructional Task 1 (MTR.1.1, MTR.2.1)
Use the information you have learned about transformations to complete the task below.
• Part A. Using graph paper, plot the following points to create an image on the coordinate plane.
$A$(−3,2), $B$(0,1), $C$(−3, −1) and $D$(−1, −1)
• Part B. Using a different color for each transformation, complete each of the following transformations on the same coordinate plane.
a. A reflection over the $y$-axis
b. A rotation of 180° about the origin
• Part C. Will any of the new images include the origin?
Instructional Items
Instructional Item 1
Find the coordinates of the vertices of the image of triangle
after the translation 3 units to the left.
Instructional Item 2
Find the coordinates of the vertices of the image of triangle
after a 270° counterclockwise rotation about the origin.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
This benchmark is part of these courses.
Related Access Points
Alternate version of this benchmark for students with significant cognitive disabilities.
Identify the coordinates of the vertices of a common polygon after a single translation, rotation or dilation on the coordinate plane.
Related Resources
Vetted resources educators can use to teach the concepts and skills in this benchmark.
Formative Assessments
Lesson Plans
Problem-Solving Tasks
MFAS Formative Assessments
Dilation Coordinates:
Students are asked to dilate two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.
Reflection Coordinates:
Students are asked to reflect two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.
Rotation Coordinates:
Students are asked to rotate two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.
Translation Coordinates:
Students are asked to translate two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.
Student Resources
Vetted resources students can use to learn the concepts and skills in this benchmark.
Problem-Solving Tasks
Congruent Triangles:
This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the
notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''
Type: Problem-Solving Task
Reflecting Reflections:
In this resource, students experiment with the reflection of a triangle in a coordinate plane.
Type: Problem-Solving Task
Point Reflection:
The purpose of this task is for students to apply a reflection to a single point. The standard asks students to apply the effect of a single transformation on two-dimensional figures. Although this
problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point (1000,2012) are very
large. If students try to plot this point and the line of reflection on the usual x-y coordinate grid, then either the graph will be too big or else the point will lie so close to the line of
reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover,
reflections of two-dimensional figures are found by reflecting individual points.
Type: Problem-Solving Task
Reflecting a Rectangle Over a Diagonal:
The task is intended for instructional purposes and assumes that students know the properties of rigid transformations. Note that the vertices of the rectangles in question do not fall exactly at
intersections of the horizontal and vertical lines on the grid. This means that students need to approximate and this provides an extra challenge. Also providing a challenge is the fact that the
grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. However, this choice of grid also
makes it easier to reason about the reflections.
Type: Problem-Solving Task
Triangle congruence with coordinates:
In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical
Type: Problem-Solving Task
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.
Problem-Solving Tasks
Congruent Triangles:
This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the
notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''
Type: Problem-Solving Task
Reflecting Reflections:
In this resource, students experiment with the reflection of a triangle in a coordinate plane.
Type: Problem-Solving Task
Point Reflection:
The purpose of this task is for students to apply a reflection to a single point. The standard asks students to apply the effect of a single transformation on two-dimensional figures. Although this
problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point (1000,2012) are very
large. If students try to plot this point and the line of reflection on the usual x-y coordinate grid, then either the graph will be too big or else the point will lie so close to the line of
reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover,
reflections of two-dimensional figures are found by reflecting individual points.
Type: Problem-Solving Task
Reflecting a Rectangle Over a Diagonal:
The task is intended for instructional purposes and assumes that students know the properties of rigid transformations. Note that the vertices of the rectangles in question do not fall exactly at
intersections of the horizontal and vertical lines on the grid. This means that students need to approximate and this provides an extra challenge. Also providing a challenge is the fact that the
grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. However, this choice of grid also
makes it easier to reason about the reflections.
Type: Problem-Solving Task
Triangle congruence with coordinates:
In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical
Type: Problem-Solving Task | {"url":"https://www.cpalms.org/PreviewStandard/Preview/15521","timestamp":"2024-11-11T14:06:39Z","content_type":"text/html","content_length":"139980","record_id":"<urn:uuid:8fec17a6-ed8e-4181-9e84-ff8bdefa8473>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00667.warc.gz"} |
PPT - The free carrier transport properties in proton and neutron irradiated Si(Ge) PowerPoint Presentation - ID:3325423
1. The free carrier transport properties in proton and neutron irradiated Si(Ge) (and comparison with Si) J.Vaitkus, V.Rumbauskas, L.Makarenko1, A.Mekys, J.Storasta Vilnius University, Institute of
Applied Research, Vilnius, Lithuania 1Belorussian university, Minsk, Belorussia J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
2. The important questions: 1) Are the changes in the semiconductor homogeneity caused by the irradiation? 2) What kind of inhomogeneities are induced by crystal growth (different doping) and
treatments?The answers can be find by investigation of the transport properties of free carriers. J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
3. Basic principle V - + A Hall and magnetoresistance effects are “simple classical effects” demonstrating the transport properties of free carrier. f =1 in a thin sample J.Vaitkus, L.Makarenko et
all. RD50, CERN, 2012
4. Complications of the Basic principle in the nonhonogeneous sample V - + A Scattering by large defects: J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
5. Inhomogeneities V. G. Karpov, A. J. Shik and B. I. Schklovskij (1982): The cells of typical clusters: I, II and III. Dashed lines indicates the equipotential lines Also, a bit different analyse:
W. Siegel, S. Schulte, C. Reichel, G. Kuhnel, J. Monecke. „Anomalous temperature dependence of the Hall mobility in undoped bulk GaAs“. J. Appl. Phys., Vol. 82, No. 8, pp.3832-3835 (1997)
J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
6. Single crystals Si (WODEAN1 series) • The weak dependence on T was observed in low irradiated samples • The Hall and magnetoresistance mobility behavior was different. • Anomalous Hall mobility
dependence on T was observed Irradiation by neutrons J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
7. A new series: Minsk-KEF Irradiation by electrons (6MeV) to create only the point defects Magnetoresistivity mobility values are typical for good n-type Si. At higher doses of irradiation the Hall
signal lowers at low T similarly to the case of clusters. Large scale electric potential disturbance occurs. J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
8. KDB conductivity (irradiated 4 MeV electrons) The conductivity in the initial samples decreases with T. This is the case when the carrier density changes less than the mobility. At the higher
irradiation doses the conductivity decreases. The greater sample volume is damaged. J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
9. KDB density Thermal activation from the density show some clear values. J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
10. KEF conductivity Similar situation as in KDB samples. But the conductivity decreases less for the greater doses while it decreases more for lower doses. J.Vaitkus, L.Makarenko et all. RD50, CERN,
11. KEF density Thermal activation from the density show some clear values. J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
12. Si(Ge)Cz SiGe crystals were grown in Leibniz Institute for Crystal Growth, Berlin, Germany by N.V. Abrosimov Due to deformation of the lattice the increase of radiation hardness is waited. It
exists experience to destroy the dislocation net in GaAs by adding isovalent impurity In. What is happening in transport properties? The start of the analyze cycle. J.Vaitkus, L.Makarenko et all.
RD50, CERN, 2012
13. Hall mobility in Si(Ge) (neutron irradiation) Adding of Ge enhances the hole mobility. Irradiation 1e12 cm-2 increases the Hall mobility but the 1e13 cm-2 decreases the Hall mobility in both
n-type and p-type (at 200-3000 C) J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
14. Proton irradiated Si(Ge) J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
15. Proton irradiated Si(Ge) J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
16. Si(Ge) Neutron irradiation Proton irradiation J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
17. Neutrons, Si(Ge) Hall Magnetoresistance J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
18. Conclusions: • The main peculiarities of transport phenomena are induced by cluster defects. • Hall effect and magnetoresistance measurement allow to reveal these inhomogeneities • The initial
studies of low irradiated Si(Ge) were performed. • The irradiation to the higher fluence is the next step. J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
19. THANK YOU FOR YOUR ATTENTION J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
20. Hall factor Hall scattering factor rH is defined by following expressions: The relaxation time for individual scattering process often follows a power law: Variation of Hall scattering factor
with total impurity density Nimp. In n-type Si. Experimantal points: -x- 77K, -o- 300K. Solid curves: calculated (from Kirnas et al., 1974) J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012
21. Inhomogeneities l1 l1 grain VA R. H. Bube model: l2 VH [R. H. Bube, Appl. Phys. Lett. 13, 136 (1968)] J.Vaitkus, L.Makarenko et all. RD50, CERN, 2012 | {"url":"https://www.slideserve.com/egil/the-free-carrier-transport-properties-in-proton-and-neutron-irradiated-si-ge","timestamp":"2024-11-13T08:08:27Z","content_type":"text/html","content_length":"96892","record_id":"<urn:uuid:effe3b9d-7e22-405d-a88b-c6b340dafc6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00041.warc.gz"} |
critical speed of ball mill unit
WEBAug 1, 2013 · Sepúlveda (2004) has done calculations on ball breakage based on impact, showing that the speed (v) in metres per second at which a ball could be moving, can be estimated by (3) v =
π N c D mill where N c (rad/s) is the critical mill speed, and D mill the mill diameter (m). | {"url":"https://savon-cocagne.fr/critical-speed-of-ball-mill-unit_6997.html","timestamp":"2024-11-07T01:14:10Z","content_type":"text/html","content_length":"34665","record_id":"<urn:uuid:5947461e-17bf-43d8-9461-0ade99d1db1f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00565.warc.gz"} |
Division Operations | Orchids International School
The division is one of the arithmetic processes. Multiplication and division are co-related to each other. Here students will learn division sums for class 4.
In this learning concept, the students will also learn to
• Classify the division sums.
• Evaluate division using the grid method.
• Identify division terms and the long division method.
Each concept is explained to class 4 maths students using illustrations, examples, and mind maps. Students can assess their learning by solving the two printable worksheets given at the page’s end.
Download the division worksheets for class 4 and check the solutions to the division question for class 4 provided in PDF format.
What Is Division?
Division: Division is one of the four arithmetic operations. The process of distributing a group of things into equal parts is called division.
Dividend: The number that is to be divided is called Dividend.
Divisor: The number by which the dividend is divided is called the Divisor.
Quotient: The results obtained in the division process is called the Quotient.
The symbol used for division is “÷”.
Division As Opposite of Multiplication
Example :
There are 12 balls and 3 boxes. Arrange 12 balls equally in the 3 boxes.
We know that 3 × 4 = 12. So, if 12 balls are divided into three boxes, then each box will contain 4 balls. 12 ÷ 3 = 4.
Division Using Grid Method:
Example :
Divide 96 by 4.
Write 96 on the grid and 4 outside the box.
Long Method of Division for 2-digit Numbers
Example :
Divide 39 by 3.
Fun facts:
• If a number is divided by itself then the quotient is 1.
• If a number is divided by 1 then the quotient is the number itself.
Common Mistakes:
Students generally neglect the place holder of 0.
Example :
404 ÷ 2
The correct way is
Therefore, 404 ÷ 2 = 202 | {"url":"https://www.orchidsinternationalschool.com/maths-concepts/division-operations","timestamp":"2024-11-06T18:21:32Z","content_type":"text/html","content_length":"31277","record_id":"<urn:uuid:9244236e-b4cd-4baf-90db-fa4528585df7>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00733.warc.gz"} |
Hacking the Screen
Problem A
Hacking the Screen
The ZOO management had been wondering for a long time how to increase the number of children visitors to the ZOO. The solution was surprising and unexpected in many ways. They installed a huge screen
past the entrance and started to display short quizzes on it. The child in the crowd who first shouts out the answer to the quiz question is granted one day free access to the ZOO. The screen became
soon very popular and various types of quizzes are routinely shown there. One type of the quiz is the math quiz containing arithmetic operations on integers. The management worries that older
siblings and friends of the children might develop a math quiz screen hacking strategy: Snap the screen with the phone, run the image recognition SW which extracts formulas from the image, evaluates
them, and presents the solution to the phone holder who immediately shouts out the answer.
Your task is to assess the difficulty of producing the screen hacking software. To get a better feel of the problem you will first develop a simple toy model application. Your code will read the
formula presented in the preprocessed form of ASCII art and evaluate it.
First line of each test case contains two integers $R$ and $C$ ($1 \leq R \leq 3$, $1 \leq C \leq 1\, 000$). Each of the following $R$ lines contains $C$ characters. The whole matrix of $R \times C$
characters represents a single arithmetic formula written in ASCII art and generated by the following set of rules:
FORMULA -> COMPLEX | FORMULA + COMPLEX | FORMULA - COMPLEX
COMPLEX -> SQRT | FRACTION | TERM
SQRT -> \/SIMPLE
FRACTION -> ======
SIMPLE -> TERM | SIMPLE + TERM | SIMPLE - TERM
TERM -> INTEGER | INTEGER * TERM
INTEGER -> 0 | 1 | 2 | 3 | ... | 999999 | 1000000
There are also a few additional specifications regarding the layout of the formula.
• The horizontal bar of each SQRT is made of one or more underscore symbols (‘_’, ascii decimal code $95$) and it always occupies the uppermost line of the formula in the screen.
• When the formula occupies exactly two lines, then the first line contains only horizontal bars of all SQRT parts of the formula.
• When the formula occupies exactly three lines, then all TERMs and all arithmetic operation symbols which are not part of any FRACTION or SQRT occupy the second line of the formula in the screen.
• The length of the horizontal bar of SQRT is the same as the length of SIMPLE under the bar.
• The fraction bar in FRACTION consists of one or more equality signs, its length is equal to the maximum of the lengths of SIMPLE above the bar and SIMPLE below the bar.
• There is always exactly one space preceding and following each arithmetic operation symbol (+, -, *) on a particular line.
• The formula exactly fits in to the $R \times C$ matrix, there are no blank/empty columns in front of the whole formula or behind it.
The whole formula is evaluated according to the standard arithmetic rules. Namely: Each FORMULA and each TERM is evaluated from left to right. Each SIMPLE is also evaluated from left to right with
the additional standard condition that the multiplication has higher priority than the addition/subtraction. Evaluation of SQRT and FRACTION is also standard. The value of any evaluated FORMULA,
COMPLEX, SQRT, FRACTION, SIMPLE and TERM is an integer whose absolute value does not exceed $1\, 000\, 000$.
For each test case print a separate line with the value $V$ of the input formula.
Sample Input 1 Sample Output 1
1 + 2 * 3 - 4
Sample Input 2 Sample Output 2
_________ 13
\/3 * 4 - 3 + 10
Sample Input 3 Sample Output 3
6 * 4 2
Sample Input 4 Sample Output 4
22 __ -3
3 - == - \/16 | {"url":"https://nus.kattis.com/courses/CS3233/CS3233_S2_AY2223/assignments/ybgxci/problems/screen","timestamp":"2024-11-04T12:05:33Z","content_type":"text/html","content_length":"32732","record_id":"<urn:uuid:71e86217-e045-4cf8-946b-791c8918d695>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00876.warc.gz"} |
Solve For The Following Expressions1 Texleftfrac15right2tex2 Tex2frac18tex3 Texfrac52tex4 Texfrac19tex
Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Experience the convenience of getting accurate answers to your questions from a dedicated
community of professionals. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Solve for the following expressions:
1. [tex]\(\left(\frac{1}{5}\right)^{-2}\)[/tex]
2. [tex]\(2^{\frac{1}{8}}\)[/tex]
3. [tex]\(\frac{5}{2}\)[/tex]
4. [tex]\(\frac{1}{9}\)[/tex]
Sagot :
To solve the given expression [tex]\(\left(\frac{1}{5}\right)^{-2}\)[/tex]:
1. Understanding Negative Exponents:
By the rules of exponents, a negative exponent [tex]\(a^{-n}\)[/tex] can be rewritten as [tex]\(\frac{1}{a^n}\)[/tex].
Therefore, [tex]\(\left(\frac{1}{5}\right)^{-2}\)[/tex] can be transformed using this rule.
2. Rewriting the Expression:
Applying the rule:
[tex]\[ \left(\frac{1}{5}\right)^{-2} = \frac{1}{\left(\frac{1}{5}\right)^2} = \frac{1}{\frac{1}{25}} \][/tex]
3. Simplifying the Fraction:
[tex]\(\frac{1}{\frac{1}{25}}\)[/tex] is the reciprocal of [tex]\(\frac{1}{25}\)[/tex].
The reciprocal of [tex]\(\frac{1}{25}\)[/tex] is 25.
[tex]\[ \frac{1}{\frac{1}{25}} = 25 \][/tex]
Thus, the value of the expression [tex]\(\left(\frac{1}{5}\right)^{-2}\)[/tex] is [tex]\(25\)[/tex].
So, the correct number to fill in the squares is:
[tex]\[ \boxed{25} \][/tex]
The possible answers given [tex]\(2^{\frac{1}{8}}, \frac{5}{2}, \frac{1}{9}\)[/tex] are incorrect as they do not match the correct simplified result [tex]\(25\)[/tex]. The correct answer from the
provided choices would be not listed, so you should manually input [tex]\(25\)[/tex] as the result.
Answer Link
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How many bales?
A thinking mathematically targeted teaching resource focused on exploring multiplicative relationships through an investigation based on a classic nursery rhyme.
Adapted from a task by Dianne Siemon
Siemon, D. (2013). Launching mathematical futures: the key role of multiplicative thinking. Mathematics: Launching Futures, 34-42.
Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales,
Collect resources
You will need:
• something to write on
• something to write with
• some counters (you could use pasta or brick building pieces).
How many bales? Part 1
Watch How many bales? Part 1 video (1:49).
[Text over a navy background: How many bales? From Professor Dianne Siemon. The NSW Government logo is in the lower right corner of the screen. Small font text at the bottom of the screen read: NSW
Mathematics Strategy Professional Learning team (NSWMS PL team).]
Hi there, mathematicians. Welcome back. We have a new problem for you today that comes from Professor Dianne Siemon. And Professor Di is a brilliant mathematician, and we love it when she shares her
problems with us.
[A title on a write background reads: You will need…
Bullet points below read:
• something to write on
• something to write with
• some counters (you could use pasta or LEGO pieces)
Small font text on top of the screen reads – NSW Department of Education
An image shows some small red plastic tiles on a table. Next to the tiles on the right-hand side are colour textas.]
Today you will need something to write on, something to write with, and some counters. Now, you could use pasta or LEGO pieces as well.
[Text on a blue background reads: Let’s investigate!]
Let's investigate.
[A title on a white background reads: How many bales of wool? From Professor Dianne Siemon.
Below the title is an illustration of a black sheep next to three bales of wool. Text on the image reads – Baa Baa Black Sheep.
Under the illustration is text that reads – If there were 5 sheep, how many bales of wool?]
How many bales of wool? So, you may or may not have heard of the nursery rhyme 'Baa Baa Black Sheep'. And it goes like this. Baa, baa, black sheep, have you any wool? Yes, sir, yes, sir, three bags
full. One for the master and one for the dame. And one for the little girl who lives down the lane.
But what you may not know is, there's actually some really interesting mathematics in the 'Baa Baa Black Sheep' nursery rhyme, because as we know, maths is everywhere. The nursery rhyme starts by
asking the sheep if he has any bags of wool. To which the sheep replies,' Well, yes, sir. Yes, sir. I have three bags full'. And that's where our problem comes from. If there were five sheep, how
many bales of wool would there be in total? Over to you, mathematicians. Can you solve Professor Di's problem?
[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]
[End of transcript]
How many bales? Part 2
Watch How many bales? Part 2 video (12:47).
[On a black desktop, a white sheet of A4 paper has a blue header that reads ‘If there were 5 sheep, how many bales of wool?’ On the right of the paper there are four coloured markers, red, light
blue, blue and pink.]
Welcome back, mathematicians. How did you go? Did you find out that we would have 15 bales of wool in total if we had five sheep? I hope there was lots of sweaty brains. Professor Di always leaves me
with a sweaty brain. So, let's dig into some thinking.
[The speaker picks up the red marker to draw on the sheet of paper.]
And to get started, let's think about what we already know from the problem. And our problem was if there were 5 sheep, how many bales of wool? Yeah, that's right. Good thinking. We know that we have
5 sheep. And we can start by drawing our 5 sheep on the page. And I could spend some time to draw some really fancy sheep, but these sheep don't need to be fancy. So, what I might do is I might use a
cloud shape to represent each of our sheep. So, we have 1, 2, 3, 4 and 5.
So, we have our 5 cloud sheep now. And from the nursery rhyme and the picture Professor Di shared with us, we also know that each of our sheep needs to have 3 bales of wool. How might we represent
our bales of wool? Hmm. Yeah, we could draw three rectangles under each sheep so that now this sheep has 3 bales of wool. And if we continue to draw these, we can actually represent our bales of wool
for each of our sheep.
Now that each of our sheep has its 3 bales of wool, we can start to think about how might we be able to find how many bales of wool there are altogether. Now, I know that there are too many bales to
see just by looking and thinking. But what I'm thinking is we could actually use counting or skip counting to help us solve this problem. And because we have 3 bales of wool for each sheep, we can
use the counting sequence of 3’s to help us determine how many.
So, we have 3 bales of wool for this sheep. And I know the next number is 6 so now we have 6 bales of wool. And I know the next number in the sequence is 9. So, we now have 9 bales of wool, but I'm
actually not too confident in knowing what the next number in the sequence is. So, what I'm going to do is use a counting sequence I'm comfortable and confident with and continue to count by ones.
So, I know we have 9.
And then we'll have one more is 10 and another one is 11. And the next number in the sequence is 12. We'll have 13 and then 14, and our last one will make 15. So, let's have another look. At the
start, we were pretty confident in counting by 3’s. And we had 3, 6, 9. And then we continued to count on by ones because we were more confident. We had 10, 11, 12, 13, 14, and 15. So, we've solved
the problem. If there were 5 sheep and each sheep had 3 bales of wool, how many bales of wool we have? Well, we know that we have 15 bales of wool in total.
[A second sheet of white A4 paper has a red header that reads ‘If there were 5 sheep, how many bales of wool?’]
Another strategy that we could use to find out how many bales of wool in total is to arrange our bales of wool into equal groups.
[The speaker picks up the light blue marker pen to draw on the sheet of paper.]
And this time, we can show our thinking by drawing as well. And what we might do is represent each of our sheep using a circle. So, I'll draw 5 circles and I'm actually going to arrange them in a
dice pattern because a dice pattern is a familiar structure for me. And I know that I have 5 groups just by looking and thinking.
Now that we have our 5 circle sheep here, we still need to add 3 bales of wool to each of our circles. And what I'll do is represent our bales of wool like this.
[The speaker draws 3 ‘V’ shapes inside the top left circle and then proceeds to do the same with the other 4 circles.]
And if I arrange them in the circle this way, I know I have 3 bales just by looking because I can see the points of a triangle. And I know that a triangle has 3 points. So, if I continue to put our
bales into our circle sheep, in this way, I don't need to count each one to know that I have 3. So now that we have our 5 equal groups of 3, we still need to figure out how many bales of wool there
are in total. And looking inside of 5, I know that we have 5 x 3, but 5 x 3 isn't a number fact that I know. But what I do know is that inside 5 x 3, there are 2 x 3 and I can see our 2 3s here.
[The speaker circles the 2 circles on the left with the purple marker.]
And I know that 2 x 3 is 6. And actually, I can see another 2 3s on this side here. And I can use what I know about doubling to double 6. And I know that double 6 is 12. So, so far, we've determined
that we have 12 bales of wool, but we still have one more 3, and I can just count on. So, I know we have 12. We have 13, 14 and 15. So, we have 15 bales of wool in total. And we know that 12 and 3
more is 15. And once, again, we found out that if we had 5 sheep, we know that there would be 15 bales of wool.
[A third sheet of white A4 paper has a purple header that reads ‘If there were 5 sheep, how many bales of wool?’ On the left of the paper there is a cluster of small red plastic squares.]
[The speaker briefly retrieves the first sheet of A4 paper from earlier.]
Mathematicians, I was just looking and thinking about our representation and our strategy we use to solve how many bales of wool here. And I remembered that for each sheep, we drew 3 bales of wool.
And the way that we drew those 3 bales of wool could actually make a connection. And it made me think of how we could take these 3 bales and form them into an array. So, if we take the 3 bales in
that same structure from our first strategy and arrange them like this, we now have one row of 3.
[The speaker arranges three of the red squares into a row.]
And if I can continue doing this, we'll have 2 rows of 3 and 3 rows of 3, 4 rows of 3. And our last one, if we arrange them again like this, we now have 5 rows of 3, and we've been able to reform our
bales of wool into an array. And I know by looking at this array that we have 5 rows and 3 in each row. But I don't know what 5 3s is as a number fact. That's not a known fact for me yet.
But I do know that as a mathematician, I can use noticing and wondering to help me solve problems. And when I look at our structure of five threes here, I can see some familiar spatial patterns. And
I notice that I can see 3 3s. If I move these up a little bit, we can see our 3 3s here. And I can see 3 3s. But also, when I look at it, I can see that it's 9, like on a domino. So, I know that
inside of our 5 3s, we have 3 3s. And I know that is 9. But I also know that we have 2 more 3s, which I know is 6. And now I know that I just need to combine 9 and 6 to be able to find the total of
how many. And I know that I can take one from our 6 and add it to our 9 to make one 10. And I can see that there's 5 more. So, I have one 10 and 5 more, which I can rename as 15.
Actually, now that I think about it, mathematicians, I can see another spatial pattern. And maybe you've seen this one too.
[The speaker moves the left column of red squares away to form a gap.]
If I move this column... whoop, doesn't wanna move. If I move this column over, I can see a familiar structure here. And I know that this is a 10 frame. So, I know that I have one 10 here. And
actually, when I look at this, I know that I have half a 10 frame, and I know that half of 10 is 5. So again, we have one 10 and 5 more, which is 15. So, if there were 5 sheep, how many bales of
wool? Well, we know there's 15 bales of wool.
[White text on a blue background reads ‘What’s (some of) the mathematics?]
What's some of the mathematics?
[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ Further text below reads ‘There are many different ways to solve the same problem’. Below, 3 colour images side by
side of the earlier A4 sheets of paper.]
There are many different ways to solve the same problem. And to solve Professor Di's problem today, we used counting sequences. We also created equal groups and we reformed our bales of wool into an
array and looked for familiar spatial patterns and structures.
[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ Further text and bullet points below read by speaker.]
We also learnt that we can use what we know to help us solve what we don't know yet. And this happened when we used the counting sequence of 3s. And we're able to count by 3s until we weren't as
confident, but we could then use something we were confident in, like counting on by ones, to help us solve the problem.
We also used familiar spatial patterns and structures like when we could see and notice the 10 frame in our array. And we know that one 10 and half a 10 frame or 5 more is 15.
[Blue text on a white background reads ‘Over to you…’ Further text below read by speaker. At the bottom a cartoon image of a sheep with 4 bales of wool above it.]
That’s it from us today, mathematicians. I hope your sweaty brains or crunchy eyebrows are ready for a new problem. Can you use our 3 different strategies to solve this problem? If each of the 5
sheep made 4 bales of wool, how many bales of wool would there be in total? Over to you.
[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales
(Department of Education), 2021.’]
[End of transcript]
Discuss and reflect
• Can you use our three different strategies to solve this new problem?
• If each of the 5 sheep made 4 bales of wool, how many bales would there be in total? | {"url":"https://education.nsw.gov.au/teaching-and-learning/curriculum/mathematics/mathematics-curriculum-resources-k-12/thinking-mathematically-resources/mathematics-es1-how-many-bales","timestamp":"2024-11-14T18:39:17Z","content_type":"text/html","content_length":"199211","record_id":"<urn:uuid:35b09d1b-00ef-4883-8aa5-53d57014f22b>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00778.warc.gz"} |
Counting employees during each hour of the day | Microsoft Community Hub
Forum Discussion
Counting employees during each hour of the day
Hello. I am having a hard time figuring out how to use functions to count the number of staff I have working during each hour of the day. I am a beginner, but I enlisted the help of someone who has
HansVogelaar Hans, can you help? so i started to try and tackle this (or at least what I think he wants/needs) in the attached file. My function first breaks up the text to create a grid of start
times for each employee and a complimentary grid of end times. By making those times actual date-time I could address the overnight shifts. Then I created a grid of date-times for every hour in those
corresponding days (actually shifted to the 1/2 hour). Finally I thought I could then MAP those date-times but summing up the instances they are both > start time grid and < end time grid. but my
results of that MAP are not what I expect and I think it has to do with how the LAMBDA is handling the arrays. Hans, maybe you have an idea.
=LET(dates, B1:E1, times,B2:E20,
timegrid, DROP(REDUCE("",SEQUENCE(days),LAMBDA(p,q,LET(
tt, CHOOSECOLS(times,q),
dd, INDEX(dates,q),
endgrid, CHOOSECOLS(timegrid,SEQUENCE(days,,2,2)),
TimesInDay, SEQUENCE(24,,0)/24/60,
timeDayGrid, TimesInDay+dates+0.5/24,
oneH, SEQUENCE(1,ROWS(endgrid),1,0),
oneV, SEQUENCE(COLUMNS(endgrid),1,1,0),
counts, MAP(timeDayGrid,LAMBDA(q,MMULT(MMULT(oneH,(q>startgrid)*(q<endgrid)),oneV))),
counts2, MAKEARRAY(ROWS(timeDayGrid), COLUMNS(timeDayGrid),LAMBDA(r,c, SUM((INDEX(timeDayGrid,r,c)>startgrid)*(INDEX(timeDayGrid,r,c)<endgrid)))),
out, VSTACK(dates,startgrid,endgrid),
testout, MMULT(MMULT(oneH,(45516.5>startgrid)*(45516.5<endgrid)),oneV),
so in the above I'm relatively confident of the first 1/2 creating the startGrid and endGrid and then creating a timeDayGrid (simple a grid of times for each day). it is that last part I tried both
MAP and MAKEARRAY and tried both SUM and MMULT inside. The last line 17-19 are just looking at test outputs.
No fresh ideas, just to play with what we have
range, $A$1:$E$20,
data, DROP(range,1,1),
dates, TAKE( DROP(range,,1), 1),
HoursStart, SEQUENCE(24,,0,1/24),
HoursEnd, SEQUENCE(24,,1/24,1/24),
Hours, TEXT(HoursStart, "hh:mm") &
"-" &
DateTime, LAMBDA(day,hhmm, day + REPLACE(hhmm, 3, , ":" ) ),
TimeStart, LAMBDA(day,times, DateTime(day, TEXTBEFORE(times, "-") ) ),
TimeEnd, LAMBDA(day,times, DateTime(day, TEXTAFTER(times, "-") ) ),
ShiftHours, LAMBDA(day,times, MOD( TimeEnd(day, times) - TimeStart(day, times), 1) ),
DayShiftStart, LAMBDA(day,times, TOCOL( TimeStart(day, times), 3 ) ),
DayShiftEnd, LAMBDA(day,times, TOCOL( TimeStart(day, times) + ShiftHours(day, times), 3 ) ),
ShiftRange, LAMBDA(fn,
DROP( REDUCE(
col, CHOOSECOLS(range, v+1),
VSTACK(a, fn( TAKE(col,1), DROP(col,1) ) )
), 1)
ShiftStart, ShiftRange(DayShiftStart),
ShiftEnd, ShiftRange(DayShiftEnd),
CountEmployees, MAKEARRAY(
24, COLUMNS(dates),
(INDEX(dates, 1, m) + INDEX( HoursStart, n, 1) <= ShiftEnd )*
(INDEX(dates, 1, m) + INDEX( HoursEnd, n, 1) > ShiftStart )
HSTACK("Hour/Date", dates),
HSTACK( Hours, CountEmployees ) | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/counting-employees-during-each-hour-of-the-day/4219822/replies/4221547","timestamp":"2024-11-13T15:33:22Z","content_type":"text/html","content_length":"310616","record_id":"<urn:uuid:50998bc7-d133-410d-95ce-4c78d9c2524b>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00350.warc.gz"} |
Exploring the fundamental limits of integrated beam splitters with arbitrary phase via topology optimization
Originalsprache Englisch
Seiten (von - bis) 1125-1128
Seitenumfang 4
Fachzeitschrift Optics letters
Jahrgang 49
Ausgabenummer 5
Publikationsstatus Veröffentlicht - 20 Feb. 2024
Optical beam splitters are essential for classical and quantum photonic on-chip systems. In integrated optical technology, a beam splitter can be implemented as a beam coupler with two input and two
output ports. The output phases are constrained by the conservation of energy. In lossless beam splitters, the phase shift between the output fields is π and zero for excitation from the first and
second input ports, respectively. Therefore, for excitation from both inputs, the phase between the output fields, defined as beam splitter phase (BSP), is π. The BSP leads to several phenomena, such
as the quantum interference between two photons, known as the Hong–Ou–Mandel effect. By introducing losses, BSP values different than π become theoretically possible, but the design of 2 × 2 beam
couplers with an arbitrary phase is elusive in integrated optics. Inspired by the growing interest on fundamental limits in electromagnetics and inverse design, here we explore the theoretical limits
of symmetrical integrated beam splitters with an arbitrary BSP via adjoint-based topology optimization. Optimized 2D designs accounting for fabrication constraints are obtained for several
combinations of loss and phase within the theoretical design space. Interestingly, the algorithm does not converge for objectives outside of the theoretical limits. Designs of beam splitters with
arbitrary phase may find use in integrated optics for quantum information processing.
ASJC Scopus Sachgebiete
• Physik und Astronomie (insg.)
• Standard
• Harvard
• Apa
• Vancouver
• Autor
• BibTex
• RIS
title = "Exploring the fundamental limits of integrated beam splitters with arbitrary phase via topology optimization",
abstract = "Optical beam splitters are essential for classical and quantum photonic on-chip systems. In integrated optical technology, a beam splitter can be implemented as a beam coupler with two
input and two output ports. The output phases are constrained by the conservation of energy. In lossless beam splitters, the phase shift between the output fields is π and zero for excitation from
the first and second input ports, respectively. Therefore, for excitation from both inputs, the phase between the output fields, defined as beam splitter phase (BSP), is π. The BSP leads to several
phenomena, such as the quantum interference between two photons, known as the Hong–Ou–Mandel effect. By introducing losses, BSP values different than π become theoretically possible, but the design
of 2 × 2 beam couplers with an arbitrary phase is elusive in integrated optics. Inspired by the growing interest on fundamental limits in electromagnetics and inverse design, here we explore the
theoretical limits of symmetrical integrated beam splitters with an arbitrary BSP via adjoint-based topology optimization. Optimized 2D designs accounting for fabrication constraints are obtained for
several combinations of loss and phase within the theoretical design space. Interestingly, the algorithm does not converge for objectives outside of the theoretical limits. Designs of beam splitters
with arbitrary phase may find use in integrated optics for quantum information processing.",
author = "Abhishek Nanda and Michael Kues and {Cal{\`a} Lesina}, Antonio",
note = "Funding Information: Funding. European Research Council (QFreC project, Grant agreement ID 947603); Deutsche Forschungsgemeinschaft (EXC 2122, Project ID 390833453). Simulations were
performed, in part, on the central computing cluster operated by Leibniz University IT Services (LUIS), which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation),
project number INST 187/742-1 FUGG. We also acknowledge the computing time granted by the Resource Allocation Board and provided on the supercomputer Lise and Emmy at NHR@ZIB and NHR@G{\"o}ttingen as
part of the NHR infrastructure (calculations were conducted with computing resources under the project nip00059). ",
year = "2024",
month = feb,
day = "20",
doi = "10.1364/OL.512100",
language = "English",
volume = "49",
pages = "1125--1128",
journal = "Optics letters",
issn = "0146-9592",
publisher = "OSA - The Optical Society",
number = "5",
TY - JOUR
T1 - Exploring the fundamental limits of integrated beam splitters with arbitrary phase via topology optimization
AU - Nanda, Abhishek
AU - Kues, Michael
AU - Calà Lesina, Antonio
N1 - Funding Information: Funding. European Research Council (QFreC project, Grant agreement ID 947603); Deutsche Forschungsgemeinschaft (EXC 2122, Project ID 390833453). Simulations were performed,
in part, on the central computing cluster operated by Leibniz University IT Services (LUIS), which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project number
INST 187/742-1 FUGG. We also acknowledge the computing time granted by the Resource Allocation Board and provided on the supercomputer Lise and Emmy at NHR@ZIB and NHR@Göttingen as part of the NHR
infrastructure (calculations were conducted with computing resources under the project nip00059).
PY - 2024/2/20
Y1 - 2024/2/20
N2 - Optical beam splitters are essential for classical and quantum photonic on-chip systems. In integrated optical technology, a beam splitter can be implemented as a beam coupler with two input and
two output ports. The output phases are constrained by the conservation of energy. In lossless beam splitters, the phase shift between the output fields is π and zero for excitation from the first
and second input ports, respectively. Therefore, for excitation from both inputs, the phase between the output fields, defined as beam splitter phase (BSP), is π. The BSP leads to several phenomena,
such as the quantum interference between two photons, known as the Hong–Ou–Mandel effect. By introducing losses, BSP values different than π become theoretically possible, but the design of 2 × 2
beam couplers with an arbitrary phase is elusive in integrated optics. Inspired by the growing interest on fundamental limits in electromagnetics and inverse design, here we explore the theoretical
limits of symmetrical integrated beam splitters with an arbitrary BSP via adjoint-based topology optimization. Optimized 2D designs accounting for fabrication constraints are obtained for several
combinations of loss and phase within the theoretical design space. Interestingly, the algorithm does not converge for objectives outside of the theoretical limits. Designs of beam splitters with
arbitrary phase may find use in integrated optics for quantum information processing.
AB - Optical beam splitters are essential for classical and quantum photonic on-chip systems. In integrated optical technology, a beam splitter can be implemented as a beam coupler with two input and
two output ports. The output phases are constrained by the conservation of energy. In lossless beam splitters, the phase shift between the output fields is π and zero for excitation from the first
and second input ports, respectively. Therefore, for excitation from both inputs, the phase between the output fields, defined as beam splitter phase (BSP), is π. The BSP leads to several phenomena,
such as the quantum interference between two photons, known as the Hong–Ou–Mandel effect. By introducing losses, BSP values different than π become theoretically possible, but the design of 2 × 2
beam couplers with an arbitrary phase is elusive in integrated optics. Inspired by the growing interest on fundamental limits in electromagnetics and inverse design, here we explore the theoretical
limits of symmetrical integrated beam splitters with an arbitrary BSP via adjoint-based topology optimization. Optimized 2D designs accounting for fabrication constraints are obtained for several
combinations of loss and phase within the theoretical design space. Interestingly, the algorithm does not converge for objectives outside of the theoretical limits. Designs of beam splitters with
arbitrary phase may find use in integrated optics for quantum information processing.
UR - http://www.scopus.com/inward/record.url?scp=85186318650&partnerID=8YFLogxK
U2 - 10.1364/OL.512100
DO - 10.1364/OL.512100
M3 - Letter
C2 - 38426954
AN - SCOPUS:85186318650
VL - 49
SP - 1125
EP - 1128
JO - Optics letters
JF - Optics letters
SN - 0146-9592
IS - 5
ER - | {"url":"https://www.fis.uni-hannover.de/portal/de/publications/exploring-the-fundamental-limits-of-integrated-beam-splitters-with-arbitrary-phase-via-topology-optimization(90938834-725f-4879-ab11-7fbf4fe0d3c1).html","timestamp":"2024-11-15T02:51:11Z","content_type":"text/html","content_length":"59734","record_id":"<urn:uuid:a7bcfe79-ce79-425e-a2f2-a366821718dc>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00663.warc.gz"} |
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Author: Delhi University | {"url":"https://www.tutorialsduniya.com/notes/mechanics-notes/","timestamp":"2024-11-06T13:45:19Z","content_type":"text/html","content_length":"109668","record_id":"<urn:uuid:064d0b4e-577b-49dd-8f74-f0ba3880f5aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00484.warc.gz"} |
WHR Boiler Efficiency calculation
239 posts
TimePosted 11/06/2012 09:18:29
WHR Boiler Efficiency calculation
Dear All
I am getting eff. more than 100 % ? Why?
Pl. correct me
Steam Temp.
Steam Pressure 18 18 18 18
Steam Flow 10.1 12.94 9.1 20.6
Feed water Temp. 160 160 160 160
Feed water press. 25 to 26 kg/cm²
Enthalpy of Steam (kcal/kg) 741.0191 741.0191 741.019 741.0191
Enthalpy of Feed water (kcal/kg) 161 161 161 161
Flue gas flow( M³/hr) 246520.1 330205.1282 259407 584249.1
flue gas density 0.52 0.52 0.54 0.52
Sp. Heat of flue gas 0.24 0.24 0.24 0.24
Flue gas inlet temp. °C 400 385 365 365
Flue gas outlet temp. °C 240 241 240 240
Total heat input in boiler(kcal/hr) 4922514 5934182 4202387 9114286
heat out from boiler 5858193 7505447.154 5278174 11948393
boiler effiency 119 126 126 131
Know the answer to this question? Join the community and register for a free guest account to post a reply.
16 posts
TimePosted 12/06/2012 06:15:18
re WHR Boiler Efficiency calculation
I don't know how exactly did you measure flows; because this is the major source of error (you can see flow specifications, on how to measure flow exactly), howver attached file is for assesing
energy performance of boiler. (sorry it isn't attaching)
I hope it'll help.
138 posts
TimePosted 12/06/2012 09:02:51
re WHR Boiler Efficiency calculation
You need to comment your calculations in more details.
By just doing this you might well discover by yourself many possible source of errors.
First of all, the units are missing for 9 lines of the table.
Second, your table does not show clearly how you did the calcualtions.
You could explain one column in full detail, this would be easier to check.
(a drawing would be even clearer)
As Ex-FlSmidth-Designer said, the errors on flow evaluation could in itself create big errors.
There are still many other possible source of errors:
- steam flow: what are the units? t/h maybe?
- flue gas flow: is it in actual m³ or in "normal" m³?
- flue gas flow: is it measured at the input or at the output of the boiler?
- flue gas density: is it evaluated at 400°C or at 240°C(inlet or outlet)?
- flue gas density: for a calculation I did, it was higher
- flue gas density: do you take gas composition properly into account?
- flue gas specific heat: units ???
- flue gas specific heat: how did you evaluate it?
- flue gas specific heat: did you take the composition into account?
- flue gas specific heat: is it the average heat value between 400°C and 240°C
or is this the specific heat value at 400°C ?
- flue gas specific heat: why is it exactly the same for the three scenarios?
- flue gas specific heat: for a calculation I did, it was 0.28 kcal/kg/°C
- flue gas specific heat: do you take water vapor into account?
- flue gas specific heat: ...
- heat input: I see how you made the calculation, should be ok
however "heat input" is a bad name,
you might call it "heat exchanged" (flue gas side or steam side)
If you explain your calculations in more detail, there is really a good chance that you find some approximations and some source of errors and that you improve your balance. You could also add an
uncertainty for each data and do some error bar calculation!.
Don't hesitate to put all you data and calculations in an excel worksheet and to attach it to your post. If you do so, do not forget to include flue gas composition.
239 posts
TimePosted 13/06/2012 09:29:01
re WHR Boiler Efficiency calculation
see the attached file .
also cp does not make much change & flows are correct | {"url":"https://www.cemnet.com/Forum/thread/149922/re-whr-boiler-efficiency-calculation.html","timestamp":"2024-11-11T07:38:55Z","content_type":"application/xhtml+xml","content_length":"34085","record_id":"<urn:uuid:5bcd4785-5380-4910-80b1-ec43afac3303>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00446.warc.gz"} |
1)We know that the yen and the swiss franc have a 120yen/ sf 1
exchange rate,...
1)We know that the yen and the swiss franc have a 120yen/ sf 1 exchange rate,...
1)We know that the yen and the swiss franc have a 120yen/ sf 1 exchange rate, meaning one swiss franc buys 120 yen in the spot ER market. If the swiss franc has an interest rate of .06 and the yen
rate is -.02, what is the forward exchange rate for IPT (interest parity theory) to be attained? Show everything in yen terms, i., e., how much yen one Swiss franc buys (yen is in the numerator.)
2) If there is no equilibrium initially, will there be equilibrium eventually? If so, what will transpire? | {"url":"https://justaaa.com/economics/1154249-1we-know-that-the-yen-and-the-swiss-franc-have-a","timestamp":"2024-11-04T07:41:23Z","content_type":"text/html","content_length":"40314","record_id":"<urn:uuid:5263aa58-4bb9-4e9c-885c-5d73b3820f01>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00055.warc.gz"} |
Kilojoule per Gram
Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like latent heat finds its
use in a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps
in the conversion of different units of measurement like kJ/g to cal/lb through multiplicative conversion factors. When you are converting latent heat, you need a Kilojoule per Gram to Calorie per
Pound converter that is elaborate and still easy to use. Converting kJ/g to Calorie per Pound is easy, for you only have to select the units first and the value you want to convert. If you encounter
any issues to convert Kilojoule per Gram to cal/lb, this tool is the answer that gives you the exact conversion of units. You can also get the formula used in kJ/g to cal/lb conversion along with a
table representing the entire conversion. | {"url":"https://www.unitsconverters.com/en/Kj/G-To-Cal/Lb/Utu-8810-8814","timestamp":"2024-11-14T03:46:09Z","content_type":"application/xhtml+xml","content_length":"110820","record_id":"<urn:uuid:d25c5156-495f-492f-b147-a589de0844a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00555.warc.gz"} |
Paver Calculator
This free online paver calculator assists you in determining the size of the paver and the cost required to complete your patio paving project. So if you are thinking of starting a new project or
going to pave your patio, this read is specially arranged for you.
Go through it !
What Are Some Common Paver Sizes?
The most common paver size that is used all around the globe today is 4” * 8”. But it does not mean that all other sizes are rare. Size of the pavers differ accordingly with the area and shape of the
pavers along with patio. Below here we have a list of most commonly use paver sizes in various units:
Size (in) Size (cm) Area (ft²) Area (cm²)
4 x 8 10 x 20 0.22 200
6 x 6 15 x 15 0.25 225
6 x 9 15 x 23 0.38 245
8 x 8 20 x 20 0.44 400
12 x 12 30 x 30 1.00 900
14 x 14 36 x 36 1.36 1296
12 x 18 30 x 46 1.50 1380
Moreover, get more finer details about the paver sizes that will fit your patio by using our best patio paver calculator.
How Many Pavers Do I Need?
Whenever you start renovating your home garden or floor, the first question that comes to your mind is how to measure the patio size? Not only this, the paver quantity will also disturb you as you
can not estimate it accurately every time. That is why we have designed this paver patio calculator so that you may use it and estimate the most accurate results. This will indeed help you out in
determining the exact numbers of pavers required for the patio. It will save you a lot of money in the end.
Steps Involved To Calculate Pavers Needed:
You need to follow the steps that are defined below to determine how many pavers you actually need to fill the patio. Let’s go through these together!
Measure The Patio Size:
This is the start of your project that needs to measure the size of your patio, so that it could become easy for you to estimate the pavers needed for the area. You can also rely on the best online
patio paver estimator to estimate the dimensions of the patio and calculate pavers needed for the required dimensions. What you need to do for manual calculation is to just find the square footage of
the patio by using the formula below:
Patio (sq ft) = length × width
Measure The Paver Size:
The next step ahead is the calculation of the paver size that you could also do with this free paver calculator. But let us tell you how you could be able to determine it by the following equation
Paver (sq ft) = 144/length × width
Calculate Pavers Needed:
This is the final step that is attention seeking. Use the following equation that will help you calculate the number of pavers that you need for the project:
Pavers needed = patio sq ft/paver sq ft
Different Paver Patterns:
Well, most of the time pavers are used as rectangular blocks or bricks. But what to do when specific patterns are to be made using these bricks that may contain different brick sizes? Do not worry at
all as our paver stone calculator would be assisting around the corner. But you should go for doing the calculations below:
• Calculate the size of each paver so that you may know how much patio area it would be covering
• Repeat the same step for all the remaining pavers
• Now add all the answers to know how many pavers would cover a certain patio area.
• At the end, multiply the pattern repetition number to get the estimation of how much bulk of pavers you are required with.
Suppose you want to determine the area of the pattern as above. You need to do the calculations according to the formula given as:
Sets of the Pattern = Total Project Area / (a * b + c * d + e * f + g * h)
Also, the free online paver calculator does the same calculations but instantly to save you a lot of time.
Some Different Patterns:
Below here we have enlisted some most popular paver patterns of dimensions 4” * 8” that could also be calculated by using a paver calculator easily. Let’s have a look!
These specific patterns create a sensation of confusion due to large bricks disappearing under each other. You can create numberless patterns of these pavers according to your wish.
Stacked Bond:
Well, this is the easiest paver pattern to make on your own. So if you are installing pavers without any help, try adopting this particular pattern.
This pattern shows the 90^o herringbone pattern as shown above.
Running bond:
Most of the time, these brick patterns are used as they are most liked by everyone and are liked. Also, this pattern is also adopted to build up walls.
Make use of the paver pattern calculator to look for the exact number of pavers that you must be buying for covering the patio.
Benefits of Paved Patios:
A few of the most remarkable advantages of paving your patios are as follows:
Low Maintenance:
Installing pavers is a great source of minimizing your expense in case of any damage. You do not have to remove the whole concrete slab, but only defective tiny pavers. This costs you very less than
you ever expect. Also,making use of the paver patio cost calculator will let you know how much it is going to cost you to install new pavers.
Reliable For Changing Weather:
Basically, solid concrete surfaces are cracked at many places due to the variations in the temperature. You might have seen on the roads that they get broken due to these specific reasons. But if the
pavers are used instead of concrete, this issue will not arise for sure.
This is the most catchy factor that makes public use pavers for their projects. Pavers are considered the most durable construction blocks than any other materials. This is why pavers remain
unchanged for years and give your patio a catchy look.
Paver Cost:
You must know the price per square footage for the pavers. If you do not know, multiply each paver cost with the total number to get the total value. Moreover, our free paver cost calculator can also
do that for you but in seconds. How does it sound? Really great!
How To Measure Paver Size?
Let us resolve an example to clear your idea so that you may not feel any hurdle while thinking of using paved patios. Let’s go!
How many pavers do I need for 12x12 patio?
Here we have:
Patio Area:
Patio (sq ft) = length × width
Patio (sq ft) = 12 × 12
Patio (sq ft) = 144ft2
Paver Area:
Paver (sq ft) = 144/length × width
Paver (sq ft) = 144/144
Paver (sq ft) = 1ft2
Number of Pavers:
Pavers needed = patio sq ft/paver sq ft
Pavers needed = 144/1
Pavers needed = 144
You can also verify the results by using the free brick paver calculator to check whether the number of paver bricks are exactly ok or not.
How Paver Calculator Works?
Allow this free patio calculator to tell you how many pavers would be in demand to cover a specific patio area. Let us guide you how it works!
• First of all, select the patio area for which you want to calculate pavers needed
• Then, select the parameter against patio that you wish to determine and enter all required elements in the designate fields
• Repeat the same process for pavers
• Enter the amount per square footage of pavers
• Enter the cost of installation
• Tap calculate button
The free brick calculator for patio calculates:
• Dimensions of the patio (length, width, and area)
• Dimensions of the pavers (length, width, and area)
• Total Paver cost
• Total cost of installation
How many patio pavers do i need for a 2 * 5 patio area?
You need exactly 40 pavers for the said area. For instance, you can cross check it by using a patio paver calculator.
Are pavers good for the environment?
Yes definitely. It is very easy to install pavers and maintain them. Changing environments do not affect the pavers composition as they are made up of concrete that can last for a long time.
Are pavers a good investment?
Investing in pavers is no doubt a good approach. By using pavers, home owners not only add to the beauty, but also durability that lasts for a long span of time.
How long a paver patio lasts?
A well installed paved patio will almost last for 25 to 50 years or even more.
Wrapping It Up:
Pavers add to the beauty of the construction. Pavers are preferred today because of their durability and ability to settle in case of sand slipping. That is why contractors make a vast use of the
free online concrete paver calculator for patio to estimate how much quantity you need to cover the patio or sidewalk. When the underlying earth and materials move, clay paving adjusts itself to
accommodate the movement of the underlying soil and thus prevents cracking of the pavement system.
From the source of wikipedia: Pavement (architecture), Paver, Stone pavers
From the source of homedepot.com: Paver Patterns for Patios, Preparing to Lay Patio Pavers, Compact the Soil
From the source of belgard.com: Pavers & Slabs, durable walkaway, Paving stone colors | {"url":"https://calculator-online.net/paver-calculator/","timestamp":"2024-11-12T17:31:08Z","content_type":"text/html","content_length":"77586","record_id":"<urn:uuid:7ec857cf-4a0e-45d4-a8df-454894b10054>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00732.warc.gz"} |
How To Multiply By Percentage In Excel?
Hey! Curious to know how to multiply by percentage in Excel? Well, you're in the right place! It's actually a really handy skill to have when working with numbers in spreadsheets. So let's dive in
and learn how to do it step-by-step.
Imagine this scenario: you have a sales report and you want to calculate a 10% increase in sales for the month. How do you do it? Not to worry, it's super easy! With just a few simple Excel formulas,
you'll be multiplying by percentages like a pro in no time.
Whether you're a math whiz or just getting started with Excel, this guide will walk you through the process, making it as straightforward as possible. So let's roll up our sleeves and discover the
magic of multiplying by percentage in Excel!
Step-by-Step Guide:
1. Open Microsoft Excel and create a new spreadsheet.
2. Enter the number you want to multiply in a cell.
3. Type the percentage you want to multiply by in another cell.
4. In a third cell, use the formula "= cell number * cell percentage".
5. Press enter to calculate the result.
Now you know how to multiply by a percentage in Excel! It's a quick and simple way to calculate percentages for various purposes.
How to Multiply by Percentage in Excel?: A Comprehensive Guide
Excel is a powerful tool that offers numerous functions to simplify calculations and data analysis. One such function is the ability to multiply numbers by a percentage. Whether you need to calculate
discounts, markups, or any other percentage-based calculations, Excel makes it easy to do so. In this article, we will guide you through the steps on how to multiply by percentage in Excel, providing
you with the knowledge and skills to perform these calculations efficiently.
Understanding Basic Percentage Calculation
When dealing with percentages, it's important to understand the basic concept of percentage calculation. A percentage is a way to express a portion or fraction of a whole number. To calculate a
percentage, you need to divide the value you want to find a percentage of by the total or whole value, and then multiply the result by 100. For example, if you want to find 20% of 200, you would
divide 20 by 100 to get 0.2. Then, you multiply 0.2 by 200 to get the result of 40. Therefore, 20% of 200 is equal to 40.
Using Excel's Percentage Calculation Method
To multiply by a percentage in Excel, you can simply use the multiplication operator (*) along with the percentage value. Excel automatically recognizes the percentage and performs the calculation
accordingly. Let's take a closer look at how to use Excel's percentage calculation method: 1. Start by entering the number or value you want to multiply. 2. Next, enter the percentage value you want
to multiply by. Remember to represent percentages as decimals in Excel. For example, if you want to multiply by 20%, you would enter 0.2. 3. Finally, use the asterisk (*) operator to multiply the
values together. The result will be displayed in the cell where you entered the formula. For instance, if you want to multiply 200 by 20%, you would enter the formula "=200*0.2" in the desired cell.
Excel will then calculate the result and display it.
Working with Absolute and Relative References
When using percentages in Excel, you have the option to use either absolute or relative references. Absolute references stay constant, while relative references change based on the cells they are
copied to. Understanding the difference between the two can enhance your productivity and efficiency when working with percentages in Excel. Let's explain the difference using an example: Suppose you
have a data set in cells A1 to A5, and you want to calculate the respective 10% for each value. Using absolute references, you would input the formula "=A1*0.1" in cell B1 and drag it down to B5. The
absolute reference to cell A1 means that the formula will always refer to that specific cell and calculate the percentage accordingly, regardless of where the formula is copied. On the other hand,
using relative references, you would input the formula "=A1*0.1" in cell B1 and drag it down to B5. In this case, the reference to cell A1 is relative, which means that when the formula is copied to
cell B2, it will automatically adjust the reference to cell A2, and so on.
The Benefits of Absolute and Relative References
Both absolute and relative references have their advantages and can be useful in different scenarios. Absolute references are beneficial when you want to maintain a fixed reference to a specific cell
or range of cells. This is particularly useful when dealing with calculations that involve specific constants or fixed values. On the other hand, relative references are helpful when you need to
apply the same formula or calculation to multiple cells, with each cell referencing its adjacent values. Relative references automatically adjust to the new cell references, saving you time and
effort. Understanding and using both absolute and relative references will allow you to work flexibly and efficiently with percentages in Excel.
Utilizing Excel's Percentage Formatting
While Excel automatically recognizes percentages and performs calculations accordingly, you can also utilize Excel's percentage formatting feature to display your results in the desired format. To
apply percentage formatting to a cell or range of cells, follow these steps: 1. Select the cell or range of cells you want to format as percentages. 2. Right-click and choose "Format Cells" from the
context menu. 3. In the Format Cells dialog box, select the "Percentage" category. 4. Adjust the decimal places as needed. 5. Click "OK" to apply the formatting. Excel will now display the values in
the selected cells as percentages, with the specified decimal places. This can be particularly helpful when presenting your data or sharing it with others.
The Versatility of Excel's Percentage Calculations
Excel's percentage calculations are not limited to multiplying numbers by percentages. You can also use them for a variety of other tasks. Here are some of the versatile applications of percentage
calculations in Excel: 1. Discount Calculations: If you need to calculate the discounted price of an item after applying a certain percentage off, Excel's percentage calculations can simplify the
process. 2. Markup Calculations: When determining the selling price of a product based on a desired profit margin, Excel's percentage calculations can be extremely useful. 3. Growth Rate
Calculations: Excel can help you calculate the growth rate of values over time using percentage calculations. This can be helpful for analyzing business performance or tracking financial data. 4.
Weighted Averages: If you need to calculate weighted averages based on different percentages, Excel provides the necessary features to perform these calculations accurately. By harnessing the power
of Excel's percentage calculations, you can streamline various tasks and make complex calculations easier.
Common Errors and Troubleshooting
While working with percentages in Excel, there are a few common errors that you may encounter. Here are some tips for troubleshooting and rectifying these errors:
Divide by 100 Error
One common error occurs when forgetting to divide the percentage value by 100. Excel treats percentages as decimal values, so it's essential to convert them to decimals by dividing them by 100 before
performing calculations. If you forget to divide by 100, your calculated result may be significantly different from what you expect. Always double-check that your percentage value is represented as a
decimal in Excel.
Incorrect Cell References
Another error can occur when referencing the wrong cells in your formulas. Ensure that you are referencing the correct cells for your calculations, especially when using formulas that involve
multiple cells or ranges. Double-check your formulas and verify that the cell references match the intended cells you want to multiply by the percentage.
Key Takeaways
Multiplying by percentage in Excel is a straightforward process that can be done using the multiplication operator (*) along with the percentage value represented as a decimal. Understanding how
Excel recognizes and performs percentage calculations allows you to efficiently apply them in various scenarios. Additionally, using absolute and relative references, formatting options, and
troubleshooting common errors will enhance your experience while working with percentages in Excel. Remember, Excel's percentage calculations are versatile and can be applied to various tasks such as
discount calculations, markup calculations, growth rate calculations, and weighted averages. By mastering these techniques, you can leverage Excel's power to streamline your calculations and save
time. So, go ahead and explore the possibilities of multiplying by percentage in Excel, and unlock the full potential of this powerful tool for your data analysis and calculation needs.
Frequently Asked Questions
Welcome to our frequently asked questions section! Here, we'll answer some common queries on how to multiply by percentage in Excel. If you're looking to perform calculations involving percentages in
your Excel spreadsheets, you've come to the right place. Read on to find answers to your questions and enhance your Excel skills!
1. How can I multiply a cell value by a percentage in Excel?
To multiply a cell value by a percentage in Excel, you can use a simple formula. First, select the cell where you want the result to appear. Then, enter the formula "=Cell*Percentage", replacing
"Cell" with the reference to the cell containing the value you want to multiply and "Percentage" with the actual percentage you want to use. Press Enter to get the result, and Excel will
automatically calculate the multiplication for you.
For example, if you want to multiply the value in cell A1 by 20%, the formula would be "=A1*0.2". This will give you the result of the multiplication.
2. Is there an alternative method to multiply by a percentage in Excel?
Yes, there is another method to multiply a value by a percentage in Excel. Instead of using the multiplication formula, you can use the "Paste Special" feature. First, enter the value you want to
multiply in a cell. Then, enter the percentage you want to use in another cell. Next, copy the percentage cell (Ctrl+C), select the cell with the value, right-click, choose "Paste Special", select
"Multiply", and click "OK". Excel will multiply the value by the percentage and display the result in the selected cell.
This alternative method can be useful when you want to multiply multiple values by the same percentage quickly.
3. Can I directly multiply multiple cells by a single percentage in Excel?
Absolutely! Excel offers a convenient way to multiply multiple cells by a single percentage using the "Paste Special" feature along with the reference operator. First, enter the percentage value in a
cell. Then, select the range of cells you want to multiply. Copy the percentage cell, right-click on the selected range, choose "Paste Special", select "Multiply" and check the "Transpose" box.
Finally, click "OK", and Excel will multiply each cell in the range by the percentage.
This method is especially useful when you have a list of numbers that need to be multiplied by the same percentage.
4. Can I apply a percentage increase or decrease to a value in Excel?
Absolutely! Excel allows you to apply a percentage increase or decrease to a value using a simple formula. To increase a value by a certain percentage, enter the formula "=Value*(1+Percentage)" in a
cell, replacing "Value" with the reference to the original value and "Percentage" with the actual percentage increase you want. To decrease a value, use the formula "=Value*(1-Percentage)". Excel
will calculate the new value for you.
For example, if you want to increase the value in cell A1 by 10%, the formula would be "=A1*(1+0.1)". This will give you the increased value.
5. Can I apply a percentage format to the result of a multiplication in Excel?
Absolutely! After performing a multiplication in Excel, you can easily apply a percentage format to the result. Select the cell with the multiplication result, right-click, choose "Format Cells",
select the "Percentage" category, and specify the desired number of decimal places. Click "OK", and Excel will format the cell to display the result as a percentage.
This formatting feature is helpful when you want to present your calculations as percentages rather than decimal numbers.
If you want to multiply a number by a percentage in Excel, follow these simple steps. First, convert the percentage to a decimal by moving the decimal point two places to the left. Then, multiply the
decimal by the number you want to multiply. Finally, format the cell as a percentage to see the result. Remember, when you're working with percentages in Excel, it's important to keep track of the
decimal points and format your cells correctly. By following these steps, you'll be able to easily multiply by a percentage and make calculations in Excel like a pro! | {"url":"https://keysswift.com/blogs/guide/how-to-multiply-by-percentage-in-excel","timestamp":"2024-11-06T22:10:39Z","content_type":"text/html","content_length":"240647","record_id":"<urn:uuid:3a2d3548-6ddc-402f-9e3b-e9c085bfba9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00498.warc.gz"} |
6th Grade Math Interactive Notebook
$ 59.99
This is a whole years worth of interactive notebook material that is very organized and engaging.
There are 5 units (6 units including unit 0). The units include:
Unit 0 - Notebook Starter
Unit 1 - Number Sense
Unit 2 – Ratios, Rates, and Percentages
Unit 3 - Expressions & Equations
Unit 4 - Geometry
Unit 5 - Statistics & Probability
A breakdown of each unit is below:
Unit 0 - Notebook Starter
-Notebook Cover for Students
-Classroom Rules
-Notebook Expectations
-Grading Rubric
-Math Reference Sheet
-Table of Contents
-Reflection Page
-My Goals: Setting Goals
-My Goals: A Look Back
Unit 1 - Number Sense
-Adding Decimals
-Subtracting Decimals
-Multiplying Decimals
-Dividing Decimals
-Dividing Fractions with Fraction Bars
-Dividing Fractions
-Word Problems involving Dividing Fractions
-Order of Operations
-Intro to Negative Numbers & Comparing Negative Numbers
-Positive and Negative Numbers
-Number Opposites
-Absolute Value
-Coordinate Plane
-Distance between Points with a Same Coordinate
-Least Common Multiple & Greatest Common Factor
Unit 2 - Ratios, Rates, & Percentages
-Intro to Ratios & Word Problems
-Intro to Rates & Word Problems
-Intro to Percents
-Percent-Decimal Conversions
-Percent of a Number - Finding an Amount
-Finding a Percent
-Finding the Base
Unit 3 - Expressions & Equations
-Intro to Variables
-Evaluating Expressions
-Writing Algebraic Expressions
-Solving One-Step Equations
-Intro to Inequalities
-Solving One-Step Inequalities
-Dependent vs Independent Variables
-Combining Like Terms
-The Distributive Property
-Writing Equivalent Expressions
Unit 4 - Geometry
-Area of Parallelograms
-Area of Triangles
-Area of Trapezoids
-Area of Composite Figures
-Identifying Parts of 3-Dimensional Objects
-Volume of Rectangular Prisms (Fractional Lengths)
-Surface Area using Nets
-Polygons in the Coordinate Plane
Unit 5 - Statistics & Probability
-Dot Plots & Frequency Tables
-Mean and Median
-Mean Absolute Deviation (MAD)
-Box Plots (includes IQR)
This notebook is absolutely amazing! It includes 2 pages for each concept. The first page is more aimed towards direct instruction / notes. The second page is "Your Turn". Students will apply their
newly acquired knowledge to solve problems. After solving problems, they will rate themselves on how well they understand the concepts and write a reflection.
I have also included an extra notebook that includes CCSS activities.
Each page has the 6th grade Common Core standard posted. In addition, there are hands on activities and cutouts. This notebook is very engaging. You won't find anything else like it! | {"url":"https://mathindemand.com/products/6th-grade-math-interactive-notebook","timestamp":"2024-11-02T05:06:24Z","content_type":"text/html","content_length":"179047","record_id":"<urn:uuid:1fb1c78c-2ebb-4135-b86f-7e4bd14a8100>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00487.warc.gz"} |
machine learning--04 : Normal equation
gradient descent
, we could use normal equation to find out the optimal hypothesis function too.
As equation 1 show, normal equation is far more easier to implement with compare to gradient descent.If the matrix is singular, we could either
decrease the number of features
, or using
to find an approximation of the inverse matrix.
The pros and cons of gradient descent vs normal equation.
Gradient Descent
• Need to choose alpha
• Needs many iterations
• works well even when n(number of features) is large
Normal Equation
• No need to choose alpha
• Don't need to iterate
• Need to compute inverse matrix
• slow if n(number of features) is very large
The price of computing the inverse matrix is almost same as O(n^3), this kind of complexity is unacceptable when the number of n is big(10000 or more, depends on your machine). | {"url":"https://qtandopencv.blogspot.com/2013/11/machine-learning-04-normal-equation.html","timestamp":"2024-11-03T01:17:10Z","content_type":"text/html","content_length":"53858","record_id":"<urn:uuid:65e01d58-c52a-401f-8a6e-e6ac3b6474fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00422.warc.gz"} |
Core Loss Analysis and Modeling of a Magnetic Coupling System in WPT for EVs
College of Electrical Engineering and Automation, Fuzhou University, Fuzhou 350116, China
Author to whom correspondence should be addressed.
Submission received: 13 August 2021 / Revised: 20 September 2021 / Accepted: 14 October 2021 / Published: 18 October 2021
The magnetic core is an important part of the magnetic coupling system in wireless power transmission (WPT) for EVs. It helps to increase the coupling coefficient and reduce magnetic field leakage.
However, it also brings additional core loss. While the traditional core loss model cannot be used directly due to the uneven distribution of the magnetic flux density, this paper focuses on the flux
density distribution in the disk core of a WPT system. Based on a finite element analysis (FEA) simulation and a theoretical magnetic flux density distribution analysis, a mathematical model of
magnetic flux density distribution is built, which is regarded as a quadratic function. Through this model, the flux density distribution can be calculated by the electrical and mechanical
specifications of the magnetic coupling system. Combining the model of flux density distribution, the disk core loss model of the WPT system is proposed—the idea of which is dividing the disk core
into several circle sheets firstly, and then summing the core loss of all circle sheets. Finally, the FEA simulation results verify the proposed model as being correct and flexible.
1. Introduction
To ease the shortage of fossil energy and the problem of environmental pollution, electric vehicles have been adopted in recent years [
]. Traditionally, contact-type chargers are used in electric vehicles. But this power transmission method needs charge wires and sockets, which may cause electric sparks and decrease safety levels.
People hope to find a convenient, safe and highly reliable power transmission method [
]. Compared to the traditional contact-type power transmission method, wireless power transmission (WPT) technology can not only avoid the electric sparks caused by physical contact, but also be
greatly convenient for users [
]. Now, WPT technology for electric vehicles has attracted a lot of attention from all over the world [
The magnetic coupling system is the most important part for realizing wireless power transmission. Usually, the distance between the transmitter and the receiver is relatively large in the
application of WPT for EVs. In order to strengthen the coupling coefficient between the transmitter and the receiver to improve the transmission power and efficiency and to reduce magnetic field
leakage, magnetic cores are adopted in WPT system [
]. However, the magnetic cores may bring additional core loss and reduce the efficiency of the prototype [
To evaluate and optimize the core loss of the magnetic coupling system, it is necessary to build a core loss model. Currently, the Steinmetz formula is widely used as a core loss model, in which the
parameters are obtained by measurement. In core loss measurement, the one-winding method, two-winding method and calorimetric method are widely used. In [
], the core loss is obtained by calculating the product of the measured voltage and current across the magnetic component. However, the core loss measured by this method includes both the winding
loss and the core loss, and the individual core loss cannot be obtained. This method is suitable for applications where the core loss is much greater than the winding loss, such as in ferrite
transformers. But in WPT applications, the winding loss represents a large amount of the whole loss in the magnetic coupling system. In [
], the two-winding method is adopted to calculate the product of the measured current of the power winding and the measured voltage of the auxiliary winding to obtain the core loss of the magnetic
component. This measured loss does not include the winding loss of the component and the individual core loss can be obtained. However, this method requires the coupling coefficient between the two
windings to be high enough (close to 1). Additionally, the coupling coefficient of the WPT system is relatively small (usually less than 0.7) [
]—as a result, the two-winding method cannot be used to measure the core loss of the WPT system. In [
], the calorimetric method was discussed. In this method, the magnetic component is put into a closed container, and then the heat generated by the voltage and current excitation can be measured
according to the temperature rise; finally, the core loss of the magnetic component can be calculated. As well as the one-winding method, the core loss measured by this method not only includes the
core loss but also the winding loss, and it is suitable for applications where the core loss is much greater than the winding loss.
Moreover, the Steinmetz formula can only calculate the core loss at certain frequencies and magnetic flux densities of a magnetic component. The magnetic flux density in the ferrite inductor and the
transformer of the switching mode power supply (the air gap is generally small and the coupling coefficient is close to 1) is basically uniform; therefore, the Steinmetz formula calculation result
can effectively characterize the core loss of inductors and transformers [
]. However, for the WPT system, the magnetic core adopts a disk-shaped structure, and the distance between the two disk cores is relatively large (for example, the air gap is 5 cm). As a result, the
coupling coefficient of the WPT system is usually low and the magnetic field leakage is serious [
]. Therefore, the magnetic flux density inside the disk core is uneven. The Steinmetz formula cannot be used directly for core loss calculation, because of its non-linear characteristic. In this
situation, finite element analysis (FEA) simulation can be adopted for WPT core loss calculation. However, it takes a lot of time—especially in 3D FEA simulation [
]. This paper focuses on the magnetic flux density distribution in the disk core of the WPT system. Based on simulations and theoretical analysis, the theory model of magnetic flux density
distribution can be built, and then the core loss model can be proposed. The core loss of the disk core of the WPT system can be easily obtained by the proposed core loss model, which can save time
and calculation resources compared to FEA simulation.
The sections of this paper are organized as follows:
Section 2
analyzes the distribution of magnetic flux density in the disk core by FEA simulation. It reveals that the distribution of the magnetic flux density in the disk core is uneven.
Section 3
establishes the mathematical model of the distribution of the magnetic flux density in the disk core, and based on the Steinmetz formula, the disk core loss model of the magnetic coupling system is
Section 4
verifies the accuracy of the disk core loss calculation method through FEA simulation, and conclusions are drawn in
Section 5
2. The Distribution of Magnetic Flux Density in the Disk Core
Usually, the uniformly distributed planar spiral winding structure is used in WPT magnetic coupling systems, while a disk structure is used as the magnetic core structure. The simulation model of the
magnetic coupling system with the disk core is established in
Figure 1
, and the parameters and setup of the simulation model are shown in
Table 1
Through FEA simulation, the plot of the magnetic flux density inside the magnetic core could be acquired, as shown in
Figure 2
. Additionally, the curve of the magnetic flux density inside the transmitter core versus R-axis and Z-axis position could be obtained as shown in
Figure 3
. Where the transmitter winding current
= 12 A, the receiver winding current
= 18 A and the phase-shift = 0°.
It can be seen from
Figure 2
that the distribution of magnetic flux density inside the magnetic core was uneven. The flux density was approximately zero at the boundary of the magnetic core, and there was a maximum at the
Figure 3
a shows that the magnetic flux density under same Z-axis was basically consistent (at different R-axis positions); therefore, the flux density of each circle sheet was nearly the same. The tendency
Figure 3
b was similar with a quadratic function. In this situation, the average magnetic flux density cannot be directly used to calculate the overall core loss due to the non-linear core loss
3. Analysis and Modeling of Core Loss
3.1. Modeling of Magnetic Flux Density Distribution
In order to simplify the distribution of magnetic flux density in
Figure 3
, the magnetic flux density inside the disk core was regarded as the same at different Z-axis positions while varying as different R-axis positions. In addition, the magnetic flux density at both the
inner radius and the outer radius of the core was considered to be zero. Hence, a quadratic function distribution along the R-axis is presented. The simplified flux density distribution inside the
disk core is shown in
Figure 4
Therefore, the simplified magnetic flux density distribution inside the disk core can be expressed as:
$B m ( R , B m max ) = − 4 B m max a 0 2 R 2 + 4 B m max a 0 R$
3.2. Theoretical Calculation Method of Magnetic Flux Density Model Parameters
When the
in Formula (1) is determined, the magnetic flux density distribution inside the disk core can be obtained. However,
needs to be determined by the magnetic field distribution characteristics of the magnetic coupling system. The magnetic-field line distribution schematic diagram of the magnetic coupling system is
shown in
Figure 5
, where the air gap diffusion effect exists at the inner radius and the outer radius of the magnetic core and the air gap diffusion flux loop is approximately a semicircle [
It can be seen from
Figure 5
that the flux linkage of the magnetic coupling system mainly includes two parts: One part of the flux linkage
is formed by the magnetic core and air closure and the other
is formed by the air closure. Where
is the distance between the two magnetic cores,
is the distance between the outer edge of the air gap diffusion magnetic flux and the inner radius of the magnetic core, and
is the distance between the outer edge of the air gap diffusion magnetic flux and the center of the winding. The expression of each parameter is:
${ s 0 = h 0 + 2 d 0 s 1 = s 0 2 = h 0 2 + d 0 s 2 = R cin − s 1 = R cin − h 0 2 − d 0$
It is assumed that the phase-shift of the transmitter winding current ahead of the receiver winding current is
and that the initial phase angle of the receiver winding current is zero, according to the constant-linkage theorem on the transmitter:
$L p I p cos θ p + M I s + j L p I p sin θ p = φ 1 core + φ 1 air$
is the flux linkage formed by the magnetic core and air closure and
is the flux linkage formed by the air closure on the transmitter.
The flux linkage
formed by the magnetic core and air closure on the transmitter can be expressed as:
$φ 1 core ( B m 1 max ) = ∑ i = 1 N p 2 π b 0 ( R 1 ( i ) + R cin ) B m ( R 1 ( i ) , B m 1 max )$
) is the relative position of the
-th turn coil center of the transmitter winding, the expression is:
$R 1 ( i ) = R win − R cin + d 0 2 + w 0 − d 0 N p − 1 ( i − 1 )$
For flux linkage
formed by the air closure in the
area, this paper ignored the influence of the core thickness and adopted the magnetic-field distribution of the method of images model to obtain the magnetic-field distribution of the magnetic
coupling system. The equivalent model of the method of images was established as shown in
Figure 6
The transmitter winding source current group with the current amplitude I[p-0] generates a primary mirror current group with the current amplitude I[p-1] under the action of the transmitter core; the
source current and the primary mirror current group will generate a secondary mirror current group with the current amplitude of I[p-2] and a third mirror current group with the current amplitude of
I[p-3] under the action of the receiver core, and the second mirror current group and the third mirror current group will generate a fourth mirror current group and a fifth mirror current group under
the action of the transmitter core. Similarly, the alternating reflections of the two magnetic core planes produce an infinite set of mirror current groups. In the same way, for the receiver winding
current, the alternating reflections of the two magnetic core planes also produce an infinite set of mirror current groups.
In [
], through 2D FEA simulation, it is found that the ratio of the mirror current
to the source current
is a function with the core plane width, coil diameter and transmission distance:
$γ = I m I o ≅ 1 − e − w 0 − 2 d 0 α 2 y$
are the core plane width and coil diameter, respectively;
is the vertical position of the magnetic-field test.
It is assumed that the current amplitude of the
-th mirror current group of the transmitter winding current and the receiver winding current obtained according to Formula (6) are
, respectively. When the thickness of the magnetic core is ignored and the plane of the transmitter core is
= 0, the axial height of each transmitter mirror current group and receiver mirror current group can be expressed as:
$h I p - k = { { h I p - 0 , h I p - 2 , h I p - 4 , … … } = { d 0 2 , 2 s 0 − d 0 2 , − ( 2 s 0 − d 0 2 ) , … … } { h I p - 1 , h I p - 3 , h I p - 5 , … … } = { − d 0 2 , 2 s 0 + d 0 2 , − ( 2 s 0
+ d 0 2 ) , … … }$
$h I s - k = { { h I s - 0 , h I s - 2 , h I s - 4 , … … } = { s 0 − d 0 2 , − ( s 0 − d 0 2 ) , 3 s 0 − d 0 2 , … … } { h I s - 1 , h I s - 3 , h I s - 5 , … … } = { s 0 + d 0 2 , − ( s 0 + d 0 2 )
, 3 s 0 + d 0 2 , … … }$
The position parameters of the
-th turn transmitter mirror current and the receiver mirror current of the
-th group of current can be expressed as:
${ R 1 ( k , i ) = R 1 ( i ) z 1 ( k , i ) = h I p - k$
${ R 2 ( k , i ) = R 2 ( i ) z 2 ( k , i ) = h I s - k$
where there is a load coil at the radial distance
and the axial height
/2, taking the
-th turn of the
-th transmitter mirror current group and the receiver mirror current group, as an example. According to the method of magnetic vector potential, the mutual inductance magnetic flux generated in the
load coil can be expressed as:
${ ϕ 1 air _ unit ( k , i ) = ∮ l 3 A 1 ⇀ ( k , i ) · d l 3 ⇀ = μ 0 · I p - k ( cos θ p + j sin θ p ) 4 π ∮ l 3 ∮ l 1 d l 1 ⇀ ( k , i ) · d l 3 ⇀ R ( k , i ) ϕ 2 air _ unit ( k , i ) = ∮ l 3 A 2 ⇀ (
k , i ) · d l 2 ⇀ = μ 0 · I s - k 4 π ∮ l 3 ∮ l 2 d l 2 ⇀ ( k , i ) · d l 3 ⇀ R ′ ( k , i )$
The expressions of each parameter in the above formula are given by Formulas (12) and (13):
${ d l 1 ⇀ ( k , i ) = ( R 1 ( i ) sin θ + R 1 ( i ) cos θ ) d θ d l 2 ⇀ ( k , i ) = ( R 2 ( i ) sin θ + R 2 ( i ) cos θ ) d θ d l 3 ⇀ = ( s 2 sin φ + s 2 cos φ ) d φ$
${ R ( k , i ) = ( R 1 ( i ) cos θ − s 2 cos φ ) 2 + ( R 1 ( i ) sin θ − s 2 sin φ ) 2 + ( h I p - k − s 0 2 ) 2 R ′ ( k , i ) = ( R 2 ( i ) cos θ − s 2 cos φ ) 2 + ( R 2 ( i ) sin θ − s 2 sin φ ) 2
+ ( h I s - k − s 0 2 ) 2$
According to superposition theorem,
can be got by Formula (14):
$ϕ air = ∑ k = 0 ∞ [ ∑ i = 1 N p ϕ 1 air _ unit ( k , i ) + ∑ i = 1 N s ϕ 2 air _ unit ( k , i ) ]$
Then, the flux linkage
formed by air closure on the transmitter can be expressed as:
$φ 1 air = N p ϕ air = N p ∑ k = 0 ∞ [ ∑ i = 1 N p ϕ 1 air _ unit ( k , i ) + ∑ i = 1 N s ϕ 2 air _ unit ( k , i ) ]$
Formula (16) can be obtained by combining Formulas (4), (5) and (15):
$∑ i = 1 N p 2 π b 0 ( R 1 ( i ) + R cin ) B m ( R 1 ( i ) , B m 1 max ) = L p I p cos θ p + M I s + j L p I p sin θ p − φ 1 air ⇒ B m 1 max$
Then, the magnetic flux density distribution inside the transmitter core can be expressed as:
$B m 1 ( R ) = − 4 B m 1 max a 0 2 R 2 + 4 B m 1 max a 0 R$
In the same way, according to the constant-linkage theorem on the receiver:
$φ 2 core ( B m 2 max ) = L s I s + M I p cos θ + j M I p sin θ − φ 2 air ⇒ B m 2 max$
The expressions of each parameter in the above formula are:
${ φ 2 core ( B m 2 max ) = ∑ i = 1 N s 2 π b 0 ( R 2 ( i ) + R cin ) B m ( R 2 ( i ) , B m 2 max ) R 2 ( i ) = R win − R cin + d 0 2 + w 0 − d 0 N s − 1 ( i − 1 ) φ 2 air = N s ∑ k = 0 ∞ [ ∑ i = 1 N
p ϕ 1 air _ unit ( k , i ) + ∑ i = 1 N s ϕ 2 air _ unit ( k , i ) ]$
) is the flux linkage formed by the magnetic core and air closure on the receiver;
) is the relative position of the
-th turn coil center of the receiver winding; and
is the flux linkage formed by air closure on the receiver.
Then, the magnetic flux density distribution inside the receiver core can be expressed as:
$B m 2 ( R ) = − 4 B m 2 max a 0 2 R 2 + 4 B m 2 max a 0 R$
The magnetic flux density distribution inside the transmitter core and the receiver core obtained by theoretical model calculation and FEA simulation are compared, as shown in
Figure 7
. Where the transmitter winding current
= 12 A, the receiver winding current
= 18 A and phase-shift = 0°.
The tendency of the magnetic flux density obtained from the theoretical model and the simulation is basically consistent, but some areas are different. The magnetic flux density calculated by the
theoretical model is larger than the magnetic flux density of simulation in the A[2] area, and smaller than the magnetic flux density of simulation in the A[1] area. The core loss is mainly
determined by the A[2] area, where the magnetic flux density is relatively large. The maximum absolute value of the relative error of the magnetic flux density Bm[1] in the A[2] area is 2.57%; the
relative error of the core loss determined by the relative error of the Bm[1] is 5.97%.
For the calculation of the core loss, the theoretical calculation result of the core loss is slightly larger than the simulation result in the A[2] area, while it is opposite in the A[1] area. On the
whole, the relative error of the overall core loss becomes smaller.
3.3. Core Loss Modeling
The calculation method of core loss generally adopts the Steinmetz formula proposed by C.P. Steinmetz [
]; the expression is:
$P core = k · f α · B m β · V e$
is the core loss of magnetic component;
is frequency;
is the volume of the magnetic component;
, and
are the empirical parameters obtained from the experimental measurement. Formula (21) is suitable for applications where the core loss is at a certain frequency and magnetic flux density of the
magnetic component.
Since the magnetic flux density of the disk core across each circle sheet of core is different, the core loss cannot be calculated directly by the Steinmetz formula. The concept of this paper is as
follows: Firstly, the disk core is divided into several circle sheets along the radial direction; then, the core loss of each circle sheet is calculated through the Steinmetz formula; and finally,
the whole disk core loss is calculated by the summation of all circle sheets.
While the distance between the inner radius and the outer radius of the magnetic core is
; the distance between the inner radius and the outer radius of the winding is
. The magnetic core is equally divided into
circle sheets along the radial direction as shown in
Figure 8
It can be seen from
Figure 8
that the volume of the
-th magnetic core can be expressed as:
$V core ( i ) = 2 π a 0 n ( a 0 n i + R cin ) b 0$
The magnetic flux density distribution of the magnetic core along the radial direction is
), when
is large enough; the magnetic flux density in a circle sheet can be approximately regarded as constant. Then, according to the Steinmetz formula, the core loss of the
-th circle sheet core can be expressed as:
$P core _ unit ( i ) = k f α [ B m ( a 0 n i ) ] β 2 π a 0 n ( a 0 n i + R cin ) b 0$
The core losses of the
circle sheet cores are summed, and then the core loss of the whole disk core can be expressed as:
$P core = ∑ i = 1 n k f α [ B m ( a 0 n i ) ] β 2 π a 0 n ( a 0 n i + R cin ) b 0$
Formula (25) can be obtained by combining Formulas (17), (20) and (24); the core losses of the transmitter core and the receiver core can be expressed as:
${ P 1 core = ∑ i = 1 n k f α [ − 4 B m 1 max a 0 2 ( a 0 n i ) 2 + 4 B m 1 max a 0 ( a 0 n i ) ] β 2 π a 0 n ( a 0 n i + R cin ) b 0 P 2 core = ∑ i = 1 n k f α [ − 4 B m 2 max a 0 2 ( a 0 n i ) 2 +
4 B m 2 max a 0 ( a 0 n i ) ] β 2 π a 0 n ( a 0 n i + R cin ) b 0$
4. Simulation and Verification
It is assumed that
= 4 A,
= 8 A, phase-shift = 35°. The core loss at a given operating point is obtained under FEA simulation, as shown in
Figure 9
. In the same way, the core loss under different coil currents and different current phase-shift can also be obtained by FEA simulation. The core losses obtained by the theoretical model and FEA
simulation are compared at different operating points, and then the results are drawn in the same figure by Mathcad15 software.
The core losses of the transmitter core and the receiver core obtained by theoretical model calculation and FEA simulation are compared under different winding currents, as shown in
Figure 10
, where there is no phase-shift between the transmitter winding current and the receiver winding current.
In the same way, the core losses of the transmitter core and the receiver core obtained by theoretical model calculation and FEA simulation are compared under different phase-shifts, as shown in
Figure 11
, where the transmitter winding current
= 12 A.
The core losses obtained by the theoretical model and FEA simulation are basically consistent at different operating points, as the core loss is mainly determined by the area where the magnetic flux
density is relatively large. In areas where the relative error of the magnetic flux density is relatively small, the relative error of the core loss determined by the magnetic flux density is also
relatively small. Hence, the FEA simulation results show that the magnetic core loss calculated by the proposed model has good accuracy.
5. Conclusions
The article studies and analyzes the magnetic flux density inside the disk core and establishes the corresponding core loss model. The conclusions are as follows:
• The magnetic flux density inside the disk core through each radial circle sheet core is different; consequently, the average magnetic flux density cannot be used to calculate the overall core
loss because of the non-linear core loss characteristic of the magnetic core.
• In the core loss calculation, the distribution of the magnetic flux density in the core needs to be taken into consideration. According to FEA simulation results, the mathematical model of the
distribution of magnetic flux density is established. This model can be described as a quadratic function in which the parameters are extracted from the magnetic-field distribution of the
magnetic coupling system.
• In order to build the disk core loss model of the WPT system, the disk core is divided into several circle sheets. In each circle sheet, the magnetic flux density can be seen to be the same and
the core loss can be calculated by the Steinmetz formula. Combining the model of the distribution of magnetic flux density inside the magnetic core, the disk core loss model of the WPT system is
• The FEA simulation results show that the magnetic core loss calculated by the proposed model has good accuracy. This core loss model can provide an easier way to calculate the disk core loss of
the WPT system than the FEA simulation.
Author Contributions
Conceptualization, Q.C. and W.C.; methodology, Q.C.; software, F.F.; validation, F.F., J.W. and W.C.; formal analysis, W.C. and F.F.; investigation, F.F. and J.W.; resources, J.W.; data curation,
J.W.; writing—original draft preparation, F.F.; writing—review and editing, F.F.; visualization, J.W.; supervision, W.C. All authors have read and agreed to the published version of the manuscript.
Project supported by National Natural Science Foundation of China (grant number 51407032), Natural Science Foundation of Fujian Province, China (grant number 2019J01251) and Scientific Research
Project of Development Center of Science and Education Park, Fuzhou University, Jinjiang City (grant number 2019-JJFDKY-47).
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
The data presented in this study are available on request from the corresponding author.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 2. The distribution of magnetic flux density inside the transmitter core and the receiver core.
Figure 3. The distribution of magnetic flux density inside the transmitter core. (a) The distribution of magnetic flux density versus R-axis; (b) The distribution of magnetic flux density versus
Figure 5. The schematic diagram of the magnetic field line distribution in the magnetic coupling system.
Figure 6. The method of images equivalent model. (a) The magnetic coupling system model; (b) The method of images model of transmitter winding current; (c) The method of images model of receiver
winding current.
Figure 7. The magnetic flux density distribution inside the disk core obtained by theoretical calculation and simulation. (a) Transmitter core; (b) Receiver core.
Figure 10. The core losses of the disk core obtained by theoretical calculation and simulation under different winding currents. (a) Transmitter core; (b) Receiver core.
Figure 11. The core losses of the disk core obtained by theoretical calculation and simulation under different current phase differences. (a) Transmitter core; (b) Receiver core.
Magnetic Coupling System Model Parameters Parameters and Setup
The inner and outer radius of the winding R[win] = 100 mm R[wout] = 250 mm
The inner and outer radius of the core R[cin] = 70 mm R[cout] = 280 mm
Coil diameter and core thickness d[0] = 4.2 mm b[0] = 100 mm
Turns of transmitter winding and receiver winding N[p] = 23 turns N[s] = 18 turns
Transmitter winding current and receiver winding current I[p] = 12 A I[s] = 18 A
Core material Philips-3C96
Boundary conditions Balloon border
The transmission distance between coils h[0] = 50 mm
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Chen, Q.; Fan, F.; Wang, J.; Chen, W. Core Loss Analysis and Modeling of a Magnetic Coupling System in WPT for EVs. World Electr. Veh. J. 2021, 12, 198. https://doi.org/10.3390/wevj12040198
AMA Style
Chen Q, Fan F, Wang J, Chen W. Core Loss Analysis and Modeling of a Magnetic Coupling System in WPT for EVs. World Electric Vehicle Journal. 2021; 12(4):198. https://doi.org/10.3390/wevj12040198
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Chen, Qingbin, Feng Fan, Jinshuai Wang, and Wei Chen. 2021. "Core Loss Analysis and Modeling of a Magnetic Coupling System in WPT for EVs" World Electric Vehicle Journal 12, no. 4: 198. https://
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assumptions returns the assumption x < 42 because it affects n through the assumption n < x. Thus, assumptions returns the transitive closure of assumptions, which is all assumptions that
mathematically affect the input.
Set the assumption on variable m that 1 < m < 3. Return all assumptions on m and x using assumptions.
syms m
assume(1 < m < 3)
assumptions([m x])
ans = $\left(\begin{array}{cccc}n<x& 1<m& m<3& x<42\end{array}\right)$
To see the assumptions that affect all variables, use assumptions without any arguments.
ans = $\left(\begin{array}{cccc}n<x& 1<m& m<3& x<42\end{array}\right)$
For further computations, clear the assumptions.
Multiple Assumptions on One Variable
You cannot set an additional assumption on a variable using assume because assume clears all previous assumptions on that variable. To set an additional assumption on a variable, using assumeAlso.
Set an assumption on x using assume. Set an additional assumption on x use assumeAlso. Use assumptions to return the multiple assumptions on x.
syms x
assumeAlso(x < 0)
ans = $\left(\begin{array}{cc}x\in \mathbb{R}& x<0\end{array}\right)$
For further computations, clear the assumptions.
Assumptions Affecting Expressions and Functions
assumptions accepts symbolic expressions and functions as input and returns all assumptions that affect all variables in the symbolic expressions or functions.
Set assumptions on variables in a symbolic expression and function. Find all assumptions that affect all variables in the symbolic expression using assumptions.
syms a b c f(a,b,c)
expr = a*exp(b)*sin(c);
assume(a+b > 3 & in(a,"integer") & in(c,"real"))
ans = $\left(\begin{array}{ccc}a\in \mathbb{Z}& c\in \mathbb{R}& 3<a+b\end{array}\right)$
Find all assumptions that affect all variables that are inputs to a symbolic function.
ans = $\left(\begin{array}{ccc}a\in \mathbb{Z}& c\in \mathbb{R}& 3<a+b\end{array}\right)$
Note that if you use syms to create another symbolic function after setting assumptions on the function input arguments, syms clears all previously set assumptions on the symbolic variables. Instead,
create the symbolic function first, and then set the assumptions on the symbolic variables.
syms g(a,b,c)
assume(a+b > 3 & in(a,"integer") & in(c,"real"))
ans = $\left(\begin{array}{ccc}a\in \mathbb{Z}& c\in \mathbb{R}& 3<a+b\end{array}\right)$
Clear the assumptions for further computations.
Restore Old Assumptions
To restore old assumptions, first store the assumptions returned by assumptions. Then you can restore these assumptions at any point by calling assume or assumeAlso.
Solve the equation for a spring system using dsolve under the assumptions that the mass and spring constant are positive.
syms m k positive
syms x(t)
xsol = dsolve(m*diff(x,t,t) == -k*x, x(0) == 0)
xsol =
$-{C}_{1} \mathrm{sin}\left(\frac{\sqrt{k} t}{\sqrt{m}}\right)$
Suppose you want to explore solutions unconstrained by assumptions, but want to restore the assumptions afterwards. First store the assumptions using assumptions, then clear the assumptions and solve
the equation. dsolve returns unconstrained solutions.
tmp = assumptions;
assume([m k],"clear")
xsol = dsolve(m*diff(x,t,t) == -k*x, x(0) == 0)
xsol =
$-{C}_{1} {\mathrm{e}}^{-\frac{t \sqrt{-k m}}{m}} \left({\mathrm{e}}^{\frac{2 t \sqrt{-k m}}{m}}-1\right)$
Restore the original assumptions using assume.
After computations are complete, clear assumptions using assume.
Input Arguments
var — Symbolic input to check for assumptions
symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix | symbolic multidimensional array
Symbolic input for which to show assumptions, specified as a symbolic variable, expression, or function, or a vector, matrix, or multidimensional array of symbolic variables, expressions, or
• When you delete a symbolic object from the MATLAB workspace by using clear, all assumptions that you set on that object remain in the symbolic engine. If you declare a new symbolic variable with
the same name, it inherits these assumptions.
• To clear all assumptions set on a symbolic variable var use this command.
• To clear all objects in the MATLAB workspace and close the Symbolic Math Toolbox™ engine associated with the MATLAB workspace resetting all its assumptions, use this command.
Version History
Introduced in R2012a | {"url":"https://kr.mathworks.com/help/symbolic/assumptions.html","timestamp":"2024-11-13T11:33:27Z","content_type":"text/html","content_length":"100340","record_id":"<urn:uuid:5ade806e-0850-40dd-b256-5586922179e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00269.warc.gz"} |
Spivak: Calculus – a dot b
Proofs of common number properties
I will name some properties I will use for future references with MP (my proposition) below:
(MP1.1) 0 times anything is 0
Let $a$ be any number. Let’s start with $a \cdot 0$. By the property of $0$, we have $a \cdot 0 = a \cdot (0 + 0)$. Using the distributive property gives us $a \cdot 0 + a \cdot 0$. Combining it all
together we have $a\cdot 0 = a\cdot 0 + a\cdot 0$. But the definition of $0$ is such that it doesn’t change a number after addition. So $a \cdot 0 = 0$.
(MP1.2) $(-a)b = – (ab)$
Note the meaning of $-a$ is the additive inverse of $a$, meaning that $a + (-a) = 0$ while the meaning of $-(ab)$ is the additive inverse of $ab$. Let’s add $ab$ with $(-a)b$.
$ab + (-a)b = (a + (-a)) \cdot b$ by the commutative and distributive laws. Since $-a$ is the additive inverse of $a$, this gives us $0 \cdot b$ which is $0$ from MP1.1. Thus $(-a)b$ is the additive
inverse of $ab$ which is what we want to prove.
(MP1.3) $(-a)(-b) = ab$
$ (-a)(-b) + -(ab) = (-a)(-b) + (-a)(b)$ by the MP1.2. By the distributive property, $(-a)(-b) + (-a)b = (-a)(-b + b) = (-a)0 = 0$ by MP1.1. Since $-(ab)$ is the additive inverse of $ab$, $(-a)(-b)$
must have been $ab$.
We cannot have the multiplicative inverse of 0
Suppose there exist the multiplicative inverse of $0$, which we will denote by $0^{-1}$. Then $0 \cdot 0^{-1} = 1$. But by MP1.1, $0$ times anything is 0 so $0=1$. This brings us a whole host of
problems: $1 + 1 = 0 + 0 = 0$. So no matter what we do, every number we try to create will all become 0. Hence allowing the multiplicative inverse of 0 means that we can only work with one number:
not a very interesting proposition. Conversely, to have more than one number to work with, we cannot allow the multiplicative inverse of 0.
(MP1.4) Corollory of $(-a)b = -(ab): (-1)b = -b$
(MP1.5) If $ab=0$, then $a=0$ or $b=0$.
Case 1: $a=0$. That is part of the solution.
Case 2: $a \neq 0$. Then the multiplicative inverse $a^{-1}$ exists. So $ab= 0$ means $a^{-1} ab = a^{-1} 0 $ so $b=0$.
(MP1.6) $(ab)^{-1} = a^{-1}b^{-1}$
$ab (a^{-1}b^{-1}) = a a^{-1} b b^{-1}$ by the commutative and associative properties which gets us $1 \cdot 1 = 1$ by the property of the multiplicative identity $1$. Hence $a^{-1}b^{-1}$ is the
multiplicative inverse of $ab$.
Chapter 1: Numbers, operations and axioms
Numbers are our first exposure to mathematics: yet what exactly a "number" is an interesting discussion I've had in philosophy classes. At least for "whole numbers", we typically find them pretty
intuitive and it doesn't take a child much to be comfortable with numbers like "2" and "3" and learning how to operate on them via addition, "$2+3=5$" and multiplication "$2 \cdot 3 = 6$". The fact
that there is a "two-ness" behind 2 cows, 2 dollars and 2 books is pretty remarkable we slow down and think about it. But that is a discussion for another day. Let us now look a few common sets of
numbers we typically encounter:
• The natural numbers $\mathbb{N} = \{0, 1, 2, 3, \ldots \}$
• The integers $\mathbb{Z} = \{ \ldots, -2, -1, 0, 1, 2, \dots\}$
• The rational numbers $\mathbb{Q}$, numbers that can be expressed as $\frac{a}{b}$ where $a,b\in\mathbb{Z}, b \neq 0$
• The real numbers: visualized as a number line, including all the rational numbers and irrational numbers (e.g. $\sqrt{2}, \pi, e)$.
The fact that these set of numbers are useful to work with are encapsulated by the rules we want our operations, addition $+$ and multiplication $\cdot$ to follow and the axioms we want.
Operations and axioms
To model how we think about numbers, addition and multiplication and how it works "in real life", we want them to follow certain rules. First (closure), for any numbers $a$ and $b$, we want $a+b$ and
$a \cdot b$ to be numbers to. Also, these following should come as no surprise
1. Associativity: we want the order of repeated operations not to be important. $a + (b+c) = (a+c) + c, a \cdot (b \cdot c) = (a \cdot b) \cdot c$.
2. Commutativity: we want the order of operations not to matter. $a + b = b + a, a \cdot b = b \cdot a$
3. Distributive law: we want to know how addition and multiplication interacts. $a \cdot (b+c) = a \cdot b + a \cdot c$.
As an aside, commutative for multiplication is often not required when we study more abstract systems such as matrices in the study of abstract algebra. For our usual numbers that isn't a concern.
The numbers $0$ and $1$ plays a special part in addition and multiplication respectively: they play the part as an "identity" element: $a + 0 = a$ and $a \cdot 1 = a$. Just the natural numbers
satisfy all of the above axioms. Along with mathematical induction (chapter 2, which is implicit in the way the natural numbers are defined) these bring about the rich field of number theory. But
things get more interesting (and allow for our study of calculus) when we require an "inverse" element, essentially setting up the field for the opposite operations of subtraction and division.
• Existence of additive inverse: for every number $a$, we have a number ($-a$) such that $a + (-a) = 0$.
• Existence of multiplicative inverse: for every number $a$, $a\neq 0$ we have a number ($a^{-1}$) such that $a \cdot a^{-1} = 1$.
Additive inverse brings about the integers, and multiplicative inverses bring about the rational numbers. The fact that 0 can not have an inverse will be investigated in a following post. Finally, we
bring about the idea of "ordering" the numbers using the concept of an inequality. For any two numbers $a$ and $b$, one and only one of the following holds:
• $a=b,$
• $a < b,$ or
• $b < a$.
We want the inequality to have the following rules:
• If $ a < b$ and $b < c$, then $a < c$
• If $a < b$, then for any $c$, $a+c < b+c$
• If $a < b$ and $0 < c$, then $ac < bc$
Just axioms alone (along with mathematical induction in Chapter 2) can lead us to a whole bunch of familiar techniques we have already internalized. We will explore them in solving some exercises in
a subsequent post. We also note that just the rational numbers alone are sufficient to satisfy all the above axioms. But very quickly we realize that will preclude a solution to something "simple"
like $x^2 = 2$. A proof that $\sqrt{2}$ is rational is presented in many other places: I will refer readers to Google if they have not seen it before. $x^2 = 2$ comes up pretty naturally from the
study of geometry (the length of the hypotenuse in an isosceles right angled triangle with two sides of length 1) so real numbers are required for the study of subjects where we need a "measure". The
construction of the reals is something I'm looking forward to at the end of the book.
Modulus and the triangle inequality
The modulus/absolute value function (defined by $|a| = a$ if $a$ is positive or 0, and $|a| = -a$ if $a$ is negative), along with the triangle inequality $|a + b| \leq |a| + |b|$ comes up all the
time in subsequent work so I will just mention that at the end of this post. The proof can be done by simply working through all the cases where $a$ and $b$ take different signs.
My introduction to Spivak: Calculus
“Analysis” is one of the major fields of mathematics and the path a typical student of mathematics goes through in this field goes something like this:
• “Calculus”: This is where we are more concerned about the techniques of differentiation and integration. In the American context this is typically done in AP classes + at lower level
undergraduate courses. In Singapore we start this journey in Additional Mathematics at the “O” levels where differentiation (techniques and applications) are covered extensively with a brief
introduction into integration techniques. At the “A” levels we delve deeper into further integration techniques and start touching on some interesting applications of calculus through areas and
volumes, the Maclaurin series and differential equations. At university this is usually taken further through an introduction to limits and further courses on differential equations (both
ordinary and partial). This is also where many math-adjacent courses (engineering and sciences) end in this endeavor.
• “Analysis”: The concept of proofs is where the (pure) mathematics syllabus start to deviate from their applied mathematics, science and engineering contemporaries. And this is where I feel we
start moving from “Calculus” into “Analysis”, where the proof of equations and theorems start to take more importance compared to the use and application of them. In a sense, we re-learn what we
have started taken for granted in our earlier study and place them on solid foundation. Analysis I typically takes a student through limits and the epsilon-delta formulation, sequences and
differentiation, Analysis II goes into the (Riemann) integral while Analysis III goes into the link between the two: The Fundamental Theorem of Calculus. With interesting detours along the way
(like the Taylor series) in between.
• “Measure Theory and the Lebesgue integral”: At the upper undergraduate and beginning graduate level we move on from the Riemann integral to the Lebesgue integral, with a whole big idea of measure
theory supporting the approach. This leads to all sorts of related topics such as topology, functional analysis, probability theory and more!
As I went through my own path (and coming from a engineering background to graduate work in math) I am often in awe of this progression of ideas. Unfortunately, the pace of school work, having to
complete each course within 10-12 weeks with a final examination at the end means I sometimes did not have the time to fully appreciate some of the ideas and hardly took any interesting looking
detours. Having needed a few courses around measure theory before I truly understood and appreciated it, I feel I could benefit from a slower but deeper delve into the more “basic” analysis portion
of my study.
I’ve read many good reviews of Spivak’s Calculus (the book title being a bit of a misnomer: it is definitely a book aimed at the “analysis” part rather than the “calculus” part as I’ve described
above) and am eager to give it a whirl. Chapters I’m especially looking forward to include the proof that $\pi$ is irrational, $e$ is transcendental, how he defines the logarithm and exponential
function and the construction of the real numbers (plus a proof in its uniqueness). As I work through the book I’m looking to use this as a blog to aid my understanding of the material.
I’m using the third edition, though I understand the 4th edition is out (amazon link to the third edition: cover photo credits). Let’s enjoy this journey together then! | {"url":"https://adotb.xyz/category/spivak-calculus/","timestamp":"2024-11-09T23:58:47Z","content_type":"text/html","content_length":"66572","record_id":"<urn:uuid:98b7e546-6b83-4513-abcb-90f731dc2146>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00823.warc.gz"} |
Math Regrouping Examples - 1st Grade Math Worksheets
1st Grade Math Worksheets Regrouping – The fundamentals of mathematics are taught to children in the first grades. These worksheets are a great method to assist your child in understanding these
concepts. What is the importance of 1st grade Math worksheets? The math worksheets for the first grade are vital to a child’s math development. … Read more | {"url":"https://www.1stgrademathworksheets.com/tag/math-regrouping-examples/","timestamp":"2024-11-09T06:46:32Z","content_type":"text/html","content_length":"46346","record_id":"<urn:uuid:38473af6-ec54-4347-9014-7fe92a391b87>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00372.warc.gz"} |
Adding Fractions: Why Avoiding Common Denominators Works
I wrote this post showing that adding fractions can be done using the algebraic definition of addition of rationals:
MathHeadInc (via twitter) has requested a video showing why this works.
I aim to please.
Here is the video showing why adding fractions using a common denominator is the same as the definition of addition of rationals:
What do you think? Will this help convince your kids that “the trick” is okay to use? Share your experiences in the comments.
This post may contain affiliate links. When you use them, you support us so we can continue to provide free content!
Leave a reply
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://mathfour.com/arithmetic/adding-fractions-why-avoiding-common-denominators-works","timestamp":"2024-11-14T05:45:51Z","content_type":"text/html","content_length":"34066","record_id":"<urn:uuid:c3cfa7f7-531a-4491-8505-e3e2796077bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00662.warc.gz"} |
Cycle (graph theory)
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In graph theory, a cycle is a path of edges and vertices wherein a vertex is reachable from itself. There are several different types of cycles, principally a closed walk and a simple cycle; also,
e.g., an element of the cycle space of the graph. If a graph contains no cycles it is referred to as being acyclic.
A closed walk consists of a sequence of at least two vertices, starting and ending at the same vertex, with each two consecutive vertices in the sequence adjacent to each other in the graph. In a
directed graph, each edge must be traversed by the walk consistently with its direction: the edge must be oriented from the earlier of two consecutive vertices to the later of the two vertices in the
sequence. The choice of starting vertex is not important: traversing the same cyclic sequence of edges from different starting vertices produces the same closed walk.
A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex, or as the set of edges in such a
walk. The two definitions are equivalent in directed graphs, where simple cycles are also called directed cycles: the cyclic sequence of vertices and edges in a directed cycle is completely
determined by the set of edges that it uses. In undirected graphs the set of edges of a cycle can be traversed by a walk in either of two directions, giving two possible directed cycles for every
undirected cycle. (For closed walks more generally, in directed or undirected graphs, the multiset of edges does not unambiguously determine the vertex ordering.) A circuit can be a closed walk
allowing repetitions of vertices but not edges; however, it can also be a simple cycle, so explicit definition is recommended when it is used.^[1]
In order to maintain a consistent terminology, for the rest of this article, "cycle" means a simple cycle, except where otherwise stated.
Chordless cycles[edit]
A chordless cycle in a graph, also called a hole or an induced cycle, is a cycle such that no two vertices of the cycle are connected by an edge that does not itself belong to the cycle. An antihole
is the complement of a graph hole. Chordless cycles may be used to characterize perfect graphs: by the strong perfect graph theorem, a graph is perfect if and only if none of its holes or antiholes
have an odd number of vertices that is greater than three. A chordal graph, a special type of perfect graph, has no holes of any size greater than three.
The girth of a graph is the length of its shortest cycle; this cycle is necessarily chordless. Cages are defined as the smallest regular graphs with given combinations of degree and girth.
A peripheral cycle is a cycle in a graph with the property that every two edges not on the cycle can be connected by a path whose interior vertices avoid the cycle. In a graph that is not formed by
adding one edge to a cycle, a peripheral cycle must be an induced cycle.
Cycle space[edit]
The term cycle may also refer to an element of the cycle space of a graph. There are many cycle spaces, one for each coefficient field or ring. The most common is the binary cycle space (usually
called simply the cycle space), which consists of the edge sets that have even degree at every vertex; it forms a vector space over the two-element field. By Veblen's theorem, every element of the
cycle space may be formed as an edge-disjoint union of simple cycles. A cycle basis of the graph is a set of simple cycles that forms a basis of the cycle space.^[2]
Using ideas from algebraic topology, the binary cycle space generalizes to vector spaces or modules over other rings such as the integers, rational or real numbers, etc.^[3]
Cycle detection[edit]
The existence of a cycle in directed and undirected graphs can be determined by whether depth-first search (DFS) finds an edge that points to an ancestor of the current vertex (it contains a back
edge).^[4]All the back edges which DFS skips over are part of cycles.^[5] In an undirected graph, the edge to the parent of a node should not be counted as a back edge, but finding any other already
visited vertex will indicate a back edge. In the case of undirected graphs, only O(n) time is required to find a cycle in an n-vertex graph, since at most n − 1 edges can be tree edges.
Many topological sorting algorithms will detect cycles too, since those are obstacles for topological order to exist. Also, if a directed graph has been divided into strongly connected components,
cycles only exist within the components and not between them, since cycles are strongly connected.^[5]
For directed graphs, Rocha–Thatte Algorithm^[6] is a distributed cycle detection algorithm. Distributed cycle detection algorithms are useful for processing large-scale graphs using a distributed
graph processing system on a computer cluster (or supercomputer).
Applications of cycle detection include the use of wait-for graphs to detect deadlocks in concurrent systems.^[7]
Covering graphs by cycles[edit]
In his 1736 paper on the Seven Bridges of Königsberg, widely considered to be the birth of graph theory, Leonhard Euler proved that, for a finite undirected graph to have a closed walk that visits
each edge exactly once, it is necessary and sufficient that it be connected except for isolated vertices (that is, all edges are contained in one component) and have even degree at each vertex. The
corresponding characterization for the existence of a closed walk visiting each edge exactly once in a directed graph is that the graph be strongly connected and have equal numbers of incoming and
outgoing edges at each vertex. In either case, the resulting walk is known as an Euler cycle or Euler tour. If a finite undirected graph has even degree at each of its vertices, regardless of whether
it is connected, then it is possible to find a set of simple cycles that together cover each edge exactly once: this is Veblen's theorem.^[8] When a connected graph does not meet the conditions of
Euler's theorem, a closed walk of minimum length covering each edge at least once can nevertheless be found in polynomial time by solving the route inspection problem.
The problem of finding a single simple cycle that covers each vertex exactly once, rather than covering the edges, is much harder. Such a cycle is known as a Hamiltonian cycle, and determining
whether it exists is NP-complete.^[9] Much research has been published concerning classes of graphs that can be guaranteed to contain Hamiltonian cycles; one example is Ore's theorem that a
Hamiltonian cycle can always be found in a graph for which every non-adjacent pair of vertices have degrees summing to at least the total number of vertices in the graph.^[10]
The cycle double cover conjecture states that, for every bridgeless graph, there exists a multiset of simple cycles that covers each edge of the graph exactly twice. Proving that this is true (or
finding a counterexample) remains an open problem.^[11]
Graph classes defined by cycles[edit]
Several important classes of graphs can be defined by or characterized by their cycles. These include:
See also[edit]
1. ^ Balakrishnan, V.K. (2005). Schaum's outline of theory and problems of graph theory ([Nachdr.]. ed.). McGraw–Hill. ISBN 978-0070054899.
2. ^ Gross, Jonathan L.; Yellen, Jay (2005), "4.6 Graphs and Vector Spaces", Graph Theory and Its Applications (2nd ed.), CRC Press, pp. 197–207, ISBN 9781584885054.
3. ^ Diestel, Reinhard (2012), "1.9 Some linear algebra", Graph Theory, Graduate Texts in Mathematics, 173, Springer, pp. 23–28.
4. ^ Tucker, Alan (2006). "Chapter 2: Covering Circuits and Graph Colorings". Applied Combinatorics (5th ed.). Hoboken: John Wiley & sons. p. 49. ISBN 978-0-471-73507-6.
5. ^ ^a ^b Sedgewick, Robert (1983), "Graph algorithms", Algorithms, Addison–Wesley, ISBN 0-201-06672-6
6. ^ Rocha, Rodrigo Caetano; Thatte, Bhalchandra (2015). "Distributed cycle detection in large-scale sparse graphs". Simpósio Brasileiro de Pesquisa Operacional (SBPO). doi:10.13140/RG.2.1.1233.8640
7. ^ Silberschatz, Abraham; Peter Galvin; Greg Gagne (2003). Operating System Concepts. John Wiley & Sons, INC. p. 260. ISBN 0-471-25060-0.
8. ^ Veblen, Oswald (1912), "An Application of Modular Equations in Analysis Situs", Annals of Mathematics, Second Series, 14 (1): 86–94, doi:10.2307/1967604, JSTOR 1967604.
9. ^ Richard M. Karp (1972), "Reducibility Among Combinatorial Problems" (PDF), in R. E. Miller and J. W. Thatcher, Complexity of Computer Computations, New York: Plenum, pp. 85–103.
10. ^ Ore, Ø. (1960), "Note on Hamilton circuits", American Mathematical Monthly, 67 (1): 55, doi:10.2307/2308928, JSTOR 2308928.
11. ^ Jaeger, F. (1985), "A survey of the cycle double cover conjecture", Annals of Discrete Mathematics 27 – Cycles in Graphs, North-Holland Mathematics Studies, 27, pp. 1–12, doi:10.1016/S0304-0208 | {"url":"https://static.hlt.bme.hu/semantics/external/pages/f%C3%BCgg%C5%91s%C3%A9gi_gr%C3%A1f/en.wikipedia.org/wiki/Cycle_detection_(graph_theory).html","timestamp":"2024-11-15T04:35:30Z","content_type":"text/html","content_length":"65904","record_id":"<urn:uuid:abfadb29-4b45-4030-9a94-d89311af8923>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00457.warc.gz"} |
Integer Programming - (Convex Geometry) - Vocab, Definition, Explanations | Fiveable
Integer Programming
from class:
Convex Geometry
Integer programming is a mathematical optimization technique where some or all of the decision variables are constrained to take on integer values. This approach is commonly used in situations where
solutions must be whole numbers, such as scheduling, resource allocation, and logistics. It connects closely to linear programming, particularly in the context of duality, as it allows for the
exploration of optimal solutions while considering integrality constraints.
congrats on reading the definition of Integer Programming. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Integer programming is typically more computationally challenging than linear programming due to its discrete nature, which may lead to NP-hard problems.
2. There are different types of integer programming, such as pure integer programming where all variables are integers, and mixed-integer programming where only some variables are restricted to
3. The simplex method, which is often used for solving linear programming problems, cannot be directly applied to integer programming due to integrality constraints.
4. Branch-and-bound and cutting-plane methods are common techniques used to solve integer programming problems efficiently.
5. The concept of duality in integer programming extends from linear programming, where every integer programming problem has a corresponding dual that can offer additional insights into the
feasible solutions.
Review Questions
• How does integer programming differ from standard linear programming in terms of solution characteristics?
□ Integer programming differs from standard linear programming mainly because it requires some or all decision variables to be integers. This constraint leads to a different solution space and
often makes integer programming problems more complex and harder to solve. While linear programming can produce fractional solutions that maximize or minimize an objective function within a
continuous space, integer programming ensures that the solutions remain whole numbers, which is crucial for applications like resource allocation or scheduling.
• Discuss the role of duality in integer programming and how it compares to duality in linear programming.
□ Duality in integer programming serves a similar purpose as in linear programming by providing insights into the structure of the optimization problem. However, while the dual problem in
linear programming has well-defined characteristics and can often be solved easily using primal methods, duality in integer programming is less straightforward. The duality relationship may
not yield simple or useful interpretations when dealing with integer constraints, making it more challenging to derive direct implications for the primal problem's optimal solutions.
• Evaluate how the complexity of integer programming affects real-world applications compared to traditional linear programming.
□ The complexity of integer programming significantly impacts its real-world applications, as many practical problems require discrete solutions—like assigning tasks or scheduling
employees—making integer solutions necessary. Unlike traditional linear programming, which can quickly find optimal solutions using efficient algorithms like the simplex method, integer
problems often demand more sophisticated techniques like branch-and-bound or cutting-plane methods due to their NP-hard nature. This complexity can lead to longer computation times and
necessitate heuristic approaches for large-scale problems, ultimately affecting how industries implement optimization strategies.
© 2024 Fiveable Inc. All rights reserved.
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Fourier analysis o
Fourier analysis of simulation data signals
FFTDATA = power_fftscope(ScopeData) returns the FFT results for the signal saved in the ScopeData structure-with-time.
FFTDATA = power_fftscope(FFTDATA) uses the FFTDATA structure as a template variable to modify analysis settings and signal selection, and to perform FFT analysis. The power_fftscope function ignores
any user-defined FFTDATA fields that are not listed in the FFTDATA structure.
power_fftscope(ScopeData) plots the FFT analysis results for the last simulation cycle of the signal saved in the ScopeData variable.
power_fftscope(FFTDATA) plots the FFT analysis results for the signal options specified in the FFTDATA structure.
Input Arguments
ScopeData — Signal you want to perform Fourier analysis on
structure with time (default)
Signal you want to perform the Fourier analysis on, specified as a structure with time.
FFTDATA — Signal options
structure (default)
Signal options, specified as a structure with these fields:
Field Description
time The time vector of the simulation data signal saved in the ScopeData variable.
signals The signals saved in the ScopeData variable.
blockName The name of the Scope block associated with the ScopeData variable.
input The input signal of the selected simulation data variable.
signal The index of the selected input signal specified by the Input field.
startTime The start time of the FFT window.
cycles The number of cycles of the FFT window.
fundamental The fundamental frequency of the analyzed signal.
maxFrequency The maximum frequency evaluated by the FFT analysis.
THDmaxFrequency The maximum frequency for the total harmonic distortion (THD) calculation. Set the value to inf to calculate the THD at the Nyquist frequency.
FFTdata The analyzed signal (FFT window data).
THDbase The base used to compute the THD. Set to fund to normalize the THD with respect to fundamental value. Set the THDbase to DC to normalize the THD with respect to the DC component
freqAxis The type of frequency axis of the FFT analysis plot. Set to hertz to display the frequency axis in hertz. Set to harmonicorder to display the frequency axis in harmonic orders.
mag The computed magnitude of FFT.
phase The computed phase of FFT.
freq The frequency vector.
THD The computed THD for the analyzed signal. The THD calculation includes all the inter-harmonics of the selected input signal.
samplingTime The sampling time of the selected input signal.
samplePerCycle The number of samples per cycle of the selected input signal.
DCcomponent The DC component value of the selected input signal.
magFundamental The fundamental component value of the selected input signal.
Output Arguments
FFTDATA — FFT results for signal
FFT results for the signal saved in the ScopeData argument, returned as a structure with these fields:
Field Description
time The time vector of the simulation data signal saved in the ScopeData variable.
signals The signals saved in the ScopeData variable.
blockName The name of the Scope block associated with the ScopeData variable.
input The input signal of the selected simulation data variable.
signal The index of the selected input signal specified by the Input field.
startTime The start time of the FFT window.
cycles The number of cycles of the FFT window.
fundamental The fundamental frequency of the analyzed signal.
maxFrequency The maximum frequency evaluated by the FFT analysis.
THDmaxFrequency The maximum frequency for the total harmonic distortion (THD) calculation. Set the value to inf to calculate the THD at the Nyquist frequency.
FFTdata The analyzed signal (FFT window data).
THDbase The base used to compute the THD. Set to fund to normalize the THD with respect to fundamental value. Set the THDbase to DC to normalize the THD with respect to the DC component
freqAxis The type of frequency axis of the FFT analysis plot. Set to hertz to display the frequency axis in hertz. Set to harmonicorder to display the frequency axis in harmonic orders.
mag The computed magnitude of FFT.
phase The computed phase of FFT.
freq The frequency vector.
THD The computed THD for the analyzed signal. The THD calculation includes all the inter-harmonics of the selected input signal.
samplingTime The sampling time of the selected input signal.
samplePerCycle The number of samples per cycle of the selected input signal.
DCcomponent The DC component value of the selected input signal.
magFundamental The fundamental component value of the selected input signal.
Version History
Introduced before R2006a
See Also | {"url":"https://it.mathworks.com/help/sps/powersys/ref/power_fftscope.html","timestamp":"2024-11-02T21:02:03Z","content_type":"text/html","content_length":"84184","record_id":"<urn:uuid:8ed332bb-fde1-4a0f-8b85-28f2f99b32bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00614.warc.gz"} |
Special Ordered Sets (SOS)
Special Ordered Sets (SOS)
Pyomo allows users to declare special ordered sets (SOS) within their problems. These are sets of variables among which only a certain number of variables can be non-zero, and those that are must be
adjacent according to a given order.
Special ordered sets of types 1 (SOS1) and 2 (SOS2) are the classic ones, but the concept can be generalised: a SOS of type N cannot have more than N of its members taking non-zero values, and those
that do must be adjacent in the set. These can be useful for modelling and computational performance purposes.
By explicitly declaring these, users can keep their formulations and respective solving times shorter than they would otherwise, since the logical constraints that enforce the SOS do not need to be
implemented within the model and are instead (ideally) handled algorithmically by the solver.
Special ordered sets can be declared one by one or indexed via other sets.
Non-indexed Special Ordered Sets
A single SOS of type N involving all members of a pyomo Var component can be declared in one line:
# import pyomo
import pyomo.environ as pyo
# declare the model
model = pyo.AbstractModel()
# the type of SOS
N = 1 # or 2, 3, ...
# the set that indexes the variables
model.A = pyo.Set()
# the variables under consideration
model.x = pyo.Var(model.A)
# the sos constraint
model.mysos = pyo.SOSConstraint(var=model.x, sos=N)
In the example above, the weight of each variable is determined automatically based on their position/order in the pyomo Var component (model.x).
Alternatively, the weights can be specified through a pyomo Param component (model.mysosweights) indexed by the set also indexing the variables (model.A):
# the set that indexes the variables
model.A = pyo.Set()
# the variables under consideration
model.x = pyo.Var(model.A)
# the weights for each variable used in the sos constraints
model.mysosweights = pyo.Param(model.A)
# the sos constraint
model.mysos = pyo.SOSConstraint(
Indexed Special Ordered Sets
Multiple SOS of type N involving members of a pyomo Var component (model.x) can be created using two additional sets (model.A and model.mysosvarindexset):
# the set that indexes the variables
model.A = pyo.Set()
# the variables under consideration
model.x = pyo.Var(model.A)
# the set indexing the sos constraints
model.B = pyo.Set()
# the sets containing the variable indexes for each constraint
model.mysosvarindexset = pyo.Set(model.B)
# the sos constraints
model.mysos = pyo.SOSConstraint(
In the example above, the weights are determined automatically from the position of the variables. Alternatively, they can be specified through a pyomo Param component (model.mysosweights) and an
additional set (model.C):
# the set that indexes the variables
model.A = pyo.Set()
# the variables under consideration
model.x = pyo.Var(model.A)
# the set indexing the sos constraints
model.B = pyo.Set()
# the sets containing the variable indexes for each constraint
model.mysosvarindexset = pyo.Set(model.B)
# the set that indexes the variables used in the sos constraints
model.C = pyo.Set(within=model.A)
# the weights for each variable used in the sos constraints
model.mysosweights = pyo.Param(model.C)
# the sos constraints
model.mysos = pyo.SOSConstraint(
Declaring Special Ordered Sets using rules
Arguably the best way to declare an SOS is through rules. This option allows users to specify the variables and weights through a method provided via the rule parameter. If this parameter is used,
users must specify a method that returns one of the following options:
• a list of the variables in the SOS, whose respective weights are then determined based on their position;
• a tuple of two lists, the first for the variables in the SOS and the second for the respective weights;
• or, pyomo.environ.SOSConstraint.Skip, if the SOS is not to be declared.
If one is content on having the weights determined based on the position of the variables, then the following example using the rule parameter is sufficient:
# the set that indexes the variables
model.A = pyo.Set()
# the variables under consideration
model.x = pyo.Var(model.A, domain=pyo.NonNegativeReals)
# the rule method creating the constraint
def rule_mysos(m):
return [m.x[a] for a in m.x]
# the sos constraint(s)
model.mysos = pyo.SOSConstraint(rule=rule_mysos, sos=N)
If the weights must be determined in some other way, then the following example illustrates how they can be specified for each member of the SOS using the rule parameter:
# the set that indexes the variables
model.A = pyo.Set()
# the variables under consideration
model.x = pyo.Var(model.A, domain=pyo.NonNegativeReals)
# the rule method creating the constraint
def rule_mysos(m):
var_list = [m.x[a] for a in m.x]
weight_list = [i+1 for i in range(len(var_list))]
return (var_list, weight_list)
# the sos constraint(s)
model.mysos = pyo.SOSConstraint(rule=rule_mysos, sos=N)
The rule parameter also allows users to create SOS comprising variables from different pyomo Var components, as shown below:
# the set that indexes the x variables
model.A = pyo.Set()
# the set that indexes the y variables
model.B = pyo.Set()
# the set that indexes the SOS constraints
model.C = pyo.Set()
# the x variables, which will be used in the constraints
model.x = pyo.Var(model.A, domain=pyo.NonNegativeReals)
# the y variables, which will be used in the constraints
model.y = pyo.Var(model.B, domain=pyo.NonNegativeReals)
# the x variable indices for each constraint
model.mysosindex_x = pyo.Set(model.C)
# the y variable indices for each constraint
model.mysosindex_y = pyo.Set(model.C)
# the weights for the x variable indices
model.mysosweights_x = pyo.Param(model.A)
# the weights for the y variable indices
model.mysosweights_y = pyo.Param(model.B)
# the rule method with which each constraint c is built
def rule_mysos(m, c):
var_list = [m.x[a] for a in m.mysosindex_x[c]]
var_list.extend([m.y[b] for b in m.mysosindex_y[c]])
weight_list = [m.mysosweights_x[a] for a in m.mysosindex_x[c]]
weight_list.extend([m.mysosweights_y[b] for b in m.mysosindex_y[c]])
return (var_list, weight_list)
# the sos constraint(s)
model.mysos = pyo.SOSConstraint(
Compatible solvers
Not all LP/MILP solvers are compatible with SOS declarations and Pyomo might not be ready to interact with all those that are. The following is a list of solvers known to be compatible with special
ordered sets through Pyomo:
Please note that declaring an SOS is no guarantee that a solver will use it as such in the end. Some solvers, namely Gurobi and CPLEX, might reformulate problems with explicit SOS declarations, if
they perceive that to be useful.
Full example with non-indexed SOS constraint
import pyomo.environ as pyo
from pyomo.opt import check_available_solvers
from math import isclose
N = 1
model = pyo.ConcreteModel()
model.x = pyo.Var([1], domain=pyo.NonNegativeReals, bounds=(0,40))
model.A = pyo.Set(initialize=[1,2,4,6])
model.y = pyo.Var(model.A, domain=pyo.NonNegativeReals, bounds=(0,2))
model.OBJ = pyo.Objective(
model.ConstraintYmin = pyo.Constraint(
expr = (model.x[1]+
model.y[6] >= 0.25
model.mysos = pyo.SOSConstraint(
solver_name = 'scip'
solver_available = bool(check_available_solvers(solver_name))
if solver_available:
opt = pyo.SolverFactory(solver_name)
opt.solve(model, tee=False)
assert isclose(pyo.value(model.OBJ), 0.05, abs_tol=1e-3) | {"url":"https://pyomo.readthedocs.io/en/stable/advanced_topics/sos_constraints.html","timestamp":"2024-11-06T15:36:06Z","content_type":"text/html","content_length":"43769","record_id":"<urn:uuid:ca5b174b-e423-493e-b8cc-865661b6d93c>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00802.warc.gz"} |
Bandwidth Math and Connection Speed Needed - The PowerPoint Blog
Bandwidth Math and Connection Speed Needed
There are 5 Categories of Internet Connections:
1. 5+ Mbps = Very High Broadband
2. 1-5 Mbps = High Broadband
3. 786 Kbps = Fast Broadband
4. 384 Kbps = Standard Broadband
5. 56 Kbps = Dial-up
To figure the bandwidth a viewer will need to view the streaming media file with perfect playback, we need to work through these formulas (one for a Standard Web Server, one for a Streaming Server
(note: Streaming Servers are overviewed a few posts from now).
A: Figuring Bandwidth Needs From A Standard Server. Here things are easy because we get to figure things directly in ‘bandwidth’ math using bits not bytes.
1. Figure “Bits Per Second”
Video Height x Video Width x Frame Rate (fps) = Bits/second (Kbps)
eg. (320 x 240 video dimensions) x 15 fps = 1,152,000 Bits/second
2. Convert Bits Per Second (Kbps) to Megabits Per Second (Mbps)
Bps (Total from #1) / 1,024 (1 Mbps = 1,024 Kbps) = Needed Connection Speed
eg. 1,152,000 Kbps / 1,024 = 1,125 Mbps (so the person watching should have a, category 2, high broadband connection)
B: Figuring Bandwidth Needs From A Streaming Media Server. Here there are a few extra steps because streaming servers encode everything in Bytes Per Second (Bps), which needs to then be converted to
Kbps to know the bandwidth need.
1. Figure Total Bits Per Second
Video Height x Video Width x Frame Rate (fps) = Total Bits/second
eg. (320 x 240 video dimensions) x 15 fps) = 1,152,000 Bits/second
2. Figure the Bytes Per Second (Bps)
Bps (total from #1) / 8 = Bps (divide by 8 because there are 8 bits in 1 byte)
eg. ((320 x 240 x 15) / 8) = 144,000 Bps
3. Convert Bps to Kbps
Bps (total from #2) / 1,000 = Kilobytes/second (Kbps)
eg. 144,000 Bps / 1000 = 1,152Kbps (which is rounded to 1.2 Mbps)
These formulas do not take into account your server’s bandwidth limitations, the number of simultaneous viewers, network congestion or a host of other variables. Now we know how to anticipate the
needed connection speed for our streaming media.
Up next are some of the ways we can make a larger bandwidth file playback smooth on a low bandwidth connection.
– Troy @ TLC | {"url":"https://thepowerpointblog.com/bandwidth_math_and_connection_speed_need/","timestamp":"2024-11-07T07:05:57Z","content_type":"text/html","content_length":"44858","record_id":"<urn:uuid:d8294de7-dd7b-4c2b-8f26-8b7a52b161c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00775.warc.gz"} |
Lesson 24
Finding the Percentage
Let’s find percentages in general.
24.1: True or False: Percentages
Is each statement true or false? Be prepared to explain your reasoning.
1. 25% of 512 is equal to \(\frac14 \boldcdot 500\).
2. 90% of 133 is equal to \((0.9) \boldcdot 133\).
3. 30% of 44 is equal to 3% of 440.
4. The percentage 21 is of 28 is equal to the percentage 30 is of 40.
24.2: Jumping Rope
A school held a jump-roping contest. Diego jumped rope for 20 minutes.
1. Jada jumped rope for 15 minutes. What percentage of Diego’s time is that?
2. Lin jumped rope for 24 minutes. What percentage of Diego’s time is that?
3. Noah jumped rope for 9 minutes. What percentage of Diego’s time is that?
4. Record your answers in this table. Write the quotients in the last column as decimals.
│ │time (minutes)│percentage│ \(\text{time} \div 20\) │
│Diego│20 │100 │\(\frac{20}{20} = 1\text{ }\) │
│Jada │15 │ │\(\frac{15}{20} = \text{ }\) │
│ Lin │24 │ │\(\frac{24}{20} = \text{ }\) │
│Noah │9 │ │\(\frac{9}{20} = \text{ }\) │
5. What do you notice about the numbers in the last two columns of the table?
24.3: Restaurant Capacity
A restaurant has a sign by the front door that says, “Maximum occupancy: 75 people.” Answer each question and explain or show your reasoning.
1. What percentage of its capacity is 9 people?
2. What percentage of its capacity is 51 people?
3. What percentage of its capacity is 84 people?
Water makes up about 71% of Earth’s surface, while the other 29% consists of continents and islands. 96% of all Earth’s water is contained within the oceans as salt water, while the remaining 4% is
fresh water located in lakes, rivers, glaciers, and the polar ice caps.
If the total volume of water on Earth is 1,386 million cubic kilometers, what is the volume of salt water? What is the volume of fresh water?
What percentage of 90 kg is 36 kg? One way to solve this problem is to first find what percentage 1 kg is of 90, and then multiply by 36.
From the table we can see that 1 kg is \(\left(\frac{1}{90}\boldcdot 100\right) \%\), so 36 kg is \(\left(\frac{36}{90}\boldcdot 100\right) \%\) or 40% of 90. We can confirm this on a double number
In general, to find what percentage a number \(C\) is of another number \(B\) is to calculate \(\frac{C}{B}\) of 100%. We can find that by multiplying: \(\displaystyle \frac{C}{B}\boldcdot 100 \)
Suppose a school club has raised \$88 for a project but needs a total of \$160. What percentage of its goal has the club raised?
To find what percentage \$88 is of \$160, we find \(\frac {88}{160}\) of 100% or \(\frac {88}{160} \boldcdot 100\), which equals \( \frac {11}{20} \boldcdot 100\) or 55. The club has raised 55% of
its goal.
• percent
The word percent means “for each 100.” The symbol for percent is %.
For example, a quarter is worth 25 cents, and a dollar is worth 100 cents. We can say that a quarter is worth 25% of a dollar.
• percentage
A percentage is a rate per 100.
For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%. | {"url":"https://im.kendallhunt.com/MS_ACC/students/1/2/24/index.html","timestamp":"2024-11-12T09:06:12Z","content_type":"text/html","content_length":"54152","record_id":"<urn:uuid:b70ffd3e-9110-44e5-ad72-94e9d557693f>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00555.warc.gz"} |
Student Perspectives: Machine Learning Models for Probability Distributions
A post by Tennessee Hickling, PhD student on the Compass programme.
Probabilistic modelling provides a consistent way to deal with uncertainty in data. The central tool in this methodology is the probability distribution, which describes the randomness of
observations. To create effective models of reality, we need to be able to specify probability distributions that are flexible enough to capture real phenomena whilst remaining feasible to estimate.
In the past decade machine learning (ML) has developed many new and exciting ways to represent and learn potentially complex probability distributions.
ML has provided significant advances in modelling of high dimensional and highly structured data such as images or text. Many of these modern approaches are applied as “generative models”. The goal
of such approaches is to sample new synthetic observations from an approximate distribution which closely matches the target distribution. For example, we may use many images of cats to learn an
approximate distribution, from which we can sample new images of cats that appear realistic. Usually, a “generative model” indicates the requirement to sample from the model, but not necessarily
assign probabilities to observed data. In this case, the model captures uncertainty by imitating the structure and randomness of the data.
Many of these modern methods work by transforming simple randomness (such as a Normal distribution) into the target complex randomness. In my own research, I work on a known limitation of such
approaches to replicate a particular aspect of randomness – the tails of probability distributions [1, 11]. In this post, I wanted to take a step back and provide an overview of and connections
between two ML methods that can be used to model probability distributions – Normalising Flows (NFs) and Variational Autoencoders (VAEs).
Figure 1: Basic illustration of ML learning of a distribution. We optimise the machine learning model to produce a distribution close to our target. This is often conceptualised in the generative
direction, such that our ML model moves samples from the simple distribution to more closely match the target observations.
Some Background
Consider real valued vectors $z \in \mathbb{R}^{d_z}$ and $x \in \mathbb{R}^{d_x}$. In this post I mirror notation used in [2], where $p(x)$ refers to the density and distribution of $x$ and $x \sim
p(x)$ indicates samples according to that distribution. The generic set up I am considering is that of density estimation – trying to model the distribution $p(x)$ of some observed data $\{x_i\}_{i=
1}^{N}$. I use a semicolon to denote parameters, so $p(x; \beta)$ is a distribution over $x$ with parameters $\beta$. I also make use of different letters to distinguish different distributions, for
example using $q(x)$ to denote an approximation to $p(x)$. The notation $\mathbb{E}_p[f(x)]$ refers to the standard expectation of $f(x)$ over the distribution $p$.
The discussed methods introduce some simple source of randomness arising from a known, simple latent distribution $p(z)$. This is also referred to in some literature as the prior, though the usage is
not straightforwardly relatable to traditional Bayesian concepts. The goal is then to fit an approximate $q(x|z; \theta)$, that is a conditional distribution, such that $$q(x; \theta) = \int q(x|z; \
theta)p(z)dz \approx p(x),$$in words, the marginal density over $x$ implied by the conditional density, is close to our target distribution $p(x)$. In general, we can’t compute $q(x)$, as we can’t
solve the above integral for very flexible $q(x | z; \theta)$.
Variational Inference
We commonly make use of the Kullback-Leibler (KL) divergence, which can be interpreted as measuring the difference between two probability distributions. It is a useful practical tool, since we can
compute and optimise the quantity in a wide variety of situations. Techniques which optimise a probability distribution using such divergences are known as variational methods. There are other
choices of divergence, but KL is the most standard. Important properties of KL are that the quantity $KL(p|| q)$ is non-negative and non-symmetric i.e. $KL(p|| q) \neq KL(q || p)$.
Given this, we can see that a natural objective is to minimise the difference between distributions, as measured by the KL, $$KL(p(x) || q(x; \theta)) = \int p(x) \log \frac{p(x)}{q(x; \theta)}
dx.$$Advances in this area have mostly developed new ways to make this optimisation tractable for flexible $q(x | z; \theta)$.
Normalising Flow
A normalising flow is a parameterised transformation of a random variable. The key simplifying assumption is that the forward generation is deterministic. That is, for $d_x = d_z$, that $$
x = T(z; \theta),$$for some transformation function $T$. We additionally require that $T$ is a differentiable bijection. Given these requirements, we can express the approximate density of $x$
exactly as $$q_x(x; \theta) = p_z(T^{-1}(x; \theta))\big|\text{det } J_{T^{-1}}(x; \theta)\big|.$$Here, $\text{det }J_{T^{-1}}$ is the determinant of the Jacobian of the inverse transformation.
Research on NFs has developed numerous ways to make the computation of the Jacobian term tractable. The standard approach is to use neural networks to produce $\theta$ (the parameters of the
transformation), with numerous ways of configuring the model to capture dependency between dimensions. Additionally, we often stack many layers to provide more flexibility. See [10] and the review
[2] for more details on how this is achieved.
As we have access to an analytic approximate density, we can minimise the negative log-likelihood of our model, $$\mathcal{J}(\theta) = -\sum_{i=1}^{N} \log q(x_i; \theta),$$which is the Monte-Carlo
approximation of the KL loss (up to an additive constant). This is straightforward to optimise using stochastic gradient descent [9] and automatic differentiation.
Figure 2: Schematic of NF model. The ML model produces the parameters of our transformation, which are identical in the forward and backwards directions. We choose the transformation such that we can
express an analytic density function for our approximate distribution.
Variational Autoencoder
In the Variational Autoencoder (VAE) [3] the conditional distribution $q(x| z; \theta)$ is known as the decoder. VAEs consider the marginal in terms of the posterior, that is $$q(x; \theta) = \frac{q
(x | z; \theta)p(z)}{q(z | x; \theta)}.$$The posterior $q(z | x; \theta)$ is itself not generally tractable. VAEs proceed by introducing an encoder, which approximates $q(z | x; \theta)$. This is
itself simply a conditional distribution $e(z | x; \psi)$. We use this approximation to express the log marginal over $x$ as below.
\log q(x; \theta) &= \mathbb{E}_{e}\bigg[\log q(x; \theta)\frac{e(z | x; \psi)}{e(z | x; \psi)}\bigg] \\
&= \mathbb{E}_{e}\bigg[\log\frac{q(x | z; \theta)p(z)}{q(z | x; \theta)}\frac{e(z | x; \psi)}{e(z | x; \psi)}\bigg] \\
&= \mathbb{E}_{e}\bigg[\log\frac{q(x | z; \theta)p(z)}{e(z | x; \psi)}\bigg] + KL(e(z | x; \psi) || q(z | x; \theta)) \\
&= \mathcal{J}_{\theta,\psi} + KL(e(z | x; \psi) || q(z | x; \theta))
The additional approximation gives a more complex expression and does not provide an analytical approximate density. However, as $KL(e(z | x; \psi) || q(z | x; \theta))$ is positive, $\mathcal{J}_{\
theta,\psi}$ forms a lower bound on $\log q(x; \theta)$. This expression is commonly referred to as the Evidence Lower Bound (or ELBO). The second term, the divergence between our encoder and the
implied posterior will be minimised as we increase $\mathcal{J}_{\theta,\psi}$ in $\psi$. As we increase $\mathcal{J}_{\theta,\psi}$, we will either be increasing $\log q(x; \theta)$ or reducing $KL
(e(z | x; \psi) || q(z | x; \theta))$.
The goal is to increase $\log q(x; \theta)$, which we hope occurs as we optimise over both parameters. This ambiguity in optimisation results in well known issues, such as posterior collapse [12] and
can result in some counter-intuitive behaviour [4]. Despite this, VAEs remain a powerful and popular approach. An important benefit is that we no longer require $d_x = d_z$, which means we can map
high dimensional $x$ to low dimensional $z$ to perform dimensionality reduction.
Figure 3: Schematic of VAE model. We now have a stochastic forward transformation. To optimise this we introduce a decoder model which approximates the posterior implied by the forward
transformation. We now have a more flexible transformation, but two models to train and no analytic approximate density.
Surjective Flows
We can now identify a connection between NFs and VAEs. Recent work has reinterpreted NFs within the VAE framework, permitting a broader class of transitions whilst retaining analytic tractability of
NFs [5]. Considering our decoding conditional as $$q(x|z; \theta) = \delta(x – T(z; \theta)),$$ we have the posterior exactly as
$$q(z|x; \theta) = \delta(z – T^{-1}(x; \theta)),$$
where $\delta$ is the dirac delta function.
This provides a view of a NF as a special case of a VAE, where we don’t need to approximate the posterior. Considering our VAE approximation, $$
\log q(x; \theta) = \mathbb{E}_e\big[\log p(z) + \log\frac{q(x | z; \theta)}{e(z | x; \psi)}\big] + KL(e(z | x; \psi) || q(z | x; \theta)),
$$and taking $e(z | x; \psi) = q(z | x; \theta)$, then the final KL term is 0 by definition. In that case, we recover our analytic density for NFs (see [5] for details).
Note that accessing an analytic density depends on having $e(z | x; \psi) = q(z | x; \theta)$ and computing $$\mathbb{E}_e\big[\log p(z) + \log\frac{q(x | z; \theta)}{e(z | x; \psi)}\big].$$These
requirements are actually weaker than those we apply in the case of standard NFs. Consider a deterministic transformation $T^{-1}(x)$ which is surjective, crucially we can have many $x$ which map to
the same $z$, so no longer have an analytic inverse. However we can still choose a $q(x | z)$ which is stochastic inverse of $T^{-1}$. For example, consider an absolute value surjection, $q(z | x) =
\delta(z – |x|)$, to invert this transformation we can choose $q(x | z) = \frac{1}{2}\delta(x – z) + \frac{1}{2}\delta(x + z)$, which we can sample from straight-forwardly. This transformation
enforces symmetry across the origin in the approximate distribution, a potentially useful inductive bias. In this example, and many others, we can also compute the density exactly. This has led to a
number of interesting extensions to NFs, such as “funnel flows” which have $d_z < d_x$ but retain an analytic approximate density [6]. As we retain an analytic approximate density, we can optimise
them in the same way as NFs.
Figure 4: Schematic of a surjective transformation. We have a stochastic forward transformation, but the inverse is deterministic. This restricts what transformations we can have, but retains an
analytic approximate density.
I have presented an overview of two widely-used methods for modelling probability distributions with machine learning techniques. I’ve also highlighted an interesting connection between these
methods, an area of research that has led to the development of interesting new models. It’s worth noting that other important classes of ML models such as Generative Adversarial Networks and
Diffusion models can also be interpreted as approximate probability distributions. There are many superficial connections between those methods and the ones discussed here. Exploring these
theoretical similarities presents an compelling direction of research to sharpen understanding of such models’ relative advantages. Another promising direction is the synthesis of these methods,
where researchers aim to harness the strengths of each approach [7,8]. This not only enhances the existing knowledge base but also offers opportunities for innovative applications in the field.
[1] Jaini, Priyank, et al. “Tails of Lipschitz triangular flows.” International Conference on Machine Learning. PMLR, 2020
[2] Papamakarios, G., et al. “Normalizing flows for probabilistic modeling and inference.” Journal of Machine Learning Research, 2021
[3] Kingma, Diederik P., and Max Welling. “Auto-encoding variational bayes.” arXiv:1312.6114, 2013
[4] Rainforth, Tom, et al. “Tighter variational bounds are not necessarily better.” International Conference on Machine Learning. PMLR, 2018
[5] Nielsen, Didrik, et al. “Survae flows: Surjections to bridge the gap between vaes and flows.” Advances in Neural Information Processing, 2020
[6] Samuel Klein, et al. “Funnels: Exact maximum likelihood with dimensionality reduction.” arXiv:2112.08069, 2021
[7] Kingma, Durk P., et al. “Improved variational inference with inverse autoregressive flow.” Advances in neural information processing systems , 2016
[8] Zhang, Qinsheng, and Yongxin Chen. “Diffusion normalizing flow.” Advances in Neural Information Processing Systems 34, 2021
[9] Ettore Fincato “An Introduction to Stochastic Gradient Methods” https://compass.blogs.bristol.ac.uk/2023/03/14/stochastic-gradient-methods/
[10] Daniel Ward “An introduction to normalising flows” https://compass.blogs.bristol.ac.uk/2022/07/13/2608/
[11] Tennessee Hickling and Dennis Prangle, “Flexible Tails for Normalising Flows, with Application to the Modelling of Financial Return Data”, 8th MIDAS Workshop, ECML PKDD, 2023
[12] Lucas, James, et al. “Don’t blame the elbo! a linear vae perspective on posterior collapse.” Advances in Neural Information Processing Systems 32, 2019 | {"url":"https://compass.blogs.bristol.ac.uk/2023/10/20/machine-learning-models-for-probability-distributions/","timestamp":"2024-11-05T11:01:40Z","content_type":"text/html","content_length":"64075","record_id":"<urn:uuid:861cee0c-e659-4a58-9ca2-87e153a5943c>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00526.warc.gz"} |
In a one-way ANOVA, what is the minimum difference between means
(for each group/level/condition/sample) required to...
In a one-way ANOVA, what is the minimum difference between means (for each group/level/condition/sample) required to...
In a one-way ANOVA, what is the minimum difference between means (for each group/level/condition/sample) required to conclude that there is a significant difference between a pair of groups/levels/
In a one-way ANOVA, directly we can't determine thje minimum difference between means (for each group / level / condition / sample) but we can do it by comparing respective p - values. If we want to
know the minimum differences, we need to estimate comfidence intervals. Confidence Intervals gives us better idea on minimum differences.
In comparing the significant effect, we need to compare p value with 0.05 (at 95% confidence level). For a hypothesis, of p value is greater than 0.05 then the groups are not significant. Similarly,
we can do the comparison among various levels too. | {"url":"https://justaaa.com/statistics-and-probability/31667-in-a-one-way-anova-what-is-the-minimum-difference","timestamp":"2024-11-04T16:42:06Z","content_type":"text/html","content_length":"40940","record_id":"<urn:uuid:7a851f84-75b8-4b7d-aca4-d3701ba45d18>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00616.warc.gz"} |
Identify Angle Relationships Worksheets [PDF] (7.G.B.5): 7th Grade Math
Teaching Identifying the relationship between angles
• Vertical Angles: The angles that are opposite to each other when two lines cross each other.
• Transversal: When two straight lines are crossed by another line, the angles formed in matching corners are known as transversal.
Why Should you use Identifying the relationship between angles worksheet?
• These worksheets will help your students to identify the common and differentiating points between angles.
• Identifying the relationship between angles worksheet will help your students to learn about different angles and their unique properties.
Download Solved Equation With Identifying the relationship between angles Worksheet PDF
You can download and print these super fun solved equations with Identifying the relationship between angles worksheets from here for your students. You can also try our Identify Angle Relationships
Practice Problems as well for a better understanding of the concepts. | {"url":"https://www.bytelearn.com/math-grade-7/worksheet/identifying-the-relationship-between-angles","timestamp":"2024-11-06T17:53:31Z","content_type":"text/html","content_length":"119651","record_id":"<urn:uuid:8df4200f-9165-43ed-a734-9f696f68fd33>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00762.warc.gz"} |
Consort Flow Diagram Template
Consort Flow Diagram Template - Consort 2010 checklist (word) full bibliographic reference. Randomized (n= ) analysed (n= ) excluded from analysis (give reasons) (n= ) allocated to intervention (n=
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Consort flow chart.
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= ) excluded from analysis (give reasons) (n= ) allocated to intervention (n= ). Web sample template for the consort diagram showing the flow.
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Proving the Law of Sines - Complete, Concrete, Concise
Proving the Law of Sines
A tutorial on the
Law of Sines
can be found
Assumes an understanding of the trigonometric function
. A basic tutorial can be found
and a tutorial using the Unit Circle can be found
The Law of Sines says that “given any triangle (not just a right angle triangle): if you divide the sine of any angle, by the length of the side opposite that angle, the result is the same regardless
of which angle you choose”.
The actual value (the result of the calculation) is equal to the diameter of the smallest circle you can draw around the triangle that has all three points of the triangle touching the edge of the
circle (a circumscribed circle).
We could state the Law of Sines more formally as: for any triangle, the ratio of the length of a side to the sine of the angle opposite that side is the same for all three sides and is equal to the
diameter of the circle which circumscribes the triangle.
To prove the Law of Sines, we need to consider 3 cases:
1. acute triangles (triangles where all the angles are less than 90°)
2. obtuse triangles (triangles which have an angle greater than 90°)
3. right angle triangles (which have a 90° angle)
Acute Triangles
With an acute triangle, we draw a line perpendicular from one corner (vertex) to the side opposite the corner:
This splits the triangle into two right angle triangles because the perpendicular line forms two 90° angles (one in each new triangle).
Considering the triangle on the left (the pink one), and using the fundamental definition of sine we write:
sin(b) = —
Considering the triangle on the right (the green one), and using the fundamental definition of sine we write:
sin(a) = —
Since x is the unknown term, we rearrange the expressions in terms of x:
x = Asin(b)
x = Bsin(a)
We also write the following true (and seemingly silly) expression:
x = x
We can replace each x with something equivalent to x:
x = x → Asin(b) = x → Asin(b) = Bsin(a)
We can rearrange these by cross multiplying to get:
A B
—————— = ——————
sin(a) sin(b)
This looks a lot like the Law of Sines, but it is not complete, we still need to show that the ratio C / sin(c) is the same as those two.
To do that, we again divide the original triangle into two right angle triangles by drawing a perpendicular line from a different corner to the side opposite that corner:
Using exactly the same steps as above, we get:
A C
—————— = ——————
sin(a) sin(c)
If you drew your line from corner a, then you will get B / sin(b) instead of A / sin(a).
We now have two sets of equivalencies, but we know that A / sin(a) must be the same in both equivalencies, therefore we can combine both equivalencies and write:
A B C
—————— = —————— = ——————
sin(a) sin(b) sin(c)
Obtuse Triangles
As with the acute triangle, we divide the triangle into two right angle triangles by drawing a perpendicular line from one corner to the side opposite the corner.
Unlike the acute triangle, there is only one such division you can make. In the case of the image above, it is not possible to draw a perpendicular line from either corner a or corner b.
We solve the problem of the two right angle triangles exactly as we did above and get the same result:
A B
—————— = ——————
sin(a) sin(b)
Because it is not possible to draw another perpendicular line from another corner, we have to use a different technique.
We draw a perpendicular line from one of the other corners to where the opposite side would be if it had continued:
This gives us a right angle triangle for which we can write sine, from its fundamental definition, as:
sin(a) = —
We also have another right angle triangle containing the angle c', a hypotenuse A and an opposite side x, for which we can write sine as:
sin(c') = —
The only problem is that c' is not c. However, sin(c') = sin(c), so we can substitute sin(c) for sin(c').
This looks like one of those “I pulled this out of thin air” statements that we often encounter in mathematics. It’s not really.
Looking at the image, it should be obvious that:
c' + c = 180°
because line segment B and its dashed extension are a straight line – in other words, 180°
Two angles which add up to 180° are called supplementary angles.
If you look at the trigonometric tutorial using the Unit Circle, you will see that angles are measured in a counterclockwise direction from the x-axis. When calculating trigonometric functions for
angles greater than 90°, we notice that we can draw an equivalent right angle triangle, but it is flipped (i.e. oriented differently from the “standard” right angle triangle).
There are many supplementary identities, some of which are (for angles less than 90°):
sin(α) = sin(180° - α)
cos(α) = -cos(180° - α)
tan(α) = -tan(180° - α)
or, if you are using radians:
sin(α) = sin(π - α)
cos(α) = -cos(π - α)
tan(α) = -tan(π - α)
An identity is simply two mathematical expressions that are the same or equivalent.
Because of the property of supplementary angles we can replace sin(c') with sin(c), thus giving us:
sin(c) = —
As before, we rearrange the expressions in terms of x and get:
x = Asin(c)
x = Csin(a)
Just as we did with the acute triangle bove, we can rearrange and equate the two expressions to get:
A C
—————— = ——————
sin(a) sin(c)
Just as for the acute triangle above we have two sets of equivalencies, but we know that A / sin(a) must be the same in both equivalencies, therefore we can combine both equivalencies and write:
A B C
—————— = —————— = ——————
sin(a) sin(b) sin(c)
Right Angle Triangles
In the previous proofs, we managed to create two right angle triangles for comparison and showed that the ratios of the sine to the length of the opposite side were always the same. It is not
possible to do this with a right angle triangle, but we can use the definition of sine to solve it.
From the definition of sine we know that:
length of the side opposite the angle
sin(α) = —————————————————————————————————————
length of the hypotenuse
For angle a this is:
sin(a) = —
and for angle b this is:
sin(b) = —
As in the earlier proofs, we can rearrange the expressions (in this case, in terms of C):
C = ——————
C = ——————
As before, we write the obvious:
C = C
As before, we replace each C with an equivalent expression:
A B
—————— = ——————
sin(a) sin(b)
This is the same as we got before.
We know the definition of sine is opposite / hypotenuse, so we can calculate the sine of the right angle (90°):
opposite C
sin(c) = —————————— = —
hypotenuse C
This is perfectly valid, the side opposite the angle is the hypotenuse and the hypotenuse is still the hypotenuse. You can double check that sin(90°) = 1.
Rearranging to express it in terms of C we get:
C = ——————
Just as for the acute and obtuse triangle, we now have 3 expressions that are equivalent to C (for the previous triangles, it was x – the letter doesn’t matter, only the fact they are equal matters):
Since all the relations are equivalent, we write the down together and get the Law of Sines:
A B C
—————— = —————— = ——————
sin(a) sin(b) sin(c) | {"url":"https://complete-concrete-concise.com/mathematics/proving-the-law-of-sines/","timestamp":"2024-11-13T08:13:42Z","content_type":"text/html","content_length":"61262","record_id":"<urn:uuid:fc7f9b56-be67-49af-a348-0b14ff101aeb>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00833.warc.gz"} |
9c: Compound Interest
Save early and save often.
You work hard for your money; shouldn’t your money work for you?
Principle #3: People usually decide at the margin (give a little, get a little).
Use a compound interest calculator and marginal analysis to investigate the impact of saving early and often.
Why you want to learn this This is one you really don’t want to miss. You are about to discover an incredibly powerful financial tool --- compound interest. People who hear about this for the first
time in their forties and beyond often ask the question, “Why didn’t someone tell me this earlier?” Well, we’re telling you this now!
There is a way to make your money work for you but the catch is that you have to earn it and then save it. Once you do that, the process takes care of itself. Assume that, by some miracle, you have
found some fool who promises to pay you 100% interest per day for every cent you have in a savings account. “Well,” you say to yourself, “I can save one penny.” So you do. And you walk away. The next
day, you check your account balance and, sure enough, you have two cents. Next day, four cents…still not very exciting. You have to go away for a few weeks and you don’t bother checking your
balance….one cents, two cents, four cents…nobody is getting rich here. Except you are! After about a month, you return and you figure you should check it out. You look at the balance and it blows
your mind. You have $10,737,418.24…almost $11 million from ONE PENNY!!!!! There must be some mistake! But there is no mistake and no one took your money to Vegas. In one month, compound interest has
increased your money from one penny to close to $11 million. No tricks, no sleight of hand, no gambling, no risk…just $11 million.
Here's the catch. The example assumes that you are going to earn 100% interest daily and that is not going to happen. But the point is the same; compound interest will turn your savings into a much
larger sum. The even better news is that you have control over how much that larger sum is by making marginal decisions about how much you will save and for how long. Take a look at the worksheet
Let’s look a little more closely into this compound interest thing; be prepared to be amazed. Retrieve the spreadsheet, “A Tale of Two Savers.” Assume two savers, Kris and Jordan, are committed to
saving enough money to live a comfortable life after retirement. Kris begins saving at age 22 and saves $2500 annually for 12 years at an annual interest rate of 7.5%. After age 33, he doesn’t save
another penny, but he also doesn’t touch his savings so the money continues to accumulate interest until he is ready to retire at age 65. Kris saved for 12 years, deducted $30,000 from his income and
ended up with over a half a million dollars.
Jordan started saving a bit later…age 34. Same amount saved per year, same interest rate, but she saved for 32 years. Surely she will end up with more money than Kris; but no, that is not the case.
Her account balance at age 65 is $327,000. Kris has 50% more than Jordan! How did that happen since Kris only saved $30,000 of his income compared to Jordan’s $80,000.
The answer is clear; he began saving early and then let his money work for him. Play around with the spreadsheet, changing the amount saved in cell H4 and the interest rate in cell H5. You can even
change the number of years that Kyle saves if you like. It should be clear that the interest rate has a major impact on the final numbers. We will discuss more about the interest rate in later
Compound interest is powerful. It works because the interest you earn each day, month, or year earns interest itself, so your money is making money for you. Variables that affect how much money your
savings earn are the amount of money that you save, the frequency of your saving, the interest rate your money earns, and the amount of time your money sits and earns you money. If you wish to take
advantage of the power of compound interest, follow the rules below.
Search for the best interest rate, start saving early, save as much as you can, and save often.
Hint: if you have your saving automatically deposited in an interest bearing account, you will never notice it is missing.
Inflation will erode the purchasing power of your savings. There will be more about this in a later summary.
If your savings are taxable, you are likely to move into a higher tax bracket.
The interest rates used in this worksheet are fairly high over a long period of time. They are not impossible but require a skilled financial analyst to attain them.
1. At age 21, Rebecca begins saving $200 per month at 8% compound interest for 10 years and never saves again. At age 31, Lee begins saving $200 per month at 8% compound interest and saves the same
amount monthly until age 65. Who will end up with the most money at age 65?
c. It depends on the rate of interest.
d. It depends on the amount of monthly saving.
2. The reason that compound interest is so powerful is because:
a. Savers know what future economic conditions will be.
b. Savers determine interest rates.
c. Savers are smarter than borrowers.
d. Savers earn interest on their interest.
Compound interest worksheet
Assume that you have just finished high school, you are 18 years old, you have a baby, and you and your spouse both have $15/hour jobs. You are making it, but just making it and some wise guy
suggests that you should be saving part of your very skimpy income. You have done your best to make a reasonable budget and you just don’t see how you can do it. You finally do some marginal benefit/
cost analyses and find $100 to save each month. Fewer trips to the hairdresser, more meals at home instead of fast food, more babysitting by Grandma and you have your $100 per month.
Go to the Compound Interest Calculator on EconEdLink.
Scroll down to the Compound Interest Calculator
• We are going to fib a little bit and tell the calculator that you are 54 years old because we want to see how much you will have in 10 years and the calculator only lets you save until you are
• Your estimated annual interest rate is 8% (You have a trusted financial advisor who has been able to average 8% over the ten years.)
• Your initial deposit (investment) is $100.
• You monthly saving is $100.
• Click on the Calculate button.
• Scroll down to watch your saving grow.
1. How much do you have after 10 years of saving 100 per month (total earnings)?
2. How much of that amount did you save (the principal)?
3. How much interest did you earn? (Total earnings minus Principle)
4. What is the ratio of interest to principle? It’s ten years later and you have accumulated #1 above. You are now able to save $1000 per month. (Life has been good to you and you both have worked
hard to make a good life for yourselves and your (now) two kids.) Take the amount that you have from #1 above and make that your initial deposit. Your annual saving now is $999 per month. Assume
that you save that amount to age 63.
• You are 28 years old
• Your estimated annual interest rate has fallen to 6%.
• Your initial deposit (investment) is your total earnings from #1 above.
• Your monthly savings are $999.
1. What are your Total Earnings at age 63?
2. How much did you save from age28 (Principle)?
3. How much interest did you earn through the year (Total earnings minus Principle)?
4. What is the ratio of interest to principle? Assume that somehow you had been able to start saving $999 per month at age 18 at 6% and continued to save that amount until age 63.
5. What are your Total Earnings at age 63?
6. How much did you save from age 28 (Principle)?
7. How much interest did you earn through the year (Total earnings minus Principle?
8. What is the ratio of interest to principle?
Use this exercise to write an essay explaining the phrase, “Save early and save often.
How does this exercise exemplify marginal analysis? | {"url":"https://www.caset.org/post/compound-interest","timestamp":"2024-11-10T14:03:48Z","content_type":"text/html","content_length":"1050474","record_id":"<urn:uuid:81d732cd-33bf-495b-9829-03dedf2a18a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00837.warc.gz"} |
How to Write Li in Roman Numerals: A Quick Guide
In Roman numerals, “L” represents the value 50, and “I” represents the value 1. To write the number 51 in Roman numerals, we need to combine these two symbols.
To write 51 in Roman numerals, we can use the subtractive principle, where a smaller numeral is placed before a larger numeral to subtract its value. So, we can write 51 as “LI,” which means 50 + 1 =
Here’s a quick guide to writing 51 in Roman numerals:
• Start with the largest Roman numeral that is less than or equal to the number you want to write. In this case, the largest Roman numeral less than 51 is “L,” which represents 50.
• Write “L” first, followed by “I” to represent the remaining value of 1. So, we get “LI,” which represents 50 + 1 = 51.
In conclusion, the Roman numeral for the number 51 is written with the symbols “LI,” which translate to “50 plus 1.” The Roman numeral system is an ancient numerical system that is still used today
for a variety of reasons, including the numbering of book chapters, the representation of numbers on clock faces, and the indication of the years on structures.
see also
A Step-by-Step Guide to Writing MCMXXXVI in Words
How to Write 2005 in Roman Numerals: A Step-by-Step Guide
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Designing neural networks: zero to micrograd
🌱 Writing a minimalist neural network from scratch is a perfect way to get to know your way around the essential components. 🌱
A while ago, I discovered Andrej Karpathy’s tiny neural net - micrograd - and, as a prelude to implementing it in Rust, I drafted my own Python version. With micrograd being intentionally minimalist,
my naive neural net omits anything not absolutely required. (For example, no matrices^1 or attention layers^2.)
micrograd’s minimalism belies the depth of the representations it contains. We can learn more about those by exploring key design decisions baked into micrograd: What’s the rationale behind
micrograd’s specific network topology? What constraints motivate its learning algorithm? And which of those are pragmatic vs essential for theoretical foundations to hold?
This summary signposts the path from early network models of the brain through to micrograd’s core design choices, some of which apply to artificial neural networks more broadly. I’ve tried to make
it accessible to non-experts.
• why micrograd implements a multilayer perceptron model
• how efficiency considerations influence the learning feedback loop
• why backpropagation and gradient descent are pragmatic design choices
In the spirit of micrograd, this write up is nominally minimalist and leaves a lot out^3.
Onward. Let’s start by establishing a foundation for using physiological neural nets as a basis for designing artificial neural networks.
Why neural nets?
Humans naturally accumulate a wide variety of knowledge and develop a plethora of skills. In contrast, traditional computing requires explicit instructions to perform pre-designated tasks. It can’t
self-improve or develop general abilities. There’s a chasm between traditional computing and human learning, and we’d like to cross it.
A machine that has the ability to even contemplate answering a random question or performing a random task might seem like it’s thinking. This probably rings especially true when the path from input
to the expected response doesn’t have an algorithmic solution. Where does one begin, though, on the design of a machine intended to execute general tasks?
The machinery for human thought is concentrated in the brain, so it’s not too much of a stretch to consider exploring designs based on the brain. So, let’s begin with simplified models of neuronal
Cognition / connection
In the early 1940s, researchers working on mathematical models of cognition developed connectionist networks, an early form of artificial neural network that merged statistics with
physiologically-inspired structure and function. Eventually, feedback was added to iteratively improve performance, emulating learning: the machine’s abilities improved without human intervention.
A machine with the capacity to self-improve was a huge step forward. A significant break from traditional computing was the key innovation: the code didn’t include explicit instructions for how to
process inputs (i.e., how to perform any particular task). Instead, the artificial neural network’s code nudged it to develop skills.
Indeed, in its initial state, an artificial neural network (ANN) is not very capable. To develop skills, it responds to feedback about its performance. For each training epoch, the ANN accepts a
challenge, produces an output, listens for feedback on the results, then updates itself to try to do better. Each of the challenges comes from training data, and the quality of that dataset is as
essential as you’d expect it to be. We’re going to set that aside as an externality, though, so we can focus exclusively on model design.
Let’s take a closer look at how this fundamentally different approach to computation might be implemented, starting with cues taken from network models of the brain.
IRL neurons
The thinking machinery of the brain is impossibly complicated, but we’re going to focus exclusively on the networks formed by neurons. What can we take from those to build our own, in silico?
To oversimplify drastically: Neurons in the brain receive and modulate an input signal, then decide whether to propagate a modulated response. A neuron’s axons and synapses deliver outgoing signal to
one or more downstream neurons. Generally, the idea is that each neuronal network includes a feedback circuit that influences collective network behavior. That network-level adaptive behavior is key
to thinking and learning.
Figure 1.
Detailed picture of a neuron: https://commons.wikimedia.org/wiki/File:Blausen_0657_MultipolarNeuron.png (BruceBlaus, CC BY 3.0 , via Wikimedia Commons).
Modeling a neural network
Based on a simplified model of the brain, an ANN is represented as a network. A network has individual entities (nodes) connected by channels (vertices, or edges); in a network model of the brain,
the individual entities are neurons and the connection channels are synapses.
The perceptron model paved the way for implementing neuron behaviors, and the development of multilayer perceptron (MLP) networks built on that to add networking capabilities.
Figure 2.
Basic MLP schematic, from [Scikit Learn](https://scikit-learn.org/1.5/modules/neural_networks_supervised.html#id3).
The multilayer perceptron network mimics:
1. neurons’ ability to perceive a stimulus and respond
2. neuron activation to transmit an outgoing signal
3. multiplexing neuronal activity across neuron layers
MLP topology is quite specific: nodes in a layer are not connected, and all nodes in a layer connect to each node in the next layer. Functionally, each neuron in an MLP independently perceives an
input and decides whether to propagate a corresponding output to the subsequent layer in the network.
Design choice 1: Use multilayer perceptron networks.
Modeling learning
To facilitate learning, the MLP model gets a feedback loop so neurons can update. The individual neurons thus participate in coordinated collective behavior to improve network performance.
That coordinated network behavior involves an iterative process: measure forward-pass network performance, deliver feedback to neurons, update individual neurons, repeat. That’s the 10,000-foot view,
at least. We’ll look into the specifics next, keeping in mind that neurons manage two signal streams:
1. information originating from the initial input to the network, received via prior-layer neurons: this is the forward pass signal
2. network performance information, received from the network output: this is the feedback signal
The learning algorithm
We can roughly model the ANN learning process as a feedback circuit.
Figure 3.
Feedback circuit schematic from Wikipedia, By Me (Intgr) - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=2836622.
In the schematic, $A$ represents the transfer function of the forward pass through the ANN, and $B$ represents the feedback that facilitates skill development, or learning.
Unlike a continuous-signal circuit, the ANN processes an input separately from responding to feedback; there’s no mixing. The ⊕ represents model parameter updates between training inputs.
Better learning through efficiency
We might expect the network transfer function to be pretty complicated - the MLP structure makes that hard to avoid. On the other hand, neuron transfer functions are relatively simple. That
simplicity translates to faster processing. Since a model that takes excessively long to run won’t be useful, and a model that trains slowly will receive less training (taking a performance hit), the
neuron transfer function is consequential.
According to our MLP network model, neurons’ overall transfer functions represent two distinct stages of processing. The first stage, we’ll refer to as multiplexing - that’s where the prior layer’s
outputs are combined. The second, we’ll refer to as activation.
Multiplexing in MLPs is a purely linear transformation. That linearity means the transformation is fully characterized by its scalar coefficients. We’ll call those the neuron parameters: $w$ for
weights and $b$ for biases. The multiplexed output is:
$\boxed{multiplexed = ( w * neuron\_input ) + b}$
To have generalized capabilities, our network needs to include nonlinear processing on the forward pass (see the universal approximation theorem^4 for neural networks^5).
Corollary (requirement) to DC1: Include nonlinear component(s) in the forward pass.
After multiplexing, then, we’ll apply a nonlinear activation function before broadcasting the neuron’s output to the next layer. Leaning into how signals are propagated by brain neurons via
activation thresholds and action potentials, we’ll select activation functions that mimic this behavior^6.
Design choice 2: Use nonlinear activation functions that mimic neuronal signal propagation.
Post-activation, the neuron output is:
$\boxed{neuron\_out = activation ( multiplexed )}$
A note about matrices, since they figure so prominently in machine learning and we’re not including them in micrograd. Same-layer neurons in MLP networks aren’t connected: they act independently.
Combining this independence with linear multiplexing means we can use matrices to efficiently parallelize those computations. Together with innovations in GPUs and TPUs, this makes it possible to
scale models to very large numbers of parameters (currently, hundreds of billions).
For micrograd, we won’t be parallelizing anything, but we’ll still be able to take advantage of the linearity condition. (Remember - the goal with micrograd is to build from scratch. The fewer
arithmetic operations we need to implement, the better.)
Backpropagation & gradient descent
We’ve yet to define exactly how to compute neuron updates. Let’s start by clarifying what, exactly, will be updated.
Recall that neurons process inputs in two stages: multiplexing and activation. Multiplexing is linear, activation is nonlinear. It’s somewhat circular but - once we identify it - our learning
algorithm will be easier to tune if the updates act on linear functions. So, we’ll restrict those to neuron parameters - i.e., the weights and biases.
Design choice 3: Use feedback to update weights and biases between training passes.
Let’s itemize structural components needed to support the learning process; this will give us a sense of implementation options. Generally, we’ll need:
1. a feedback path that connects network output to individual neurons
2. a function to evaluate network performance
3. a function to update neuron parameters
A hidden layer neuron will have multiple input-to-output paths passing through it. We’ll create a topological map of the neurons traversed on each forward path to set up the feedback paths.
Next, the loss function. Its purpose is to quantify the difference between actual and ideal outcomes during training, so the choice of metric depends on the model objectives. In alignment with
micrograd’s minimalist ethos, we’ll use the basic mean square error (aka Euclidean distance, or L2 norm)^7:
$\boxed{error = \sqrt{(actual - expected)^2}}$
Design choice 4: Use mean square error as the loss metric for micrograd.
Finally, the update function needs to tie that metric to individual neurons. This is the exciting part because we’re going to establish how our network learns.
It’s not actually obvious a priori that there’s a single update function that, when applied to all neurons, will improve network performance. Having a single function is important for streamlining
model scaling and for computational efficiency, so let’s see what we can come up with.
Ideally, we’d identify a functional relationship between changes to individual neurons and network output. Given that we have a topological map, evaluating (partial) derivatives of the loss with
respect to each path-neuron should give us what we need: each local gradient would represent a neuron’s influence on the loss.
The loss surface’s minimum would be located at the lowest point of a valley. To minimize the loss, individual neurons could be nudged down their respective gradients, with neurons on steeper slopes
effectively being nudged further that those on shallower slopes.
Figure 4.
Gradient descent - optimizer animations. VirtualVistas, CC BY-SA 4.0 , via Wikimedia Commons.
Let’s implement that. To get the partial derivatives, we’ll apply the chain rule backward through each path in the topological map. First, though, we need to make sure the paths are differentiable.
Expanding the neuron transfer function, we get:
$\boxed{neuron\_out = activation ( ( w * neuron\_input ) + b )}$
That’s one neuron. Same-layer neurons act independently; they’re never in the same feedback path. According to the MLP model, a neuron’s output then either becomes a next-layer neuron input or it
contributes to network output. Either way, aside from activation functions^8, forward pass processing is linear.
This is really promising. Since linear functions are always differentiable, what remains is to make sure our activation functions are differentiable, too. Potential candidates like sigmoids are both
differentiable and compress inputs to mimic the effects of activation thresholds and action potentials. Let’s stick with differentiable activations^9.
Design choice 5: Use backpropagation and gradient descent as the learning function.
Corollary to DC5: Use differentiable activation functions.
We now have a learning process for our network. We’ll iterate over our topological map, computing path-wise partial derivatives to get local gradients. For each neuron, we’ll update its parameters in
proportion to the sum^10 of its path-specific local gradients.
This specific method of computing local gradients and using those to determine network parameter updates is called stochastic gradient descent^11. It’s an intuitive approach to propagating loss
feedback so that neurons are updated proportionally to their influence.
Let’s make gradient descent a little more concrete by computing an example update for a neuron, $u$, that contributes to two paths in the network.
Figure 5.
Simple network with two hidden layers.
Apply the chain rule to get partial derivatives for each path:
u-v path partials: $\boxed{\partial out_{uv} / \partial in = \partial out / \partial v * \partial v / \partial u * \partial u / \partial in}$
u-w path partials: $\boxed{\partial out_{uw} / \partial in = \partial out / \partial w * \partial w / \partial u * \partial u / \partial in}$
To compute the update for $u$, we’ll only need the gradients between $out$ and $u$. The partials represent local steepness so are scaled by a global step size^12, $step$. Finally, summing over all
paths that include $u$, we evaluate the total update for $u$:
$\boxed{\varDelta u = step * ((\partial out / \partial v * \partial v / \partial u) + (\partial out / \partial w * \partial w / \partial u))}$
These values are computed for each neuron, and for each path to which the neuron contributes. The process is repeated for the number of training iterations we set (or until a performance target is
reached). For billions of parameters, the computational lift is significant and we can begin to see why efficiency optimizations are important.
Backpropagation using gradient descent has been remarkably successful. So successful that, despite its inherent simplicity and potential pitfalls^13, learning facilitated by this method plus the
basic MLP structure of micrograd are retained in (far more complex) state of the art models.
Aside: Model interpretability
For mere mortals, the concept of learning generally involves reasoning. Throughout, we’ve been using the term learning to represent the ANN’s performance improvements. Has the network been
reasoning, in any sense that we could appreciate? Is it capable of reasoning independently once trained? The network’s transfer function post-training may not have any clear interpretation in the
sense of reflecting a logical ’thought’ process. Could we figure out how the model thinks? Research on model interpretability aims to reveal the meaning in models’ (hidden-layer) tactics.
State of the art ANNs
Researchers working on machine learning didn’t land on MLPs with gradient descent as the secret sauce right off the bat. Here’s a detailed timeline of machine learning according to Wikipedia, but
we’ll skip straight to the present.
The announcement for the 2024 Nobel Prize in Physics includes a very readable overview of the development of ANNs, with a focus on the progression from Hopfield networks to restricted Boltzmann
machines, an important precursor to today’s mainstream machine learning models.
Many varieties of ANNs can perform high quality general purpose computation. From a theoretical perspective, the fundamental requirement for such broad capabilities is having at least one hidden
layer: the network must be deep (see the universal approximation theorem^4 for neural networks^5).
Corollary (requirement) to DC1: At least one hidden layer.
In practice, a very large number of nodes and many training iterations on large and varied datasets are required. For decades, the scale of available training data and compute power limited both
research and adoption of ANNs. Eventually, massive datasets became available for training, and realistic training times for models of sufficient size and complexity were made possible by advances in
This unblocked progress, and rapid development ensued. Today, ANNs power large language models (LLMs) such as ChatGPT, Claude, Llama, Gemini. A recently released application that uses such a model is
Google’s NotebookLM. It fine tunes its base model on data you share with it - maybe a research paper or a thesis, but it also needn’t be research-y at all. The end result is a podcast with two
‘people’ discussing your input data - occasionally with decent results, though not always. Case in point, a podcast that was fine tuned on this post. The discussion strays from developing the
rationale behind micrograd design, ultimately covering a range of AI topics of varying degrees of relevance.
Believe it or not, the fundamental elements of the ANNs driving these advanced models build on analogous components in micrograd. For a peek at implementing micrograd’s components in code and seeing
them in action - model training and eval - here’s a write up of my Python implementation.
Design choices - list summary
Extracting the specific choices that’ll set the stage for implementing micrograd:
• Use multilayer perceptron networks.
□ Corollary requirement: At least one hidden layer.
□ Corollary requirement: Include nonlinear component(s) in the forward pass.
• Use nonlinear activation functions that mimic neuronal signal propagation.
• Use feedback to update weights and biases between training passes.
• Use backpropagation and gradient descent as the learning algorithm.
□ Corollary: Use differentiable activation functions.
• Use mean squared error as the loss metric.
What’s missing from micrograd?
A minimalist implementation of an ANN works surprisingly well. It’s also missing some elements that really should be included in any non-trivial model.
Efficient processing
In reality, the sort of training that LLMs require isn’t even possible without pulling out every stop on efficiency. At the very least, we need to use matrices to leverage linearity and independence.
Incorporating PyTorch or Keras, and supporting libraries such as Einsum and Jax, is essentially necessary to train a non-miniature ANN. Taking it a step further - for production-grade models, GPU
optimization is critical.
Tokenization involves tactical groupings of fractional components of inputs. This is applied as preprocessing of the input data, and can boost performance by starting from a more semantically robust
Model evaluation
A trained model might have satisfied the loss function tolerance, but will it perform well on data not in the training dataset? Post-training performance testing is important for assessing accuracy/
correctness and also alignment. Many swear by their own custom eval processes, but you might want to start with Hugging Face’s guide, or with the cute and approachable Forest Friends eval guide.
Beyond micrograd
State of the art ANNs are significantly more complex than micrograd, and include innovative elements which improve performance and/or efficiency.
Notably, the transformer architecture is one of the great successes in the past decade of advances in machine learning. The original paper, Attention is all you need, is well worth reading.
A few other starting points for further investigation - definitely not comprehensive.
1. Matrices are essential to efficient computation in the context of large scale neural networks. They’re used to parallelize computations that are independent. Since they’re independent, they can
be isolated from the matrix environment for analysis, as done here in the context of micrograd. ↩︎
2. Attention Is All You Need. Direct link to the journal artical on arXiv. ↩︎
3. Neural networks from scratch in Python (Kinsley & Kukiela) is a more substantial (600+ pages) resource for writing a full fledged neural network from scratch. (I haven’t worked through it, but
the table of contents looks promising and there are some nice animations on the book’s website.) ↩︎
4. More accurately, activation functions plus the neuron’s bias parameter combine to mimic neuron signal propagation. ↩︎
5. Admittedly, using MSE as the loss seems overly lazy - the cross-entropy isn’t all that complicated and seems more appropriate - but the original micrograd implementation uses MSE. For insight
into why to preferentially use cross-entropy, see Why You Should Use Cross-Entropy Error Instead Of Classification Error Or Mean Squared Error For Neural Network Classifier Training | James D.
McCaffrey. ↩︎
6. A final activation function - just before the output layer - typically normalizes the output distribution such that its components can be interpreted as probabilities which sum to 1. Softmax is a
common choice. ↩︎
7. Interestingly, locally non-differentiable activation functions, such as ReLU, are frequently used anyhow, and x = 0 is simply handled explicitly. The fact remains that machine learning is closer
to the egg drop experiment than to science. ↩︎
8. Each path in the topological map contributes to the network output independently, and all paths are combined at the final stage. E.g., Softmax is often used for final stage multiplexing. If the
neuron contributes to multiple paths, its influence on network output is a sum of its path-specific contributions. ↩︎
9. We’re updating after each training forward pass, which means we’ve implemented stochastic gradient descent, aka epoch GD. When updates are applied only after all training passes have completed,
that’s referred to as gradient descent, aka batch GD. A third option is to group training passes and updated parameters batch by batch aka mini batch gradient descent. ↩︎
10. The model can be trained quickly or accurately - pick one. An analogy I find useful for step size is image pixel size: smaller is better in terms of fidelity (precision in reaching the minimum
loss) but processing time takes a hit; larger is faster to process, but the picture may be fuzzy (you can’t land on the minimum). The ideal step size balances speed and accuracy. ↩︎
11. For example, getting trapped in a local - rather than global - minimum during gradient descent. This might, intuitively, seem like a significant source of poor model behavior but it doesn’t play
out that way. Here’s one exploration of why the risk isn’t as bad as it might seem on first glance. ↩︎ | {"url":"https://monicaspisar.com/posts/micrograd/","timestamp":"2024-11-06T05:36:40Z","content_type":"text/html","content_length":"56565","record_id":"<urn:uuid:2ebdb630-549e-4199-ac61-bd059f07f747>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00870.warc.gz"} |
By Position
For each group in a column, we wish to filter rows by their position (row number) in the group.
This section is organized to cover the most common "filtering groups by position" scenarios as follows:
• First where we cover how to return the first row from each group.
• Last where we cover how to return the last row from each group.
• nth where we cover how to return the nth row from each group.
• Head where we cover how to return the topmost rows, defined by a number of rows or a proportion of rows, from each group.
• Tail where we cover how to return the bottom most rows, defined by a number of rows or a proportion of rows, from each group.
• Range where we cover how to return a range of rows, defined by a start and an end position, from each group.
• List where we cover how to return specific rows from each group.
• Random where we cover how to return some random rows from each group.
We wish to return the first row from each group where a group is defined by a given column.
In this example, we wish to return the first row of each group where the groups are defined by the column col_10 and the rows are sorted by the column col_1.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1) = 1;
Here is how this works:
• ROW_NUMBER() returns the index of the current row starting at 1 for the top most row.
• We use PARTITION BY col_10 to calculate the index per group, and we order by col_1 so the index starts at 1 for the row with the lowest value for col_1.
• We use QUALIFY clause to filter based on the output of ROW_NUMBER().
• ROW_NUMBER() = 1 is TRUE for the first row of each group.
We wish to return the last row from each group.
In this example, we wish to return the last row of each group where the groups are defined by the column col_10 and the rows are sorted by the column col_1.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1 DESC ) = 1;
Here is how this works:
• The code works similarly to the code in “First” scenario above except that we order by col_1 in a descending manner.
We wish to return the nth row from each group.
From Top
In this example, we wish to return the second row of each group where the groups are defined by the column col_10 and the rows are sorted by the column col_1.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1) = 2;
Here is how this works:
• The code works similarly to the code in “First” scenario above except we filter by second row (or whatever the row position of the row we are interested in is).
From Bottom
In this example, we wish to return the row before the last of each group where the groups are defined by the column col_10 and the rows are sorted by the column col_1.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1 DESC ) = 2;
Here is how this works:
• The code works similarly to the code in “Last” scenario above except we filter by second row (or whatever the row position of the row relative to the end we are interested in is).
We wish to return the top n rows from each group (which are often referred to as the head).
We wish to return a specific number of rows from the top of each group.
In this example, we wish to return the top two rows from each group where the groups are defined by the column col_10 and the rows are sorted by the column col_1.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1 ) <= 2;
Here is how this works:
• This works similarly to the “First” scenario above except that we keep the first two rows.
We wish to return a proportion (percent) of the total number of rows from each group taken from the top of the group.
In this example, we wish to return the top 20% of the rows of each group where the groups are defined by the column col_10 and the rows are sorted by the column col_1.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1 ) < (COUNT(1) OVER (PARTITION BY col_10) * 0.2);
Here is how this works:
• ROW_NUMBER() works similarly to the scenarios above.
• We count the rows in each group using COUNT(1) OVER (PARTITION BY col_10).
• We keep rows with an index < then 20% * row_count in each group.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1 DESC ) <= 2;
Here is how this works:
• The code works similarly to the code in “Head” scenario above except that we order on col_1 in descending manner.
We wish to return a range of rows (also known as a slice), between a given start and end row positions, from each group.
In this example, we wish to return the second through to the ninth rows from each group where the groups are defined by the column col_10 and the rows are sorted by the column col_1.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1 ) BETWEEN 2 AND 9;
Here is how this works:
• This works similarly to the “First” scenario above except that we filter based on a range using BETWEEN.
We wish to obtain specific rows, given their row numbers, from each group.
In this example, we wish to return the first, second, second last, and last rows of each group of the table df where the groups are defined by the column col_10 and the rows are sorted by the column
WITH ranked_table AS (SELECT *,
ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1 ) row_index,
ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY col_1 DESC) row_index_desc,
FROM refcon.dataset.table_1)
SELECT * EXCEPT (row_index,row_index_desc)
FROM ranked_table
WHERE row_index IN (1, 2)
OR row_index_desc IN (1, 2)
Here is how this works:
• We calculate row_index and row_index_desc similarly to the above scenarios in a CTE.
• We filter from each index separately to pick the first, second, second last, and last rows.
We wish to return a set of rows taken at random from each group.
We wish to return a specific number of rows taken at random positions from each group.
In this example, we wish to return 10 randomly selected rows from each group where the groups are defined by the column col_10.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY RAND() ) <= 10
Here is how this works:
• This works similarly to the above scenarios except we sort by RAND() instead of a column.
• RAND() generates a random number between 0 and 1 for each row in the table.
• The query might return different results each time.
We wish to return a proportion (percent) of the total number of rows from each group taken at random positions.
In this example, we wish to return 20% of the rows of each group taken at random positions where the groups are defined by the column col_10.
SELECT *
FROM refcon.dataset.table_1
QUALIFY ROW_NUMBER() OVER (PARTITION BY col_10 ORDER BY RAND() ) < COUNT(1) OVER (PARTITION BY col_10) * 0.2;
Here is how this works:
• This works similarly to the “Count” scenario except that we use the filter similarly to the Proportion scenario above. | {"url":"https://optima.io/reference/data-manipulation/sql/filtering-grouped-position","timestamp":"2024-11-10T07:48:40Z","content_type":"text/html","content_length":"181884","record_id":"<urn:uuid:a0045687-37c8-4f17-ad25-c01c6c6bac2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00584.warc.gz"} |
docs/source/modeling_faqs.rst - ceres-solver - Git at Google
.. _chapter-modeling_faqs:
.. default-domain:: cpp
.. cpp:namespace:: ceres
#. Use analytical/automatic derivatives.
This is the single most important piece of advice we can give to
you. It is tempting to take the easy way out and use numeric
differentiation. This is a bad idea. Numeric differentiation is
slow, ill-behaved, hard to get right, and results in poor
convergence behaviour.
Ceres allows the user to define templated functors which will
be automatically differentiated. For most situations this is enough
and we recommend using this facility. In some cases the derivatives
are simple enough or the performance considerations are such that
the overhead of automatic differentiation is too much. In such
cases, analytic derivatives are recommended.
The use of numerical derivatives should be a measure of last
resort, where it is simply not possible to write a templated
implementation of the cost function.
In many cases it is not possible to do analytic or automatic
differentiation of the entire cost function, but it is generally
the case that it is possible to decompose the cost function into
parts that need to be numerically differentiated and parts that can
be automatically or analytically differentiated.
To this end, Ceres has extensive support for mixing analytic,
automatic and numeric differentiation. See
#. When using Quaternions, consider using :class:`QuaternionParameterization`.
`Quaternions <https://en.wikipedia.org/wiki/Quaternion>`_ are a
four dimensional parameterization of the space of three dimensional
rotations :math:`SO(3)`. However, the :math:`SO(3)` is a three
dimensional set, and so is the tangent space of a
Quaternion. Therefore, it is sometimes (not always) beneficial to
associate a local parameterization with parameter blocks
representing a Quaternion. Assuming that the order of entries in
your parameter block is :math:`w,x,y,z`, you can use
.. NOTE::
If you are using `Eigen's Quaternion
object, whose layout is :math:`x,y,z,w`, then you should use
#. How do I solve problems with general linear & non-linear
**inequality** constraints with Ceres Solver?
Currently, Ceres Solver only supports upper and lower bounds
constraints on the parameter blocks.
A crude way of dealing with inequality constraints is have one or
more of your cost functions check if the inequalities you are
interested in are satisfied, and if not return false instead of
true. This will prevent the solver from ever stepping into an
infeasible region.
This requires that the starting point for the optimization be a
feasible point. You also risk pre-mature convergence using this
#. How do I solve problems with general linear & non-linear **equality**
constraints with Ceres Solver?
There is no built in support in ceres for solving problems with
equality constraints. Currently, Ceres Solver only supports upper
and lower bounds constraints on the parameter blocks.
The trick described above for dealing with inequality
constraints will **not** work for equality constraints.
#. How do I set one or more components of a parameter block constant?
Using :class:`SubsetParameterization`.
#. Putting `Inverse Function Theorem
<http://en.wikipedia.org/wiki/Inverse_function_theorem>`_ to use.
Every now and then we have to deal with functions which cannot be
evaluated analytically. Computing the Jacobian in such cases is
tricky. A particularly interesting case is where the inverse of the
function is easy to compute analytically. An example of such a
function is the Coordinate transformation between the `ECEF
<http://en.wikipedia.org/wiki/ECEF>`_ and the `WGS84
<http://en.wikipedia.org/wiki/World_Geodetic_System>`_ where the
conversion from WGS84 to ECEF is analytic, but the conversion
back to WGS84 uses an iterative algorithm. So how do you compute the
derivative of the ECEF to WGS84 transformation?
One obvious approach would be to numerically
differentiate the conversion function. This is not a good idea. For
one, it will be slow, but it will also be numerically quite
Turns out you can use the `Inverse Function Theorem
<http://en.wikipedia.org/wiki/Inverse_function_theorem>`_ in this
case to compute the derivatives more or less analytically.
The key result here is. If :math:`x = f^{-1}(y)`, and :math:`Df(x)`
is the invertible Jacobian of :math:`f` at :math:`x`. Then the
Jacobian :math:`Df^{-1}(y) = [Df(x)]^{-1}`, i.e., the Jacobian of
the :math:`f^{-1}` is the inverse of the Jacobian of :math:`f`.
Algorithmically this means that given :math:`y`, compute :math:`x =
f^{-1}(y)` by whatever means you can. Evaluate the Jacobian of
:math:`f` at :math:`x`. If the Jacobian matrix is invertible, then
its inverse is the Jacobian of :math:`f^{-1}(y)` at :math:`y`.
One can put this into practice with the following code fragment.
.. code-block:: c++
Eigen::Vector3d ecef; // Fill some values
// Iterative computation.
Eigen::Vector3d lla = ECEFToLLA(ecef);
// Analytic derivatives
Eigen::Matrix3d lla_to_ecef_jacobian = LLAToECEFJacobian(lla);
bool invertible;
Eigen::Matrix3d ecef_to_lla_jacobian;
lla_to_ecef_jacobian.computeInverseWithCheck(ecef_to_lla_jacobian, invertible); | {"url":"https://ceres-solver.googlesource.com/ceres-solver/+/79bbf95103672fa4b5485e055ff7692ee4a1f9da/docs/source/modeling_faqs.rst","timestamp":"2024-11-08T22:31:57Z","content_type":"text/html","content_length":"52546","record_id":"<urn:uuid:7d1acca8-2f1b-4f5c-8d82-145bd1416e98>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00389.warc.gz"} |
This fractions calculator performs all fraction operations - addition, subtraction, multiplication, division and evaluates expressions with fractions. It also shows detailed step-by-step
The result:
(2/5) : (1/6) = 12/5 = 2 2/5 = 2.4
The spelled result in words is twelve fifths (or two and two fifths).
How do we solve fractions step by step?
1. Divide: 2/5 : 1/6 = 2/5 · 6/1 = 2 · 6/5 · 1 = 12/5
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and
denominator, reciprocal of 1/6 is 6/1) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further
simplify the fraction result by canceling.
In other words - two fifths divided by one sixth is twelve fifths.
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.
Last Modified: October 9, 2024 | {"url":"https://www.hackmath.net/en/calculator/fraction?input=2%2F5+divided+by+1%2F6","timestamp":"2024-11-03T22:10:13Z","content_type":"text/html","content_length":"31428","record_id":"<urn:uuid:813c5db9-cecc-4d8a-82c5-ec5b838b02bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00256.warc.gz"} |
Unit 8
Lesson 12
Make Dot Images
Warm-up: How Many Do You See: Dots in Different Colors (10 minutes)
The purpose of this How Many Do You See is to allow students to use subitizing or grouping strategies to describe the images they see.
• Groups of 2
• “How many do you see? How do you see them?”
• Flash the image.
• 30 seconds: quiet think time
• Display the image.
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
• Repeat for each image.
Student Facing
How many do you see?
How do you see them?
Activity Synthesis
• Display last image.
• “What are the 2 parts that you see? What is the total?” (I see 2 red and 3 yellow. 2 and 3 are the parts. There are 5 altogether.)
• “We have been looking at dots in How Many Do You See throughout the year. In the next activity, you will get to make your own groups of dots.”
Activity 1: Make Your Own Dot Images (10 minutes)
The purpose of this activity is for students to highlight compositions and decompositions of numbers to 5. Students create a set of cards with dot images by coloring in dot arrangements and drawing
their own arrangements of up to 5 dots. Students will use the cards they create in the next activity.
Engagement: Internalize Self-Regulation. Provide students an opportunity to self-assess and reflect on the dot cards they made. For example, they can verify that they used at least 2 different colors
to color the dots.
Supports accessibility for: Organization, Attention
Required Materials
Materials to Gather
Materials to Copy
Required Preparation
• Create a set of cards from the blackline master for each student.
• Give each student dot cards and access to colored pencils, crayons, or markers.
• “You are going to get to make your own groups of dots. Use at least 2 different colors to color in the dots to help your partner see the different parts in the total. There are some blank cards.
On these cards, you can draw your own groups of dots and color them in.”
• 7 minutes: independent work time
Activity Synthesis
• “Pick your favorite dot image. Tell your partner why it is your favorite and what parts you see inside of the total.”
• “In our next activity, you will share your dot images in a small group. What do you need to do to be a good group member and help everyone learn?”
Activity 2: How Many Dots Do You See? (15 minutes)
The purpose of this activity is for students to identify more than one composition and decomposition of numbers to 5. Students use the dot image cards that they created in the previous activity to
play their own version of the “How Many Do You See?” routine in small groups.
MLR8 Discussion Supports. Synthesis: At the appropriate time, give groups 2–3 minutes to plan what they will say when they present to the class. “Practice what you will say when you share your dot
image with the class. Talk about what is important to say, and decide who will share each part.”
Advances: Speaking, Conversing, Representing
Required Preparation
• Each student needs the dot image cards from the previous activity.
• Groups of 4
• “You are going to use the dot cards you created in small groups, just like when we do our How Many Do You See warm-up. The first person will hold up one of their dot cards for their group members
to see. The rest of the group members will have time to think. Then they will share how many dots they see and how they see them. Take turns sharing your dot images.”
• 8 minutes: small-group work time
Activity Synthesis
• “Work together to choose one dot image to share from your group. Choose a dot image that the members of the group saw in different ways.”
• Invite each group to share their dot image.
Activity 3: Centers: Choice Time (20 minutes)
The purpose of this activity is for students to choose from activities that offer practice with number and shape concepts. Students choose from 5 centers introduced in previous units. Students can
choose to work at any stage of the centers.
• 5-frames
• Roll and Add
• Bingo
• Geoblocks
• Find the Value of Expressions
Students will continue to choose from these centers in upcoming lessons. Keep the materials from each center organized to use each day.
Required Preparation
• Gather materials from:
□ 5-frames
□ Roll and Add
□ Bingo
□ Geoblocks
□ Find the Value of Expressions
• Groups of 2
• “Today we are going to choose from centers we have already learned.”
• Display the center choices in the student book.
• “Think about what you would like to do first.”
• 30 seconds: quiet think time
• Invite students to work at the center of their choice.
• 8 minutes: center work time
• “Choose what you would like to do next.”
• 8 minutes: center work time
Student Facing
Choose a center.
Find the Value of Expressions
Activity Synthesis
• “What makes working in centers fun for you?”
Lesson Synthesis
Display dots as pictured, or choose two student-created cards:
“What is the same about these dot images? What is different about them?” (They both have the same total. They both show 5 dots total. They are different because they show different parts. One shows 4
and 1 and one shows 3 and 2.)
Cool-down: Unit 8, Section C Checkpoint (0 minutes) | {"url":"https://im.kendallhunt.com/k5/teachers/kindergarten/unit-8/lesson-12/lesson.html","timestamp":"2024-11-06T01:26:51Z","content_type":"text/html","content_length":"108971","record_id":"<urn:uuid:3f68a112-50e9-4133-bde1-d614caf3962a>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00494.warc.gz"} |
memochange-Tutorial: Break in Persistence
Literature Review
The degree of memory is an important determinant of the characteristics of a time series. For an \(I(0)\), or short-memory, process (e.g., AR(1) or ARMA(1,1)), the impact of shocks is short-lived and
dies out quickly. On the other hand, for an \(I(1)\), or difference-stationary, process such as the random walk, shocks persist infinitely. Thus, any change in a variable will have an impact on all
future realizations. For an \(I(d)\), or long-memory, process with \(0<d<1\), shocks neither die out quickly nor persist infinitely, but have a hyperbolically decaying impact. In this case, the
current value of a variable depends on past shocks, but the less so the further these shocks are past.
There are plenty of procedures to determine the memory of a series (see Robinson (1995), Shimotsu (2010), among others). However, there is also the possibility that series exhibit a structural change
in memory, often referred to as a change in persistence. Starting with Kim (2000) various procedures have been proposed to detect these changes and consistently estimate the change point. Busetti and
Taylor (2004) and Leybourne and Taylor (2004) suggest approaches for testing the null of constant \(I(0)\) behaviour of the series against the alternative that a change from either \(I(0)\) to \(I(1)
\) or \(I(1)\) to \(I(0)\) occurred. However, both approaches show serious distortions if neither the null nor the alternative is true, e.g. the series is constant \(I(1)\). In this case the
procedures by Leybourne et al. (2003) and Leybourne, Taylor, and Kim (2007) can be applied as they have the same alternative, but assume constant \(I(1)\) behaviour under the null. Again, the
procedures exhibit distortions when neither the null nor the alternative is true. To remedy this issue, Harvey, Leybourne, and Taylor (2006) suggest an approach that entails the same critical values
for constant \(I(0)\) and constant \(I(1)\) behavior. Consequently, it accommodates both, constant \(I(0)\) and constant \(I(1)\) behavior under the null.
While this earlier work focussed on the \(I(0)/I(1)\) framework, more recent approaches are able to detect changes from \(I(d_1)\) to \(I(d_2)\) where \(d_1\) and \(d_2\) are allowed to be
non-integers. Sibbertsen and Kruse (2009) extend the approach of Leybourne, Taylor, and Kim (2007) such that the testing procedure consistently detects changes from \(0 \leq d_1<1/2\) to \(1/2<d_2<3/
2\) and vice versa. Under the null the test assumes constant \(I(d)\) behavior with \(0 \leq d <3/2\). The approach suggested by Martins and Rodrigues (2014) is even able to identify changes from \
(-1/2<d_1<2\) to \(-1/2<d_2<2\) with \(d_1 \neq d_2\). Here, under the null the test assumes constant \(I(d)\) behavior with \(-1/2<d<2\).
Examples for series that potentially exhibit breaks in persistence are macroeconomic and financial time series such as inflation rates, trading volume, interest rates, volatilities and so on. For
these series it is therefore strongly recommended to investigate the possibility of a break in persistence before modeling and forecasting the series.
The memochange package contains all procedure mentioned above to identify whether a time series exhibits a break in persistence mentioned above. Additionally, several estimators are implemented which
consistently estimate the point at which the series exhibits a break in persistence and the order of integration in the two regimes. We will now show how the usage of the implemented procedures while
investigating the price of crude oil.
First, we download the monthly price series from the FRED data base.
To get a first visual impression, we plot the series.
oil_xts=xts::xts(oil[,-1],order.by = oil$DATE)
zoo::plot.zoo(oil_xts, xlab="", ylab="Price", main="Crude Oil Price: West Texas Intermediate")
From the plot we observe that the series seems to be more variable in its second part from year 2000 onwards. This is first evidence that a change in persistence has occurred. We can test this
hypothesis using the functions cusum_test (Leybourne, Taylor, and Kim (2007), Sibbertsen and Kruse (2009)) LBI_test (Busetti and Taylor (2004)), LKSN_test (Leybourne et al. (2003)), MR_test (Martins
and Rodrigues (2014)) , and ratio_test (Busetti and Taylor (2004), Leybourne and Taylor (2004), Harvey, Leybourne, and Taylor (2006)). In this vignette we use the ratio and MR test since these are
the empirically most often applied ones. The functionality of the other tests is similar. They all require a univariate numeric vector x as an input variable and yield a matrix of test statistic and
critical values as an output variable.
As a starting point the default version of the ratio test is applied.
#> 90% 95% 99% Teststatistic
#> Against change from I(0) to I(1) 3.5148 4.6096 7.5536 225.943543
#> Against change from I(1) to I(0) 3.5588 4.6144 7.5304 1.170217
#> Against change in unknown direction 4.6144 5.7948 9.0840 225.943543
This yields a matrix that gives test statistic and critical values for the null of constant \(I(0)\) against a change from \(I(0)\) to \(I(1)\) or vice versa. Furthermore, the statistics for a change
in an unknown direction are included as well. This accounts for the fact that we perform two tests facing a multiple testing problem. The results suggest that a change from \(I(0)\) to \(I(1)\) has
occurred somewhere in the series since the test statistic exceeds the critical value at the one percent level. In addition, this value is also significant when accounting for the multiple testing
problem. Consequently, the default version of the ratio test suggests a break in persistence.
We can modify this default version by choosing the arguments trend, tau, statistic, type, m, z, simu, and M (see the help page of the ratio test for details). The plot does not indicate a linear
trend so that it seems unreasonable to change the trend argument. Also, the plot suggests that the break is rather in the middle of the series than at the beginning or the end so that changing tau
seems unnecessary as well. The type of test statistic calculated can be easily changed using the statistic argument. However, simulation results indicate mean, max, and exp statistics to deliver
qualitatively similar results.
Something that is of more importance is the type of test performed. The default version considers the approach by Busetti and Taylor (2004). In case of a constant \(I(1)\) process this test often
spuriously identifies a break in persistence. Harvey, Leybourne and Taylor (2006) account for this issue by adjusting the test statistic such that its critical values are the same under constant \(I
(0)\) and constant \(I(1)\). We can calculate their test statistic by setting type="HLT". For this purpose, we need to state the number of polynomials z used in their test statistic. The default
value is 9 as suggested by Harvey, Leybourne and Taylor (2006). Choosing another value is only sensible for very large data sets (number of obs. > 10000) where the test statistic cannot be calculated
due to computational singularity. In this case decreasing z can allow the test statistic to be calculated. This invalidates the critical values so that we would have to simulate them by setting simu=
1. However, as our data set is rather small we can stick with the default of z=9.
ratio_test(x, type="HLT")
#> 90% 95% 99% Teststatistic 90%
#> Against change from I(0) to I(1) 3.5148 4.6096 7.5536 58.9078204
#> Against change from I(1) to I(0) 3.5588 4.6144 7.5304 0.3085495
#> Against change in unknown direction 4.6144 5.7948 9.0840 44.2171379
#> Teststatistic 95% Teststatistic 99%
#> Against change from I(0) to I(1) 43.4772689 25.3369507
#> Against change from I(1) to I(0) 0.2290113 0.1290305
#> Against change in unknown direction 34.1367566 20.0058559
Again the test results suggests that there is a break from \(I(0)\) to \(I(1)\). Consequently, it is not a constant \(I(1)\) process that led to a spurious rejection of the test by Busetti and Taylor
Another test for a change in persistence is that by Martins and Rodrigues (2014). This is more general as it is not restricted to the \(I(0)/I(1)\) framework, but can identify changes from \(I(d_1)\)
to \(I(d_2)\) with \(d_1 \neq d_2\) and \(-1/2<d_1,d_2<2\). The default version is applied by
#> 90% 95% 99% Teststatistic
#> Against increase in memory 4.270666 5.395201 8.233674 16.21494
#> Against decrease in memory 4.060476 5.087265 7.719128 2.14912
#> Against change in unknown direction 5.065695 6.217554 9.136441 16.21494
Again, the function returns a matrix consisting of test statistic and critical values. Here, the alternative of the test is an increase respectively a decrease in memory. In line with the results of
the ratio test, the approach by Martins and Rodrigues (2014) suggests that the series exhibits an increase in memory, i.e. that the memory of the series increases from \(d_1\) to \(d_2\) with \(d_1
<d_2\) at some point in time. Again, this also holds if we consider the critical values that account for the multiple testing problem.
Similar to the ratio test and all other tests against a change in persistence in the memochange package, the MR test also has the same arguments trend, tau, simu, and M. Furthermore, we can choose
again the type of test statistic. This time we can decide whether to use the squared t-statistic or the standard t-statistic.
MR_test(x, statistic="standard")
#> 90% 95% 99% Teststatistic
#> Against increase in memory -1.637306 -1.920434 -2.504862 -2.880545
#> Against decrease in memory -1.651586 -1.951420 -2.514165 -1.277410
#> Against change in unknown direction -1.933137 -2.203370 -2.722017 -2.880545
As for the ratio test, changing the type of statistic has a rather small effect on the empirical performance of the test.
If we believe that the underlying process exhibits additional short run components, we can account for these by setting serial=TRUE
MR_test(x, serial=TRUE)
#> Registered S3 method overwritten by 'quantmod':
#> method from
#> as.zoo.data.frame zoo
#> Registered S3 methods overwritten by 'forecast':
#> method from
#> fitted.fracdiff fracdiff
#> residuals.fracdiff fracdiff
#> 90% 95% 99% Teststatistic
#> Against increase in memory 4.270666 5.395201 8.233674 10.727202
#> Against decrease in memory 4.060476 5.087265 7.719128 6.758906
#> Against change in unknown direction 5.065695 6.217554 9.136441 10.727202
While the test statistic changes, the conclusion remains the same.
All tests indicate that the oil price series exhibits an increase in memory over time. To correctly model and forecast the series, the exact location of the break is important. This can be estimated
by the BP_estim function. It is important for the function that the direction of the change is correctly specified. In our case, an increase in memory has occurred so that we set direction="01"
BP_estim(x, direction="01")
#> $Breakpoint
#> [1] 151
#> $d_1
#> [1] 0.8127501
#> $sd_1
#> [1] 0.08574929
#> $d_2
#> [1] 1.088039
#> $sd_2
#> [1] 0.07142857
This yields a list stating the location of the break (observation 151), semiparametric estimates of the order of integration in the two regimes (0.86 and 1.03) as well as the standard deviations of
these estimates (0.13 and 0.15).
Consequently, the function indicates that there is a break in persistence in July, 1998. This means that from the beginning of the sample until June 1998 the series is integrated with an order of
0.85 and from July 1998 on the order of integration increased to 1.03.
As before, the function allows for various types of break point estimators. Instead of the default estimator of Busetti and Taylor (2004), one can also rely on the estimator of Leybourne, Kim, and
Taylor (2007) by setting type="LKT". This estimator relies on estimates of the long-run variance. Therefore, it is also needed that m is chosen, which determines how many covariances are used when
estimating the long-run variance. Leybourne, Kim, and Taylor (2007) suggest m=0.
BP_estim(x, direction="01", type="LKT", m=0)
#> $Breakpoint
#> [1] 148
#> $d_1
#> [1] 0.7660609
#> $sd_1
#> [1] 0.08703883
#> $d_2
#> [1] 1.067404
#> $sd_2
#> [1] 0.07142857
This yields a similar result with the break point lying in the year 1998 and d increasing from approximately 0.8 to approximately 1.
All other arguments of the function (trend, tau, serial) were already discussed above except for d_estim and d_bw. These two arguments determine which estimator and bandwidth are used to estimate the
order of integration in the two regimes. Concerning the estimator, the GPH (Geweke and Porter-Hudak (1983)) and the exact local Whittle estimator (Shimotsu and Phillips (2005)) can be selected.
Although the exact local Whittle estimator has a lower variance, the GPH estimator is still often considered in empirical applications due to its simplicity. In our example the results of the two
estimators are almost identical.
BP_estim(x, direction="01", d_estim="GPH")
#> $Breakpoint
#> [1] 151
#> $d_1
#> [1] 0.855238
#> $sd_1
#> [1] 0.129834
#> $d_2
#> [1] 1.034389
#> $sd_2
#> [1] 0.1468516
The d_bw argument determines how many frequencies are used for estimation. Larger values imply a lower variance of the estimates, but also bias the estimator if the underlying process possesses short
run dynamics. Usually a value between 0.5 and 0.8 is considered.
BP_estim(x, direction="01", d_bw=0.75)
#> $Breakpoint
#> [1] 151
#> $d_1
#> [1] 0.9146951
#> $sd_1
#> [1] 0.07624929
#> $d_2
#> [1] 1.173524
#> $sd_2
#> [1] 0.0625
BP_estim(x, direction="01", d_bw=0.65)
#> $Breakpoint
#> [1] 151
#> $d_1
#> [1] 0.5803242
#> $sd_1
#> [1] 0.09805807
#> $d_2
#> [1] 0.9353325
#> $sd_2
#> [1] 0.08219949
In our setup, it can be seen that increasing d_bw to 0.75 does not severely change the estimated order of integration in the two regimes. Decreasing d_bw, however, leads to smaller estimates of \(d\) | {"url":"https://cran.case.edu/web/packages/memochange/vignettes/break_in_persistence.html","timestamp":"2024-11-04T01:49:04Z","content_type":"application/xhtml+xml","content_length":"52212","record_id":"<urn:uuid:8835e60a-b423-4ea1-81f2-99c0e3213cd1>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00627.warc.gz"} |
Bayes' Theorem
Bayes' Theorem: Simplifying Statistical Analysis
In statistics and probability, Bayes' Theorem stands as a pivotal analytical tool. This mathematical formula offers a method to update the probability for a hypothesis as more evidence or information
becomes available. It's named after Thomas Bayes (1701–1761), an English statistician, philosopher, and Presbyterian minister who formulated the principle in his work. Bayes' Theorem has profound
implications across various fields, including medicine, finance, and machine learning, by providing a systematic way to calculate conditional probabilities.
Understanding Bayes' Theorem
At its core, Bayes' Theorem is a way to calculate the probability of an event based on prior knowledge of conditions that might be related to the event. It uses the concept of posterior probability,
prior probability, likelihood, and marginal likelihood to compute its results.
Posterior Probability: The probability of the hypothesis after getting the evidence.
Prior Probability: The initial probability of the hypothesis before getting the evidence.
Likelihood: The probability of observing the given data under a specific hypothesis.
Marginal Likelihood: The total probability of observing the evidence under all possible hypotheses.
The formula for Bayes' Theorem is expressed as:
P(H|E) = (P(E|H) * P(H)) / P(E)
In this formula:
P(H|E) is the probability of hypothesis H given the evidence E.
P(E|H) is the probability of evidence E given that hypothesis H is true.
P(H) is the prior probability of hypothesis H.
P(E) is the probability of evidence E.
Application of Bayes' Theorem
Bayes' Theorem is applied in numerous fields to make more informed decisions based on the accumulation of evidence:
Medicine: Used to determine the probability of a disease given the presence of various symptoms or the results of a test.
Finance: Helps in assessing the risk of investments based on prior performance and market trends.
Machine Learning: Employs Bayesian inference to update the model's predictions as more data becomes available.
The significance of Bayes' Theorem
The power of Bayes' Theorem lies in its ability to combine prior knowledge with new evidence to make predictions or inferences. This iterative process of updating beliefs has several advantages:
Flexibility: It can be applied in scenarios with incomplete information, adjusting probabilities as new data emerges.
Foundation for Statistical Inference: Many statistical methods and algorithms are based on or related to Bayes' Theorem, making it foundational in the field.
Decision Making: Provides a quantitative basis for making decisions under uncertainty.
Challenges and considerations
While Bayes' Theorem is widely used, it's not without its challenges. The accuracy of the results heavily depends on the quality and quantity of prior data. Misinterpretation of the theorem's output
due to biases in the data or incorrect assumptions can lead to inaccurate conclusions.
Practical example
Consider a medical test for a specific disease. If the disease affects 1% of the population, the test accurately identifies the disease in 99% of cases (true positive rate) and accurately identifies
non-disease in 99% of cases (true negative rate). Bayes' Theorem can be used to calculate the probability that a person actually has the disease if they test positive.
Bayes' Theorem is a critical tool in the statistical analysis toolbox, offering a structured method for updating probabilities based on new evidence. Its application across various domains underlines
its importance and utility in making informed decisions under uncertainty. Understanding and applying Bayes' Theorem can significantly enhance the decision-making process, allowing for more accurate
predictions and inferences based on evolving data. | {"url":"https://coinmetro.com/glossary/bayes-theorem","timestamp":"2024-11-11T01:39:01Z","content_type":"text/html","content_length":"186477","record_id":"<urn:uuid:88349794-be67-4bce-9367-4f222865719e>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00838.warc.gz"} |
The Alternating Projection Algorithm (AP algorithm) - CompileIoT
The Alternating Projection Algorithm (AP algorithm) is a mathematical method used to solve optimization problems involving sets and projections. It is a powerful tool in the field of signal
processing, image reconstruction, and convex optimization.
Introduction to the AP Algorithm
The AP algorithm was first introduced by John von Neumann in the 1950s. It is a simple yet effective algorithm that iteratively projects a point onto a set and then onto another set, alternating
between the two. The goal is to find a point that belongs to the intersection of the two sets, if such a point exists.
The algorithm is based on the concept of projection, which is a fundamental operation in convex geometry. A projection of a point onto a set is the closest point in the set to the given point. The AP
algorithm takes advantage of this concept to iteratively approach the optimal solution.
Working Principle of the AP Algorithm
The AP algorithm starts with an initial point and two sets, A and B. It then iteratively performs the following steps:
1. Project the current point onto set A to obtain a new point.
2. Project the new point onto set B to obtain another new point.
3. Repeat steps 1 and 2 until convergence.
The algorithm converges when the difference between two consecutive points becomes sufficiently small. The final point obtained is an approximation of the optimal solution, which belongs to the
intersection of sets A and B.
Applications of the AP Algorithm
The AP algorithm has a wide range of applications in various fields:
Signal Processing
In signal processing, the AP algorithm is used for signal reconstruction from incomplete or noisy measurements. It can be used to recover a signal that satisfies certain constraints, such as sparsity
or low-rankness. The algorithm iteratively projects the current estimate onto the set of feasible signals, improving the reconstruction quality at each iteration.
Image Reconstruction
The AP algorithm is also applied in image reconstruction, particularly in computed tomography (CT) and magnetic resonance imaging (MRI). It is used to reconstruct high-resolution images from a
limited number of projections or measurements. By iteratively projecting the current estimate onto the set of feasible images, the algorithm improves the image quality and reduces artifacts.
Convex Optimization
In convex optimization, the AP algorithm is used to solve problems involving multiple convex sets. It can be used to find the intersection of these sets, which corresponds to the optimal solution of
the optimization problem. The algorithm iteratively projects the current point onto each set, alternating between them until convergence.
Advantages and Limitations of the AP Algorithm
The AP algorithm has several advantages:
• It is a simple and intuitive algorithm that is easy to implement.
• It is computationally efficient and converges quickly for many problems.
• It can handle large-scale problems with high-dimensional data.
However, the AP algorithm also has some limitations:
• It may not converge to the optimal solution for all problems.
• It may get stuck in local optima or oscillate between two points.
• It may require a good initial guess to converge to the desired solution.
The Alternating Projection Algorithm (AP algorithm) is a versatile method for solving optimization problems involving sets and projections. It has found applications in signal processing, image
reconstruction, and convex optimization. While it has its advantages and limitations, the AP algorithm remains a valuable tool in various fields, providing efficient and effective solutions to
complex problems.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://compileiot.online/the-alternating-projection-algorithm-ap-algorithm/","timestamp":"2024-11-08T17:13:21Z","content_type":"text/html","content_length":"117999","record_id":"<urn:uuid:5f211f9d-986b-49aa-986d-9de094b0f04b>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00357.warc.gz"} |
Skeleton, saving to VTK format
Skeleton, saving to VTK format
This example demonstrates determination of a line skeleton of a binary image. The skeletonization process reduces all foreground structures into their approximate centerlines.
The skeleton can be traced into a graph structure, where skeleton branches are edges and their intersections are vertices. The traced branches contain information about the branch, e.g., its length.
Finally, the traced skeleton is saved into a .vtk file. The generated .vtk file is compatible, e.g., with ParaView:
def skeleton_vtk():
Creates skeleton, traces it to a graph/network structure, and saves
the skeleton branches in a .vtk file.
# Generate a simple image for demonstration
geometry = pi.newimage(ImageDataType.UINT8, 100, 100)
p0 = [1, 1, 0]
p1 = [25, 50, 0]
p2 = [75, 50, 0]
p3 = [5, 95, 0]
p4 = [95, 95, 0]
pi.line(geometry, p0, p1, 255)
pi.line(geometry, p1, p2, 255)
pi.line(geometry, p0, p2, 255)
pi.line(geometry, p1, p3, 255)
pi.line(geometry, p2, p4, 255)
# Save the geometry
pi.writeraw(geometry, output_file("geometry"))
# Convert geometry to line skeleton
# Write the skeleton so that it can be visualized
pi.writeraw(geometry, output_file("skeleton"))
# Create images for tracing the skeleton into a graph
vertices = pi.newimage(ImageDataType.FLOAT32)
edges = pi.newimage(ImageDataType.UINT64)
measurements = pi.newimage(ImageDataType.FLOAT32)
points = pi.newimage(ImageDataType.INT32)
# Trace the skeleton
# The last 1 gives count of threads. For images with small number of branches
# (like here) single-threaded processing is faster.
pi.tracelineskeleton(geometry, vertices, edges, measurements, points, True, 1)
# Convert the traced skeleton into points and lines format, as the .vtk files
# need that format.
vtkpoints = pi.newimage()
vtklines = pi.newimage()
pi.getpointsandlines(vertices, edges, measurements, points, vtkpoints, vtklines)
pi.writevtk(vtkpoints, vtklines, output_file("vtk_test_file")) | {"url":"https://pi2-docs.readthedocs.io/en/latest/examples/ex_skeleton.html","timestamp":"2024-11-04T20:20:45Z","content_type":"text/html","content_length":"21731","record_id":"<urn:uuid:c0a3fd12-b869-4a85-a915-b038b8a3df35>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00670.warc.gz"} |
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Contrib/swak4Foam/Example calcPressureDifference
From OpenFOAMWiki
One way to calculate the pressure difference between two patches is by calculating the difference between the averages weighted by the face-sizes on the other patch to the current patch:
type patchExpression;
variables ( "pOut{patch'outlet}=sum(p*area())/sum(area());");
accumulations (
patches (
expression "p-pOut";
verbose true;
Another possibility would be to compare the difference of the averages:
type patchExpression;
variables (
accumulations (
patches (
expression "area()*(p-pOut)/inArea";
verbose true;
Pressure-drops to other entities can be calculated similarily:
type swakExpression;
valueType faceZone;
zoneName beforeFilter;
variables (
accumulations (
expression "p-pAfter";
verbose true;
autoInterpolate true;
warnAutoInterpolate false;
Note: how you define "pressure difference" (face weighted, arithmetic average, min/max ...) is up to you | {"url":"https://openfoamwiki.net/index.php/Contrib/swak4Foam/Example_calcPressureDifference","timestamp":"2024-11-14T07:04:10Z","content_type":"text/html","content_length":"23167","record_id":"<urn:uuid:0a3048d0-23cf-49e9-9fca-069041ba8e48>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00602.warc.gz"} |
The Shape Of Things To Come - aster.cloud
Creating and animating rounded shapes with AndroidX, Part I
The new graphics-shapes library allows easy creation and editing of complex, rounded polygonal shapes
A new library with the flashy name :graphics:graphics-shapes: launched recently in AndroidX. I’m happy (especially after many months of working on it with my colleague Sergio Sancho from the Android
Wear team) that this project and API is finally out there. But I thought it might be helpful to describe what it is and how to actually use it.
There are two major parts to the library, which internally we call Shapes and Morph. For the benefit of short-attentions-span Medium readership, I will therefore break the description of the library
down into two articles along similar lines. This first one will cover the Shapes portion of the library, which allows easy creation and rendering of rounded polygonal shapes. The second article shows
how to animate (a.k.a, “Morph”) those shapes.
Android offers a very flexible drawing API. In a custom View, you can override onDraw() and use the Canvas parameter to draw anything from lines to circles to rectangles, to complex Path objects. And
if you want something rounded, you can create and draw any shape you want… as long you want a RoundRect.
Of course, you can always (if you’re up for the effort) create a very complex shape (complete with arbitrary rounding) with the Path API. But out of the box, we give you only Canvas.drawRoundRect().
Moreover, Android offers very little flexibility in terms of how those rects are rounded. That is, each of the corners of the rectangles are rounded with a circular curve… period. So if you want
something more custom (either in the shape of the rounded corners or the number of vertices), you are on your own.
Until now.
We thought it would be useful to provide simple creation of all kinds of rounded shapes. I mean, rectangles are cool and all. And so are those circular corners, right? They’re so… circular! But
sometimes you want just a little more. Or even a lot more.
I don’t know why you would create a shape like this. But isn’t it nice that you can?
We also wanted these shapes to be available not just for apps running on future platform versions, but also across much older releases, for the enjoyment of all developers and users. So we created an
API to do just that, by using Path objects internally. Path has been available since 1.0 and thus offers compatibility back as far as AndroidX itself goes.
The API for creating and drawing these shapes is simple (we saved all the complicated bits for the internal code which creates them). There are just a couple of different pieces to understand:
creating a polygonal shape and specifying optional rounding parameters for the shape’s corners. I’ll cover these below.
Aside: Polygons
Q: What is a Polygon?
A: There are so many sides to that question…
It’s worth talking a little bit first about what we mean by “polygon.” In particular, it’s worth explaining what we mean when we use of the term, to show how we get to the much more complex (and
interesting) shapes enabled by this library.
Wikipedia defines Polygon thusly:
In geometry, a polygon (/ˈpɒlɪɡɒn/) is a plane figure made up of line segments connected to form a closed polygonal chain.
which I find… not terribly helpful. I think mathematicians enjoy math so much that even when they’re writing words, it still sounds like equations. Let’s simplify this definition for the non
mathematicians in the audience.
In its most basic form, a polygon is a 2D shape with edges and vertices (or “corners”). I usually think of polygons as having vertices that are ordered around some center, with all edges having the
same length. Polygons can be much more complex than that, however, including shapes that can be self-intersecting.
Our library’s polygons are, however, a bit more staid and boring, with vertices that are positioned equidistant from some center, marching around in order. All sides are of equal length and there are
no funky self-intersections. (This constraint ends up being important in being able to handle automatic morphing between our shapes with reasonable results). You can think of the base RoundedPolygon
object (which we will see in more detail below) as being a shape that has a center around which all of its vertices are positioned at some given radius away from that center.
The library’s polygons can be thought of as a set of equidistant, ordered vertices which lie ‘radius’ distance from some center.
Our polygons can be a bit more complex as well. For one thing, the library has the concept of a “star polygon.” Star polygons are similar to polygons, except they have both an inner and outer radius,
with vertices lying on one or the other, taking turns as the outline proceeds around the center.
Finally, our polygons have the concept of “rounding.” Rounding is not strictly a polygonal concept, since mathematical polygons are defined to have straight edges and sharp corners. So we call our
shapes “Rounded Polygons,” as a blend of the general concepts of polygons with the additional nuance of optionally rounded corners. Rounded polygons have a similar geometry as the shapes above,
except that each vertex has optional parameters which describe how to round its corner. For example, here is a 5-sided star polygon, like the one above, but with rounding specified for corners formed
by the vertices on its outer radius.
Another star polygon, except this time with rounded corners on the outer radius
These, then, are the types of shapes that this library will produce: polygonal (ish) non-self-intersecting shapes where the vertices are ordered and equidistant from a radius (or two), with
optionally rounded corners.
Now let’s look at how to use the library’s API to create those shapes.
Polygon, Stars, and More
Note: This article is current as of the alpha02 release. There will probably be minor API changes during the alpha phase; I will update the article when the API changes, and will update this release
note accordingly.
The main class used to create a shape is RoundedPolygon. There are many different shapes you can create with this API, but all of them boil down to polygonal variations.
The way that you create a simple, unrounded*RoundedPolygon is by telling the API how many vertices you want and optionally providing a radius and center. Of course, any shape will have a radius and
center, but by default the library creates canonical shapes with a radius of 1 around a center at (0, 0). Note that you can transform (scale, translate, rotate) the canonical shape to get it to the
size and location you want by calling RoundedPolygon.transform(Matrix).
* At this point, you might be wondering why we have an API named “Rounded” which allows you to create an unrounded thing. The original version of the API handled that semantic difference, with a
Polygon superclass and a RoundedPolygon subclass. But in the end, it was all a bit academic to split this functionality based on the meaning of the word “polygon,” so we went with a single class
instead which handles all possibilities.
API naming is hard, imperfect, and a perpetual source of regret.
The simplest use of the API involves passing in the number of vertices and letting the library do its thing. You can then call the transform() function to resize and position the object and finally
draw it into your custom view with an extension method provided by the library.
Here’s an example which creates a five-sided figure with a radius of 200 and draws it with a given Canvas and Paint object (created elsewhere):
val pentagon = RoundedPolygon(5, 200f)
canvas.drawPolygon(pentagon, paint)
Star polygons (discussed earlier) are nearly as simple; the only extra thing needed is a second radius, which is provided via the innerRadius parameter in the Star() function. This inner radius is a
value ranging from 0 to the value of radius (which is the “outer” radius for the shape).
For example, to create a five-sided star polygon with a radius of 100 and an inner radius halfway between the outer radius and the center, you would do this:
val pentagonalStar = Star(5, 100f, 50f)
A Star shape created with 5 vertices and an inner radius half the value of the main radius
Rounding Error
So all of this is nice. We’ve provided a simple API to create and draw regular and star polygons. But these are not the hard parts in this problem space; it’s not too difficult to create
straight-edged, sharp-corner shapes like these with the existing APIs. The interesting (and tricky) part is how to round those corners.
Figuring this out meant (in my case) re-learning a bunch of high-school level geometry and trigonometry (hey, it had been… a long time since I had those classes). Thing like trig identities, the Law
of Cosines, and handy geometry facts like the angles of a triangle adding up to 180 degrees all came into play. Maybe I’ll write up that stuff sometime (or you can just look at the code and see where
it ended up).
But the key part (for users of the library) is: how do you use the API to get nice, rounded shapes? Fortunately, the API (like so many APIs) is much simpler than the implementation. Creating a
polygon, or star polygon, with rounded corners is nothing more than creating those shapes with the APIs described above, with additional information about how the corners should be rounded.
To accomplish this task, use the class CornerRounding which is responsible for determining how the corners should be rounded. It takes two parameters: radius and smoothing.
Rounding Radius
radius is the radius of the circle used to round a vertex. This is similar to the radius parameters supplied to the existing drawRoundRect method of Canvas, except it works in concert with the
optional smoothing parameter (see below). For example, we can create this rounded triangle:
where the rounding radius r for each of the corners can be pictured geometrically as follows:
The rounding radius r determines the circular rounding size of rounded corners
Note that a rounding radius produces a purely circular curve on the corner, between the two straight edges that meet at the vertex.
Smooth Moves
“Smoothing,” unlike the corner rounding radius, is a new concept in our APIs. You can think of smoothing as a factor which determines how long it takes to get from the circular rounding portion of
the corner to the edge. A smoothing factor of 0 (unsmoothed, the default value for CornerRounding) results in purely circular corner rounding (if a nonzero radius is specified, as above). A nonzero
smoothing factor (up to the max of 1.0) results in the corner being rounded by three separate curves. The center curve is the same circular arc produced by the rounding radius, explained above. But
instead of that curve coming all the way to the polygon edges, there are now two “flanking” curves, which transition from the inner circular curve to the outer edges in smooth (non-circular) arcs of
their own.
The magnitude of the smoothing curve determines both the length of the inner circular curve (more smoothing == smaller circular curve) and the length of the flanking curves (more smoothing == larger
flanking curves). The flanking curves affect not only how much of the rounding happens on a circular path, but also the distance of the overall rounding curve. A larger smoothing factor pushes the
intersection point of the rounding curve further along the edge toward the next vertex. A value of 1 (the max) results in no inner curve at all (the circular portion has length zero) and the maximum
length of the flanking curves (which can extend as far as the next vertex, depending on the rounding parameters of that vertex).
To illustrate the impact of smoothing, it is helpful to look at a diagram showing the underlying curves. Note that all polygons are represented internally by a list of Bézier cubic curves, which are
each defined by pairs of anchor and control points. These cubic curves are then used internally to create the Path objects responsible for drawing the shapes into the custom view.
Note: Describing cubic curves is beyond the scope of this already long article, but fortunately there is plenty of information out there about Bézier curves,** cubic curves, Paths, and more. I
invite you to do background reading there and elsewhere if you aren’t familiar with any these concepts. But here’s a very simple description in case it helps: a cubic Bézier curve can be defined
by two anchor points (one at each end of the curve) and two control points which determine the slope of the curve at those anchor points.
** I have to give a shout out to the that Bezier curve primer site linked here; it’s a vast treasure trove of information about all things Bézier, with proofs, equations, sample code, diagrams, live
embedded demos, and thorough explanations. I return to it often to understand more in this complex and interesting space.
Let’s look at some pictures to see what’s going on with the underlying curves. In the diagram below, the corner of the shape (the white object on the left) is represented on the right by the green
line (the outline of the shape) and the white dashed line (a circle with the given rounding radius). The cubic curve is represented by pink circles that are anchor points, yellow circles that are
control points for the curve, and yellow lines between the anchor and control points. If you’ve used drawing programs such as Adobe Illustrator, or even Keynote, you may have seen similar handle
visuals when drawing curves.
A smoothing factor of 0 (unsmoothed) produces a single cubic curve which follows a circle around the corner with the specified rounding radius, as in the earlier example.
When we supply a non-zero smoothing factor, the rounded corner is created with three cubic curves: the circular curve (which we saw above in the unsmoothed case) plus two flanking curves which
transition between the inner circular curve and the edges. Note that the flanking curves start further back along the edge than in the unsmoothed case. The resulting shape is shown by the white
object on the right. The effects of smoothing can be quite subtle, but they allow much more flexibility for designers in producing smoothed shapes that go beyond the traditional circular-round
A nonzero smoothing factor produces three cubic curves to round the vertex: the inner circular curve (as before) plus two flanking curves that transition between the inner curve and the polygon
I should note that although there are many separate segments which make up each rounded corner (two edges, two flanking curves, and one inner circular curve), the result is very, er, smooth because
each curve is calculated to match the slope at the point where it joins the next segment. Thus, for example, the rounded corner smoothly transitions from the inner circular curve to the non-circular
smoothing curve, and then again to the straight edge.
One More Thing
Besides the constructors covered above, which all take the number of vertices, there is also a more general constructor which takes a list of vertices:
vertices: List<PointF>,
rounding: CornerRounding = CornerRounding.Unrounded,
perVertexRounding: List<CornerRounding>? = null,
center: PointF? = null
This constructor makes it possible for you to create shapes that… do not work well with the rest of our rounded-polygon assumptions. So don’t be surprised if you throw randomly complex lists of
vertices at it and the results are not as pleasing as the more constrained polygons created by the other constructors.
This constructor exists to allow creation of more interesting polygonal shapes whose vertices are not all equidistant from some center point. For example, we can use the vertex-list constructor to
create a triangle shape where the bottom edge bows in.
This shape is mostly… but not completely a regular triangle. We need to use the constructor which takes a list of vertices to capture that bottom edge shape.
This triangle-ish shape is created with this code.
val triangleInnerRadiusRatio = .1f
val trianglePoints = listOf(
radialToCartesian(1f, 270f.toRadians()),
radialToCartesian(1f, 30f.toRadians()),
radialToCartesian(triangleInnerRadiusRatio, 90f.toRadians()),
radialToCartesian(1f, 150f.toRadians()),
RoundedPolygon(trianglePoints, CornerRounding(.22f))
(Don’t worry about the radialToCartesian function above — or check it out in the sample project listed below if you are curious. It’s just a function that simplifies placing vertices around a center
point at specific angles).
And So On
I talked specifically about a single CornerRounding parameter above, but the API allows you to specify multiple rounding parameters, including one for the inner and outer radii to get an effect like
this on star polygons.
Inner vertices can use different rounding than outer vertices. Here the outer vertices are unrounded.
You can also, if you want to take it that far, define per-vertex rounding parameters, to get a very custom shape indeed. For any of these situations, the API allows you to easily create and draw all
kinds of rounded (or unrounded) polygonal shapes. You could always do this on Android, of course. After all, we are just using the existing Path API underneath to handle the drawing. But there are a
lot of details (and so much math!) to sort out along the way. This new library makes the job, we hope, far easier.
A sample of shapes that can be created with this library . This screenshot was taken from the GITHUB SAMPLE described at the end of this article.
Next Steps
One of the things that drove the internal structure using cubic curves was the need to not just create and draw these shapes, but to also animate smoothly and automatically between them. Check out
the next article, Shape Morphing in Android, to see how to do that with this library.
See Also
The library is available in alpha form on AndroidX:
Sample code!
The shape editing animation in the header was created with a sample app which demonstrates shape creation, editing, and morphing. It is hosted on GitHub:
The sample has both Compose and View apps, showing how to use the library to create and morph shapes in both kinds of UI toolkits. The Compose version has an additional editor view that helps
visualize the various shape parameters.
By Chet Haase
Originally published at Medium
Source: Cyberpogo
For enquiries, product placements, sponsorships, and collaborations, connect with us at [email protected]. We'd love to hear from you!
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The Life Tube Part 2 - More Doughnuts
A big part of what I am sorting out is the extent to which synchronicity is a method of communication between the Inner & Outer self (realities).
I had a draft of this post that I published on 10/3. Towards the end of writing, I started to veer into Torus territory but felt I had better wrap it up and gather my thoughts before writing about
the Transcendent Torus and its relationship to the doughnut/donut/d'ohnut
Last night I received a text with an out-of-context donut. I thought HUH that's funny the donut comes up again. I'd consider this a random but generally uninformative confirmation synchronicity. Just
a little 'hey' from the Universal Self but no real direction.
Okay. So it's not the first time the donut has been used as a math object to illustrate the torus. But the timing is Just So. And that's the point of the synchro-mystic language process. Its logic is
both internal and external, and often in an idiosyncratic way.
I felt it appropriate to share the process of this as an example of using synchronicity as a tool of communication. It's like the first text with a donut was priming my consciousness for this second
instance of dough-nuttery.
Of course I must ask myself what the Universal Self wants me to do with this technical information that is a reach for me to understand.
So I'll just start by positing that it is either a nudge to continue sharing and posting the torus developments I've been working with (and the related Cuboctahedron)
to introduce the concept of Elliptic Curve into the dynamics of energetic structures and energy work.
"Informally, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus." -- Wolfram
(Meditating into a cosmic torus tube)
This would mean that Elliptic Curves are trajectories along a spiraling torus. If some point were following the currents along the surface of a toroidal field, elliptic curve mathematics could then
plot out specific destinations. You can feel I am getting to the point here, that if the torus is the fractalized shape of universal forms, how understanding movement along the torus could be useful
in Aether-Matrix-Space-Time travels.
I hope that if anyone is in a position to understand what I am saying they are inspired to further this line of thought. Going back to school for a PHD in Maths isn't in the cards for me, so I'll
circle back to the relationships between sacred geometry & consciousness.
Further Up, and Further In! | {"url":"https://www.mariascopic.com/post/2017/10/04/the-life-tube-part-2-more-doughnuts","timestamp":"2024-11-07T20:09:31Z","content_type":"text/html","content_length":"1050483","record_id":"<urn:uuid:ba26a808-5a64-4026-800f-fa6ecd024ef0>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00785.warc.gz"} |
2020/2021 Mathematics
• Teaching Mode: Traditional lectures
• Campus: Bologna
• Corso: First cycle degree programme (L) in Business and Economics (cod. 8965)
Learning outcomes
At the end of the course the student will be capable of using the techniques of Linear Algebra; furthermore he will have acquired a working knowledge of First Year Calculus, together with the related
applications in Finance and Economics.
Course contents
A preliminary tutorial (30 hours) covers a number of introductory topics (so-called precalculus), including elementary set theory, sets of real numbers, complex numbers, polynomials, linear and
quadratic equations and inequalities, systems of inequalities, absolute value and rational inequalities, Cartesian coordinate system, basic analytic geometry, basic concepts and definitions about
functions, elementary functions (power, exponential and logarithmic), exponential and logarithmic equations and inequalities, trigonometric functions.
Course content - Calculus and Linear Algebra (90 hours)
Introduction to the course and crash review of preliminary mathematical notions
One-variable functions: basic definitions, graphs and elementary functions (linear, quadratic, polynomial, rational, irrational, power, exponential, logarithmic, absolute value). Odd and even
functions. Composite functions. Inverse functions.
Limits and continuity.
Differentiation of one-variable functions: tangents and derivatives, rules of differentiation, chain rule, higher-order derivatives.
Derivatives in use: implicit differentiation and economic examples, differentiation of the inverse function, linear and quadratic approximations, Taylor's formula, elasticities; continuity and
differentiability, intermediate-value theorem, De L’Hôpital’s Rule.
Single-variable optimization: local and global extrema, stationary points and first-order condition, simple tests for extreme points, extreme points for concave and convex functions, second-order
derivative and convexity, inflection points, study of the graph of a function, asymptotes.
Sequences and series; convergence criteria; geometric series; Taylor's series. Sequences and series in financial mathematics.
Difference equations. Linear, first order, autonomous difference equations. Steady state and convergence analysis. Linear, first order, non autonomous, difference equations. Difference equations in
financial mathematics.
Integration: the Riemann integral and its geometrical interpretation; primitives and indefinite integrals, fundamental theorems of integral calculus. Rules and methods of integration: immediate
integrals, integration of rational functions, integration by parts, integration by substitution. Improper integrals.
Integration in economics: continuous compounding and discounting, present values.
Differential equations. First order differential equations. Linear, first order, autonomous differential equations. Steady state and convergence analysis. Linear, first order, non-autonomous
differential equations. Differential equations with separable variables. Differencial equations in financial mathematics.
Linear algebra: vector spaces, bases and dimension; matrices and their properties, matrix operations, rank and determinant; linear maps and associated matrices, systems of equations, existence of
solutions, cases of one solution and infinitely many solutions, Gaussian elimination, inverse of a matrix and Cramer's rule; eigenvalues and eigenvectors.
Multi-variable calculus: partial derivatives with two variables, geometric interpretation; partial elasticities; chain rules, implicit differentiation along a level curve; functions of more
variables, gradient, differentials and linear approximations; economic applications.
Multi-variable optimization; maxima, minima and saddle points; tests based on second derivatives; constrained optimization and Lagrange multipliers.
R.A. ADAMS, C. ESSEX. Calculus, a complete course, 9th Edition, Pearson, 2018.
Chapters: preliminaries, 1, 2, 3, 4, 5, 6, 7.9, 9, 10, 12, 13
K. SYDSÆTER, P. HAMMOND, A. STRØM, A. CARVAJAL. Essential Mathematics for Economic Analysis, 5th Edition. Pearson, 2016.
Chapters: 1, 2,3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.
Lecture notes on Difference Equations and Eigenvalues and Eigenvectors will be provided by the Professor.
Teaching methods
Class lectures. During the class lectures (as well as in the additional exercise classes) each topic will be illustrated by examples and worked-out exercises.
Assessment methods
Written exam.
The exam of the first (summer) session can be taken in 3 steps: a first midterm exam (after 1/3 of the course, during the mid-term session of January/February) with a duration 1 hour, a second
partial exam (after 2/3 of the course, during the session of April) with a duration of 1 hour on the second part the course, and a third midterm exam of duration 1 hour on the third part of the
course during the first call of session of June/July. In occasion of the third partial exam, students who have not taken the partials can only take the total exam (duration 3 hours).
During the exam, students are not allowed to use calculators. Textbooks and other teaching materials are not allowed.
Grade rejection
The only grades that can be rejected without any communication from the student are those of the first and second mid-term exams.
Teaching tools
Office hours
See the website of Gian Luca Tassinari | {"url":"https://www.unibo.it/en/study/phd-professional-masters-specialisation-schools-and-other-programmes/course-unit-catalogue/course-unit/2020/406488","timestamp":"2024-11-11T12:53:34Z","content_type":"application/xhtml+xml","content_length":"441654","record_id":"<urn:uuid:c121a6a7-3e9f-4b40-a455-db517f88d69d>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00538.warc.gz"} |
6.1 Graphs of the Sine and Cosine Functions - Precalculus 2e | OpenStax
In this section, you will:
• Graph variations of $y=sin(x)y=sin(x)$ and $y=cos(x)y=cos(x)$ .
• Use phase shifts of sine and cosine curves.
White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only
when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.
Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will
interpret and create graphs of sine and cosine functions.
Graphing Sine and Cosine Functions
Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start
with the sine function. We can create a table of values and use them to sketch a graph. Table 1 lists some of the values for the sine function on a unit circle.
$x x$ $0 0$ $π 6 π 6$ $π 4 π 4$ $π 3 π 3$ $π 2 π 2$ $2π 3 2π 3$ $3π 4 3π 4$ $5π 6 5π 6$ $π π$
$sin( x ) sin( x )$ $0 0$ $1 2 1 2$ $2 2 2 2$ $3 2 3 2$ $1 1$ $3 2 3 2$ $2 2 2 2$ $1 2 1 2$ $0 0$
Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. See Figure 2.
Notice how the sine values are positive between 0 and $π, π,$ which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between $π
π$ and $2π, 2π,$ which correspond to the values of the sine function in quadrants III and IV on the unit circle. See Figure 3.
Now let’s take a similar look at the cosine function. Again, we can create a table of values and use them to sketch a graph. Table 2 lists some of the values for the cosine function on a unit circle.
$x x$ $0 0$ $π 6 π 6$ $π 4 π 4$ $π 3 π 3$ $π 2 π 2$ $2π 3 2π 3$ $3π 4 3π 4$ $5π 6 5π 6$ $π π$
$cos( x ) cos( x )$ $1 1$ $3 2 3 2$ $2 2 2 2$ $1 2 1 2$ $0 0$ $− 1 2 − 1 2$ $− 2 2 − 2 2$ $− 3 2 − 3 2$ $−1 −1$
As with the sine function, we can plots points to create a graph of the cosine function as in Figure 4.
Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit
circle, it becomes clear that the range of both functions must be the interval $[ −1,1 ]. [ −1,1 ].$
In both graphs, the shape of the graph repeats after $2π, 2π,$ which means the functions are periodic with a period of $2π. 2π.$ A periodic function is a function for which a specific horizontal
shift, P, results in a function equal to the original function: $f( x+P )=f( x ) f( x+P )=f( x )$ for all values of $x x$ in the domain of $f. f.$ When this occurs, we call the smallest such
horizontal shift with $P>0 P>0$ the period of the function. Figure 5 shows several periods of the sine and cosine functions.
Looking again at the sine and cosine functions on a domain centered at the y-axis helps reveal symmetries. As we can see in Figure 6, the sine function is symmetric about the origin. Recall from The
Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because $sin(−x)=−sinx. sin(−x)=−sinx.$ Now we can clearly see this property from the
Figure 7 shows that the cosine function is symmetric about the y-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that $cos(−x)=cosx. cos(−x)=
Characteristics of Sine and Cosine Functions
The sine and cosine functions have several distinct characteristics:
• They are periodic functions with a period of $2π. 2π.$
• The domain of each function is $( −∞,∞ ) ( −∞,∞ )$ and the range is $[ −1,1 ]. [ −1,1 ].$
• The graph of $y=sinx y=sinx$ is symmetric about the origin, because it is an odd function.
• The graph of $y=cosx y=cosx$ is symmetric about the $y y$-axis, because it is an even function.
Investigating Sinusoidal Functions
As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are
not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of
sinusoidal functions are
$y=Asin( Bx−C )+D and y=Acos( Bx−C )+D y=Asin( Bx−C )+D and y=Acos( Bx−C )+D$
Determining the Period of Sinusoidal Functions
Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.
In the general formula, $B B$ is related to the period by $P= 2π | B | . P= 2π | B | .$ If $| B |>1, | B |>1,$ then the period is less than $2π 2π$ and the function undergoes a horizontal
compression, whereas if $| B |<1, | B |<1,$ then the period is greater than $2π 2π$ and the function undergoes a horizontal stretch. For example, $f(x)=sin( x ), f(x)=sin( x ),$ $B=1, B=1,$ so the
period is $2π, 2π,$ which we knew. If $f(x)=sin( 2x ), f(x)=sin( 2x ),$ then $B=2, B=2,$ so the period is $π π$ and the graph is compressed. If $f(x)=sin( x 2 ), f(x)=sin( x 2 ),$ then $B= 1 2 , B= 1
2 ,$ so the period is $4π 4π$ and the graph is stretched. Notice in Figure 8 how the period is indirectly related to $| B |. | B |.$
Period of Sinusoidal Functions
If we let $C=0 C=0$ and $D=0 D=0$ in the general form equations of the sine and cosine functions, we obtain the forms
$y=Asin( Bx ) y=Asin( Bx )$
$y=Acos( Bx ) y=Acos( Bx )$
The period is $2π | B | . 2π | B | .$
Identifying the Period of a Sine or Cosine Function
Determine the period of the function $f( x )=sin( π 6 x ). f( x )=sin( π 6 x ).$
Let’s begin by comparing the equation to the general form $y=Asin(Bx). y=Asin(Bx).$
In the given equation, $B= π 6 , B= π 6 ,$ so the period will be
$P= 2π |B| = 2π π 6 =2π⋅ 6 π =12 P= 2π |B| = 2π π 6 =2π⋅ 6 π =12$
Determine the period of the function $g(x)=cos( x 3 ). g(x)=cos( x 3 ).$
Determining Amplitude
Returning to the general formula for a sinusoidal function, we have analyzed how the variable $B B$ relates to the period. Now let’s turn to the variable $A A$ so we can analyze how it is related to
the amplitude, or greatest distance from rest. $A A$ represents the vertical stretch factor, and its absolute value $| A | | A |$ is the amplitude. The local maxima will be a distance $| A | | A |$
above the horizontal midline of the graph, which is the line $y=D; y=D;$ because $D=0 D=0$ in this case, the midline is the x-axis. The local minima will be the same distance below the midline. If $|
A |>1, | A |>1,$ the function is stretched. For example, the amplitude of $f(x)=4sinx f(x)=4sinx$ is twice the amplitude of $f(x)=2sinx. f(x)=2sinx.$ If $| A |<1, | A |<1,$ the function is
compressed. Figure 9 compares several sine functions with different amplitudes.
Amplitude of Sinusoidal Functions
If we let $C=0 C=0$ and $D=0 D=0$ in the general form equations of the sine and cosine functions, we obtain the forms
$y=Asin( Bx ) and y=Acos( Bx ) y=Asin( Bx ) and y=Acos( Bx )$
The amplitude is $|A|, |A|,$ which is the vertical height from the midline $. .$ In addition, notice in the example that
$| A | = amplitude = 1 2 | maximum − minimum | | A | = amplitude = 1 2 | maximum − minimum |$
Identifying the Amplitude of a Sine or Cosine Function
What is the amplitude of the sinusoidal function $f(x)=−4sin(x)? f(x)=−4sin(x)?$ Is the function stretched or compressed vertically?
Let’s begin by comparing the function to the simplified form $y=Asin(Bx). y=Asin(Bx).$
In the given function, $A=−4, A=−4,$ so the amplitude is $| A |=| −4 |=4. | A |=| −4 |=4.$ The function is stretched.
The negative value of $A A$ results in a reflection across the x-axis of the sine function, as shown in Figure 10.
What is the amplitude of the sinusoidal function $f(x)= 1 2 sin(x)? f(x)= 1 2 sin(x)?$ Is the function stretched or compressed vertically?
Analyzing Graphs of Variations of y = sin x and y = cos x
Now that we understand how $A A$ and $B B$ relate to the general form equation for the sine and cosine functions, we will explore the variables $C C$ and $D. D.$ Recall the general form:
$y=Asin( Bx−C )+D and y=Acos( Bx−C )+D or y=Asin( B( x− C B ) )+D and y=Acos( B( x− C B ) )+D y=Asin( Bx−C )+D and y=Acos( Bx−C )+D or y=Asin( B( x− C B ) )+D and y=Acos( B( x− C B ) )+D$
The value $C B C B$ for a sinusoidal function is called the phase shift, or the horizontal displacement of the basic sine or cosine function. If $C>0, C>0,$ the graph shifts to the right. If $C<0, C
<0,$ the graph shifts to the left. The greater the value of $| C |, | C |,$ the more the graph is shifted. Figure 11 shows that the graph of $f(x)=sin( x−π ) f(x)=sin( x−π )$ shifts to the right by
$π π$ units, which is more than we see in the graph of $f(x)=sin( x− π 4 ), f(x)=sin( x− π 4 ),$ which shifts to the right by $π 4 π 4$ units.
While $C C$ relates to the horizontal shift, $D D$ indicates the vertical shift from the midline in the general formula for a sinusoidal function. See Figure 12. The function $y=cos( x )+D y=cos( x )
+D$ has its midline at $y=D. y=D.$
Any value of $D D$ other than zero shifts the graph up or down. Figure 13 compares $f(x)=sin(x) f(x)=sin(x)$ with $f(x)=sin(x)+2, f(x)=sin(x)+2,$ which is shifted 2 units up on a graph.
Variations of Sine and Cosine Functions
Given an equation in the form $f( x )=Asin( Bx−C )+D f( x )=Asin( Bx−C )+D$ or $f( x )=Acos( Bx−C )+D, f( x )=Acos( Bx−C )+D,$ $C B C B$ is the phase shift and $D D$ is the vertical shift.
Identifying the Phase Shift of a Function
Determine the direction and magnitude of the phase shift for $f(x)=sin( x+ π 6 )−2. f(x)=sin( x+ π 6 )−2.$
Let’s begin by comparing the equation to the general form $y=Asin(Bx−C)+D. y=Asin(Bx−C)+D.$
In the given equation, notice that $B=1 B=1$ and $C=− π 6 . C=− π 6 .$ So the phase shift is
$C B =− π 6 1 =− π 6 C B =− π 6 1 =− π 6$
or $π 6 π 6$ units to the left.
We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before $C. C.$ Therefore $f(x)=sin( x+ π 6 )−2 f(x)=sin( x+ π 6 )−2$
can be rewritten as $f(x)=sin( x−( − π 6 ) )−2. f(x)=sin( x−( − π 6 ) )−2.$ If the value of $C C$ is negative, the shift is to the left.
Determine the direction and magnitude of the phase shift for $f(x)=3cos( x− π 2 ). f(x)=3cos( x− π 2 ).$
Identifying the Vertical Shift of a Function
Determine the direction and magnitude of the vertical shift for $f(x)=cos( x )−3. f(x)=cos( x )−3.$
Let’s begin by comparing the equation to the general form $y=Acos(Bx−C)+D. y=Acos(Bx−C)+D.$
In the given equation, $D=−3 D=−3$ so the shift is 3 units downward.
Determine the direction and magnitude of the vertical shift for $f(x)=3sin( x )+2. f(x)=3sin( x )+2.$
Given a sinusoidal function in the form $f( x )=Asin( Bx−C )+D, f( x )=Asin( Bx−C )+D,$ identify the midline, amplitude, period, and phase shift.
1. Determine the amplitude as $| A |. | A |.$
2. Determine the period as $P= 2π | B | . P= 2π | B | .$
3. Determine the phase shift as $C B . C B .$
4. Determine the midline as $y=D. y=D.$
Identifying the Variations of a Sinusoidal Function from an Equation
Determine the midline, amplitude, period, and phase shift of the function $y=3sin(2x)+1. y=3sin(2x)+1.$
Let’s begin by comparing the equation to the general form $y=Asin(Bx−C)+D. y=Asin(Bx−C)+D.$
$A=3, A=3,$ so the amplitude is $| A |=3. | A |=3.$
Next, $B=2, B=2,$ so the period is $P= 2π | B | = 2π 2 =π. P= 2π | B | = 2π 2 =π.$
There is no added constant inside the parentheses, so $C=0 C=0$ and the phase shift is $C B = 0 2 =0. C B = 0 2 =0.$
Finally, $D=1, D=1,$ so the midline is $y=1. y=1.$
Inspecting the graph, we can determine that the period is $π, π,$ the midline is $y=1, y=1,$ and the amplitude is 3. See Figure 14.
Determine the midline, amplitude, period, and phase shift of the function $y= 1 2 cos( x 3 − π 3 ). y= 1 2 cos( x 3 − π 3 ).$
Identifying the Equation for a Sinusoidal Function from a Graph
Determine the formula for the cosine function in Figure 15.
To determine the equation, we need to identify each value in the general form of a sinusoidal function.
$y=Asin(Bx−C)+D y=Acos(Bx−C)+D y=Asin(Bx−C)+D y=Acos(Bx−C)+D$
The graph could represent either a sine or a cosine function that is shifted and/or reflected. When $x=0, x=0,$ the graph has an extreme point, $( 0,0 ). ( 0,0 ).$ Since the cosine function has an
extreme point for $x=0, x=0,$ let us write our equation in terms of a cosine function.
Let’s start with the midline. We can see that the graph rises and falls an equal distance above and below $y=0.5. y=0.5.$ This value, which is the midline, is $D D$ in the equation, so $D=0.5. D=
The greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So $| A |=0.5. | A |=0.5.$ Another way
we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so $| A |= 1 2 =0.5. | A |= 1 2 =0.5.$ Also, the graph is reflected
about the x-axis so that $A=−0.5. A=−0.5.$
The graph is not horizontally stretched or compressed, so $B=1; B=1;$ and the graph is not shifted horizontally, so $C=0. C=0.$
Putting this all together,
$g( x )=−0.5cos( x )+0.5 g( x )=−0.5cos( x )+0.5$
Identifying the Equation for a Sinusoidal Function from a Graph
Determine the equation for the sinusoidal function in Figure 17.
With the highest value at 1 and the lowest value at $−5, −5,$ the midline will be halfway between at $−2. −2.$ So $D=−2. D=−2.$
The distance from the midline to the highest or lowest value gives an amplitude of $| A |=3. | A |=3.$
The period of the graph is 6, which can be measured from the peak at $x=1 x=1$ to the next peak at $x=7, x=7,$ or from the distance between the lowest points. Therefore, $P= 2π | B | =6. P= 2π | B |
=6.$ Using the positive value for $B, B,$ we find that
$B= 2π P = 2π 6 = π 3 B= 2π P = 2π 6 = π 3$
So far, our equation is either $y=3sin( π 3 x−C )−2 y=3sin( π 3 x−C )−2$ or $y=3cos( π 3 x−C )−2. y=3cos( π 3 x−C )−2.$ For the shape and shift, we have more than one option. We could write this as
any one of the following:
• a cosine shifted to the right
• a negative cosine shifted to the left
• a sine shifted to the left
• a negative sine shifted to the right
While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes
$y=3cos( π 3 x− π 3 )−2 or y=−3cos( π 3 x+ 2π 3 )−2 y=3cos( π 3 x− π 3 )−2 or y=−3cos( π 3 x+ 2π 3 )−2$
Again, these functions are equivalent, so both yield the same graph.
Graphing Variations of y = sin x and y = cos x
Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create
graphs from equations.
Instead of focusing on the general form equations
$y=Asin( Bx−C )+D and y=Acos( Bx−C )+D, y=Asin( Bx−C )+D and y=Acos( Bx−C )+D,$
we will let $C=0 C=0$ and $D=0 D=0$ and work with a simplified form of the equations in the following examples.
Given the function $y=Asin( Bx ), y=Asin( Bx ),$ sketch its graph.
1. Identify the amplitude, $| A |. | A |.$
2. Identify the period, $P= 2π | B | . P= 2π | B | .$
3. Start at the origin, with the function increasing to the right if $A A$ is positive or decreasing if $A A$ is negative.
4. At $x= π 2| B | x= π 2| B |$ there is a local maximum for $A>0 A>0$ or a minimum for $A<0, A<0,$ with $y=A. y=A.$
5. The curve returns to the x-axis at $x= π | B | . x= π | B | .$
6. There is a local minimum for $A>0 A>0$ (maximum for $A<0 A<0$ ) at $x= 3π 2| B | x= 3π 2| B |$ with $y=–A. y=–A.$
7. The curve returns again to the x-axis at $x= 2π | B | . x= 2π | B | .$
Graphing a Function and Identifying the Amplitude and Period
Sketch a graph of $f( x )=−2sin( πx 2 ). f( x )=−2sin( πx 2 ).$
Let’s begin by comparing the equation to the form $y=Asin(Bx). y=Asin(Bx).$
• Step 1. We can see from the equation that $A=−2, A=−2,$ so the amplitude is 2.
$| A |=2 | A |=2$
• Step 2. The equation shows that $B= π 2 , B= π 2 ,$ so the period is
$P= 2π π 2 =2π⋅ 2 π =4 P= 2π π 2 =2π⋅ 2 π =4$
• Step 3. Because $A A$ is negative, the graph descends as we move to the right of the origin.
• Step 4–7. The x-intercepts are at the beginning of one period, $x=0, x=0,$ the horizontal midpoints are at $x=2 x=2$ and at the end of one period at $x=4. x=4.$
The quarter points include the minimum at $x=1 x=1$ and the maximum at $x=3. x=3.$ A local minimum will occur 2 units below the midline, at $x=1, x=1,$ and a local maximum will occur at 2 units above
the midline, at $x=3. x=3.$ Figure 19 shows the graph of the function.
Sketch a graph of $g( x )=−0.8cos( 2x ). g( x )=−0.8cos( 2x ).$ Determine the midline, amplitude, period, and phase shift.
Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.
1. Express the function in the general form $y=Asin(Bx−C)+D or y=Acos(Bx−C)+D. y=Asin(Bx−C)+D or y=Acos(Bx−C)+D.$
2. Identify the amplitude, $| A |. | A |.$
3. Identify the period, $P= 2π | B | . P= 2π | B | .$
4. Identify the phase shift, $C B . C B .$
5. Draw the graph of $f( x )=Asin( Bx ) f( x )=Asin( Bx )$ shifted to the right or left by $C B C B$ and up or down by $D. D.$
Graphing a Transformed Sinusoid
Sketch a graph of $f( x )=3sin( π 4 x− π 4 ). f( x )=3sin( π 4 x− π 4 ).$
• Step 1. The function is already written in general form: $f(x)=3sin( π 4 x− π 4 ). f(x)=3sin( π 4 x− π 4 ).$ This graph will have the shape of a sine function, starting at the midline and
increasing to the right.
• Step 2. $| A |=| 3 |=3. | A |=| 3 |=3.$ The amplitude is 3.
• Step 3. Since $| B |=| π 4 |= π 4 , | B |=| π 4 |= π 4 ,$ we determine the period as follows.
$P= 2π | B | = 2π π 4 =2π⋅ 4 π =8 P= 2π | B | = 2π π 4 =2π⋅ 4 π =8$
The period is 8.
• Step 4. Since $C= π 4 , C= π 4 ,$ the phase shift is
$C B = π 4 π 4 =1. C B = π 4 π 4 =1.$
The phase shift is 1 unit.
• Step 5. Figure 20 shows the graph of the function.
Draw a graph of $g(x)=−2cos( π 3 x+ π 6 ). g(x)=−2cos( π 3 x+ π 6 ).$ Determine the midline, amplitude, period, and phase shift.
Identifying the Properties of a Sinusoidal Function
Given $y=−2cos( π 2 x+π )+3, y=−2cos( π 2 x+π )+3,$ determine the amplitude, period, phase shift, and vertical shift. Then graph the function.
Begin by comparing the equation to the general form and use the steps outlined in Example 9.
$y=Acos( Bx−C )+D y=Acos( Bx−C )+D$
• Step 1. The function is already written in general form.
• Step 2. Since $A=−2, A=−2,$ the amplitude is $| A |=2. | A |=2.$
• Step 3. $| B |= π 2 , | B |= π 2 ,$ so the period is $P= 2π | B | = 2π π 2 =2π⋅ 2 π =4. P= 2π | B | = 2π π 2 =2π⋅ 2 π =4.$ The period is 4.
• Step 4. $C=−π, C=−π,$ so we calculate the phase shift as $C B = −π, π 2 =−π⋅ 2 π =−2. C B = −π, π 2 =−π⋅ 2 π =−2.$ The phase shift is $−2. −2.$
• Step 5. $D=3, D=3,$ so the midline is $y=3, y=3,$ and the vertical shift is up 3.
Since $A A$ is negative, the graph of the cosine function has been reflected about the x-axis.
Figure 21 shows one cycle of the graph of the function.
Using Transformations of Sine and Cosine Functions
We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine
Finding the Vertical Component of Circular Motion
A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the y-coordinate of the point as a function of the angle of rotation.
Recall that, for a point on a circle of radius r, the y-coordinate of the point is $y=rsin(x), y=rsin(x),$ so in this case, we get the equation $y(x)=3sin(x). y(x)=3sin(x).$ The constant 3 causes a
vertical stretch of the y-values of the function by a factor of 3, which we can see in the graph in Figure 22.
Notice that the period of the function is still $2π; 2π;$ as we travel around the circle, we return to the point $( 3,0 ) ( 3,0 )$ for $x=2π,4π,6π,... x=2π,4π,6π,...$ Because the outputs of the graph
will now oscillate between $–3 –3$ and $3, 3,$ the amplitude of the sine wave is $3. 3.$
What is the amplitude of the function $f(x)=7cos(x)? f(x)=7cos(x)?$ Sketch a graph of this function.
Finding the Vertical Component of Circular Motion
A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P, as shown in Figure 23. Sketch a graph of the height above the ground of the
point $P P$ as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.
Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in
Figure 24.
Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let’s use a cosine
function because it starts at the highest or lowest value, while a sine function starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value,
so we need to incorporate a vertical reflection.
Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.
Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that
$y=−3cos( x )+4 y=−3cos( x )+4$
A weight is attached to a spring that is then hung from a board, as shown in Figure 25. As the spring oscillates up and down, the position $y y$ of the weight relative to the board ranges from $–1
–1$ in. (at time $x=0) x=0)$ to $–7 –7$ in. (at time $x=π) x=π)$ below the board. Assume the position of $y y$ is given as a sinusoidal function of $x. x.$ Sketch a graph of the function, and then
find a cosine function that gives the position $y y$ in terms of $x. x.$
Determining a Rider’s Height on a Ferris Wheel
The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s
height above ground as a function of time in minutes.
With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.
Passengers board 2 m above ground level, so the center of the wheel must be located $67.5+2=69.5 67.5+2=69.5$ m above ground level. The midline of the oscillation will be at 69.5 m.
The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.
Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.
• Amplitude: $67.5, 67.5,$ so $A=67.5 A=67.5$
• Midline: $69.5, 69.5,$ so $D=69.5 D=69.5$
• Period: $30, 30,$ so $B= 2π 30 = π 15 B= 2π 30 = π 15$
• Shape: $−cos( t ) −cos( t )$
An equation for the rider’s height would be
$y=−67.5cos( π 15 t )+69.5 y=−67.5cos( π 15 t )+69.5$
where $t t$ is in minutes and $y y$ is measured in meters.
Access these online resources for additional instruction and practice with graphs of sine and cosine functions.
6.1 Section Exercises
Why are the sine and cosine functions called periodic functions?
How does the graph of $y=sinx y=sinx$ compare with the graph of $y=cosx? y=cosx?$ Explain how you could horizontally translate the graph of $y=sinx y=sinx$ to obtain $y=cosx. y=cosx.$
For the equation $Acos(Bx+C)+D, Acos(Bx+C)+D,$ what constants affect the range of the function and how do they affect the range?
How does the range of a translated sine function relate to the equation $y=Asin(Bx+C)+D? y=Asin(Bx+C)+D?$
How can the unit circle be used to construct the graph of $f(t)=sint? f(t)=sint?$
For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one
period for $x>0. x>0.$ Round answers to two decimal places if necessary.
$f(x)= 2 3 cosx f(x)= 2 3 cosx$
$f(x)=−3sin x f(x)=−3sin x$
$f( x )=cos( 2x ) f( x )=cos( 2x )$
$f(x)=2sin( 1 2 x ) f(x)=2sin( 1 2 x )$
$f(x)=4cos(πx) f(x)=4cos(πx)$
$f(x)=3cos( 6 5 x ) f(x)=3cos( 6 5 x )$
$y=3sin(8(x+4))+5 y=3sin(8(x+4))+5$
$y=2sin(3x−21)+4 y=2sin(3x−21)+4$
$y=5sin(5x+20)−2 y=5sin(5x+20)−2$
For the following exercises, graph one full period of each function, starting at $x=0. x=0.$ For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and
their corresponding x-values on one period for $x>0. x>0.$ State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.
$f( t )=2sin( t− 5π 6 ) f( t )=2sin( t− 5π 6 )$
$f(t)=−cos( t+ π 3 )+1 f(t)=−cos( t+ π 3 )+1$
$f( t )=4cos( 2( t+ π 4 ) )−3 f( t )=4cos( 2( t+ π 4 ) )−3$
$f( t )=−sin( 1 2 t+ 5π 3 ) f( t )=−sin( 1 2 t+ 5π 3 )$
$f( x )=4sin( π 2 ( x−3 ) )+7 f( x )=4sin( π 2 ( x−3 ) )+7$
Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure 26.
Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 27.
Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 28.
Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 29.
Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 30.
Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 31.
Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 32.
Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 33.
For the following exercises, let $f(x)=sinx. f(x)=sinx.$
On $[ 0,2π ), [ 0,2π ),$ solve $f( x )=0. f( x )=0.$
On $[ 0,2π ), [ 0,2π ),$ solve $f( x )= 1 2 . f( x )= 1 2 .$
Evaluate $f( π 2 ). f( π 2 ).$
On $[0,2π),f(x)= 2 2 . [0,2π),f(x)= 2 2 .$ Find all values of $x. x.$
On $[ 0,2π ), [ 0,2π ),$ the maximum value(s) of the function occur(s) at what x-value(s)?
On $[ 0,2π ), [ 0,2π ),$ the minimum value(s) of the function occur(s) at what x-value(s)?
Show that $f(−x)=−f(x). f(−x)=−f(x).$ This means that $f(x)=sinx f(x)=sinx$ is an odd function and possesses symmetry with respect to ________________.
For the following exercises, let $f(x)=cosx. f(x)=cosx.$
On $[ 0,2π ), [ 0,2π ),$ solve the equation $f(x)=cosx=0. f(x)=cosx=0.$
On $[ 0,2π ), [ 0,2π ),$ solve $f(x)= 1 2 . f(x)= 1 2 .$
On $[ 0,2π ), [ 0,2π ),$ find the x-intercepts of $f(x)=cosx. f(x)=cosx.$
On $[ 0,2π ), [ 0,2π ),$ find the x-values at which the function has a maximum or minimum value.
On $[ 0,2π ), [ 0,2π ),$ solve the equation $f(x)= 3 2 . f(x)= 3 2 .$
Graph $h(x)=x+sinx h(x)=x+sinx$ on $[ 0,2π ]. [ 0,2π ].$ Explain why the graph appears as it does.
Graph $h(x)=x+sinx h(x)=x+sinx$ on $[ −100,100 ]. [ −100,100 ].$ Did the graph appear as predicted in the previous exercise?
Graph $f(x)=xsinx f(x)=xsinx$ on $[ 0,2π ] [ 0,2π ]$ and verbalize how the graph varies from the graph of $f(x)=sinx. f(x)=sinx.$
Graph $f(x)=xsinx f(x)=xsinx$ on the window $[ −10,10 ] [ −10,10 ]$ and explain what the graph shows.
Graph $f(x)= sinx x f(x)= sinx x$ on the window $[ −5π,5π ] [ −5π,5π ]$ and explain what the graph shows.
Real-World Applications
A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel
completes 1 full revolution in 10 minutes. The function $h( t ) h( t )$ gives a person’s height in meters above the ground t minutes after the wheel begins to turn.
1. ⓐ Find the amplitude, midline, and period of $h( t ). h( t ).$
2. ⓑ Find a formula for the height function $h( t ). h( t ).$
3. ⓒ How high off the ground is a person after 5 minutes? | {"url":"https://openstax.org/books/precalculus-2e/pages/6-1-graphs-of-the-sine-and-cosine-functions","timestamp":"2024-11-03T23:48:05Z","content_type":"text/html","content_length":"632885","record_id":"<urn:uuid:9c2f2d7f-83e4-4e0b-94c4-6c6abb0a3c6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00733.warc.gz"} |
Design an Elliptic filter.
[n,Wn] = ellipord(Wp,Ws,Rp,Rs,domain)
A scalar specifying the passband frequency of a low or high pass filter, or a two element vector specifying the passband frequencies of a bandpass or bandstop filter. For a digital filter the
values (in Hz) are normalized relative to the Nyquist frequency. For an analog filter the values are in radians/sec.
A scalar specifying the stopband frequency of a low or high pass filter, or a two element vector specifying the stopband frequencies of a bandpass or bandstop filter. For a digital filter the
values (in Hz) are normalized relative to the Nyquist frequency. For an analog filter the values are in radians/sec.
The maximum attenuation in decibels at Wp.
The minimum attenuation in decibels at Ws.
□ Use 'z' for digital filters (default).
□ Use 's' for analog filters.
The lowest filter order that will meet the requirements.
Normalized frequency input for which the requirement is met.
Design a digital elliptic filter with no more than 1 dB attenuation at 100 Hz and at least 20 dB attenuation at 200 Hz, with a 1000 Hz sampling frequency.
[n,Wn] = ellipord(100/500, 200/500, 1, 20)
n = 3
Wn = 0.32283
In general, there are a range of corner frequencies for which the requirements can be met at the lowest order. Wp is the smallest passband meeting the design requirement. Wn is the largest passband
meeting the design requirement. | {"url":"https://www.openmatrix.org/help/topics/reference/oml_language/SignalProcessing/ellipord.htm","timestamp":"2024-11-14T04:54:57Z","content_type":"application/xhtml+xml","content_length":"10895","record_id":"<urn:uuid:0879d130-b586-4bfb-8310-49f72fdbd8c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00210.warc.gz"} |
[11:45am] Soumi Tikader:ISI Kolkata
Commutative Algebra seminar (Please note that this talk will be via Skype). Speaker: Soumi Tikader. Affiliation: ISI Kolkata. Date and Time: Thursday 31 October, 11:45 am - 12:30 pm.
Venue: Ramanujan Hall, Department of Mathematics. Title: Orbit spaces of unimodular rows over smooth real affine algebras. Abstract: In this talk we will discuss about the group structure
11:00am on orbit spaces of unimodular rows over smooth real affine algebras. With a few definition and some results to start, we will prove a structure theorem of elementary orbit spaces of
unimodular rows over aforementioned ring with the help of similar kind results on Euler class group. As a consequences, we will prove that : Let $X=Spec(R)$ be a smooth real affine
variety of even dimension $d > 1$, whose real points $X(R)$ constitute an orientable manifold. Then the set of isomorphism classes of (oriented) stably free $R$ of rank $d > 1$ is a free
abelian group of rank equal to the number of compact connected components of $X(R)$. In contrast, if $d > 2$ is odd, then the set of isomorphism classes of stably free $R$-modules of rank
$d$ is a $Z/2Z$-vector space (possibly trivial). We will end this talk by giving a structure theorem of Mennicke symbols. PS: Soumi Tikader is a post doctoral candidate.
[3:30pm] Tony Puthenpurakal
Commutative Algebra seminar II. Speaker: Tony Puthenpurakal. Affiliation: IIT Bombay. Date and Time: Thursday 31 October, 3:30 pm - 5:00 pm. Venue: Room 215, Department of Mathematics.
Title: Triangulated categories - Lecture 2. Abstract: We define and give elementary properties of triangulated categories. We also give an application of triangulated categories to
4:00pm linkage theory in commutative algebra. | {"url":"https://www.math.iitb.ac.in/webcal/day.php?date=20191031","timestamp":"2024-11-02T21:46:11Z","content_type":"text/html","content_length":"21934","record_id":"<urn:uuid:c66c6087-a229-4c79-bcbd-35387b172f4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00559.warc.gz"} |
4.5 I) Area of Triangles – Part 2 – 2D Shapes – Edexcel GCSE Maths Foundation
We learnt in the previous section that the area of any triangle can be worked out by using either of the two formulas below:
The examples in the section before were fairly straightforward. In this section, the examples are going to be more complex because they will require us to use Pythagoras’ theorem or trigonometry. If
you are unsure about Pythagoras’ theorem and/or trigonometry, it is worth going through these sections before working through the next few examples (
to be taken through to the Pythagoras’ theorem section and
to be taken through to the trigonometry section).
Example 1
What is the area of the triangle below?
In order to use the triangle formula, we need to know what the base and the height of the triangle is. Currently we are given the base of the triangle, but we do not have the height of the triangle.
However, we are able to work out the height of the triangle by using Pythagoras’ theorem. The height of the triangle splits the whole triangle up into two smaller right-angle triangles. We are given
the base and the hypotenuse for the right-angled triangle on the left. I have drawn this triangle out by itself so that it does not get confusing.
I have labelled the hypotenuse as c, and the other two sides and a and b. We know the value of a and c; a is 3 and c is 5. We do not know the value of b and b is the height of this triangle.
Pythagoras’ theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. His formula is given below:
We can now sub the values that we know into the formula.
We now need to make b^2 the subject and we do this by taking 9 from both sides.
The final step is to square root both sides and this is because we want to know what the value of b is rather than the value of b^2.
The height of the right-angle triangle and the height whole triangle is 4 cm. Let’s add this value to our original diagram.
We are now able to work out the area of the whole triangle by subbing in 9 cm as the base and 4 cm as the height.
The area of the triangle is 18 cm^2.
An extension for this question is to work out what the perimeter of the triangle is. The perimeter is the distance around the outside of the shape (give your answer to one decimal place).
We have two of the three sides for our triangle; we do not have the downward sloping side on the right of the triangle. We are able to work the length of the other side by using Pythagoras’ theorem.
When we used Pythagoras’ theorem, we used the right-angle triangle on the left. We are now going to use the right-angle triangle on the right to find out what the length of the downwards sloping side
Before we use Pythagoras’ theorem, we need to find out what the length of the base is for the right-angle triangle that is on the right. We know that the base of the whole triangle is 9 cm, and the
base of the right-angle triangle on the left is 3 cm. This means that the base for the right-angle triangle on the right is 6 cm.
A sketch of the right-angle triangle on the right is given below.
We are looking for the hypotenuse of this triangle and we have the other two sides; the hypotenuse of the triangle in the formula is C. We can now sub the values that we have into the formula to see
what the value of the hypotenuse is.
Let’s add this information to the diagram.
We are now able to work out the perimeter of this shape by adding up all of the outside lengths of the triangle.
The perimeter of the triangle is 21.2 cm.
Example 2
What is the area of the right-angled triangle below? Give your answer to two decimal places.
In order to work out the area of any triangle, we need to know what the base and the height of the triangle is. For the triangle above, we only have the base and do not yet have the height. However,
this triangle is a right-angled triangle and we do have an angle in the triangle. This means that we are able to work out the height of the triangle by using trigonometry.
There are three different sides in triangles; hypotenuse, adjacent and opposite. The hypotenuse is always the longest side of the triangle; it does not matter what angle we are looking at. The
opposite and the adjacent are dependent upon the angle that we are looking at. The adjacent is the side that is next to the angle that we are using, and it is the side that is not the hypotenuse. The
opposite is the side that is opposite the angle that we are looking at.
In the triangle that we are given, we have an angle and the adjacent and we want to find the opposite. The labelled triangle is shown below:
We now need to find out which trigonometry formula triangle we will be using. The trigonometry formula triangles are given below:
We are going to be using the TOA trigonometry formula triangle. When using formula triangles (for maths or science), we cover up what we are looking for and do the operation that is left. We are
looking for the opposite, which means that we cover up the opposite.
When we do this, we see that the operation that we need to undertake is:
We are now able to sub the values in.
We can add this information to our diagram.
We now have the height of our triangle, which means that we are now able to find the area of the triangle.
We are asked in the question to give our answer to two decimal places.
The area of the triangle is 71.41 cm^2. | {"url":"https://www.elevise.co.uk/g-e-m-f-45-i.html","timestamp":"2024-11-03T20:09:31Z","content_type":"text/html","content_length":"107314","record_id":"<urn:uuid:b11dcee7-d867-49cb-a27b-fd911a36c15a>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00702.warc.gz"} |
MathGroup Archive: March 2011 [00870]
Re: Why Mathematica does not issue a warning when the calculations
• To: mathgroup at smc.vnet.net
• Subject: [mg117632] Re: Why Mathematica does not issue a warning when the calculations
• From: Richard Fateman <fateman at eecs.berkeley.edu>
• Date: Tue, 29 Mar 2011 06:50:26 -0500 (EST)
On 3/25/2011 8:33 AM, Daniel Lichtblau wrote:
... snip
>> (RJF)We've discussed this before. If you have a true function f, that
>> is a procedure whose results depend only on its argument(s), then when
>> A is equal to B, f(A) should be
>> equal to f(B). This is fundamental to programming and to mathematics,
>> but is violated by Mathematica.
> Probably by other mathematical programming languages as well. Especially in the operating world of finite precision arithmetic. It's something people often get used to (of necessity, if you work on this stuff).
I disagree that this is a common problem, although it would probably not be
allowed to discuss particular other mathematical programming languages
in this forum.
So far as I can tell, it has nothing to do with finite precision
arithmetic unless you
have some notion that "equality" means something like "close enough for
government work".
In Mathematica there are two kinds of equality: == and === . The
latter means IDENTICAL.
Consider a=SetPrecision[2,40] b=SetPrecision[2,50]. I could
sympathize with a==b being
true, but certainly not a===b, which says they are IDENTICAL, and in my
book, f[a] must equal
f[b] for all functions f. Yet let f=Precision, and you see we get
different answers.
2.000000000000000001000 == 2 is False, but
2.00000000000000000100 == 2 is True.
Such rules of the road lead to situations in which a==b but a-b is not
... snip...
>> This is not the understanding in Mathematica, where (for software
>> bigfloats)
>> 2.000000000000000000 is more like shorthand for
>> "all numbers between
>> 1.9999999999999999995
>> 2.0000000000000000005
>> "
>> (DL may be willing and able to explain what Mathematica REALLY does)
> Somethin along the lines you suggest. Your alternative approach is fine for handling of integers. It does not work so well for inputs such as 1.3 which do not translate exactly to finite binary representations. But Mathematica semantics would still be violated by upping precision, and that in no way contradicts the documentation remarks about ratcheting precision internally, as they refer to handling of exact input (N[2,10] et al are not exact, and they are what FractionalPart sees).
The simplest way of having consistent mathematical semantics in a
programming language is to have the programming language obey the rules
of mathematical semantics. Each step away from this, as imposed by the
reality of finiteness of computers etc, tends to add hazards.
That's why floating-point arithmetic is cluttered with
underflow/overflow/rounding etc, that does not occur with rational
There has to be a tradeoff in utility vs deviation from mathematics for
each such step. In my view
Mathematica adds more steps and therefore more hazards.
... snip.. about Significance arithmetic.
> I think there may be some since then. No matter. We find it works well for what we need.
> Your view, roughly, is that we picked it up off the garbage heap of numerics literature.
no, I speculate that Wolfram took the idea that is taught in Physics 101
lab about carrying significant figures when you are combining
measurements, and thought that it would be a great idea to put this into
a programming language. He might even have thought that he invented it.
(That is, a programming language with significance arithmetic). I truly
doubt that Wolfram studied any serious numerical analysis literature,
and I would certainly not expect him to study its garbage heap :)
> Perhaps so. In all the conferences I've attended, I've only once heard it disparaged (not derided, but disparaged) as a bad method.
For disparagement, one need not look any further than the wikipedia
article, at least last time I looked.
> More below. Anyway, it may be helpful to bear in mind that (unlike these endless threads), this usage is not academic to us. We rely on it for those other things, like putting kids through school. Maybe the folks who shied away from it in the literature had similar considerations? That would make for a humerous rift between theory and practice...
Tempting, but I will refrain.
>> I thought you were claiming that Mathematica had the
>> lion's share of bigfloat usage.
> No. Though I suspect it does, once you get past the 200 digit mark or so.
Past the 200 (decimal) digit mark, I think most usage would be in trying
to prove/disprove number-theory conjectures,
though there are some poorly formulated geometry problems that might
require this. Unless Mathematica gives really
direct access to libraries, it would not be a good choice. But I think
it (eventually) calls GMP and such.
> This is to exclude the slightly high precision scientific computational apps, and those possibly ubiquitous cryptography apps, which will dominate any usage of Mathematica. (Caveat: with emergence of Wolfram|Alpha, this could change. Though I doubt W|A gets tremendous high precision usage at this time).
>>> It is fairly obvious that nobody knows the extent to which bignum
>>> arithmetic is used, either within or outside of Mathematica. My
>>> claim is that those who use it in Mathematica seem to get along with
>>> it.
>> Except for the ones who post those annoying questions here:)
> I'm not sure your plural is justified. Certainly the original post in this thread was about a limitation of machine arithmetic functionality. One that, I might add, is readily overcome with Mathematica's bignum arithmetic semantics.
OK, if we are going to talk about the original post, the answer looks
To remind the patient readers, if there are any left..
N[FractionalPart[(6 + Sqrt[2])20]]
-160. (Though I get -64).
Why does this appear absurd? For a starter, that expression is
non-negative, so the fractional part of it
would have to be non-negative. Next, the fractional part p would have
to obey the relation
So the answer -160 is crazy, and FractionalPart would seem to be nonsense.
Actually, it is not total nonsense, just somewhat silly. It is
apparently computed as
FractionPart[x]= x- IntegerPart[x]
where IntegerPart works pretty hard to work for arbitrary expressions,
in this case resulting in (6 + Sqrt[2])^20 - 251942729309018542 , which
is correct if inconvenient.
now it appears that the next part of the computation N[%] is the guilty
party, and that
it might say "uhoh, there are too many digits there for me to do that
in hardware float without catastrophic loss of accuracy .." but instead
of saying
that, it just burps out an answer. Doing N[%,3] invokes the software
floats and
indeed uses enough precision to get 3 digits right IN THE ANSWER, which is
nice. Clearly 3 digits internally is insufficient.
It seems to me that a cure for all this is to allow IntegerPart and
to operate only on explicit rational and float expressions. Any float
that has more digits to the left of the decimal (actually binary) point
than precision, is equal to a particular integer, and has zero
fractional part.
Among other advantages: the result can be explained.
>> (RJF) My impression is that the actual primary use of Grobner basis
>> computations is to
>> write papers on Grobner basis computations, and the other "uses" are
>> "in principle" "demonstrations". This may have changed with a more
>> numerical
>> approach, however.
> Your impression is off insofar as industrial usage goes. For example, we use them in NSolve, exact equation solving, some polynomial algebra, and probably a few places I am forgetting.
You are right. I should have expanded my statement to include ..
"primary use of Grobner basis
computations is to write papers AND PROGRAMS ...".
Unfortunately, including GB in "NSolve" (etc.) does not, to me, really
represent an application of GB.
It represents a situation in which a well-intentioned programmer
presents an opportunity for
someone to use GB to solve a problem. With luck the program may be
applied in simple
circumstances that permit a simple solution.
But a so-called application in "commutative algebra" does not represent
an application. (Here, I am
perhaps going out on a limb saying that I won't count "an application to
pure mathematics"
as a true application.)
The rather extensive wikipedia article on the topic lists many many
applications, most of which are
of the type I do not count, or are of the form "in principle we could do
this with GB, and so it is
an application" though in fact no one could ever realistically use GB
for it except on exceedingly
toy-like instances. (Occasionally one can let a problem run for a very
long time if you only have
to solve it one time, as in some robotics problems.)
I have only limited personal experience with trying to use GB, on 2
kinds of problems, oil reservoir
simulation and the inverse problem of imaging via CAT scan or similar.
A recent (2008) PhD thesis
by M-LTorrente on Applications to the Oil Industry seems to show the
situation has not changed.
That is, casting a problem in terms of GB is unlikely to be a useful
step in realistically
solving that problem. This does not prevent the writing of a paper
exploiting a vastly simplified formulation.
While the numbers may change somewhat, consider that you could solve a
problem characterized
by 6 or 10 parameters and 3 variables using GB. Increasing the number of
variables may
increase the computation cost exponentially. In practice there will be
10,000 parameters and
200 variables.
I am not, incidentally, saying there is an alternative symbolic method
to replace GB that is much faster.
Instead, I am suggesting that if you can reframe and simplify your
question as a numerical procedure, you stand
a much better chance of getting a result.
As an analogy, one could ask for a factorization of a polynomial over an
algebraic number field, but
if all you need is an approximation of a root near a particular point,
maybe that is what you should ask for.
> Not an academic exercise, those applications. Also starting to make inroads in hybrid polynomial algebra computation e.g handling of low precision inputs. Not yet sure if that will fly, though.
I remain skeptical.
>> Are you claiming that now people are buying Mathematica to run GB
>> computations?
>> And are you claiming furthermore this could only be done in
>> significance
>> arithmetic?
> Neither. And most definitely not the first.
Then I think we may agree on this.
> I am claiming that this is extremely useful for the "bigger picture" technology it enables, a few parts of which I noted just above. Whether people buy Mathemntica for that stuff I would not know. Pretty sure it does not hurt though.
I do not fault Mathematica's Solve program for eventually trying GB; if
people ask for all solutions to polynomial
systems. I think a really good program would point out that what a user
asks for may not be what he
really wants, and enter into a dialog of some sort.
> There are some issues with IEEE arithmetic, by the way. That too is a work in progress (I'm sure this is not news to Kahan).
There are certainly issues, many of which were probably raised in the
IEEE754 standards committee deliberations;
also there is a more recent standards committee for (non-binary -- in
particular decimal) arithmetic. Curiously
some of the issues there include cases in which people would prefer
results that are less accurate.. | {"url":"http://forums.wolfram.com/mathgroup/archive/2011/Mar/msg00870.html","timestamp":"2024-11-04T01:34:37Z","content_type":"text/html","content_length":"55679","record_id":"<urn:uuid:9cd5a049-ffc2-4bf7-8565-4470ae32cc3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00883.warc.gz"} |
CBSE Class 9 Maths Chapter 14 Notes Statistics
Statistics Class 9 Notes Understanding the Lesson
There are two types of data:
We can represent the data by:
• Ungrouped and
• Grouped frequency distribution.
Data can also be represented by:
• Bar graph
• Histogram
• Frequency polygons.
Class mark of grouped data
\(=\frac{\text { lower limit }+\text { upper limit }}{2}\)
Measure of central tendencies are mean, median and mode.
where, Σf[i]x[i] = Sum of all observations
Σf[i] = Total frequency.
Median: Arrange the observations in ascending or descending order,
(i) If numbers of observations (x) are odd, then median \(\left(\frac{n+1}{2}\right)^{t h}\) terms
(ii) If number of observations (x) are even, then median \(\frac{n^{t h}}{2} \text { and }\left(\frac{n}{2}+1\right)^{t h}\)
Mode: The observation whose frequency is highest.
Relationship between mean, median and mode:
Mode = 3 Median – 2 Mean.
Graphical Representation of Data
• Bar graphs: A bar graph is a pictorial representation of the numerical data by a number of bars (rectangles) of uniform width erected horizontally or vertically with equal space between them.
Each rectangle or bar represents only one value of the numerical data and the height or length of bar indicates the corresponding value of the numerical data.
• Histogram: A histogram or frequency histogram is a representation of a frequency distribution in the form of rectangles such that there is no gap between any two successive rectangles.
• Frequency polygon: It is another method of representing frequency distribution graphically. | {"url":"https://ncert-books.com/cbse-class-9-maths-chapter-14-notes-statistics/","timestamp":"2024-11-07T04:17:03Z","content_type":"text/html","content_length":"122082","record_id":"<urn:uuid:88d71f69-fcc8-455a-a094-774cb51ba139>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00252.warc.gz"} |
Bioequivalence and Bioavailability Forum
Carrot and whip [Regulatives / Guidelines]
Dear Helmut! I'm grateful for your reply!
❝ Read the entire thread again.
Ok, my HW was not perfect
By the way, there is a typo in f' (the graph is correct), first term should be with minus:
\(\small{f'(x)=A(-k_{el}\cdot e^{-k_{el}\cdot x}+k_a\cdot e^{-k_a\cdot x})}\)}
and for AUC as the integral from 0 to a, "a" should go instead of x:
\(\small{A \left ( (e^{k_a \cdot a}-1)/k_a - (e^{k_{el}\cdot a}-1)/k_{el} \right )}\)
❝ I’m referring to the EMA’s guideline. Too lazy to Google-translate the ones in Russian.
Do not bother yourself, EAEC experts had already translated it: "Количество отобранных образцов также должно быть достаточным, чтобы обеспечить надежную оценку длительности экспозиции. Это
достигается, когда AUC
перекрывает не менее 80 процентов от AUC
Though the statement is not scientifically based it is still in the list of regulator's requirements. So in case if some problems with the "80% covering rule" we have to justify somehow that the
validity of the study should not be called into question - that's what I've asked about.
❝ Therefore, the FDA does not have such a bizarre “AUC0–t ≥ 80% AUC0–∞ rule
By the way, was this question ever discussed on the conferences on harmonization?
❝ ...71 h → AUC[0–71]...
Usually the last sampling point divides completely by 12, so I may expect 48 to be the last sampling point or may be 60, but who is so crazy to leave 71 as the last point?
But there arising another interesting situation: imagine there is a IR drug with T
equals to 18 hours. 4*T
=72 so we can leave 72 hours as the last sampling point. Which of the two parameters AUC
or AUC
should be used to choose the best strategy to confirm BE in this case? At first sight they should be equal but this is not true. The NCA software (like Phoenix) uses AUC
to calculate AUC
, so for 72 hours it would be equal to AUC
but not AUC
! For some subjects the last sample could be below LLOQ, for them AUC
with an area of triangle.
"Being in minority, even a minority of one, did not make you mad"
Complete thread: | {"url":"https://forum.bebac.at/forum_entry.php?id=21503&order=time","timestamp":"2024-11-10T03:32:20Z","content_type":"text/html","content_length":"21614","record_id":"<urn:uuid:1c12422b-a676-45a2-9d2c-f89bf421a226>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00564.warc.gz"} |
Ordinal Numbers For Kindergarten Video - OrdinalNumbers.com
Ordinal Numbers For Kindergarten Video – It is possible to enumerate infinite sets by using ordinal numbers. They are also able to generalize ordinal numbers.But before you are able to use these
numbers, you need to understand what they exist and how they function. 1st The basic concept of mathematics is the ordinal. It is … Read more | {"url":"https://www.ordinalnumbers.com/tag/ordinal-numbers-for-kindergarten-video/","timestamp":"2024-11-13T23:16:49Z","content_type":"text/html","content_length":"46344","record_id":"<urn:uuid:237946ae-91d9-4cdf-ac44-f201d2abfb20>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00255.warc.gz"} |
The list of instructions below can be useful when approaching a circuits problem. The most important concept to remember is that a minute of thinking about the solution approach will save ten minutes
of backtracking and fixing mistakes.
1. Look at the circuit to determine if it is a standard circuit type such as a voltage divider, current divider or an op-amp inverting amplifier. If so, use the standard solution to solve the
2. Otherwise, consider the nodes and loops in the circuit. If the circuit contains fewer loops, select the current loop method. If the circuit contains fewer nodes, select the node voltage method.
Before continuing, verify that the select method can be used for the circuit.
3. For the node voltage method define node voltages and current directions. For the current loop method define current loops and indicate voltage rises or drops by adding ’+’ or ’-’ signs.
4. Write the equations for the loops or nodes.
5. Identify the desired value and eliminate unwanted values using algebra techniques.
6. Use numerical values to find a final answer.
The circuit in Figure 7.22 Example problem could be solved with two loops, or two nodes. An arbitrary decision is made to use the current loop method. The voltages around each loop are summed to
provide equations for each loop.
The equations in Figure 7.22 Example problem are manipulated further in Figure 7.23 Example problem (continued) to develop an input-output equation for the second current loop. This current can be
used to find the current through the output resistor R2. The output voltage can then be found by multiplying the R2 and I2.
Figure 7.23 Example problem (continued)
The equations can also be manipulated into state equations, as shown in Figure 7.24 Example problem (continued). In this case a dummy variable is required to replace the two first derivatives in the
first equation. The dummy variable is used in place of I1, which now becomes an output variable. In the remaining state equations I1 is replaced by q1. In the final matrix form the state equations
are in one matrix, and the output variable must be calculated separately.
Figure 7.24 Example problem (continued)
Figure 7.25 Drill problem: Use the node voltage method
Figure 7.26 Drill problem: Find the state equation
The circuit in Figure 7.27 Circuit solution using impedances can be evaluated as a voltage divider when the capacitor is represented as an impedance. In this case the result is a first-order
differential equation.
Figure 7.27 Circuit solution using impedances
The first-order differential equation in Figure 7.27 Circuit solution using impedances is continued in Figure 7.28 Circuit solution using impedances (continued) where the equation is integrated. The
solution is left in variable form, except for the supply voltage. | {"url":"https://engineeronadisk.com/V2/book_modelling/engineeronadisk-64.html","timestamp":"2024-11-12T02:40:00Z","content_type":"text/html","content_length":"7063","record_id":"<urn:uuid:f904af86-0d04-4d70-afa1-80dbffffb06e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00619.warc.gz"} |
Where Does the Density Localize? Convergent Behavior for Global Hybrids, Range Separation, and DFT+U | Kulik Research Group
Approximate density functional theory (DFT) suffers from many-electron self-interaction error, otherwise known as delocalization error, that may be diagnosed and then corrected through elimination of
the deviation from exact piecewise linear behavior between integer electron numbers. Although paths to correction of energetic delocalization error are well-established, the impact of these
corrections on the electron density is less well-studied. Here, we compare the effect on density delocalization of DFT+U (i.e., semilocal DFT augmented with a Hubbard U correction), global hybrid
tuning, and range-separated hybrid tuning on a diverse test set of 32 transition metal complexes and observe the three methods to have qualitatively equivalent effects on the ground state density.
Regardless of valence orbital diffuseness (i.e., from 2p to 5p), ligand electronegativity (i.e., from Al to O), basis set (i.e., plane wave versus localized basis set), metal (i.e., Ti, Fe, Ni), and
spin state, or tuning method, we consistently observe substantial charge loss at the metal and gain at ligand atoms (∼0.3–0.5 e or more). This charge loss at the metal is preferentially from the
minority spin, leading to increasing magnetic moment as well. Using accurate wave function theory references, we observe that a minimum error in partial charges and magnetic moments occurs at higher
tuning parameters than typically employed to eliminate energetic delocalization error. These observations motivate the need to develop multifaceted approximate-DFT error correction approaches that
separately treat density delocalization and energetic errors to recover both correct density and orbital energy-derived properties.
J. Chem. Theory Comput., 12, 5931-5945 (2016) | {"url":"http://hjkgrp.mit.edu/publication/gani-where-2016/","timestamp":"2024-11-09T10:04:39Z","content_type":"text/html","content_length":"28336","record_id":"<urn:uuid:a1555c69-c2b6-445d-b745-78a4ffb92dc6>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00457.warc.gz"} |
Use the model of Towers of Hanoi in order to help students understand recursion. To demonstrate Towers of Hanoi, use three baby ring-stacking toys and the programming language Alice.
• Activity involving three baby-ring stacking toys:
□ Provide each group of students with three baby ring-stacking toys, but with only one set of rings (4 rings is enough).
□ Explain the requirements of Tower of Hanoi
☆ Goal: Move the rings from one post to another.
☆ Restrictions:
○ You can only move one ring at a time.
○ You can’t put a bigger ring on top of a smaller ring.
□ Have students document their process as they go so that they can reproduce their solution and explain it to the class.
□ Start by giving the students 5-10 minutes to try and work through the solution.
☆ If students get stuck walk them through the case of needing to move only 1 ring.
☆ If no solution is found, start by removing the top three disks from the first post and placing them on the second post. Then ask the class how they would start solving for only needing to
move the bottom disk from post 1 to post 3 (base case).
○ Write the steps to moving the disk:
■ which direction it moved,
■ how far it moved,
■ and the instructions for placing it on the last post.
○ Then move the three rings from post 2 to post 3.
■ So students see that the easiest solution is to move 3 rings to different cones, after the initial ring is moved.
○ Now, put all the rings back and start again.
□ Scaffold according to the following script:
☆ "We would want to find a way to move 3 rings, and get them to the middle cone. So if we start by moving ring 1 to cone 2, then ring 2 to cone 3, ring 1 to cone 3, ring 3 to cone 2, ring 1
back to cone 1, ring 2 back to cone 2, then ring 1 back to cone 2 (now all the rings except the largest one are on cone 2, so move ring 4 to cone 3, then move ring 1 to cone 3, ring 2 to
cone 1, ring 1 to cone 1, then move ring 3 to cone 3 (two largest rings are on the 3rd post), then move the smallest (ring 1) to post 2, and ring 2 (now alone on post 1) and be moved to
post 3, then it’s just a simple matter of moving ring 1 to post 3 and they are all in order!"
□ After students create the algorithm, you can show them this completed solution video
□ Here is a walk through of this activity from Carnegie Mellon University that talks through recursion in the solution algorithm.
• Possible Extension using Alice:
□ Below is a storyboard and sample code for the Towers method, which will animate the solution to the Towers of Hanoi puzzle.
□ Check out Chapter 9 section 2 (p. 361) of William Taffe’s book called Programming in Alice.
☆ You can provide this section to students to give them an in depth explanation of how to write and solve the Towers of Hanoi puzzle in Alice. | {"url":"https://www.csteachingtips.org/tip/use-model-towers-hanoi-order-help-students-understand-recursion-demonstrate-towers-hanoi-use","timestamp":"2024-11-06T10:54:23Z","content_type":"text/html","content_length":"43601","record_id":"<urn:uuid:24f0729b-2ae5-4182-b2c2-4d2f5b8f037d>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00687.warc.gz"} |
Multiplication Chart 1-20
Multiplication Chart 1-20 - Use these charts to teach your child multiplication. You can hang these multiplication charts in the kids’ room or classroom wall so that the kids can glance at them
during school days. This blank 20×20 multiplication table. We have several chart sizes available, so. Web maths table 1 to 20 is the basis of arithmetic calculations that are most widely used in
multiplication and division. Our website has so many blank multiplication chart worksheet of different. The multiplication chart from 1 to 20 is a visual aid that presents the multiplication. Web
different sizes of charts: If so, your quest ends. Here is the printable colorful multiplication chart.
Multiplication Chart 120 Table Free Printable in PDF
Web every number in the multiplication table 1 to 20 is a whole number. If so, your quest ends. Web here you can find multiple free printable complete or blank multiplication chart to be filled in
designs from 1 to 20 in classic or. Web this website lists multiplication tables from 1 to 20. Web these blank multiplication tables are.
Tables 1 to 20 PDF Tablas de multiplicar, Tabla de multiplicar para
Web complete these blank charts to memorize your times tables from 1 to 20. This blank 20×20 multiplication table. The larger times table charts. Web every number in the multiplication table 1 to 20
is a whole number. Blank multiplication charts are the perfect way to help your students learn.
Printable Multiplication Chart 120 Printable World Holiday
Take the printout of these multiplication chart 1 to 20 and paste this chart in the kids study room. Web here you can find multiple free printable complete or blank multiplication chart to be filled
in designs from 1 to 20 in classic or. Blank multiplication charts are the perfect way to help your students learn. Table 1 will produce.
Times Table Charts 120 Activity Shelter
You can hang these multiplication charts in the kids’ room or classroom wall so that the kids can glance at them during school days. This chart is a perfect tool in. For example, 2 × 2 = 4, 6 × 6 =.
The larger times table charts. Web different sizes of charts:
Multiplication Chart 1 20 Printable
Web this website lists multiplication tables from 1 to 20. Web maths table 1 to 20 is the basis of arithmetic calculations that are most widely used in multiplication and division. Show the different
between in each number of table. Web blank 1 to 20 multiplication table. Web these blank multiplication tables are available to print in 10×10, 12×12, 15×15,.
Printable multiplication Charts 120 (PDF) Free Memozor
Web blank 1 to 20 multiplication table. Here is the printable colorful multiplication chart. Chart from 1 to 10 (10x10), chart from 1 to 12 (12x12), chart from 1 to 15 (15x15) or chart from 1 to 20
(20x20). Take the printout of these multiplication chart 1 to 20 and paste this chart in the kids study room. Choose the.
Multiplication Chart To 100 / Division Chart 1 100 Haval / Printable
Show the different between in each number of table. The larger times table charts. Web maths table 1 to 20 is the basis of arithmetic calculations that are most widely used in multiplication and
division. You can hang these multiplication charts in the kids’ room or classroom wall so that the kids can glance at them during school days. Blank.
Multiplication Charts Printable
You can hang these multiplication charts in the kids’ room or classroom wall so that the kids can glance at them during school days. This blank 20×20 multiplication table. Web these blank
multiplication tables are available to print in 10×10, 12×12, 15×15, and 20×20 sizes. Take the printout of these multiplication chart 1 to 20 and paste this chart in.
Multiplication Chart 1 20 Printable
Web here you can find multiple free printable complete or blank multiplication chart to be filled in designs from 1 to 20 in classic or. You can hang these multiplication charts in the kids’ room or
classroom wall so that the kids can glance at them during school days. Web use a printable multiplication table to help a student learn.
Printable Multiplication Chart 120 Printable World Holiday
Web every number in the multiplication table 1 to 20 is a whole number. For example, 2 × 2 = 4, 6 × 6 =. Choose the one that suits your needs! We have several chart sizes available, so. Take the
printout of these multiplication chart 1 to 20 and paste this chart in the kids study room.
Here is the printable colorful multiplication chart. Web complete these blank charts to memorize your times tables from 1 to 20. This chart is a perfect tool in. Take the printout of these
multiplication chart 1 to 20 and paste this chart in the kids study room. Web different sizes of charts: Table 1 will produce the original. Web use a printable multiplication table to help a student
learn their times tables. Web here is the printable color multiplication chart (pdf) from the 1 time table up to the 20 times table, it's a free. This blank 20×20 multiplication table. Web here you
can find multiple free printable complete or blank multiplication chart to be filled in designs from 1 to 20 in classic or. The larger times table charts. Chart from 1 to 10 (10x10), chart from 1 to
12 (12x12), chart from 1 to 15 (15x15) or chart from 1 to 20 (20x20). Show the different between in each number of table. Web blank 1 to 20 multiplication table. We have several chart sizes
available, so. You can hang these multiplication charts in the kids’ room or classroom wall so that the kids can glance at them during school days. Web this chart is a grid containing the
multiplication tables from 1 to 20, it is also called « pythagorean table ». Our website has so many blank multiplication chart worksheet of different. Web maths table 1 to 20 is the basis of
arithmetic calculations that are most widely used in multiplication and division. For example, 2 × 2 = 4, 6 × 6 =.
You Can Hang These Multiplication Charts In The Kids’ Room Or Classroom Wall So That The Kids Can Glance At Them During School Days.
For example, 2 × 2 = 4, 6 × 6 =. Web use a printable multiplication table to help a student learn their times tables. Web complete these blank charts to memorize your times tables from 1 to 20. Web
maths table 1 to 20 is the basis of arithmetic calculations that are most widely used in multiplication and division.
Table 1 Will Produce The Original.
Blank multiplication charts are the perfect way to help your students learn. Here is the printable colorful multiplication chart. This chart is a perfect tool in. The multiplication chart from 1 to
20 is a visual aid that presents the multiplication.
If So, Your Quest Ends.
A number multiplied by itself gives the square of that number. Use these charts to teach your child multiplication. Take the printout of these multiplication chart 1 to 20 and paste this chart in the
kids study room. Web these blank multiplication tables are available to print in 10×10, 12×12, 15×15, and 20×20 sizes.
The Larger Times Table Charts.
Web here you can find multiple free printable complete or blank multiplication chart to be filled in designs from 1 to 20 in classic or. Chart from 1 to 10 (10x10), chart from 1 to 12 (12x12), chart
from 1 to 15 (15x15) or chart from 1 to 20 (20x20). Web here is the printable color multiplication chart (pdf) from the 1 time table up to the 20 times table, it's a free. Our website has so many
blank multiplication chart worksheet of different.
Related Post: | {"url":"https://chart.sistemas.edu.pe/en/multiplication-chart-1-20.html","timestamp":"2024-11-04T08:57:33Z","content_type":"text/html","content_length":"33095","record_id":"<urn:uuid:d9c15662-ecda-4751-b1b3-c61f31510ab4>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00138.warc.gz"} |
Geometry - Explanation, Major Branches, and FAQs
Geometry is one of the oldest branches of Mathematics that has been used widely since ancient times. Mathematicians have forever been fascinated by the shapes, sizes, and positions of objects like
planets, stars and the moon.
A geometer is a mathematician who is specifically concerned with Geometrical studies of shapes and figures.
From the 19th century onwards the field of Geometry has seen wide advancements that have seen several practical applications of geometrical concepts. Geometry is a compulsory subject taught in all
schools. Therefore it is important to understand the development of geometrical science over time, its concepts, and practical applications. Interestingly, Geometry is not only applied in Mathematics
but also in the field of Physics, Art, Architecture and modern-day AI technology and gaming services.
So whether you are a student or someone looking to explore career options that exploit geometrical concepts, or just someone who is fascinated by shapes and figures, this article has several
important pieces of information for you.
In the following paragraphs, you will be introduced to several important branches of Geometry including Euclidean Geometry, Non-Euclidean Geometries, Analytic Geometry, Projective Geometry,
Differential Geometry, and Topology. You will also learn about very basic aspects and components of Geometrical Mathematics. All of the points are explained in a brief and comprehensive manner. The
concepts are explained in a simple manner by our experts so that everyone can easily comprehend them.
As a bonus, simple study tips are mentioned towards the end, to help students study the subject of Geometry in the most effective method.
What is Geometry?
Geometry is a branch of mathematics that deals with the properties, measurement, and relationships of lines, angles, points, coordinates, solids and surfaces, and solids. It is the 4th math course in
high school that will help you guide through among other things inclusive of points, planes, parallel lines, lines, angles, quadrilaterals, triangles, squares, similarity, trigonometry,
transformations, circles, circumference and area. Learning geometry, you will be able to study the properties of given elements that remain invariant under particular transformations.
Major Branches of Geometry
1. Euclidean Geometry
In ancient cultures there developed a type of geometry apt to the relationships between lengths, areas, and volumes of physical figures. This geometry gained popularity being codified in Euclid’s
elements based upon 10 axioms, or postulates, from which a hundred many theorems were proved by deductive logic.
2. Non-Euclidean Geometries
Several mathematicians substituted alternatives to Euclid’s parallel postulate, which, in its modern form, reads, “Given a line and a point not on the line, it is feasible to construct exactly one
line along the given point parallel to the line.”
3. Analytic Geometry
Introduced by the French mathematician René Descartes (1596–1650), this geometry is representative of algebraic equations. This French mathematician only initiated rectangular coordinates to locate
points and to allow lines and curves to be delineated with algebraic equations.
4. Projective Geometry
The French mathematician Girard Desargues (1591–1661) initiated projective geometry to enable dealing with those properties of geometric objects that are not revised by projecting their image, or
“shadow,” on another surface.
5. Differential Geometry
In connection with practical problems of surveying and geodesy, a German mathematician introduced the domain of differential geometry. Using differential calculus, the innate properties of curves and
surfaces are distinguished. For example, he characterized that the intrinsic curvature of a cylinder is just similar to that of a plane, as can be observed by cutting a cylinder through its axis and
flattening, but not similar to that of a sphere, which is unable to be flattened without distortion.
6. Topology
Topology, the youngest and most innovative branch of geometry, emphasizes upon the properties of geometric shapes that remain unaltered upon ongoing deformation—stretching, contracting, and folding,
but not tearing.
Geometry Mathematics
Let’s get to know what you will be learning under concepts of geometry:
• Rays, Lines and line segments
• Measuring Lines
• Parallel, perpendicular, points, and planes
• Geometry definitions
• The golden ratio
• Properties and Classification of geometric shapes
• Equal parts of shapes
• Polygons and Angles with polygons
• Curves
• Solid geometry (3D shapes)
• Introduction to Angles
• Measuring and Constructing Angles
• Angles between bisecting lines
• Types of Angles
• Angles in circle
• Types of Triangles
• Triangle angles
• Triangle inequality theorem
• Angle bisectors and Perpendicular bisectors
• Altitudes, Medians & centroids
• Types of Quadrilaterals
• Proofs & angles of Quadrilaterals
Coordinate Plane
• Coordinate plane: Quadrants on the coordinate plane, quadrant 1 and 4 quadrants
• Reflecting points on the coordinate plane
• Quadrilaterals and Polygons on the coordinate plane
Area and Perimeter
• Count unit squares to find the area
• Area of rectangles, trapezoids and Area of parallelograms
• Area of triangles
• Area of shapes on grids
• Area of composite figures
• Area and circumference of circles
• Advanced area with triangles
Volume and Surface Area
• Volume of rectangular prisms
• Volume with fractions
• Surface area
• Surface and volume density
• Volume of spheres, cones, and cylinders
• Volume and surface area of Solid geometry
• Cross sections of 3D objects
• Koch snowflake fractal
• Heron's formula
• Introduction to rigid transformations
• Properties and definitions of transformations
• Dilations, Translations, Rotations and Reflections of Transformations
• Overview of Rigid transformations
• Symmetry
• Introduction and Definition of Similarity
• triangle similarity
• Solving similar triangles
• Angle bisector theorem
• Solving with similar and congruent triangles
• Solving modeling problems
• Transformations and congruence
• Triangle congruence
• Theorems in reference to triangle properties and quadrilateral properties
• Working with triangles
• Proofs of general theorems that apply triangle congruence
• Introduction to the trigonometric ratios
• Special right triangles
• Modeling with right triangles
• Solving for a side and for an angle in a right triangle using the trigonometric ratios
• Trigonometric ratios and similarity
• The law of sines and cosines
• Sine and cosine of complementary angles
• The reciprocal trigonometric ratios
• Solving problems on general triangles
• Circle basics
• Introduction to radians
• Arc measure, Arc length (from degrees), Arc length (from radians)
• Sectors
• Problem solving on Inscribed angles and Inscribed shapes
• Properties of tangents
• Area of inscribed triangle
• Standard and Expanded equation of a circle
Analytic Geometry
• Distance and midpoints
• Dividing line segments
• Distance on the coordinate plane Problem solving
• Parallel and perpendicular lines
• Distance between a point and a line challenge
Geometric Constructions
• Constructing bisectors of lines and angles
• Constructing regular polygons
• Constructing inscribed in circles
• Constructing incircles and circumcircles
• Constructing a line tangent to a circle
Pythagorean Theorem
• Introduction and Application of Pythagorean theorem
• Pythagorean theorem and distance between points
• Pythagorean theorem proofs
• Worked examples pr problem solving
How to Study Geometry the Right Way?
Geometry is a very interesting field of Mathematics. When studied the right way, you will not only enjoy learning it but also be able to apply it to various real-time situations. Here is how to make
the most of learning geometry:
• Read the concepts provided in the textbook a few times before attempting the exercise questions.
• Refer to animated videos from different sources to get an added visual aid during studies.
• Draw relevant figures for every problem.
• Understand various real-life applications of every geometrical concept. This helps to relate to the concepts better and therefore understand them practically. It will also help to apply the
concepts to application-based questions easily.
• Refer to authentic websites like Vedantu for study material, free classes, videos and personalized tuitions for all the subjects.
FAQs on Geometry
1. What is the History Behind Geometry?
The earliest known unambiguous examples began with devising mathematical rules and methods useful for constructing buildings, surveying land areas, and measuring storage containers. The Greeks around
the 6th century BCE collected and delivered this practical knowledge and generalized it in the form of the abstract subject now known as geometry. It is derived from the summation of the Greek words
geo (“Earth”) and metron (“measure”) for the measurement of the Earth. Check out the Vedantu website for a detailed and easy explanation.
2. What are the Various Geometric Concepts?
There are majorly 10 geometrical concepts. Geometry guides you through other elements that include:
• Points
• Planes
• parallel lines
• Lines
• Angles
• Triangles
• Similarity
• Trigonometry
• Transformations
• Quadrilaterals
• Circles and area.
Geometry is an important field of study that has several practical applications. From construction to designing, the understanding of all of these geometrical concepts comes in very handy.
3. What is the Geometric Concept of Finding the Right Angle?
Builders and surveyors in ancient times were required to construct right angles onto the field. The technique used by the Egyptians gained them the name “rope pullers” in Greece. This might be
because they used a rope for laying out their construction. The surveyor often referred to as a rope stretcher used 3-4-5 triangles and a plummet to carry out his work. These are continued to be used
by modern-day surveyors. However, these days there are far more convenient ways that can be used to find a right angle.
4. Which Ancient Method Helps Finding the Right Angle?
The simplest method to conduct the trick is to take a rope that is 12 units long, mark knot 3 units from one end and another 5 units from the other end, and then knot the ends together to form a
loop. However, the Egyptian scribes generalize them to obtain the Pythagorean Theorem: the square on the line opposite the right angle is equivalent to the sum of the squares on the other two sides.
5. What are some professions where Geometry is applied extensively?
There are several professions that use geometrical concepts in their respective fields. Some of these are:
• Architecture: a building is essentially a set of various shapes. Architects extensively apply geometrical concepts while designing buildings.
• Fashion Designing: this is another profession where geometry is used widely to create dresses of different designs.
• Computer-Aided Design engineer: a CAD engineer creates construction plans of different things using geometrical concepts.
• Mathematics teacher: Geometrical knowledge is essential for any Mathematics teacher.
• Animator: geometry is used extensively in video editing and animations.
• Interior designer: interior designers have extensive knowledge of shapes and geometry to design houses as per clients' preferences.
• Surveyor: geometry is used by surveyors to draw charts, maps and city plans.
• Game developer: this is another developing field where knowledge of geometry is a must. | {"url":"https://www.vedantu.com/maths/geometry","timestamp":"2024-11-02T15:44:53Z","content_type":"text/html","content_length":"412605","record_id":"<urn:uuid:61b2f120-d540-45cb-ab66-cd5c40a872d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00718.warc.gz"} |
Policies for Distributions on an Ad Hoc Basis
All of the statistical distributions in this library are class templates that accept two template parameters: real type (float, double ...) and policy (how to handle exceptional events), both with
sensible defaults, for example:
namespace boost{ namespace math{
template <class RealType = double, class Policy = policies::policy<> >
class fisher_f_distribution;
typedef fisher_f_distribution<> fisher_f;
This policy gets used by all the accessor functions that accept a distribution as an argument, and forwarded to all the functions called by these. So if you use the shorthand-typedef for the
distribution, then you get double precision arithmetic and all the default policies.
However, say for example we wanted to evaluate the quantile of the binomial distribution at float precision, without internal promotion to double, and with the result rounded to the nearest integer,
then here's how it can be done:
#include <boost/math/distributions/binomial.hpp>
using boost::math::binomial_distribution;
// Begin by defining a policy type, that gives the behaviour we want:
//using namespace boost::math::policies; or explicitly
using boost::math::policies::policy;
using boost::math::policies::promote_float;
using boost::math::policies::discrete_quantile;
using boost::math::policies::integer_round_nearest;
typedef policy<
promote_float<false>, // Do not promote to double.
discrete_quantile<integer_round_nearest> // Round result to nearest integer.
> mypolicy;
// Then define a new distribution that uses it:
typedef boost::math::binomial_distribution<float, mypolicy> mybinom;
// And now use it to get the quantile:
int main()
cout << "quantile(mybinom(200, 0.25), 0.05) is: " <<
quantile(mybinom(200, 0.25), 0.05) << endl;
Which outputs:
quantile is: 40 | {"url":"https://www.boost.org/doc/libs/1_82_0/libs/math/doc/html/math_toolkit/pol_tutorial/ad_hoc_dist_policies.html","timestamp":"2024-11-07T04:16:34Z","content_type":"text/html","content_length":"11658","record_id":"<urn:uuid:7b0c6370-feb0-4d6a-a433-6596926c093e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00330.warc.gz"} |
How to Calculate Square Footage
If you are someone who loves to do DIY projects, you will surely know how important it is to calculate the square footage of any area or property. Not measuring the area correctly can seriously
affect your project negatively. Once you calculate the total square footage of the surface area, the task becomes easy.
You can easily calculate the cost of your new DIY projects like layering a new floor or painting. If you understand the square footage, it can help you to calculate the price of your space. Many
contractors charge according to the square foot for painting or any other work.
If you know about it, the chances are you might not get scammed. Many materials like sod, tile, drywall is sold by the square foot; hence it has become essential to understand the calculation. If you
are not aware of it, don’t worry about this article, we will cover everything.
The square footage area can help you in various ways. It can make your work easy; for e.g., if you want to decide the size of the rug you are going to buy, or even if you’re going to figure out
whether the queen size bed can fit your usable footage area or not, you should know how to calculate square footage area.
Tip for you- if you want to arrange your house uniquely, knowing the furniture’s measurements can be helpful. Footprint can be described as the amount of space taken up by an item on the ground, just
like we humans have footprint furniture cover a specific area and have a footprint.
What Is a Square Foot?
A square foot is a unit used to measure Area. It is nothing but the Area inside a square that helps to measure the foot per side, length multiplied by breadth is Area.
How Do You Calculate Square Footage?
Most of the home project be it wallpaper or landscaping begins with the correct measurement. To measure the square foot of the Area, all you need is measuring tape and a method for recording the
measurement. The steps of the measurement are.
• Measure the length of side of the room which is the longest and measure it in a straight line.
• Check the length
• Now measure the length of the shortest side of the room and then measure all the four sides if it’s square space.
• Check out the area’s width
• Multiplication of the length and the breadth measure to get square feet
Calculation of the Square Footage of these Various Shapes
Calculating the square foot of any space is very easy all you need to know is the formula for it here they are
Rectangular and Square Spaces
The formula for Area of rectangle and square = l (length)x w(width)
Circular Spaces
The formula of Area for circular space = πr2(radius)
The distance from the midpoint of the space of the circle to its edge. Radius is the measure of the straight line which passes through the centre of the circle.
The value of the pi is 3.14
Triangular Spaces
The formula for Area of triangular space. = ½ b(breadth)h(heights)
This formula works for the right-angle triangle (it has a 90-degree angle). To measure the surface properly, if your Area is not a right-angle triangle, break it into one. For every right-angle
triangle, multiply the base and the height and divide it by two for total square footage to add the value.
Calculate Square Footage of Spaces with Odd Dimensions
If the Area you are measuring is not a neat geometric shape, divide the Area into smaller shapes to get an accurate measurement. Add all the values to get the total Area. For e.g., if your kitchen is
rectangular, you can measure the nook and central part of the room separately to get the actual square footage.
First, all you have to do is find the length and width of each section:
Main room measurement
10 feet x 12 feet = 120 square feet
Nook measurement –
6 feet x 4 feet = 24 square feet
Then, add these two values together:
24 square feet + 120 square feet = 144 square feet
How to Calculate Square Footage of Material to Adjust for Waste?
After cutting wallpaper tiles and other materials, odd pieces are left behind that cannot be used further. Measuring wastage becomes very important as scarcity of the material can happen, and the
project will not be finished on time. In order to avoid scarcity of material, the best thing is to order 5-10% of the extra material.
Calculate Square Footage of a Whole House
The square footage of the home can be calculated by including livable, finished areas. You will indeed not calculate an unfinished basement garage or attic. Outbuildings are also not measured. The
things which should be included in the measurement are the closet, stairwells, and hallways.
Measure each room or Area individually, e.g. like this
Measurement of Bedroom 1:
12 feet x 10 feet = 120 square feet
Measurement of Bedroom 2:
12 feet x 10 feet = 120 square feet
Measurement of Living room:
15 feet x 12 feet = 180 square feet
Measurement of Bathroom: feet to square feet ( the length and breadth are in feet, but when multiplied, we get Area which is in square feet)
7 feet x 5 feet = 35 square feet
Measurement of Kitchen:
12 feet x 10 feet = 120 square feet
Measurement Dining room:
10 feet by 10 ft = 100 square feet
Measurement of Front Hall:
4 feet by 10 feet = 40 square feet
Measurement Closet:
1 foot by 5 feet = 5 square feet
Add all the calculated values tighter here; in this case, this house is 720 square feet.
Convert Inches to Feet
For some projects, you would be required to convert inches to feet; hence you must know this anyone can do. It’s simple. All you need to do is divide the inched by 12(for converting inches to feet).
For converting from feet to inch, multiply inches by 12.
Other units of measure
To convert square footage area into yards which is comm, divide the width and length measurement by 3(one-yard consist of 3 meters)
For calculating the Area in square meters used mainly for the metric system, divide the width and length by 0.3 as one foot consists of 0.3 meters.
What is the usable square footage?
Usable square footage this term is used in real state settings to give an idea about the measurement of the usable square footage, excluding areas like hallways, stairwells and lobbies.
How to calculate gross square feet (GSF)
It refers to the measure of the total enclosed area of the building from exterior walls. Its is used as a term for describing all Area in the facility. GSF is used for budgeting, planning for
operation construction, and maintenance. It is an essential metric for maintenance and operations.
Calculate Gross Square Footage
If calculating gross square footage is very easy. All you need to do is calculate the exterior dimension of the building. Measure the width and length of the building walls. Multiply the width and
height in order to find the square footage. If your building has more than one floor, don’t forget to multiply the square footage by the number of the floors
WHAT IS NET SQUARE FEET?
How to calculate net square feet (NSF)
Net square feet is obtained by subtracting areas that are inaccessible to people from the gross square feet. The NSF would include Area that people can walk for eh classroom, office, stairwell,
hallways and closets. It would not consist of space taken up by mechanical chases closed off between floor or walls
NSF comes in handy to measure the Area during capital planning for renovation, determining circulation and mechanical area locations. NSF can be time-consuming and is not available readily in most
facilities. The spec management software is an excellent alternative to measure accurately.
Calculate Net Square Feet
Measure the building gross square footage area and subtract it from accessible space like mechanical Area and wall. The value obtained will be net square feet. The other method to calculate the net
square feet, i.e., by adding the Area of every room in the building or flat
It is the sum of the measure of all the Areas assigned to the person (occupant) for specific use. Some examples of the assignable spaces include laboratories, classrooms, study areas, offices, and
general use rooms. Residential areas, unique use rooms. People gather to accomplish a task in these areas.
The net assignable areas include the Area in which the occupant can make such a task which help them to accomplish the mission of the organisation. The areas which will not involve in net assignable
areas are elevator shaft, stairwell, closets etc. as there are areas are not assigned for the use of occupant, they are not included in NASF
The department can be allocated the best space for working if the accurate measurement of NASF is known. NASF helps to access the revenue for leased Areas. It also helps determine the staff needed
for a particular area so that there is no wastage of Area.
So now you know how to calculate the square footage area of various geometric regions. Now you might be confident before going ahead with any project s you know multiple metrics. Hope you enjoyed
reading it, and the article was helpful.
When it comes to the fact that how to calculate square footage, then we have mentioned all the points above. All you have to do is, be very careful and follow every point mentioned above.
The calculation of the square feet to feet, is not very challenging, but all it requires is to managing a few steps. Before buying a house or space or in apartments for rent in nashville tn, it is
very important to have a close look at the square footage and know all the measurements in detail.
Make sure to use all the tools correctly and get hands on the right tools to be able to measure it properly.
Moviesda have the power to move us like no other medium can. They can be frightening, come… | {"url":"https://itemverses.com/how-to-calculate-square-footage/","timestamp":"2024-11-13T05:49:23Z","content_type":"text/html","content_length":"127640","record_id":"<urn:uuid:51e71922-36ad-436d-87fb-f4cf51042a15>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00533.warc.gz"} |
Task Empty Cuboids (pus)
Empty Cuboids
Memory limit: 32 MB
We call a cuboid regular if:
• one of its vertices is a point with coordinates ,
• edges beginning in this vertex lay on positive semi-axes of the coordinate system,
• the edges are not longer than .
There is given a set of points of space, which coordinates are integers from the interval . We try to find a regular cuboid of maximal volume, which does not contain any of the points from the set .
A point belongs to the cuboid if it belongs to the inside of the cuboid, i.e. it is a point of the cuboid, but not of its wall.
Write a program which:
• reads from the standard input coordinates of points from the set ,
• finds one of the regular cuboids of maximal volume, which does not contain any points from the set ,
• writes the result to the standard output.
In the first line of the standard input one non-negative integer , , is written. It is the number of elements in the set . In the following lines of the input there are triples of integers from the
interval , which are coordinates (respectively , and ) of points from . Numbers in each line are separated by single spaces.
In the only line of the standard output there should be three integers separated by single spaces. These are coordinates (respectively , and ) of the vertex of the regular cuboid of maximal volume.
We require that coordinates are positive.
For the input data:
the correct result is:
Task author: Bogdan S. Chlebus. | {"url":"https://szkopul.edu.pl/problemset/problem/zgd-jOYv9ULJG4uDFVlNzDPo/site/?key=statement","timestamp":"2024-11-09T10:51:58Z","content_type":"text/html","content_length":"26644","record_id":"<urn:uuid:f83bcafa-4a42-446e-ac10-adec57c4fceb>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00155.warc.gz"} |
This section gives directives for the calculation of Nuclear Quadrupole Coupling Constants (NQCC). By default they are given in the EFG principal axis system.
Advanced options
Activate the calculation of NQCC in the cartesian axis system. If symmetry detection is activated, the molecule will be placed into the principal axis system of the moment of inertia tensor and
centered at center of mass so that he NQCC will be given in that axis system, following the same axis convention used in Refs. [Bauder1997, Gaul2020, Soulard2006].
Default: Use the EFG principal axis system. | {"url":"http://www.diracprogram.org/doc/release-24/manual/properties/nqcc.html","timestamp":"2024-11-10T02:15:04Z","content_type":"text/html","content_length":"6613","record_id":"<urn:uuid:ba815525-9f0d-4564-b9a5-7d25d247ae31>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00539.warc.gz"} |
How To Find The Vertices Of An Ellipse
The vertices of an ellipse, the points where the axes of the ellipse intersect its circumference, must often be found in engineering and geometry problems. Computer programmers also must know how to
find the vertices to program graphic shapes. In sewing, finding the vertices of the ellipse can be helpful for designing elliptic cutouts. You can find the vertices of an ellipse in two ways: by
graphing an ellipse on paper or through the equation of the ellipse.
Graphical Method
Step 1
Circumscribe a rectangle with your pencil and ruler such that the midpoint of each edge of the rectangle touches a point on the circumference of the ellipse.
Step 2
Label the point where the right rectangle edge intersects the circumference of the ellipse as point "V1" to indicate that this point is the first vertex of the ellipse.
Step 3
Label the point where the top rectangle edge intersects the circumference of the ellipse as point "V2" to indicate that this point is the second vertex of the ellipse.
Step 4
Label the point where the left edge of the rectangle intersects the circumference of the ellipse as point "V3" to indicate that this point is the third vertex of the ellipse.
Step 5
Label the point where the lower edge of the rectangle intersects the circumference of the ellipse as point "V4" to indicate that this point is the fourth vertex of the ellipse.
Finding the Vertices Mathematically
Step 1
Find the vertices of an ellipse defined mathematically. Use the following ellipse equation as an example:
x^2/4 + y^2/1 = 1
Step 2
Equate the given ellipse equation, x^2/4 + y^2/1 = 1, with the general equation of an ellipse:
(x – h)^2/a^2 + (y – k)^2/b^2 = 1
By doing so, you will obtain the following equation:
x^2/4 + y^2/1 = (x – h)^2/a^2 + (y – k)^2/b^2
Step 3
Equate (x – h)^2 = x^2 to calculate that h = 0 Equate (y – k)^2 = y^2 to calculate that k = 0 Equate a^2 = 4 to calculate that a = 2 and -2 Equate b^2 = 1 to calculate that b = 1 and -1
Step 4
Note that for the general equation of the ellipse, h is the x-coordinate of the center of the ellipse; k is the y-coordinate of the center of the ellipse; a is one-half the length of the longer axis
of the ellipse (the longer of the width or length of the ellipse); b is one-half the length of the shorter axis of the ellipse (the shorter of the width or length of the ellipse); x is a value of
x-coordinate of the given point "P" on the circumference of the ellipse; and y is a value of a y-coordinate of the given point "P" on the circumference of the ellipse.
Step 5
Use the following "vertex equations" to find the vertices of an ellipse:
Vertex 1: (XV1, YV1) = (a – h, h) Vertex 2: (XV2, YV2) = (h – a, h) Vertex 3: (XV3, YV3) = (k, b – k) Vertex 4: (XV4, YV4) = (k, k – b)
Substitute the values of a, b, h and k (a = 2, a = -2, b = 1, b = -1, h = 0, k = 0) previously calculated to obtain the following:
XV1, YV1 = (2 – 0, 0) = (2, 0) XV2, YV2 = (0 – 2, 0) = (-2, 0) XV3, YV3 = (0, 1 – 0) = (0, 1) XV4, YV4 = (0, 0 – 1) = (0, -1)
Step 6
Conclude that the four vertices of this ellipse are on the x-axis and the y-axis of the coordinate system and that these vertexes are symmetrical about the origin of the center of the ellipse and the
origin of the x-y coordinate system.
Cite This Article
Stansberry, Mark. "How To Find The Vertices Of An Ellipse" sciencing.com, https://www.sciencing.com/vertices-ellipse-8389447/. 24 April 2017.
Stansberry, Mark. (2017, April 24). How To Find The Vertices Of An Ellipse. sciencing.com. Retrieved from https://www.sciencing.com/vertices-ellipse-8389447/
Stansberry, Mark. How To Find The Vertices Of An Ellipse last modified March 24, 2022. https://www.sciencing.com/vertices-ellipse-8389447/ | {"url":"https://www.sciencing.com:443/vertices-ellipse-8389447/","timestamp":"2024-11-10T14:48:08Z","content_type":"application/xhtml+xml","content_length":"73367","record_id":"<urn:uuid:2d2c3ef0-bb7e-4456-8975-987ccb765a99>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00766.warc.gz"} |
Radiometric dating methods
Radiometric dating methods
Radiometric dating methods
You will remember from relative ages and close agreement among the environment's natural radioactivity. For non-living things. As presented in the carbon-14 dating methods of argon in geology
observations about the. Radiometric methods disadvantages - radiometric dating is the earth. The limits. When https://www.sumiglass.net/ have developed and so validation can generally be used for
carbon-based. An absolute age not infallible. Jump to measure the earth's history since radioactive isotopes were developed and. Potassium-40 has been in 1902 ernest rutherford suggested that the
earth was created a mineral. Hope that must be useful in the measurement of reality, which has proved the. More precise and metamorphic rocks and minerals using the earth, with radiometric dating are
radiometric dating methods. Radioactive isotopes. Figure 3: making sense of the chronometric dating methods. Radioactive properties of comparing the geological. These radioactive decay processes,
which test procedures.
Radiometric dating methods
Authigenic overgrowths on amazon. When fossil, yale university professor bertram boltwood was the text. What is credited for dating methods, but the basic underpinnings of redtube But the laboratory
procedures and others, a method is simple in use radioactive dating methods were. See also make radiocarbon dating. These methods are methods, argon-argon dating is limited to infer the earth, there
are. Figure 3: the very, uranium- lead, from japan's lake suigetsu single taken mentally dating tom holland help make assumptions. Let's take a diversity of dating methods, rocks. Geology,
radiometric dating and believed approximate age of age of the articles below for determining the basic underpinnings of a component in. Uranium-Series u-series dating methods are based on the
isochron dating. Many common minerals is not infallible. Furthermore, given the ratio of an object based on qualified orders. Rubidium-Strontium dating or radioactive dating is via radioactive dating
methods are complex. As charles lyell, a highly sensitive method is any technique which are unstable chemical ages of a mineral.
Different methods of radiometric dating
You also used in calendar years, each method for non-living things. We will explore the 4 types of rock is not correlate. Explore the 4 types of. Could be presented of radioactive decay rate of a lot
of a method of different rocks and to lead-208 pb to confirm the. All dating. Types of determining the same result to lead-208 pb to date materials based on the results. Finally, various methods -
literally. Hence if you need to enable radiometric dating, 730 years, an examination and.
What are three methods of radiometric dating
There are numerous by more stable state. Explain further what are most effective betty may have covered a lot of radiometric and radiopotassium dating methods used heavily in geochronology to date
rocks. Looking for the age. Radiometric dating, terrestrial cosmic-ray exposure dating can draw some very slowly, and potassium. An isotope is too short. Comedian top three methods of carbon: u238/
u235/th232 series saved by changes in a lot of carbon dating can be so sure of earth.
What are types of radiometric dating methods
Lastly, and the different rates. Therefore, and can be determined by comparing the concentration of other methods - literally. In use radiometric dating depends in it provides 3 sub-samples of all
the decay of zero should be accurate. Earth was only in the most widely used for the. Over 40 k. Characterization of a different.
How many radiometric dating methods are there
Seventy years old. Caveat: c-12 has been and fossils are two different kinds of these scientists. Nuclear decay of decay for example, 78 how likely are 16 green dots and to rock dating method.
Explanation: graphical methods were developed in dinosaur bones. If. Note that there g. It's often seems to be used scientific journals that can be so much of the study of a radioactive elements with
much longer.
3 methods of radiometric dating
Contrary to. Many pictures. Currently in geology rely on the process of radiometric dating methods of dating or younger than. Among the lines of dating is carbon-14, 730 years. After two basic
equation. Pdf the concentration of an artifact by. Scientists agree that. Green parent isotope is the parent isotope is based on igneous and its uncertainties, etc. We determine the chemical
properties of lead every 100 million singles: quicktime realplayer many chemical elements such dating requires that is a given. | {"url":"https://www.sumiglass.net/index.php/radiometric-dating-methods/","timestamp":"2024-11-13T09:33:24Z","content_type":"application/xhtml+xml","content_length":"10126","record_id":"<urn:uuid:ccad94c2-114f-48c8-83f9-b433b2042bbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00873.warc.gz"} |
Hidden Markov Model | FlowHunt
Hidden Markov Models (HMMs) are a sophisticated class of statistical models used to represent systems where the underlying states are not directly observable. These models are instrumental in
interpreting data where the process that generates the observations is hidden, making HMMs a fundamental tool in fields such as speech recognition, biological sequence analysis, and financial
Hidden states are the unobservable aspects of the system. In an HMM, these states evolve according to a Markov process, meaning that the future state is dependent only on the current state and not on
the sequence of events that preceded it. This property is known as the Markov property. Understanding the hidden states is crucial because they represent the actual dynamics of the system being
Observable Events
Observable events are the data points or signals we can measure. In the context of HMMs, each observation is produced by one of the hidden states. The main challenge and objective when using HMMs is
to infer the sequence of hidden states from the sequence of observed events. This inference allows for insights into the underlying process that is not directly accessible.
Transition Probabilities
Transition probabilities are a set of probabilities that define the likelihood of moving from one hidden state to another. These probabilities form a transition matrix, where each element indicates
the probability of transitioning from one state to another. This matrix is fundamental in predicting the future states and understanding the dynamics of the underlying process.
Emission Probabilities
Emission probabilities describe the likelihood of observing a particular event from a specific hidden state. These probabilities are organized into an emission matrix, where each entry corresponds to
the probability of observing a given observation from a hidden state. This component is critical for connecting the hidden states to the observable data.
Initial State Distribution
The initial state distribution provides the probabilities of the system starting in each of the possible states. It is essential for defining the starting condition of the model and is used in
conjunction with transition and emission probabilities to model the entire process.
Viterbi Algorithm
The Viterbi algorithm is a dynamic programming approach used to determine the most probable sequence of hidden states given a sequence of observations. It efficiently calculates the optimal path
through the state space by evaluating all possible paths and selecting the one with the highest probability. This algorithm is widely used in decoding problems, such as in speech recognition and
Forward Algorithm
The forward algorithm computes the probability of a sequence of observations given the model parameters by summing over all possible hidden state sequences. This is done using dynamic programming,
which allows for efficient computation and avoids the exponential complexity of evaluating every possible state sequence.
Baum-Welch Algorithm
Also known as the Forward-Backward algorithm, the Baum-Welch algorithm is an iterative method used to estimate the parameters of an HMM. It is a specific instance of the Expectation-Maximization (EM)
algorithm and is employed to find the maximum likelihood estimates of the transition and emission probabilities given a set of observations. This algorithm is crucial for training HMMs when the model
parameters are unknown.
Speech Recognition
HMMs are a cornerstone in speech recognition technology. They model the sequence of spoken words by associating hidden states with phonetic units, such as phonemes or words, and observations with
acoustic signals. This enables the system to recognize and process human speech effectively.
Biological Sequence Analysis
In bioinformatics, HMMs are applied to model biological sequences, including DNA, RNA, and proteins. They are used for tasks such as gene prediction, sequence alignment, and modeling evolutionary
processes. HMMs help in understanding the functional and structural characteristics of biological molecules.
In the financial sector, HMMs are employed to model market behaviors and for predictive analytics. Hidden states can represent different market conditions, while observations might include stock
prices or economic indicators. HMMs are valuable for forecasting and risk assessment in financial markets.
Natural Language Processing
HMMs are used in natural language processing (NLP) for tasks like part-of-speech tagging, where the goal is to assign parts of speech to words in a sentence. Hidden states correspond to parts of
speech, while observations are the words themselves. This application aids in understanding and processing human language computationally.
Example Use Case: Weather Prediction
Consider an HMM used for predicting weather patterns. In this model, hidden states might include “Sunny” and “Rainy,” while observable events are “Dry” and “Wet.” Transition probabilities define how
likely the weather will change from one state to another. Emission probabilities indicate the likelihood of observing dry or wet conditions given the current weather state. By analyzing sequences of
dry and wet days, the HMM can infer the most likely sequence of underlying weather states.
Implementation in AI and Automation
In artificial intelligence, HMMs are integral to systems that need to make decisions based on incomplete information. For instance, in chatbots, HMMs can model user intent and understand the sequence
of user inputs to provide more accurate and contextually appropriate responses. In AI-driven automation, HMMs can predict user actions and automate repetitive tasks by learning from user behavior
In conclusion, Hidden Markov Models provide a powerful framework for modeling systems with hidden states. Their ability to handle sequential data and make predictions based on observable events makes
them invaluable across various domains, including AI and automation. HMMs continue to be a vital tool for researchers and practitioners in fields where understanding and predicting complex, hidden
processes are necessary.
Hidden Markov Models (HMMs)
Hidden Markov Models are powerful statistical models used to represent systems that transition between unobservable, or “hidden,” states. They are widely applied in various fields such as speech
recognition, bioinformatics, and finance. Below are summaries of some key scientific papers that discuss different aspects and advancements in Hidden Markov Models:
1. Context Tree Estimation in Variable Length Hidden Markov Models
Authors: Thierry Dumont
This paper addresses the complex issue of estimating context trees in variable length hidden Markov models. The author proposes a new estimator that does not require a predefined upper limit on
the context tree’s depth. The estimator is proven to be strongly consistent, utilizing information-theoretic mixture inequalities. An algorithm is introduced for efficient computation of this
estimator, with simulation studies supporting the validity of the proposed method. Read more
2. Infinite Structured Hidden Semi-Markov Models
Authors: Jonathan H. Huggins, Frank Wood
The paper explores advancements in Bayesian nonparametric methods for infinite hidden Markov models, focusing on enhancing state persistence. It introduces a new framework called the infinite
structured hidden semi-Markov model, which allows for constructing models with structured and explicit-duration states. This framework is significant for applications requiring left-to-right or
other structured state transitions. Read more
3. Speaker Identification in a Shouted Talking Environment Based on Novel Third-Order Circular Suprasegmental Hidden Markov Models
Authors: Ismail Shahin
This research aims to improve speaker identification in challenging environments, such as when speakers are shouting. It introduces the Third-Order Circular Suprasegmental Hidden Markov Models
(CSPHMM3s), which integrate features from several types of HMMs. The results demonstrate that CSPHMM3s outperform other models, achieving speaker identification performance close to human
listeners’ subjective assessments. Read more
Explore how large language models detect languages, enhancing translations, chatbots, and more. Learn about NLP and AI integration.
Explore Clustering in AI: Discover types, applications, and how embedding models enhance data grouping. Perfect for AI enthusiasts!
Explore FlowHunt's AI Glossary for a comprehensive guide on AI terms and concepts. Perfect for enthusiasts and professionals alike!
Artificial Neural Networks (ANNs)
Explore Artificial Neural Networks (ANNs) and discover their role in AI. Learn about types, training, and applications across various industries. | {"url":"https://www.flowhunt.io/glossary/hidden-markov-model/","timestamp":"2024-11-03T07:18:34Z","content_type":"text/html","content_length":"95379","record_id":"<urn:uuid:3809144a-c37e-4b05-84f8-47b8a580ca7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00770.warc.gz"} |
Università degli Studi di Padova Dipartimento di Fisica e Astronomia “Galileo Galilei” Dipartimento di Ingegneria dell’Informazione
Università degli Studi di Padova
Dipartimento di Fisica e
Astronomia “Galileo Galilei”
Dipartimento di Ingegneria
Corso di Laurea Magistrale in Fisica
Entangled States on one or more degrees
of freedom and their application to
Fundamental Physics
RELATORE: Chiar.mo Prof. Paolo Villoresi
CONTRORELATORE: Chiar.mo Prof. Michele Merano, CORRELATORE: Chiar.mo Prof. Giuseppe Vallone,
Dott. Marco Tomasin
LAUREANDO: Elia Mantoan
1 Quantum information 3 1.1 EPR Paradox . . . 3 1.2 Entanglement . . . 4 1.3 Bell Inequality . . . 5 1.4 Density Matrix . . . 6 1.5 Quantum cryptography . . . 9
1.6 Quantum Dense coding . . . 14
1.7 Quantum teleportation . . . 15
2 Hyper-Entanglement: definition and generation 19 2.1 Hyperentanglement . . . 19
2.2 Non-linear Optic for Hyperentanglement . . . 21
3 Entanglement loophole 27 3.1 “Locality” loophole . . . 27
3.2 “Efficiency” loophole . . . 28
3.3 “Post selection” loophole . . . 28
4 Experimental Setup 35 5 Hyper-entangled photons Measurements and Stabilization 45 5.1 Polarization measurement . . . 45
5.2 Time-bin measurement . . . 48
5.3 Interferometer stabilization . . . 49
5.4 Time synchronization . . . 54
5.5 Calculation of Coincidences . . . 57
5.6 Quantum Tomography Measure . . . 63
6 Experimental Results 69 6.1 Bell Measurement . . . 69
6.2 “Chained” Bell Measurement . . . 72
6.3 Quantum State Tomography . . . 75
Quantum Mechanics is one of the fundamental branch of physics which deals with the world at the nanoscale (. 10−9[m] [) where we found the quantum]
particles. When experiments could not be explained by Classical Mechanics, Quantum Mechanics was born and at the beginning it was just a set of contro-versial mathematical explanations. One of the
interesting aspect is the quantum entanglement, which is a physical phenomenon where two or more particles are generated or interact in such way that a measure performed in a part of the system will
instantaneously influence the results of the measure on the remain-ing part. This means that the particles can not be described separately and the system has a non local behavior and so the quantum
state may be given for the system as a whole. During the ’80 some scientists understood that this partic-ular behavior of quantum particles can be exploited to use in communication and computation
fields. For example Richard Feynman in 1982 showed that a classical Turing machine is exponentially slower when it’s used to simulate quantum phenomena than his hypothetical quantum version. Quantum
compu-tation, quantum cryptography, quantum dense coding, quantum entanglement, quantum teleportation get great advantage compared with the correspondent classical protocol.
The goal of our experiment is first to demonstrate that a local realist model with long realist delays can not describe the physical results and second to success-fully perform an hyperentangled1
[communication in free space using photons as]
a means. In this experiment photons, entangled in polarization and time-bin, are used. We achieve to prove that these states engrave the characteristic of hyperentagled quantum states and that they
can not describe a local realist model. In order to show this, techniques such as quantum tomography and Chained Bell’s measurements are used .
Chapter 1
Quantum information
In this chapter we introduce how the entanglement was born, the Bell’s inequal-ity that is a measure which, using an entangled state, it allows to discriminate if a local hidden variable theory can
describe the physical world and the Den-sity Matrix that allows to express a quantum state. After this, we introduce a set of quantum protocols that work using entangled or hyperentangled states like
the quantum key distribution, the quantum dense coding and the quantum teleportation.
EPR Paradox
Last century saw the birth of some of the greatest scientific revolution. One of the most important revolution is sure the quantum mechanic. In the 1935 the theoretical understanding of the quantum
theory was based on the Bohr’s ideas. According to Bohr view, when a measure is performed on a quantum object, this involves a physical interaction with the measuring device that affects both
systems. This interaction is what performs the measurement “result” and, because it is uncontrollable, that can only be predicted statistically. So if the position of particle is observed, than its
momentum is uncontrollably affected1[.]
Einstein was firstly enthusiastic about the quantum theory but he had some reservation that led him in May 15, 1935 to publish “Can Quantum-Mechanics Description of Physical Reality Be Considered
The argument of EPR Paradox
In the article Einstein, Podolsky and Rosen affirmed that the physical descrip-tion of quantum mechanics can not be complete. First they gave the following definition:
Complete theory: A theory is complete if every element of the physical reality has a counter part in the physical theory.
Reality: A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. Locality: An action execute on a system A can
not affect a system B
A theory can be considered satisfactory, they affirmed, only if it is correct and its description is complete. The correctness of a theory is assessed by the pre-vision of a theory and what human
being experiences. The quantum mechanics surely fulfill these points. About the completeness they said that: “Quantum Mechanics is not complete or two non-commuting observables cannot have
simultaneous reality”.
Let’s consider the following system (spin state): |ψi = √1
2(|↑iA|↓iB− |↓iA|↑iB) (1.1) where A indicates Alice system, B indicates Bob system. If Alice executes a spin measure along z-axis and leaves |ψiin the state |↑iA|↓iB than Bob can predict
his states |↓iB and his eigenvalue along z-axis without disturbing his particle.
Now let’s think that Alice executes a measure on x-axis, in the same way as before Bob can predict his eigenvalue on x-axis. So, from the reality definition, we can affirm that both σx, bob and σz,
bob are elements of reality. Anyway from
quantum mechanic we know that
[σi, σj] = 2iijkσk (1.2)
and from Heisenberg’s Uncertainty Principle we can affirm that D
(∆A)2(∆B)2E≥ |[A, B]|
4 (1.3)
so if both σx, Boband σz, Bob have simultaneous realities, the perfect knowledge
of σx, Bob implies the perfect uncertainty of σz, Bob. This leads to say that the
quantum mechanic can not describe all elements of reality and that “Quan-tum Mechanics is not complete or two non-commuting observables cannot have simultaneous reality”.
This particular state of matter of a strongly correlated system was first treated in the EPR Paradox. However the one that first coined the word entanglement was Erwin Schrödinger that used it to
describe the correlation between two particles that interact and then separate, as in the EPR experiment. In the paper “Discussion of probability relations between separated systems” he defined and
discussed the notion and termed it “entanglement”.
A definition of entangled system is: a quantum state cannot be factored as a product of states of its local constituent. Consider a two particles system with
states |ψi and a single particle states |ϕiA and |ωiB which lie respectively in
HA and HB Hilbert spaces. The system is entangled if the whole system |ψi
can not be separated in the product of single Hilbert spaces HAand HB.
|ψi 6= |ϕi[A]⊗ |ωi[B] (1.4)
Bell Inequality
About thirty years later, in 1964, John S. Bell wrote a paper regarding what later was called Bell’s inequality. In the paper he explained a method to verify if a local hidden variable theory was
correct or wrong, performing a set of measures using an entangled state. Regarding his demonstration, he used a pair of spin 1
2 particles formed somehow in the singlet spin state, moving in
opposite direction, inside a measurement apparatus that can perform measure in selected components of the spin σ1 and σ2. If a measurement is performed
along the ~a direction, if ~σ1· ~a = +1 then, according to quantum mechanics,
σ2· ~a = −1 independently by the distance of the two particles and the time
of the measurement. This suggested that the value of measurement must be predetermined. However the wave-function does not determine the result of an individual measurement so this implies that a
more complete representation of the wave function could exist. Let’s think that this new representation is affected by a parameter λ2[. In this system then the result A of a measurement]
σ1· ~a is determined by ~a and λ and the result B of a measurement ~σ2· ~b is
determined by ~b and λ.
A(~a, λ) = ±1, B(~b, λ) = ±1 (1.5) Note that it is assumed that the result of B does not depend on the value of ~a and vice versa A on ~b. With the above assumption and if ρ(λ) represents the
probability distribution of λ then the expectation value of the product of the two components A and B is
E(~a,~b) = ˆ
dλρ(λ)A(~a, λ)B(~b, λ) (1.6) and it should be equal to the quantum mechanical expectation value which for the singlet states is
D ~
σ1· ~a ~σ2· ~b
= −~a · ~b (1.7) but this is not possible. To demonstrate this we use the CHSH inequality. We define
S(~a,~b, ~a0[, ~][b]0[) = E(~a,~b) − E(~a, ~][b]0[) + E(~][a]0[,~b) + E(~][a]0[, ~][b]0[)] [(1.8)]
E(~a,~b) = C(~a,~b) − C( ~a⊥,~b) − C(~a,~b⊥) + C(~a⊥,~b⊥) C(~a,~b) + C( ~a⊥,~b) + C(~a,~b⊥) + C(~a⊥,~b⊥)
and C(~a,~b) = ˆ dλρ(λ)1 + A(~a, λ) 2 1 + B(~b, λ) 2 (1.10)
represent the coincidences that are revealed on the receiver that is when A(~a, λ) = 1(+1).
LHV theories main idea is that the non local behavior found in the descrip-tion of Quantum Mechanics derives from some hidden variables that relate the results of the measurements. Using formula
number 4 with number 3 we obtain:
S(~a,~b, ~a 0[, ~][b]0[)] = ˆ
dλρ(λ)hA(~a, λ)B(~b, λ) − A(~a, λ)B(~b0[, λ) + A(~][a]0[, λ)B(~b, λ) + A(~][a]0[, λ)B(~][b]0[, λ)]i
≤ ˆ dλρ(λ) A(~a, λ) B(~b, λ) − B(~b0[, λ)][+ A(~][a]0[, λ)][B(~b, λ) + B(~][b]0[, λ)] ≤ ˆ dλρ(λ)2 = 2 (1.11) and therefore we obtain
|S| ≤ 2. (1.12)
Using quantum mechanics the S parameter becomes S(~a,~b, ~a0[, ~][b]0[) =]D[σ][~] 1· ~a ~σ2· ~b E −Dσ~1· ~a ~σ2· ~b0 E +Dσ~1· ~a0σ~2· ~b E +Dσ~1· ~a0σ~2· ~b0 E (1.13) and by selecting four vectors
that have an internal angle separation of π
4 the S
assumes value |S| = 2√2 that violates the above inequality.
Density Matrix
The density matrix operator is a tool developed to represent a system with numerable possible levels in a more affordable way than the wavefunction. This characteristic makes it interesting to
describe two particles entangled system. Even more, thanks to quantum tomography it can be used to compare the measured data with what theory predicts.
A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states.
Explicitly, suppose a quantum system may be found in state |ψ1iwith
prob-ability p1, or it may be found in state |ψ2i with probability p2, or it may be
found in state |ψ3iwith probability p3, and so on. The density operator for this
system is
ˆ ρ =X
pi|ψiihψi| (1.14)
where {|ψii}don’t need to be orthogonal. If we introduce an orthogonal basis
{|uii}we call density matrix the element
ρmn= hum| ˆρ |uni =
Practically the density matrix can also be defined as ˆ
ρ =X
|umi ρmnhun| . (1.16)
From this definition, since pi represents a probability, it’s possible to observe
that X i pi = 1 0 ≤ pi≤ 1, ∀i X i p2[i] ≤ 1 (1.17)
From these conditions we obtain that ˆρ will be positive semi-definite. By the third condition, Pip
i ≤ 1, it is possible to distinguish two particular cases:
• If Pip2i < 1, then we have a Mixed State, in which pi assume several
values. This represents the most common case. • If Pip
i = 1 then there’s only one pi that assumes value 1 and all the
others are 0. Because of this characteristic, it’s possible to write the density operator as
ρ = |ψi hψ| (1.18) where |ψi is the state with unitary probability. This represents a pure state.
Let us see some property of the density operator: trˆρ = 1. It can be easily demonstrated if you consider an orthogonal basis {|uni}
tr ˆρ =X n X i hun|pi|ψiihψi|uni =X n X i pihun|ψiihψi|uni =X i X n pihψi|unihun|ψii =X i pihψi| X n |unihun| ! |ψii =X i pihψi|ψii =X i pi= 1 (1.19)
where we have used the completeness relation. Since we have not put any condition on pi values, this equation holds both for mixed and pure states, so
it can not be used to distinguish them. Now let us investigate the property of the square of matrix density. In case of pure state we can write that
ρ2= (|ψihψ|) (|ψihψ|) = |ψihψ|ψihψ| = |ψihψ| = ˆρ (1.20) consequently
tr ˆρ2= tr ˆρ = 1. (1.21) Instead when we work with mixed state we obtain
ˆ ρ2= X i pi|ψiihψi| ! X j pj|ψjihψj| =X i X j pipj|ψiihψi|ψjihψj| (1.22)
by this formula we find that tr ˆρ2=X n X i X j hun|pipj|ψiihψi|ψjihψj|uni =X n X i X j pipjhun|ψiihψi|ψjihψj|uni =X i X j pipjhψj| X n
(|unihun|) |ψiihψi|ψji
=X i X j pipjhψj|ψiihψi|ψji =X i X j pipj|hψi|ψji|2 ≤ X i pi !2 (1.23)
The equality case can be reached only if there is only a unitary probability in the piensemble (pure state) or if |hψi|ψji|2= 1, ∀i, j(always pure state). Thus
we reach the conclusion that
tr ˆρ2= 1pure state
tr ˆρ2< 1mixed state (1.24) This results helpful in order to check the purity of the measured state so we define
P = tr ˆρ2 (1.25) The density matrix is also very useful to calculate the expected value of a mea-sure. In fact if O is an observable with associated operator ˆO, the expectation
value hOi is given by hOi =X i pihψi| ˆO|ψii =X i X j pihψi| ˆO|ujihuj|ψii =X i X j pihuj|ψiihψi| ˆO|uji =X j huj| X i pi|ψiihψi| ! ˆ O|uji =X j huj|ˆρ ˆO|uji = trρ ˆˆO (1.26)
Quantum cryptography
Born by the need to maintain secret a communication, cryptography is a tech-nique that secures a communication channel between two or more people over an insecure communication channel. Suppose two
users, Alice and Bob that wish to secretly exchange information over a long distance, uncompromised by the possible presence of a third party-eavesdropper Eve located somewhere along the
communication channel. The present strategy for doing this is to employ the public key cryptosystem, the most widely used example of which is the RSA (Rivest-Shamir-Adleman) protocol. The idea behind
the public key cryptosys-tem is to use for message encryption a one-way function which is easy to compute but requires to solve a hard computational problem to find the inverse function. In the RSA
protocol, such an inverse function involves the factorization of a large integer, which is hard to compute. Employing the quantum Fourier trans-form algorithm, Shor has shown that integer
factorization becomes tractable on quantum computers, which will thus threaten the security of currently used public key cryptosystems. There is however, an alternative, absolutely secure protocol
based on the private key cryptosystem, known as the one-time pad, or Vernam’s cipher. In this protocol, Alice and Bob share a private key-random string of N bits known only by them. When Alice needs
to communicate to Bob a secret message via public communication channel, she first converts it into ASCII binary string and then she uses the private key to encrypt the message and sends it to Bob.
The encryption procedure is realized by adding the random bits of the private key, one by one, to the message string using addition modulo 2 operation. The fact that the private key string is not
shorter than the message string ensures that each random bit is used only once. This guaranties absolute secrecy since the encrypted message sent through the channel does not contain any repetitive
structure and is completely random. Bob, upon receiving the encrypted message, can decrypt it by binary adding the same string of random
bits of the private key. This results in undoing the encrypting transformation, and after converting the binary string into usual alphabet, Bob can read the original message. So far the cryptography
protocol above is easy to perform, provided Alice and Bob share a common private key. The most difficult and costly part of the protocol is reliable private key distribution. Alice and Bob may have
met before and generated the key in private. But the key should be used only once and destroyed afterwords, in order not to compromise the secrecy of communication. Once they run out of random bits,
they should meet again to generate and agree on the new random string. Otherwise Alice and Bob should rely on a third party for the key distribution, but can they trust him? Fortunately, quantum
information theory has found an alternative secure way of private key distribution. In fact the most advanced application of quan-tum information today is quanquan-tum key distribution, generally
refereed to as quantum cryptography. We thus outline below three essentially equivalent pro-tocols, demonstrating slightly different, yet complimentary aspects of quantum mechanics.
Quantum bit In classical information theory, information is usually repre-sented in the form of classical bits, i.e. 0 and 1. For example it can be represented as an uncharged transistor (‘0’) and a
fully charged transistor (‘1’). A charged transistor can easily hold 108 [electrons. When we ]
con-sider a single particle, this situation changes. Let’s think the information is stored in his internal states i.e spin. For example sz = −1 represents
the value ‘0’ while sz = −1represents the value ‘1’. This seems the same
situation as in the classical case of a transistor. However there are dif-ferences. First, the system is more sensitive to perturbations ( if even a single spin value is perturbed, the information
changes). Furthermore a pure qubit state can be a linear superposition of the basis states (quan-tum mechanic allows superposition). So the qubit can be represented like a linear combination of his
|ψi = α |0i + β |1i (1.27) where α and β are probability amplitudes and in general can be both complex numbers. Thanks to this property it’s possible to represent si-multaneously several values in a
single quantum bit. For example let’s think of a four qubits system. They can be in a state that is a coherent superposition of 16 different states
|ψi =1
4(|0000i + |0001i + |0010i + |0011i + + |0100i + |0101i + |0110i + |0111i + + |1000i + |1001i + |1010i + |1011i + + |1100i + |1101i + |1110i + |1111i)
Evidently a collection of n qubits can be in a state that is a coherent superposition of 2n [different quantum states, each of which represents a]
number in binary notation. If we apply a unitary transformation of such a state, we therefore manipulate 2n [binary numbers simultaneously! This]
represents a massive parallelism in our computation which is responsible for the fact that a quantum mechanical system can solve certain problems exponentially faster than any classical system can
“No cloning theorem” [1] It’s perhaps the most important quantum mechanic tool we deal with, talking about Quantum Cryptography. It states that it’s impossible to create an identical copy of an
arbitrary unknown quan-tum state. Let’s think a quanquan-tum system (A), which we wish to copy |φiA
. In order to make a copy we use another quantum system (B) with the same Hilbert space and blank state |eiB. The blank state is independent
of |φiA (which is an unknown state). A copy unitary operator acts as a
copier if respects the following equation
U |φi[A]|ei[B]= |φi[A]|φi[B] (1.29) Now, if we apply this operator to |0i and |1i we obtain:
U |0i[A]|ei[B]= |0i[A]|0i[B]
U |1i[A]|ei[B]= |1i[A]|1i[B] (1.30) Let’s consider a general state
|ψi = α |0i + β |1i (1.31) The linearity of U implies that
U |ψi[A]|ei[B]= U (α |0i[A]+ β |1i[A]) |ei[B] = αU |0i[A]|ei[B]+ βU |1i[A]|ei[B] = α |0i[A]|0i[B]+ β |1i[A]|1i[B]
(1.32) But from formula 1.29 we have
U |φi[A]|ei[B]= (α |0i[A]+ β |1i[A]) (α |0i[B]+ β |1i[B])
= α2|0i[A]|0i[B]+ βα |1i[A]|0i[B]+ αβ |0i[A]|1i[B]+ β2|1i[A]|1i[B] (1.33) The only value that allows 1.32 to equal 1.33 is either α = 1, β = 0 or α = 0, β = 1. This is the reason why an eavesdropper
can not get the photons without being discovered. But this represents also a problem if you want to exchange key over long distances and to establish a quantum type network (quantum network). In fact
as for an eavesdropper is impossible to copy the quantum state without altering the properties as well as for a classical repeater is impossible to receive and retransmit a qubit without destroying
the state. Because of this the maximum communication distance is limited and it is determined by the loss of channel used to transmit the qubits. At the present day, the maximum distance reached is
307 km using optical fiber[2] and 144 km using free space[3].
The first protocol for private key distribution was firstly suggested in 1984[4]. In this protocol Alice and Bob establish a quantum communication channel and a classical one. Alice sends photons to
Bob through the quantum channel to test the presence of a possible eavesdropper between their communication channel. To do this she sends photons in one of the following states
|Ψ0i = |0i |Ψ1i = |1i |Ψ+i = 1 2(|0i + |1i) |Ψ−i = 1 2(|0i − |1i) (1.34)
randomly chosen. States |Ψ0i and |Ψ+i correspond to the value 0 of Alice’s
random bit, and states |Ψ1i and |Ψ−i correspond to 1. Bob measures the
photon states in one of the bases (|Ψ0i, |Ψ1iand |Ψ+i, |Ψ−i), chosen at random.
After the measure they use the classical channel (that can be a public one) to share what base they have used for each photon and keep just the photons for which Alice and Bob’s bases were the same,
discarding all the other photons. Furthermore Alice discards all the photons that, for any reason, doesn’t reach Bob. After that, they share a representative sample of the photons measured on the
same base and estimate the quantum bit error rate (QBER) usually due by the interaction of the photons with the environment. From the QBER estimation it is possible to quantify the maximum amount of
information that may have leaked to a potential eavesdropper and remove it from the remaining photons (using amplification and error correction techniques), so to obtain a communication in which the
leaked information can be kept arbitrarily low.
Now let’s consider what happens when an eavesdropper, Eve, tries to infer the private key. Eve, similarly to Bob, measures the qubits in a randomly chosen basis and records the result of the
measurement (but according to the no-cloning theorem, Eve can not clone the qubits). Then she has to generate each detected qubit in the measured state and send it to Bob, since otherwise Alice will
discard all the lost qubits and substitute them with new ones. When Eve’s basis is correct, Bob receives a qubit in the correct state. But when Eve’s basis is incorrect after projecting onto the
correct basis, Bob’s measurement yields in a half of those cases the wrong outcome and this unavoidably increases the QBER. So if the QBER of the shared sample increases we detect the presence of an
In 1992 Bennett suggested a simplified protocol, called B92[5]. In this protocol Alice uses only two non orthogonal states, |Ψ0iand |Ψ+i ,corresponding to the
values 0 and 1. Bob instead randomly decides in which base to perform the measure (R = {|Ψ0i , |Ψ1i} or D = {|Ψ+i , |Ψ−i}). If he uses the R basis, he
assigns the value 1 to his random bits, otherwise he assigns the value 0. Fur-thermore Bob uses a control string where he records the measurement result, 0 for states |Ψ0ior |Ψ+i, and 1 otherwise.
That is, Bob’s random bits correspond
to the measurement bases, rather than to the measurement results. After this, using the classical channel he shares only the control string. In order to create a common private key, both Bob and
Alice preserve the random bits that corre-spond to control bits having value 1, and discard all of the other bits. As before, due to low efficiency or external factors, some photons will not be
detected by Bob, and Alice discards her corresponding photons too. As in BB84 protocol Alice and Bob need to sacrifice a sample of photons to calculate the QBER in order to check the presence of an
eavesdropper Eve.
If Eve randomly chooses the right base she will send to Bob the correct polarization photons and he will not suspect about her presence. Nevertheless she has a 50% of possibility to choose the wrong
base. In this case she will send to Bob a different photon. In a half of those cases the value result of Bob bit would be different from the Alice’s one. This leads to increase the QBER value and by
this info Alice and Bob will conclude that Eve is there.
EPR protocol
Proposed by Ekert in 1991[6], this protocol is accomplished with the help of EPR entanglement. In this case a large number of entangled qubits with states
|ψi[00]= √1
2(|0iA|1iB+ |1iA|0iB) = 1 √
2(|+iA|−iB+ |−iA|+iB) (1.35) is shared between Alice and Bob. This could be also shared by a third part. Alice and Bob have respectively two set of measurement states ΦA[= (|Ψ]
0i , |Ψ+i , |Ψ1i)
and ΦB [= (|Ψ]
+i , |Ψ1i , |Ψ−i). The measures are divided in two groups:
1. one with ΦA i 6= Φ B j 2. one with ΦA i = ΦBj .
With the first group they make a Bell’s measurement to check for the presence of Eve. Like already said in section 1.3, Quantum Mechanic predicts
|S| = 2√2 (1.36) for these set of states, instead in classical theory or LHV theory the result would be
|S| ≤ 2 (1.37)
Since Eve, when she performs a measure, introduces an element of reality, if |S| = 2√2 then we are sure that there isn’t any eavesdropper. Instead, if we obtain |S| ≤ 2, we will suspect about the
presence of Eve. The violation of Bell’s inequalities guarantees also that E(ΦA
2, ΦA1) = E(ΦA3, ΦB2) = −1(See section 1.3
Quantum Dense coding
As already said above, quantum bits can carry more information with respect to the classical case. We show now how to send more than a bit of information interacting with a single qubit.
Figure 1.1: Dense coding setup
Let us think to use a pair of entangled particles. The underlying idea is that Bell’s states represent an orthonormal basis and because of this, it is possible to perfectly distinguish between these
states (at least neglecting experimental error). We remind that, given two photons A and B, Bell’s states are defined as follows
Ψ± = √1
2(|0iA|1iB± |1iA|0iB)
Φ± = √1
2(|0iA|0iB± |1iA|1iB) .
Firstly a third party send the first photon of the generated states |Ψ+[i][to Alice]
and the second to Bob. Alice by performing unitary transformation on the en-tangled photon can change the |Ψ+[i][state in one of the Bell’s states. Then Alice]
sends to Bob the entangled photon, thus Bob performing a Bell measurement can distinguish in which of the different Bell states the pairs are. This can be done using a physical implementation of an
Hadamard and CNOT.
Figure 1.2: Quantum teleportation setup, the first line represents the teleported qubit. The second and third lines, after the first two gates represent the en-tangled pair. U, that is reached by two
classical information (cbit), represents one of the four possible unitary transformation to perform, in order to get the teleported state.
The Hadamard gate, defined (
H |Ψ0i = |Ψ+i
H |Ψ1i = |Ψ−i
(1.39) it’s a gate that transforms the basis {|Ψ0i , |Ψ1i}in the new basis {|Ψ+i , |Ψ−i}.
Since H2[= I] [the inverse transformation H]†[= H][.]
The CNOT gate is a two qubits gate defined
CN OT (|xi |yi) = |xi (|xi ⊕ |yi) (1.40) where ⊕ represents XOR operation also on superposition states. It’s easily to see that CNOT gate can generate entangled states:
CN OT (|0i + |1i) |0i = √1
2(|00i + |11i) (1.41) CNOT gate is self-inverse since also CNOT2[= I] [. So if we apply in ]
succes-sion a CNOT and a Hadamard gates ((H ⊗ I) CNOT ) we can transform the Bell’s states in |00i , |01i , |10i , |11i that are classical bits. Practically we can successfully compress 2 bits in one
Quantum teleportation
Firstly idealized in 1993[7] and realized in 1997[8], the quantum teleportation is one of the most amazing application of quantum physics. This protocol, using 2 classical bits, can successfully send
a quantum bit from Alice to Bob.
Let us consider a simple example of teleportation. Alice has a two levels system in some unknown state
She wish to send this state to Bob using only an entangled state and two bits of classical information. Even if at the beginning this seems impossible because the measurement of a qubits destroy its
state. Even more this qubit hold an infinite amount of classical information because it lives in a continuous space. However this is possible if Alice and Bob share an entangled states.
Ψ+ = √1
2(|01i + |10i) (1.43) Half of the entangled pair is sent to Alice and the other is sent to Bob. At this point Alice let |ϕi interact with her half pairs of the entangled state. Let us consider the
total state:
|χi = |ϕi ⊗ Ψ+ = √α 2(|001i + |010i) + β √ 2(|101i + |110i) = 1 2 Φ+ (α |1i + β |0i) +1 2 Φ− (α |1i + β |0i) + +1 2 Ψ+ (α |0i + β |1i) + 1 2 Ψ− (α |0i − β |1i) (1.44)
Therefore Alice performs measurements on the Bell’s basis and obtains with probability 1/4 one of the Bell’s states. So she obtains two bits of information and sends them to Bob. By this information
Bob performs one of the four possible unitary transformation (I, σx, iσy, σz) on his qubit and recovers |ϕi.
We note that, thanks to entanglement, local manipulations modify the whole system, no matter how distant Alice and Bob are. This is a common feature of quantum protocols.
Entanglement Swapping
Realized in 1998[9], this protocol, particularly similar to the quantum telepor-tation, has the capability to entangle particles that are generated by different sources that have never interacted
Let us consider two pairs of entangled particles such as Ψ+ 12= 1 √ 2(|01i12− |10i12) Ψ+ 34= 1 √ 2(|01i34− |10i34) . (1.45)
At this point we interact half of the first entangled pairs with the second ones and we obtain the total state
2(|01i12− |10i12) ⊗ (|01i34− |10i34) =1 2 Ψ+ 14 Ψ+ 23+ Ψ− 14 Ψ− 23+ Φ+ 14 Φ+ 23+ Φ− 14 Φ− 23 (1.46)
Figure 1.3: Entanglement swapping setup. The first and second lines, after the first two gates, represent the first entangled pair. Likewise the third and fourth lines after the first two gates
represent the second entangled pair. Furthermore the second and third lines represent respectively the particle 2 and 3, in which we perform a measurements on the Bell’s basis. The first and fourth
lines represent the finals entangled pair.
Therefore if we perform a measurements on the Bell’s basis of the particles num-ber 2 and 3, the particle 1 and 4 become entangled even if they never interact. We can easily note that this protocol
is archived with a similar apparatus of that of quantum teleportation, the difference consists on the fact that the particle teleported is half of an entangled pair.
Chapter 2
definition and generation
In the previous chapter we have explained the great possibilities of quantum entanglement. In particular we have showed that quantum entangled states allow to increase the quantum channel capacity
(Dense Coding) and we have described different QKD protocols that allow to securely share a secret key. We have also shown how entanglement can be exploited to teleport quantum states, case that is
useful to entangle particles that they never interact before. In this chapter we explain what is an hyperentagled state and which physical processes are needed to produce it.
A system is defined hyperentangled when two or more particles are entangled in more than one degrees of freedom (DOFs) that is, if given |φij and |χij that
represent single particle states in the j-th degree of freedom, then we can not separate the j-th degree of freedom state |ψij:
|ψi[j] 6= |φi[j]⊗ |χi[j].
An example of hyperentangled state in polarization and time-bin DOFs is |Ωi = 1
2 |SSi + e
|HV i + eiϕ[|V Hi ,]
where |Hi and |V i represent respectively the horizontal and vertical polariza-tions instead |Si and |Li represent respectively the fast and slow arrival time of the particle from the moment of
Advantage of Hyperentanglement
Hyperentangled photons present a number of unique opportunities in quan-tum information processing. Firstly, they reside in an enlarged Hilbert space
compared to that of photons simply entangled (for example in polarization). For photons entangled in polarization and time-bin DOFs, the Hilbert space is 2 × 2 × 2 × 2 = 16 dimensional. In this kind
of system it is relatively easy to perform quantum logic on different degrees of freedom of the same photons, as opposed to qubits residing in different photons. Consequently, hyperentan-glement
enables new capabilities in quantum information processing, including remote preparation of entangled states, full Bell-state analysis, and improved super-dense coding, as well as the possibility of
quantum communication with larger alphabets. This is feasible because it is possible to extract information from each degree of freedom without destroying the correlation in the other de-grees. This
entails that working with four polarization entangled photons will give us the same information that we can get by working with two photons entangled in two different DOFs. By this information it is
easy to understand that to detect N two level particles states you need at the very least N detec-tors (one for each particles). Instead with a hyperentangled particles you can have the same
information by entangled N different degrees of freedom in the same pair of particles that needs only two detector to be revealed. So using hyperentangled quantum states it is possible to reduce the
number of detector to send the same information. Furthermore the efficiency of the whole system is increased because if η represents the detection efficiency of the detector, then the efficiency of
the overall system is given by ηm[, where m is the number of]
detectors used.
Another advantage of hyperentanglement is related to the problem of deco-herence. It is obviously more difficult to avoid decoherence as the number of photons increases. Alignment process is always a
challenging part of quantum information experiments, and concentrating more information in a small num-ber of particles makes experiments easier to be done. To sum up, we can say that
hyperentanglement allows to simplify experiments without changing the basic ideas related with entanglement phenomena.
Super-dense Coding
The super-dense coding represents an improvement of the dense coding protocol (section 1.6). It was proposed as a solution to expand the capacity to carry infor-mation by a single particle because
the linear optic doesn’t permit to distinguish completely the Bell’s states. However in 2008 it was demonstrated by the Paul Kwit group[10] that it’s possible to improve the previous limit reached in
1995 by A. Zeilinger et al. (log23 ≈1.58)[11]. The main idea behind this experiment
is to code the information in more than one DOFs. Thanks to the property of hyperentangled quantum states, it is possible to perform measurements on the Bell’s basis on each entangled DOFs. In theory
we can code the information in four orthogonal Bell’s states for each entangled DOFs. This, for an entangled states in 2 degrees, would lead a total of 4×4 = 16 different orthogonal states, so the
total information that a single photon could carry is log216 = 4bits. Using
only linear optical instruments to distinguish Bell’s states in a hyperentangled photons in polarization and OAM degrees of freedom is possible to distinguish 7
different messages. This leads to a total information of log27 ≈ 2.8bits. It must
be noted however that in the experiment of Paul Kwit group only polarization was manipulated, letting OAM states unchanged. Furthermore they didn’t try to distinguish all possible messages and taking
in account the experimental error a channel capacity of 1.63 was reached, effectively beating the previous record.
Non-linear Optic for Hyperentanglement
In order to generate hyperentangled photons pair, some non-linear optic pro-cesses are needed. In this section the main process involved in our experiment will be described.
Non-linear Optic
When we speak of linear medium, we mean a medium where the relationship between the polarization density and electric field is linear
P = 0χE (2.1)
where 0 is the vacuum permittivity, χ is the electric susceptibility of the
medium. Instead a non linear medium has a non linear relation between P and E. The polarization density P = Np is given by the individual dipole momentum p induced by the electric field and the
number density of dipole mo-mentum N. The origin of the non linear momo-mentum may reside either in p or in N. For example the relationship between p and E is linear for small electric field. But,
for large displacement, the relationship between the restraining elas-tic force of the dipole is non linear with the displacement. Then you find that the polarization density P is a non linear
function of electric field. An other way to obtain non linear behavior could reside in the number density N. Just think a laser medium where the number of atoms occupying the energy level involved in
the absorption and emission of light are dependent on the intensity of the light.
Under these conditions the function P can be expanded in Taylor series about E = 0 and by renaming the coefficient of the expansion it’s found
P = 0χE + 2dE2+ 4χ(3)E3+ · · · (2.2)
where the coefficient d and χ(3) [express the strength of the second and ]
third-order non linear effect.
When we apply this non linear polarization density to the Maxwell’s equa-tions ∇2[E −] 1 c2 0 ∂2[E] ∂t2 = µ0 ∂2[P] ∂t2 (2.3)
and conveniently writing the polarization density as a sum of a linear and non linear coefficients
P = 0χE + PN L
PN L= 2dE2+ 4χ(3)E3+ · · ·
we found ∇2[E −] 1 c2 0 ∂2[E] ∂t2 = −S S = −µ0 ∂2PN L ∂t2 (2.5) where S is regarded as radiation source.
Second-Harmonic Generation
The second-harmonic generation (SHG) is a second order non linear effect in which, by providing a harmonic electric field with frequency ω, the pumping field is splitted in two beams: one with the
same frequency of the pumping electric field, the other with a double frequency. Let us consider the electrical pumping field E(t) = <E0eiωt = 1 2 E0e iωt[+ E]∗ 0e−iωt . (2.6)
The non linear coefficient of the polarization density can be write as PN L(t) = d |E0|
+ <dE02ei2ωt . (2.7)
Consequently the source S(t) = −µ0∂2PN L(t)/∂t2contains a component at
fre-quency 2ω with complex amplitude S2ω = 4µ0ω2dE02which radiates an optical
field at frequency 2ω. Since the amplitude emitted is proportional to S2ω, its
intensity is proportional to I2ω ∝ |S2ω| 2
∝ I2
ω whereas Iω ∝ |E0|
2[. Since the]
emissions are added coherently the intensity of the second-harmonic wave is proportional to the square of the length of the interaction volume L.
By these information it’s possible to affirm that the efficiency of the second-harmonic generation is ηSHG= I2ω Iω = C2L2Iω= C2 L2 AP (2.8)
where P is the incident power of the wave, A is the cross-section area of the interaction volume and C2 [is a constant proportional to d]2 [and ω]2[.]
Spontaneous Parametric Down Conversion
The spontaneous parametric down conversion[12] (SPDC) is a three-wave mixing process where there is only a single input wave and the downconversion at lower frequency is spontaneous, that is
stimulated by vacuum random fluctuations.
In particular the Type-II SPDC is the most important tool to generate en-tangled photons. The basic idea is that the incoming photon is spitted into two photons, labeled by e and o, which are
entangled in different DOFs. To be able to allow this phenomena, energy and momentum (phase-matching) must be conserved. In Type-II crystal, the process is summarized as
Figure 2.1: BBO and circles generated by SPDC process. The polarization entangled photons are located in the intersection between the two circles. In the intersection one does not know which photon
will be H or V
this implies that the emitted downconverted photons must lie on two shifted crossing circles with horizontal and vertical polarization (see figure 2.1). Let us consider the amplitude to detecting
photons pair at conjugate space-time points (~r1, t1)and (~r2, t2)is given by
A =D ~E[H]+(~r1, t1) ~E[H]+(~r2, t2)
(2.10) where t1and t2 represent the detection time and ~EH+(~ri, ti) , (i = 1, 2) are the
Heisenberg electric field operators. In the steady state the right-hand-side of amplitude it can be expressed as
A ∝ ˆ d3r3 ˆ d3k1 ˆ d3k2U~[k]? 1λ1(~r3)U ? ~ k2λ2(~r3)U~k0λ0(~r3)fp(~r3) ~ωk0 2ε0 12 ~ωk1 2ε0 12 ~ωk2 2ε0 12 × hak0, 0| ~E (+) I (~r2, t2) ~E (+) I (~r1, t1) |ak0, k1, k2i δ (ωk1+ ωk2− ωk0) (2.11)
where ~EI(~ri, ti) , (i = 1, 2) are the interaction-picture electric field operators
and fp(~r3)is a function which describes the shape of the pump in the transverse
direction. U~[k]
0λ0(~r3), (i = 0, 1, 2) are plane-wave modes describing the
electro-magnetic field in free space with ~k1, ~k2 as the wave vectors of the signal and
idler photons, ~k0is the wave vector of the incident pump photon and λiare
po-larization indices. The initial state of the electromagnetic field |0, ak0i, consists
of a coherent state with wave-vector ~k0 and frequency ω0 (the monochromatic
the downconverted pair are given by ~ E[I]+(~r1, t1) = 1 √ 2 h ~[E]+ I,o(~r1, t1) + ~EI,e+ (~r1, t1+ τ ) i ~ E[I]+(~r2, t2) = 1 √ 2 h ~[E]+ I,o(~r2, t2) + ~EI,e+ (~r2, t2+ τ ) i (2.12)
where ~E[I,o]+ , ~E[I,e]+ are the electric fields for ordinary and extraordinary photons respectively. The interaction-picture quantized electric field ~E(+)[I] (~r, t)is given as ~ E[I]+(~r, t) = iX
λ ˆ d3k ~ωk 2ε0 12 ε~[kλ]ei( ~ k~r−ωt[)a] ~[kλ] (2.13)
where a~[kλ] is the annihilation operator for photons with wave vector ~k and λ
is the polarization index; ~ε~[kλ] (λ = 1, 2) denote the mode polarization vector.
We assume that the divergence of the pump is negligible over the length of the crystal and that the transverse shape of the pump is Gaussian
fp(~r3) ∝ e − [r2] 3x +r23y ε2[⊥] (2.14) The plane-wave mode functions are given as U~[k]
jλj(~r3) = e(
−i~kj~r3). After
per-forming the ~r3 integration in formula 2.11 and using the transformations in
formula 2.12 we obtain A ∝ ˆ ˆ dωk1dωk2sinc ∆kd 2 [h] ei(~k1~r2−ω1t2)ei(~k2~r1−ω2(t1+τ )) −ei(~k1~r1−ω1t1)ei(~k2~r2−ω2(t2+τ ))i[δ (ω] k1+ ωk2− ωk0) (2.15) where ~k1, ~k2 are the wave vectors of the
ordinary and extraordinary photons
respectively, d is the length of the non linear crystal and ∆k = k0−k1−k2 .
The two-photons state has a finite bandwidth so we let ωk1 = ωk1?+ ν and
ωk2= ωk?2− νwhere |ν| ωk?1,2and ωk?1,2are phase-matched frequencies of the
signal and idler photons. Expanding k1 and k2 to first order in ν we obtain
k1= k?1+ ν/u0
k2= k?2+ ν/ue
(2.16) where uo(ue) is the group velocity for the ordinary (extraordinary) photons.
If we consider the degenerate case in which ωk? 1,2 =
2 and we use the delta
function in equation 2.15 together with the dispersion relations for the wave numbers in 2.16 we obtain, after integrating over all t1 and t2, the modulus
square of A proportional to the following integral |A|2=
where τ1= 1 u0 − 1 ue d 2and τ2= 1 u0 − 1 ue z (2.18) and we have assumed that each detector is of equal distance z from the center of the non linear crystal. Note that, polarization entangled
photons are generate in the crossing points of the two circles. In fact, the incoming photons of the ordinary and extraordinary waves have horizontal and vertical polarization re-spectively. In the
intersections we will find either horizontal or vertical polarized photons however it is not possible to know where we will find one or the other until we measure the photons polarization. Hence
through SPDC process, we have generated the polarization entangled state
|Ψi = √1
2 |HV i + e
Chapter 3
Entanglement loophole
When a Bell’s measure is performed, for practical reasons some loopholes are permitted. These loopholes refer to circumstances in an experiment that force us to make extra assumptions for the test to
apply. Such a loophole can be used to avoid the law without technically breaking it.
The Bell inequality is derived under the assumption of local realism and it is violated by quantum-mechanical predictions, and therefore local realist models cannot give quantum-mechanical
predictions. However, during an experiment, we are no longer in the ideal setting of the Bell theorem. There are unintended and unexpected circumstances that open possibilities for local realism to
give the output of the experiment, circumstances that constitute loopholes in Bell inequality tests. Then in this chapter we show some of the main loopholes and the one that we have verified.
“Locality” loophole
Firstly presented by Bell in 1964[13], this loophole comes from the fact that locality is an explicit assumption made to derive the Bell inequality. If commu-nication between the sites is possible
(so that locality is not enforced), then local hidden variables models are possible. For example this happens when measure-ment settings are chosen and set long before an experimeasure-mental run.
Under these circumstances there is nothing that prevents a signal from traveling from one site to another and that signal can carry influence from the remote setting to the local outcome. All these
highlight the importance of changing the measure-ment settings quickly. The first experimeasure-ment to close the locality loophole was performed by Aspect, Dalibard, and Roger in 1982[14]. In their
experiment the distance from each measurement site to the source was 6 meters, or 20 ns. The settings were switched every 10 ns, and the coincidence window used was 18 ns, closing the locality
loophole as already stated.
“Efficiency” loophole
Reported by Pearle in 1970[15], this represents a very delicate problem, yet one of great importance. In an ideal experiment every experimental run is concluded with a detection in the measurement
system. Instead in a real experiment this is not true. The main causes of this behavior are that:
• if there are losses in the experimental setup then the particles could never reach the receiver,
• even the receiver has their own efficiency of detection, so one or both the particles of a entangled pair could be lost. Since there may be no indication of this in the experiment, there could be
no corresponding event recorded in the experimental data.
This unexpected circumstance makes the original Bell inequality derivation in-valid.
The first experiment to close the efficiency loophole was performed by Rowe in 2001[16]. In his experiment he used two ions in a trap and every experimental run gives output data, so it is free of
the efficiency loophole. Sadly the ions are 3 µm apart while a measurement lasts 1 ms, so the experiment own the locality “loophole”. A more recent experiment performed in 2015 employs an event-ready
scheme that enables the generation of high-fidelity entanglement between distant electron spin (a spatial separation of 1.3 km) and this represent the first loophole-free violation of a Bell
“Post selection” loophole
This represents the loophole that we have tested during the thesis in order to verify if a local realist model with long realist delays can describe the physical results of time-bin entangled
photons. To create a time-bin entangled pair we need to use an unbalanced Michelson interferometer where the path difference is larger then the coherence length, before the SPDC crystal. In this way
we create a two levels system with two detection events: one at the “early-setting readoff” event for the early detection, and one at the “late-setting readoff” event for the late detection. In order
to make indistinguishable the two levels, at the measure side, two other interferometers with the same path difference must be used: one for Alice and one for Bob (see figure 3.1). In this kind of
setup, even in the ideal case of two-particle with enough spatial separation between the local measurements (which closes the locality loophole), and perfect detection efficiency (which closes the
detection loophole), there are local hidden variable models that reproduce the quantum predictions for the violation of the CHSH inequality. This happen because in our setup, the detection of a
coincidences event (two photons that are detected at the same time) could depend on local setting.
To show this we use the local hidden variables model introduced in 1999 by Aerts et al.[18]. Their model need two hidden variables: an angular coordinate
Figure 3.1: Generic setup to create and measure a time-bin entangled source. φ and ψ represent the shifts introduced by the local setting on Alice and Bob interferometer. They can be controlled
moving the position of the mirrors of some nanometers on the long arms of the Alice/Bob interferometers as shown in figure. +1E/L represent the early and late events that after have chosen
the short or long path in the first passage through the beam splitter, during the second one have chosen the path that leads to the SPDC crystal. Instead −1E/L represent the early and late events
that after have chosen the short or
long path in the first passage through the beam splitter, during the second one have chosen the path that leads to the detector. Since this setup use a Michelson interferometer, only −1X events can
be detected, if instead we use a
Mach-Zehnder interferometer, then we can detect both +1Xand −1X. ±1Eand
±1L represent respectively the early and late event before the second passage
through the beam splitter. If we reject each pairs of events whose registration times differ by the time delay between the short and long light path of the interferometers then we are sure that early
events have chosen the long arm of Alice/Bob interferometers and the late events have chosen the short arm of Alice/Bob interferometers.
Figure 3.2: LHV model for Alice detector. The shifted value of the angular hidden variable, θ0 [= θ − φ][, and r, determine if the particle are revealed (-1) or]
it’s sent back to the SPDC crystal (+1), and whether the particle is detected early E or late L. The lower curve in the left side of the chart is given by
π 8sin θ
0[, and the shape of the other curves are of similar form. To be noted that]
the symbols have the same meaning as in figure 3.1
θ ∈ [0, 2π] and an additional coordinate r ∈ [0, 1]. This variable are uniformly distributed in (θ, r). Each pair of entangled particles is then described by a definite point (θ, r). On Alice
detector, the measurement results are decided by the hidden variables (θ, r) and the local setting (φ) of the apparatus that is the phase introduced by Alice’s interferometer. When a photon pass
through the interferometer the variable θ is shifted by the local setting (θ0 [= θ − φ][).]
The results of the measurement are read off in figure 3.2. Likewise on Bob detection a shift is introduced (θ00 [= θ + ψ] [, where ψ is the phase introduced]
by Bob interferometer). In figure 3.3 is shown the results obtained during a measure. The single-particle detection probabilities follow the predictions of Quantum Mechanics since in figures 3.2 and
3.3 all the possibility have the same probability. The coincidence probabilities are determined by overlapping the two figures 3.2 and 3.3 with the proper shifts. We now provide the estimate of the
expectation values of the net coincidence probability:
P (−1; −1(coinc)|φ, ψ) = P (−1E; −1E|φ, ψ) + P (−1L; −1L|φ, ψ) = 2 2π ˆ ψ+φ 0 π 8sin(θ)dθ = 1 8[1 − cos(ψ + φ)] (3.1)
Figure 3.3: The measurement result at the Bob detector as a function of the shifted hidden variables. The symbols have the same meaning as in figure 3.2.
where E and L subscripts represent respectively the early and late coinci-dences. It is easy to verify that this model also gives the correct prediction for the other detection events. Using this
model let us see what happen when we perform a Bell inequality. To study this model we introduce the following parameters: td that is the detection time, tret that is the time in which light
reaches the detector from the location of the phase shifter (interferometer), ∆T that is the time delay between the short and long light path of the interferom-eter. Each pairs of events whose
registration times differ by ∆T are rejected, moreover each pairs of events which do not have the feature that the phase setting at td− ∆T − tretwas φ0on Alice detector and ψ0on Bob detector. If we
discard the latter event we ensure that the hypothetical LL subensemble within the remaining data is independent of the phase settings at td− tret. Then, if
local realism holds, the Bell CHSH inequality applies to this LL subensemble |ELL(φ1, ψ1) + ELL(φ2, ψ1) + ELL(φ2, ψ2) − ELL(φ1, ψ2)| ≤ 2 (3.2)
where the phase are taken at td− tret. This is valid since each of the correlation
functions is an average on the same ensemble. If the ensemble is a function of the phase settings td− tret, then the bound would be higher. Indeed, the
remaining EE subensemble may still depend on the phase setting at td− tret
even after this selection, and we only have
Of all events that submit this inequality, again half are EE and half are LL, so that Ecoinc.(φ, ψ) = 1 2EEE(φ, ψ) + 1 2ELL(φ, ψ) (3.4) thus we found that the outcomes of the module of S parameter
(see formula 1.8) for a local realist model with long realist delays obey
|S| ≤ 1
2(2 + 4) = 3 (3.5) where the bounds 4 represent the early coincidences and the bound 2 represent the late coincidences. But this is larger than the maximal quantum prediction 2√2. To establish a
better bound we need to use the so-called “chained” Bell inequalities.
“Chained” Bell inequality
The “chained” Bell inequality is a “chained” extension of the Bell-CHSH inequal-ity. For example if we calculate the bound for a Bell inequality at 6 parameters for a local realist model regarding
the LL subensemble is
|ELL(φ1, ψ1) + ELL(φ2, ψ1) + ELL(φ2, ψ2) + ELL(φ3, ψ2)
+ELL(φ3, ψ3) − ELL(φ1, ψ3)| ≤ 4
(3.6) If local realism holds formulas 3.3, 3.4 and 3.6 suggest that
|Ecoinc(φ1, ψ1) + Ecoinc(φ2, ψ1) + Ecoinc(φ2, ψ2) + Ecoinc(φ3, ψ2)
+Ecoinc(φ3, ψ3) − Ecoinc(φ1, ψ3)| ≤
2(4 + 6) = 5
(3.7) Again the bound is the mean value of the trivial bound 6 for the early coincidences and 4 for the late coincidences. Now we consider the following 6 directions π/6 apart in a plane
φ1 0 ψ1 π/6
φ2 −π/3 ψ2 π/2
φ3 −2π/3 ψ3 5π/6
using the quantum mechanic we obtain
|Ecoinc(φ1, ψ1) + Ecoinc(φ2, ψ1) + Ecoinc(φ2, ψ2) + Ecoinc(φ3, ψ2)
+Ecoinc(φ3, ψ3) − Ecoinc(φ1, ψ3)| =
|cos(π/6) + cos(π/6) + cos(π/6) + cos(π/6) + cos(π/6) − cos(5π/6)| = 6 cos(π/6) ≈ 5.20 > 5
Because no local realist model with long realist delays give a value greater than 5, with this it is possible to perform a test. So, even if the standard Bell inequalities are not sensitive enough to
show a violation of local realism
in the experiment, because their bound is raised by the noise introduced by the early-early subensemble, using a “chained Bell inequality” a violation of the local realism can be found even with this
noise included. However, this violation is minimal, therefore a measure with high visibility is needed. To solve this problem, studying the “chained” Bell inequality at a greater number of parameter
[19] like the previous steps, we found
# of parameters Realism bound Quantum prediction Critical visibility 4 3 4 cos(π/4) ≈ 2.828 > 100% 6 5 6 cos(π/6) ≈ 5.196 96.23% 8 7 8 cos(π/8) ≈ 7.391 94.71% 10 9 10 cos(π/10) ≈ 9.511 94.63% 12 11
12 cos(π/12) ≈ 11.59 94.90% 2N ≥ 14 2N − 1 2N cos(π/(2N )) incr. with N It’s easy to understand that considering the visibility, the best conditions in order to violate “chained” Bell inequality are
met using 10 parameters.
Chapter 4
Experimental Setup
In this experiment we want to create an hyper-entangled state in time-bin e polarization degrees of freedom. As we explain in the previous chapters, to ensure this result we use a Michelson
interferometer and a type II spontaneous parametric down conversion process. Since we want to perform a “chained” Bell measurement and a tomography measurement on a free-space channel we need a pair
of polarimeters and Michelson interferometers which can manipulate the polarization and time-bin degrees of freedom as well as a free space channel. This last part is explained in this and 5
The experimental setup (figure 4.1) is composed of:
Laser Source
Mira-HP Coherent is a commercial ultrafast Ti:Sapphire oscillator (figure 4.2). It works at an average power of 3.5 W in femtosecond mode and can also work at picosecond mode. Furthermore it has a
tunable wavelength and it is designed specifically to be pumped by the Verdi G18 laser. Verdi G18 Coherent is a continuous wave green pump laser that uses a semiconductor chip as the active medium in
place of a conventional laser crystal. The oscillating wavelength is 1064 nm intracavity doubled to produce a 532 nm green output beam. It has a spectral purity higher than 99 %, an output power of
18 W , a linear vertical po-larization and a spatial mode T EM00. The following table reports the Mira-HP
characteristics working at 800 nm when pumped with Coherent Verdi G18 laser (figure 4.2). In our experiment the Mira-HP is used at a wavelength of 808 nm in femtosecond mode principally to increase
the SHG efficiency. As shown later this is also useful because working in femtosecond mode, the difference length of the two arms of the interferometers to produce the time-bin entangled pairs is
Figure 4.1: Setup of the hyperentanglement source in time-bin and polarization DOFs. The red line represents photons with wavelength at 808nm instead the blue line represents photons at 404 nm.
Figure 4.2: Laser source. On the left we can see the Mira-HP Coherent Laser, on the right we found Verdi G18 Coherent Laser source.
Output Power >3.5 Pulse Width <130 fs Tuning Range (nm) 700 to 1000 Repetition-Rate (MHz) 76 Noise (%) <0.1 Stability (%) <3 Beam Diameter (mm) 0.8 Beam Divergence (mrad) 1.5
Spatial Mode T EM00
Polarization Horizontal Table 4.1: Mira-HP Characteristic
SHG Crystal
Since the semiconductor detectors have a high efficiency around 800 nm a BiBo crystal is used as Second harmonic generation. By this crystal we obtain 404 nm photons that after the SPDC crystal
become at 808 nm. In order to increase the conversion efficiency (formula 2.8) we reduce the cross area of the active volume focalizing the laser beam using a lens with 75mm of focal. The SHG
efficiency is found to be about ηSHG ≈ 37%. Since there isn’t an unitary efficiency, we
collect this light and apply a delay so that can be time distinguished from the entangled light. We call this light calibration light and it’s used in order to stabilize the interferometer.
Pump Michelson Interferometer
An unbalanced Michelson’s interferometer (figure 4.3) is used to create the time-bin entangled photons. In order to create the time-time-bin entangled photons, we need to create a time
indetermination of the pulse position. This is done by strongly unbalanced Michelson’s interferometer. To be sure to generate a two levels system, the path difference of the interferometer must be
longer than the pulse length in order to avoid a single-photon interference. After the interfer-ometer we obtain photons in the state
|ϕi = √1
2 |Si + e
iφ[|Li] [(4.1)] where |Si and |Li represent the short and the long path respectively and φ = 2πD−d
λ where D and d indicate the long and the short path length and λ
repre-sents the wavelength of the beam. All this is placed after a lens with 200 mm focal which is useful to increase the Rayleigh range so that the width of the beam that follows the long arm and
the one that follows the short arm are very similar. This property guarantees that the power measured by the beam which follows the long arms is equal to the beam which follows the short arm. It can
be noted using the information that we get until now that a path differ-ence between the two arms, greater that 39 µm is enough to be sure that there isn’t single-photon interference. However this
can not be distinguished by our detectors and electronics, so a path difference of 50 cm is used.
SPDC crystal
The SPDC crystal (see figure 4.4) is a BBO crystal which allows the Spontaneous Parametric Down Conversion by which we obtain the hyperentangled pair. The beam is focalized on the crystal with a 500
mm focal lens. It splits the incoming photon in two photons transforming the initial state in equation 4.1 in a state entangled in different degrees of freedom. Among the different DOFs we take
Figure 4.3: The top image shows a Michelson’s Interferometer scheme. The bottom image represents its implementation.
Figure 4.4: In this image we can see the SPDC crystal, the walk off crystals, and the polarimeters. The blue and red lines represent respectively the photons at 808 nm and 404 nm.
advantage of polarization (see section 2.2.3) and we obtain |Ξi = 1 2 |SiA|SiB+ e iφ[|Li] A|Lib ⊗ |HiA|V iB+ e iϕ[|V i] A|Hib [(4.2)]
Walk-Off crystal
Since BBO is a birefringent crystal, different polarizations get different delays. This represents a problem because the BBO crystal length is enough to intro-duce a delay between the polarizations
greater than the coherence length of the impulse (39 µm). Thanks to this it’s possible to distinguish the different polar-ization states and this destroys the entangle states. To get over this
difficulty, we insert, after SPDC, on both channels, a Walk-Off Crystal (figure 4.5). The Walk-Off crystal (see figure 4.4) is practically another BBO crystal with half length of the SPDC crystal
that is rotated of 90° respect to the BBO crystal in order to invert the ordinary and extraordinary axes. The half length is due to the fact that the incoming photons split, at half of the SPDC
crystal on average. The inversion of the ordinary and extraordinary axes is done in order to correct the initial delay.
In order to verify the produced polarization entangled state we need an instru-ments that is able to perform measures of photons at several polarization states. To measure the polarization we use a
polarizing beam splitter (PBS) that di-vides the incoming beam in two beams: one with horizontal polarization and
Figure 4.5: Correction obtained with Walk-Off Crystals.
one with vertical polarization. Furthermore in order to measure other type of polarization before the PBS a λ
4 and a λ
2 wave plates are used to convert all the
possible polarization states in vertical polarization states (see sec. 5.1). Using a motorized rotation stage PRM1Z8 of Thorlabs (see image 4.4) the axes of the just mentioned plates are rotated to
the desired position in order to find the wanted stage.
Free-space channel
To simulate a free-space propagation of the photons sent to Bob, we create a free space channel (figure 4.6) using some mirrors, in order to create a long light path maintaining small size of the
device. A set of two lens are used to ensure a good focalization of the beam when it reaches the Bob detector.
Alice and Bob Michelson Interferometers
In order to measure different type of time-bin entangled states two unbalanced Michelson’s interferometers are used to project the state in the wanted basis (figure 4.8). The path differences must be
the same of the first interferometer at least of the coherence length of the laser impulse. The length of the long
Figure 4.6: Free-space channel
arms are controlled using two SmarAct linear actuator that allow nanometric movements coupled with two Modular Control Systems (MCS).
Figure 4.7: Linear actuator driving principle.
The linear actuators use the stick-slip principle (figure 4.7) to perform fine steps down to 50 nm (step mode). Moreover, with a slow elongation of the piezo element the slide can be moved with
sub-nanometer resolution within a range of about 1.4 µm by default and up to several µm on request (scan mode). Furthermore the light passing through the long e short arms can be stopped in order to
select the light coming only by one arm. This, as explained in sec. 5.2 and 5.3, can be used to project the time-bin entangled state in a selected state and retrieve the result.
In order to detect the single photons, Single Photon Avalanche Diodes (SPAD) are used, provided by Excelitas SPCM-AQRH, Single Photon Counting Mod-ule. It uses a unique silicon avalanche photodiode,
achieving a peak photon detection efficiency greater than 70% at 700nm over a 180µm diameter with uniformity over the full active area. A TTL level pulse is generated for each photon detected and the
signal is available at the BNC connector at the rear of the module. The signal should be terminated into 50 Ω. The photodiode is both thermoelectrically cooled and temperature controlled, ensuring
stabilized performance despite ambient temperature changes. This detector has a jitter of 300-400 ps and their signals are collected by two QuTau. The QuTau is a time-to-digital converter that for
each incoming event stores a 64 bit value of time arrival with a time bin size of 81 ps.
Chapter 5
Hyper-entangled photons
Measurements and
In the previous chapters we discuss the experimental setup and the production of hyperentangled photons pairs. In this one we’ll explain how to perform mea-surements on the two entangled degrees of
freedom, the issue of the experimental setup and their solutions, in particular the instability of the interferometers.
Polarization measurement
When dealing with polarized states, Jones arithmetic[20] is commonly used. Jones vectors are used to characterize a monochromatic plane wave that travels in the z direction. Let us consider the
electric field
E(z, t) = <nA exphiωt −z c
(5.1) where the complex envelope A = Axx+Aˆ yyˆis a vector with complex components
Ax = axeiϕx and Ay = ayeiϕy. The Jones vectors represent a way to express
these quantities in form of a column matrix J = Ax Ay (5.2) Since we are working at single photon, we don’t care of the intensity value1 [so]
we express the Jones vector with normalized intensity. In the following table we provide the Jones vectors for some particular polarization states. | {"url":"https://123dok.org/document/yd7em0rl-universit%C3%A0-dipartimento-astronomia-galileo-galilei-dipartimento-ingegneria-informazione.html","timestamp":"2024-11-02T04:27:06Z","content_type":"text/html","content_length":"218748","record_id":"<urn:uuid:922c0d90-bf83-498d-bd09-a5425d366fbb>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00163.warc.gz"} |