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A primer to understanding statistical models such as the Oxford coronavirus study 3A primer to understanding statistical models such as the Oxford coronavirus study 3 This is Part 3 of a series of posts on understanding statistical models. This series is motivated by the media frenzy over the Oxford study (link) that claimed the majority of Britons have already been infected with the novel coronavirus by March 19, 2020. In Part 2, we learned that the type of model built by the Oxford research team includes unknowable quantities, such as the date the virus first appeared in the UK, and the cumulative proportion of people already infected. These are unknowable in the sense that no ground truth will ever emerge. Because the values of these quantities are used in decision-making, we introduce mathematical modeling to estimate what are likely values. As Part 2 ended, I was in the middle of drawing connections between the conclusions of the study and the outputs of the statistical modeling. Before answering that question, I must first explain the Bayesian approach of modeling, which is the main topic of this post. Here is our full program: Part 1: Does the model explain or does it predict? (link) Part 2: Overcoming the inutility of raw data (link) Part 3: What is a Bayesian model? (this post) Part 4: How is modeling vulnerable? (link) Part 5: Models = Structure + Assumptions + Data (link) Part 6: Key takeaways (link) EXTRA: Commentary on the data graphics in the Oxford study (link) - New 4/15/2020 Part 3: What is a Bayesian model? This is a good place to describe a Bayesian model. The Oxford modeling strategy uses an input for the date of introduction, which is when the novel coronavirus first landed in the UK. It also harnesses several other inputs, such as the proportion of people at risk of severe disease and the reproduction number (R0), which is how many people are infected by the average infectious person at the start of the epidemic. A set of equations links all the inputs to the output, in this case, the cumulative death counts. The only quantity directly measured is the output, the daily cumulative death counts. We have official statistics for these. The equations then allow us to compute the output if we provide the inputs. But none of the inputs have available values. (Other available data such as cases or severe cases are not utilized in the Oxford study.) What to do? If we think of these inputs as buttons and levers on a machine, we try applying different settings, and evaluating the outputs to see if they meet acceptance requirements. The goal of modeling is to find which combination of inputs can generate the observed death counts, if only approximately (assuming blatantly that the set of equations linking them holds up). For Bayesians, getting the process started with an initial setting is known as setting the "priors". A prior on the input is an assumption of its value based on our best knowledge. (Strictly for the nerds, it's an assumption on the distribution of values, not just on the average value.) For the reproduction number (R0), the grey model assumes a prior average of 2.25 (a normal distribution with mean 2.25 and standard deviation 0.025). The researchers cited a few scientific papers as the basis for this assumption. As a reminder, what I call the grey model is the one that produced outputs that were widely publicized by the Financial Times (link). Throughout the preprint's charts, this model is given a grey Now, while the model is being fitted to replicate the death counts, all inputs will be varied from their initial settings. When the grey model is complete, we look at the R0 and learn that its final setting (posterior mean) is still around 2.25. [The researchers didn't provide the actual numbers but this can be inferred from Figure 1(C).] Think of the posterior value of R0 as a compromise between the existing state of knowledge and the wisdom of the new data. The lack of movement suggests that the observed data adequately line up with the initial guess of R0. Now back to the question: How does the modeling results support the study's conclusions? Recall that the Financial Times report focused on two key metrics computed by the Oxford models: the date of introduction, and the cumulative proportion of infected. For the date of introduction, the prior setting was chosen to be random (a uniform distribution). By setting this prior, the researchers admitted no special knowledge of this input. The technical term is a flat or uninformative prior. So here the observed data did all the work, and when the grey model was completed, the researchers found the date of introduction to be roughly 4 days before the first report of infection. When the press breathlessly reported that the coronavirus has been spread around quietly for over a month before the first reported death, that's the number from the Oxford study to which they are reacting. That's a silly way to frame it; it's more appropriate to say four days before the first reported case. Even if the Oxford study didn't exist, the official statistics depict a gap of 34 days between the first case and the first death. It is quite a stretch to claim the coronavirus has been spreading around "quietly" in these 34 out of those 38 days. Oops, I may have killed the fun of sensational journalism yet again. As with anything statistic, there is a margin of error associated with the number 4. Again, the preprint didn't contain actual numbers; judging from Figure 1(E), I think a reasonable range is from 0 to 8 days. (If you are looking at the chart, I will get to the red and green models eventually. Just focus on the grey model for now.) Notice that the cumulative proportion of infected - the quantity in the Financial Times headline - is hyper-sensitive to this date of introduction because the grey model suggests an extremely fast propagation through the population. Figure 1(B) shows how half the country got infected in a mere 20 days. Shifting the start of this curve by a few days changes the story quite a bit. If the curve is shifted forward by 4 days, then of course, the 50% mark would be reached four days later but four days prior, the proportion of infected would be 30% lower than claimed! [As you can see, the lines drawn on these charts are so terrifyingly thick that one can't get precise in discussing the findings.] In Part 3, I explained how the Oxford team constructed their Bayesian models, and how the two key quantities - the date of introduction and the cumulative proportion of infected - arose from the modeling, and plugged straight into the Financial Times report. In Parts 4 and 5 (link), I point out where this type of statistical modeling is vulnerable. I'm not bashing statistical modeling - I'm a believer and an insider. But I think you should be a smart consumer of such models, and I like to give you some pointers. Continue to Part 4: How is modeling vulnerable? You can follow this conversation by subscribing to the comment feed for this post. Recent Comments	{"url":"https://junkcharts.typepad.com/numbersruleyourworld/2020/04/a-primer-to-understanding-statistical-models-such-as-the-oxford-coronavirus-study-3.html","timestamp":"2024-11-13T12:45:52Z","content_type":"application/xhtml+xml","content_length":"65659","record_id":"<urn:uuid:f22419d7-e0e8-45d1-8e67-1bad8013b776>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00312.warc.gz"}
A Quick Review of Pre-Calculus - Avidemia René Descartes (1595–1650), left, and Pierre de Fermat (1607–1665), right. Before the seventeenth century, geometry and algebra were two very distinct branches of mathematics. In 1637, Descartes made a huge impact on the development of mathematical knowledge by unifying these two branches of mathematics. His approach is now called analytic geometry. Independently of Descartes, Fermat discovered analytic geometry. Fermat developed a method for finding the tangent to curves and determining their maximum and minimum points that is equivalent to differential calculus. In a letter, Sir Isaac Newton stated that his own ideas about calculus were inspired by Fermat’s way of drawing tangents and applying it to abstract equations. Fermat is also called the founder of the modern number theory and co-founder of probability. In this book, we quickly review what we will need to start learning calculus. Specifically, we review high-school algebra, analytic geometry, functions, elementary transcendental functions, and trigonometry. There are more than 100 fully solved examples. Table of Contents 1. Arithmetic and geometric progressions	{"url":"https://avidemia.com/review-of-precalculus/","timestamp":"2024-11-10T05:09:33Z","content_type":"text/html","content_length":"86712","record_id":"<urn:uuid:d0d83b02-d275-43c7-aa03-bf0665ba9d49>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00083.warc.gz"}
Albert Girard (French pronunciation: [alˈbɛʁ ʒiˈʁaʁ]) (11 October 1595 in Saint-Mihiel, France − 8 December 1632 in Leiden, The Netherlands) was a French-born mathematician. He studied at the University of Leiden. He "had early thoughts on the fundamental theorem of algebra"^[1] and gave the inductive definition for the Fibonacci numbers.^[2] He was the first to use the abbreviations 'sin', 'cos' and 'tan' for the trigonometric functions in a treatise.^[1] Girard was the first to state, in 1625, that each prime of the form 1 mod 4 is the sum of two squares.^[3] (See Fermat's theorem on sums of two squares.) It was said that he was quiet-natured and, unlike most mathematicians, did not keep a journal for his personal life. In the opinion of Charles Hutton,^[4] Girard was ...the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation. This had previously been given by François Viète for positive roots, and is today called Viète's formulas, but Viète did not give these for general roots. In his paper,^[4] Funkhouser locates the work of Girard in the history of the study of equations using symmetric functions. In his work on the theory of equations, Lagrange cited Girard. Still later, in the nineteenth century, this work eventuated in the creation of group theory by Cauchy, Galois and others. Girard also showed how the area of a spherical triangle depends on its interior angles. The result is called Girard's theorem. He also was a lutenist and mentioned having written a treatise on music, though this was never published.^[5] 1. ^ ^a ^b O'Connor, John J.; Robertson, Edmund F., "Albert Girard", MacTutor History of Mathematics Archive, University of St Andrews 2. ^ Dickson, Leonard Eugene (1919). "Ch. XVII: Recurring series; Lucas' u[n], v[n]". History of the Theory of Numbers, Vol. I. Washington, D.C.: Carnegie Institution of Washington. p. 393. 3. ^ Dickson, Leonard Eugene (1920). "Ch. VI: Sum of two squares". History of the Theory of Numbers, Vol. II. Washington, D.C.: Carnegie Institution of Washington. pp. 227–228. 4. ^ ^a ^b Funkhouser, H. Gray (1930). "A short account of the history of symmetric functions of roots of equations". Amer. Math. Monthly. 37 (7): 357–365. doi:10.2307/2299273. JFM 56.0005.02. JSTOR 5. ^ The Galileo Project: Girard, Albert	{"url":"https://www.knowpia.com/knowpedia/Albert_Girard","timestamp":"2024-11-03T19:08:37Z","content_type":"text/html","content_length":"78750","record_id":"<urn:uuid:216eef7d-d11c-47ac-a19a-5a6c3afc39f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00305.warc.gz"}
Comparing Bubble and Bucket Sorting Algorithms Published on Friday, August 11, 2023 Imagine you’re building an app and need to sort a massive list of data – maybe product prices, customer names, or high scores. Choosing the right sorting algorithm can make a huge difference in performance. Today, we’ll pit two popular contenders against each other: bubble and bucket. Before we dive into the code, let’s briefly explore the basics of both algorithms. If you’re eager to see the action, feel free to jump straight to the code comparison here. Bubble Sort The best fact about the bubble sorting algorithm is, arguably, the speculation on when it was invented. It was first described in 1955 (published 1956) by Edward Harry Friend, he called this a sorting exchange algorithm. This went unnoticed for years, until Kenneth E. Iverson found it in 1962 and coined the name Bubble sort. The worst-case performance of this is $O(n^2)$. The Bubble Sort is a simple, yet often inefficient, sorting algorithm. It’s a classic choice for beginners due to its straightforward logic, but it’s not the best option for large datasets. A Brief History The bubble sort was first described in 1955 (published in 1956) by Edward Harry Friend, who referred to it as a “sorting exchange algorithm.” However, it remained relatively unknown until Kenneth E. Iverson rediscovered it in 1962 and coined the name “Bubble Sort.” How It Works The bubble sort works by repeatedly stepping through the list, comparing adjacent pairs of elements and swapping them if they are in the wrong order. This process is repeated until no swaps are needed, indicating that the list is sorted. Time Complexity The worst-case time complexity of the bubble sort is O(n^2), which means it’s not suitable for large datasets. In the best case, when the list is already sorted, the bubble sort has a time complexity of O(n). However, this is rare in practice. Advantages and Disadvantages • Simple to understand and implement • Can be useful for small datasets or nearly sorted lists • Inefficient for large datasets • Slow compared to other sorting algorithms When to Use Bubble Sort While bubble sort is not the most efficient sorting algorithm, it can be a good choice for: • Small datasets: When the number of elements is small, the overhead of more complex algorithms might not be justified. • Nearly sorted lists: If the list is almost sorted, bubble sort can be efficient. • Educational purposes: It’s a good algorithm to learn from due to its simplicity. Bucket Sort Bucket Sort, also known as Bin Sort, is a sorting algorithm that’s particularly efficient when dealing with data that’s uniformly distributed. It leverages a clever technique called Scatter-Gather to divide and conquer the sorting process. How It Works 1. Create Buckets: Determine the number of buckets needed based on the range of values in the input array. 2. Scatter: Distribute elements from the input array into the appropriate buckets based on their values. 3. Sort Buckets: Sort each individual bucket using a suitable sorting algorithm (often insertion sort). 4. Gather: Concatenate the sorted buckets to form the final sorted array. Time Complexity The time complexity of bucket sort depends on the distribution of the input data and the choice of sorting algorithm used for the buckets. In the worst case, when all elements end up in the same bucket, bucket sort degenerates to $O(n^2)$. This can happen when the data is not uniformly distributed or when the number of buckets is too Advantages and Disadvantages • Efficient for uniformly distributed data • Can be faster than comparison-based sorting algorithms in the best case • Can be implemented in-place • Less efficient for non-uniform data • Requires knowledge of the data distribution • May not be suitable for all types of data When to Use Bucket Sort Bucket sort is a good choice for: • Uniformly distributed data: When you know that the data is evenly spread across a certain range. • Large datasets: It can be faster than comparison-based sorting algorithms for large, uniformly distributed datasets. • Applications where space efficiency is important: Bucket sort can be implemented in-place, reducing memory usage. In conclusion, bucket sort is a valuable sorting algorithm that can be very efficient for certain types of data. Understanding its strengths and limitations can help you make informed decisions when choosing a sorting algorithm for your specific use case. The Clash We put both algorithms to the test with a battlefield of 3500 random numbers. Now, let’s see who emerges victorious! Now that we have some data to test on, we want to add the algorithm for the bubble sort. This goes as follows. And of course the bucket sort as well, otherwise we won’t have anything to compare against. Now, let’s test the two against one another. Delve deeper: For even more sorting options, explore our collection of sorting algorithms. Want to get your hands dirty with the code? Head over to bubble sort VS. bucket sort Implementation. The Winner Brace yourselves! The benchmark revealed that the bucket sort is a staggering 14.02x faster than its competitor! That translates to running the bucket sort almost 15 times in the time it takes the bubble sort to complete once! The A.I. Nicknames the Winners: We consulted a top-notch AI to give our champion a superhero nickname. From this day forward, the bucket sort shall be known as The Bucket Wrangler! The bubble sort, while valiant, deserves recognition too. We present to you, The Bubble Buster! The Choice is Yours, Young Padawan So, does this mean the bucket sort is the undisputed king of all sorting algorithms? Not necessarily. Different algorithms have their own strengths and weaknesses. But understanding their efficiency (which you can learn more about in the Big-O Notation post) helps you choose the best tool for the job! This vast world of sorting algorithms holds countless possibilities. Who knows, maybe you’ll discover the next champion with lightning speed or memory-saving magic! This showdown hopefully shed light on the contrasting speeds of bubble and bucket sorting algorithms. Stay tuned for more algorithm explorations on the blog.	{"url":"https://hasty.dev/blog/sorting/bubble-vs-bucket","timestamp":"2024-11-06T04:23:43Z","content_type":"text/html","content_length":"27413","record_id":"<urn:uuid:099f93fa-741e-4f76-a118-091b825feefc>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00380.warc.gz"}
Bio-data of Prof. R. N. Iyengar Raja Ramanna Fellow at IISc, Bangalore. Tel:080-22932818(o); Fax:080-23600404; E.mail: rni@civil.iisc.ernet.in; aareni@yahoo.com Retired in July 2005 as K.S.I.D.C Chair Professor, Department of Civil Engineering Also at Center for Atmospheric & Oceanic Sciences, Indian Institute of Science, Bangalore 560012. Formerly Director (April 1994 - April 2000) Central Building Research Institute, CSIR, Roorkee - 247 667 Date of Birth 2^nd June 1943 Fields of Research Disaster Mitigation, Structural Dynamics, Random Vibration, Nonlinear Systems. Earthquake Engineering, Railway Track-dynamics Rainfall Modelling, History of Science Educational B.E. (Civil) MysoreUniversity, 1962 M.Sc. (Engg.)Indian Institute of Science, 1966 Thesis : ‘Vibration of Beam and Slab Bridges'. Ph.D. Indian Institute of Science, 1970 Thesis : 'Nonstationary Random Process Model for Earthquakes’ Honours and Distinctions Fellow, Indian Academy of Sciences, Bangalore. Fellow, Indian National Academy of Engineering, New Delhi. Fellow, National Academy of Sciences, Allahabad. Member, Asia-Pacific Academy of Materials. *Member, European Academy of Sciences, Brussels Alexander Von Humboldt Fellowship (Senior), Germany 1978 to 80, 1992, 1997 First Prize of the Railway Board for the paper 'Identification of Damping in Railway Vehicles under Running Condition’ 1989-90. National Representative, General Assembly of IUTAM 1987-92 Plenary Speaker GAMM Conference, on Applied Mechanics, Leipzig, Germany, 1992 Distinguished Schmidt Visiting Chair, Florida Atlantic University, Boca Raton, U.S.A. 1995, Shelter Award by the Shelter Promotion Council of India, Calcutta, 1996 Sir M. Visvesvaraya Award for Senior Scientists for life time Contribution to Science and Technology ; Govt. of Karnataka, 1996. Narayanan Memorial lecture Award, Acoustical Society India, 1996. 18th ISET Annual Lecture Indian Society of Earthquake Technology, 1997 A.K. Sen Memorial Lecture Award, IIT, Kharagpur, 1998. Chairman and Organizer IUTAM Symposium on ‘Nonlinearity and Stochastic Structural Dynamics’, 4-8 January 1999, IIT Madras. Invited Sectional Lecture, ICTAM, Chicago, USA. August 2000, Swamy Rao Memorial Lecture Award, JNTU, Kakinada, 2001. NRDC-Technology Day Award for ‘Pollution Control Device for Brick Kilns’ (Shared), 2001 Member, Editorial Board, J. of Probabilistic Engineering Mechanics, Eisevier Applied Sciences, U.K.(1994-2004) Member, Editorial Board, Sadhana, Indian Academy of Sciences, Bangalore. Member, Editorial Board, J. of Indian Soc. of Earthquake Technology. Associate Editor, J. of Indian Institute of Science, Bangalore (1987-92). Membership of Professional Societies Fellow, Institution of Engineers (India) Life Member, Indian Science Congress Life Member, Indian Society of Earthquake Technology. Life Member, Indian Nuclear Society, Mumbai. Life Member, Indian Geotechnical Society. Professional Record Raja Ramanna Fellow, Indian Institute of Science, August 2005-Continuing. Visiting Professor, VIT (Deemed University), Vellore Professor, Indian Institute of Science, Bangalore, 1986-2005 Director, Central Building Research Institute, (CSIR) Roorkee, 1994-2000 Associate Professor, Indian Institute of Science, 1981-1986. Assistant Professor, Indian Institute of Science, 1974-1981. Lecturer, Indian Institute of Science, 1969-1974. Visiting Assistant Professor, Dept. of Aeronautics and Engineering Sciences, Purdue University, W. Lafayette, USA, 1970–1971. Visiting Scholar, Dept. of Electrical Engineering, Brooklyn Polytechnic, New York, USA, Feb-March 1971. Research Associate, Dept. of Civil Engineering, Columbia University, New York, USA, April-July 1971. Senior Alexander Von Humboldt Fellow: Univ. of Hannover, Germany, Oct 1978-Dec1979, May-July 1980 and April-June1992, Reinvited: 1997 and 2000. Research Publications A. Structural Dynamics, Random Vibration, Earthquake Engineering ( Refereed Journals) 1. R.N. Iyengar (1964), Analysis of Arcades, J. Inst. of Engrs (India), XIIV, 11, 722-730. 2. R.N. Iyengar (1965), On the Application of the Reciprocal Theorem to the Vibration of Continuous Beams. Bull. Indian Society of Earthquake Technology, 17 53-60. 3. K.T.S. Iyengar and R.N.I. (1967), Free Vibration of Beam and Slab Bridges, Publns, IABSE, 1- 14. 4. K.T.S. Iyengar & R.N.I (1967), Orthotropic Plate Parameters of Stiffened Plates and Grillages in Free Vibration, Appl. Scientific Research 1 7, 422-438. 5. K.T.S. Iyengar, K.S. Jagadish and R.N.I (1967), Vibration of Beam and Slab Bridges, Proc. Inst. of Engrs. (India), IXVIII, 1, 141-169. 6. R.N.I & K.T.S. Iyengar (1969), A Nonstationary Random Process Model for Earthquake Accelerograms, Bull. Seismological Society of America. 59.3, 1163-1188. 7. R.N.I and K.T.S. Iyengar (1970), Probabilistic Response Analysis to Earthquake J.Engg. Mechanics, (ASCE), 96, EM3, 207-225. 8. R.N.I & M. Shinozuka (1972), Effect of self Weight and Vertical Acceleration on Tall Structures during Earthquakes, Intnl. J. of Earthquake Engineering and Structural Dynamics, 1, 1, 69-72. 9. R.N.I (1972), Worst Inputs and a Bound on the Highest Peak Statistics of a class of Nonlinear System’, J. of Sound and Vibration, 25, 1, 29-37. 10. R.N.I (1973), First Passage Probability during Random vibration, J. of Sound and Vibration, 31, 2, 185-193. 11. Ramaprasad & R.N.I (1975), Free Vibration of a Reservoir-Foundation System, J. of Sound and Vibration, 39, 2, 21 7-227. 12. R.N.I (1975), 'Random Vibration of a Second Order Nonlinear Elastic System’, J. of Sound and Vibration, 40, 2, 155-165. 13. R.N.I & K.B. Athreya (1975), ‘A Diffusion Process Approach to a Random Eigen Value Problem’, J. of Indian Institute of Science, 57, 5, 185-191. 14. R.N.I & P.N. Rao (1975), ‘Free Vibration of Elastic Medium', Bull. Indian Society of Earthquake Technology, 12, 147-154. 15. R.N.I & P.K. Dash (1976), ‘Random Vibration Analysis of Stochastic Time varying Systems’, J. of Sound and Vibration, 45, 1, 69-89. 16. R.N.I & P.K. Dash (1977), ‘Highest Peak Distribution In Time-varying Systems’, J. of Engg. Mech. Div. (ASCE), 103, EM5, 869-898. 17. R.N.I & K.J. Iyengar (1978), ‘ Stochastic Analysis of Yielding System', J.of Engg. Mech.Div. (ASCE), 104, EM2, 383-398. 18. R.N.I & P.K. Dash (1978), ‘Study of Random Vibration of Non-linear Systems by the Gaussian Closure Technique’, J. of Applied Mech. (ASME), 45, 393-398. 19. R.N.I & P.K. Dash (1978), 'Random Vibration In Spacecraft Structures: A review', J. of Aeronautical Society of India, 30, 1,2, 1-21. 20. R.N.I & P.N. Rao (1979), ‘Generation of Spectrum Compatible Accelerograms', lntnl. J. of Earthquake Engg. and Structural Dynamics, 7, 253-263. 21. R.N.I (1979), 'Inelastic Response of Beams under Sinusoidal ad random Loads', J. of Sound and Vibration, 64, 2, 161-1 72. 22. R.N.I & O. Mahrenholtz (1982), ‘Nonlinear Oscillation of a Vortex Excited Cylinder in Wind’, Solid Mechanics Archives, 7, 411-432. 23. P.K. Dash and R.N. Iyengar (1982), ‘Analysis of Randomly Time Varying Systems by Gaussian Closure Technique', Journal of Sound and vibration, 83, 2, 241 -151. 24. R.N.I & K.C. Prodhan(1983), ‘Classification and Rating of Strong-Motion Records', Journal of Earthquake Engineering and Structural Dynamics, 11, 415-426. 25. R.N.I. (1984), 'Estimation of Yield Damage in Plates and Shells during Random Vibration', J. Aero. Soc. of India, 36, 2. 26. R.N.I. & K. Meera (1984), ‘Earthquake in South India on 20th March,1984’. Bull. Ind. Soc. of Earthquake Technology, 21, 2, 49-61. 27. R.N.I. & M.R. Pranesh (1985), ‘Dynamic Response of a Beam on a Foundation of Finite Depth', Indian Geotechnical Journal, 1 5, 53-63. 28. R.N.I. (1986), 'A Nonlinear System under Combined Random and Periodic Excitation’. J. of Statistical Physics, 44, 5 1 6, 907-920. 29. R.N.I., N. Anantanarayana and D. Deepak (1986), 'Track Classification and Track Index', Rail International, 39-44. 30. R.N.I. and C.S. Manohar (1987),’Nonstationary Random Critical Seismic Excitation', J. of Engg. Mech., 113, 4, 529-541. 31. R.N.I. & K.C. Prodhan (1987), 'Internal Consistency of Smooth Design Response Spectra', Bull. ISET, 24, 161-173. 32. R.N. Iyengar (1988), 'Higher order Linearization in Nonlinear Random Vibration', Intnl. J. of Nonlinear Mech. 23, 5, 385-391. 33. R.N. Iyengar (1988), ‘Stochastic Response and Stability of the Duffing Oscillator under Narrow band Excitation', J. of Sound and Vibration, 120,2, 255-263. 34. R.N. Iyengar and G.V. Rao (1988), ‘Free Vibration and Parametric Instability of a Laterally Loaded Cable’, J. of Sound and Vibration, 127,2, 231-243. 35. R.N. Iyengar, K.R. Reddy and C.S. Manohar (1989), ‘Chaotic Response of Some simple Nonlinear Oscillators', J. Aero. Soc. of India, 41, 1, 1-6. 36. R.N. Iyengar and C.S. Manohar (1989), ‘Probability Distribution of the Eigen-values of the Random String Equation’, J. of Applied Mechanics, 56, 202-207. 37. R.N. Iyengar (1989), 'Response of Nonlinear System Systems under Narrow Band Excitation'. J. of Structural Safety, 6, 177-188. 38. G.V. Rao & R.N. Iyengar (1990), ‘Nonlinear Planar Response of a Cable under Random Excitation’, J.of Probabilistic Engineering Mechanics, 5,4, 182-191. 39. R.N. Iyengar & C.S. Manohar (1991), ‘Narrowband Random Excitation of a Limit Cycle System', Archive of Applied Mech., 61,133-141, 40. R.N. Iyengar & G.V. Rao (1991), ‘Internal Resonance and Nonlinear Response of a Cable under Harmonic Excitation’, J.of Sound and Vibration. 149, 1, 25-41 41. R.N. Iyengar (1991), 'Stochastic Characterization of Chaos in a Nonlinear System', Physics Letters-A, 154, 7-8, 357-360. 42. G.V. Rao & R.N. Iyengar (1991), ‘Seismic Response of a Long Cable', Int. J. Earthquake Engineering and Structural Dynamics, 20, 243-258. 43. C.S. Manohar & R.N. Iyengar (1991), 'Entrainment in Vander Pol's Oscillator In the presence of Noise', Int. J. of Nonlinear Mech., 26, 5, 679-686. 44. R.N. I.& C.S. Manohar (1991), 'Rocking Response of Rectangular Rigid Blocks under Random Noise Base Excitation’, Invited paper, Special Issue. Int. J. of Nonlin Mech., 26, 6. 45. R.N. Iyengar & O.R. Jaiswal (1993), ‘Dynamic Response of a Beam on Elastic Foundation of Finite Depth under a Moving Force', Acta Mechanica, 96, 67-83. www.springerlink.com/index/ 46. R.N. Iyengar & O.R. Jaiswal (1993), ‘A New Model for Non-Gaussian Random Excitation', J. of Probabilistic Engg.Mechanics, 8, 280-287. 47. C.S. Manohar & R.N. Iyengar (1993), 'Probability Distribution of the Eigen values of Systems governed by the Stochastic Wave Equation', J. Probabilistic Engg.Mech.8, 57-64. 48. R.N.I. (1993), 'Chaotic Behaviour in Non-linear oscillators', ZAMM,73, 4-5 T46-T53. 49. R.N.Iyengar & O.R. Jaiswal (1994), ‘Random Process Modelling of Railway Track Unevenness', Indian Railway Technical Bulletin. 50. C.S. Manohar & R.N. Iyengar (1993), ‘Free Vibration Analysis of Stochastic Strings', J. of Sound and Vibration. 51. R.N. Iyengar & D. Roy (1994), ‘Nonlinear Dynamics of a Rigid Block on a Rigid Base', J. of Applied Mechanics, 63, pp 55-61. 52. R.N. Iyengar, C.S. Manohar & O.R. J. Jaiswal (1994), 'Field Investigation of the 30th Sept.1993 Earthquake in Maharashtra', Current Science, .67, 5, 53. R.N. Iyengar & O.R. Jaiswal (1995), 'Random Field Modelling of Railway Track Irregularities', J. Transport Engg., ASCE, 121, 4, 303-308. 54. R.N. Iyengar & D. Roy ( 1996), ‘Conditional Linearization In Random Vibration', J. Engg. Mech, ASCE. 122, 31,197-200. 55. R.N.Iyengar (1997) ‘Strong motion: Analysis, Modelling and Applications’ Bull. ISET, Paper no. 370, 34, 4, 171-208 56. R.N. Iyengar & D. Roy (1998) ‘ New Approaches for the Study of Non linear Oscillators’ J. of Sound and Vibration, 211,5, 843-875. 57. R.N. Iyengar and D. Roy (1998), ’Extension of the Phase Space Linerization Technique for Nonlinear Oscillators’ J. of Sound and Vibration, 211,5, 877-906. 58. R.N. Iyengar and S.K. Agarwal (1999) ‘Statistical Analysis of ensemble of strong motion records’, Current Science, 76, 5, 684-687. 59. R.N.Iyengar (2000) ‘Seismic Status of Delhi Megacity’ Current Science.78, 568-574. 60.R.N.Iyengar and S.K. Agrawal (2001) Earthquake Source Model using Strong Motion Displacement as Response of Finite Elastic Media, Proc. Indian Acad. Sci., J. Earth and Planetary Sciences.110,1, 9-23. www.ias.ac.in/epsci/mar2001/E1363.pdf 61. R.N.Iyengar and S.T.G.Raghukanth (2002) Strong ground motion at Bhuj City during the Kutch Earthquake, Current Science, 82, 11, 1366-72. 62.R.N.Iyengar and S.T.G.Raghukanth (2004) Attenuation of strong ground motion in Peninsular India, Seismological Research Letters, 75, 4, 530-540. 63. RNI and S.Ghosh, (2004), Microzonation of earthquake hazard in Greater Delhi area, Current Science, 87,9, 1193-1202. www.ias.ac.in/currsci/nov102004/1193.pdf 64. RNI and S.T.G.Raghukanth (2006) Strong ground motion estimation during the Kutch, India earthquake, Pure and Applied Geophysics 163, pp 153-173. 65. RNI and B.Basak (2005) Investigation of a Nonlinear System Under Partially Prescribed Random Excitation, Int. J of Nonlinear Mechanics, 40, pp 1102-1111. 66. RNI and S.T.G.Raghukanth (2005) Attenuation of Seismic Spectral Acceleration in Peninsular India, under review, J. of Earthsystem Sciences 67. RNI and S.T.G.Raghukanth (2005) Estimation Seismic Hazard for Mumbai City, Accepted for Publication in Current Science. 68. RNI and B.P.Radhakrishna (2005) Evolution of the western Coastline of India and the Probable Location of Dwaraka of Krishna: Geological Perspectives, J of Geological Soc. of India, 66, Sept. pp 285-292. http://abob.libs.uga.edu/bobk/maha/dwaraka/ B.Atmospheric Sciences (Refereed Journals) 1. S.Gadgil and R.N. Iyengar (1980), ‘Cluster Analysts of Rainfall Stations of the Indian Peninsula', Quarterly J. Royal Meteorological Society, 106, 450, pp 873-886. 2. R.N. Iyengar (1982), 'Stochastic Modelling of Monthly Rainfall’, Journal of Hydrology, 57, pp 375-387. 3. R.N. Iyengar (1987), 'Analysis of Weekly Rainfall Time Series', Mausam. 38,4,453-458. 4. R.N. Iyengar (1991), 'Variability of Rainfall through Principal component Analysis', J.of Earth and Planetary Sciences, Proc. of Indian Acd. of Sciences, 100,2, 105-126. 5. R.N. Iyengar & P. Basak (1994), ‘Regionalization of Indian Monsoon Rainfall and Long-term Variability Signals’, Intnl. J.of Climatology, 1 4, 1095-1114. www.rmets.org/publication/IJC/ijc94.php 6. R.N.Iyengar and S.T.G.Raghukanth (2003) ‘Empirical Modelling and Forecasting of Indian Monsoon Rainfall’ Current Science, 85,8, 1189-1201. 7. R.N.Iyengar and S.T.G.Raghukanth (2004) Intrinsic mode functions and a strategy for forecasting Indian monsoon rainfall. J of Met. and Atmos. Physics. July-Online. 8. RNI (2004) Description of rainfall variability in Brihat-samhita of Varaha-mihira, Current Science, 87, 4, 531-533. www.ias.ac.in/currsci/aug252004/531.pdf 9. R.N.Iyengar and S.T.G.Raghukanth (2006) Forecasting of Seasonal Monsoon Rainfall at Subdivision Level. Current Science, 10^th August. pp. 350-356. C.History of Science (in Refereed Journals) 1. R.N. Iyengar & D. Sharma (1996) Some Earthquakes of Kashmir from Historical Sources Current Science, 71,4, 330-331. 2. R.N. Iyengar (1999), Earthquakes in Ancient India, Current Science, 77, 6, 827-829. www.ias.ac.in/currsci/sep25/articles32.htm 3. R.N.Iyengar, D.Sharma and J.M.Siddiqui (1999) Earthquake History of India in Medieval Times, Ind. J. Hist. of Science, 34 (3),181-237. 4. R.N.Iyengar and D.Sharma (1999) Some Earthquakes of the Himalayan Region from Historical Sources, J Himalayan geology, 20,1,81-85. www.himgeology.com/pubvolumns/volume20(1).html 5. R.N.Iyengar (2003) Internal Consistency of Eclipses and Planetary Positions in Mahabharata, Ind. J.of. Hist. Science, 38, 2, 77-115. www.hindunet.org/saraswati/rniyengar.pdf 6. RNI (2003) Historicity of Celestial Observations of Mahabharata, Q. J of Mythic Society, XCIV, 1-2, 150-186. 7. R.N.Iyengar (2004) Profile of a Natural Disaster in Ancient Sanskrit Literature, Ind. J. Hist. of Science, 39,1, 11-49. http://abob.libs.uga.edu/bobk/maha/skandapuranadisaster.pdf 8. RNI (2004) Extra Terrestrial Impacts, J. of Geological Soc. of India, (Correspondence) 64, 6, 826-827. 9. RNI (2005) Eclipse period number 3339 in Rigveda, Ind. J. Hist. of Science, 40, 2, June, pp.139-152. http://abob.libs.uga.edu/bobk/maha/rigvedaeclipse/ 10. RNI (2006) Some Comet observations of ancient India, J.Geological Soc. Ind. 67, March, pp289-294. http://abob.libs.uga.edu/bobk/maha/cometindia/ 11. RNI (2006) Celestial observations associated with Krishna-lore, Ind. J. Hist. of Science, 41.1, pp1-13. D.General Articles (Some in Refereed Journals) 1. R.N. Iyengar (1993), 'How Safe is Tehri Dam to Earthquakes', Current Science, Vol.65. No.5, 383-392. 2. R.N. Iyengar (1994), 'Dynamic Analysis and Seismic Specification of Tehri Dam' Earthquake Hazard & Large Dams in the Himalaya, INTACH, Bharatiyam, Nizamudiin, New Delhi, pp 137-152. 3. R.N. Iyengar (1994), 'Armenia : A Travelogue’, Current Science, Vol.67, No.3, pp 157-161. 4. R.N. Iyengar (1994), 'History of Earthquakes in South India', The Hindu, January 23, 1994. 5. R.N. Iyengar (1994), 'Earthquake Experimentation using Railway Track', The Hindu, September 28, 1994. 6. R.N. Iyengar (1996), ’Fire Engineering: Research and Development at Central Building Research Institute', IBC News (Special Issue), Vol. 3, No. 2, April-June, 1996. 7. R.N. Iyengar (1996), ’Rural Housing: A Discussion’, Kurukshetra (Special Issue), Vol. XLIV, Nos. 8, 9, May-June, 1996. 8. R.N. Iyengar (1997), ‘Natural Disaster Mitigation Research at CBRI’, Journal of Indian Building Congress, 3rd Annual Convention and Seminar on Built Environment and Natural Hazards, Feb. 7-8,1997. 9. R.N.Iyengar (2001) ‘Is the seismic zonation map of India correct?’ The Hindu, 17^th April. E.Publications in Proceedings of Seminars and Symposia 1. A.J. Schiff, R.N. Iyengar, J.R. Duan & J.L. Bogdanoff (1970), 'On the Behaviour of Walls under Seismic disturbances’, Proc. Conf. on Earthquakes and Structures, Jassy, Rumania. 2. R.N. Iyengar & K.T.S. Iyengar (1973), ’Stochastic Process Modelling of Earthquake and Probabilistic Response Analysis’, Proc. Symposium o Earth and Earth Structures subjected to Earthquakes, Roorkee, India, Vol. 1, pp 223-231. 3.R.N. Iyengar, K.T.S. Iyengar, V. Ramachandran & Ramaswamy (1974), 'Seismic Analysis of the Ventilation Stack for the Madras Atomic Power Plant’, Proc. 5th Symposium on Earthquake Engineering, Roorkee, pp 39-46. 4.R.N. Iyengar & K.S. Jagadish (1974), ‘Probability of Failure of Structures under Earthquake Excitations', Proc. 5th symposium on Earthquake Engineering, Roorkee, pp 253-258. 5.R.N. Iyengar & P.N. Rao (1975), ‘Dynamic Amplification In Two Dimensional Soil Media', Proc. 5th Asian Regional Conference on soil Mechanics and Foundation Engg, Bangalore, pp 279-282. 6. R.N. Iyengar, D.V. Reddy & S. Vedula (1976), ‘A Random Process Model for Monthly River Flows', Proc. Symposium on Mathematical Modelling of Water Resources Problems, Patna, India. 7.R.N. Iyengar, A.S. Reddy (1977), ‘Response of a Soil-pile System during Earthquakes’, Proc. Intnl. Symp. on soil Structure Interaction, Roorkee, pp 355- 360. 8.R.N. Iyengar & T.K. Saha (1977), ’Effect of Vertical Ground Motion on the Response of Cantilever Structures’, Proc.6th WCEE, New Delhi, pp 3-193-198. 9.R.N. Iyengar & K.C. Prodhan (1982), ’Equivalent single Degree Systems and Approximate Seismic Response’, Proc. VII Symp. on Earthquake Engineering, Roorkee, Vol. I 10.R.N. Iyengar & C.S. Manohar (1984), ‘Extreme Seismic Excitation, Proc. Symp. on Earthquake Effects on Plant and Equipment, Hyderabad, pp 65-69. 11.R.N. Iyengar & K.C. Prodhan (1984), 'Engineering Classiflcation of Earthquake Records', Proc. Symp. on Earthquake Effect on Plant and Equipment, Hyderabad, pp 71-75. 12.R.N. Iyengar & K. Meera (1984), ‘Overturning of Rigid Bodies during Earthquakes', Proc. Symp. on Earthquake Effects on Plant ad Equipment, Hyderabad, pp 129-133. 13.R.N. Iyengar (1984), ‘Stochastic Process Modelling of Rainfall’, Proc.Colloquium on Precipitation Analysis and Flood Forecasting, Ind. Inst. of Tropical Meteorology, Pune. 14.R.N. Iyengar (1984), 'The Vortex Oscillator as a Non-conservative System', Proc. Euromech Colloquium 190, University of Hamburg - Harburg, pp 36-38. 15.R.N. Iyengar & C.S. Manohar (1985), ‘System Dependent Critical Stochastic Seismic Excitation'. Transactions of the 8th Int. Conf. on Structural Mechanics in Reactor Technology, SMIRT, Brussels, M 15/6, pp 147-1 51. 16.R.N. Iyengar (1985), 'Some Studies in Random Vibration', First Indo-Soviet symposium on Mechanics', Tashkent, USSR. 17.R.N. Iyengar, V. Ramachandran and U.S.P. Verma (1987) 'Uplift of a Power Plant Founded on rock under Seismic Excitation', Trans. of the 9th Intnl.Conf. on SMIRT, Lausanne. 18.R.N. Iyengar, A. Krishnan and K.S. Dixit (1987), 'Dynamic Analysis of Coolant Channel Assembly under Seismic Excitation, 9th Intnl. Conf. SMIRT, Lausanne. 19.R.N. Iyengar & C.S. Manohar (1987), ‘Vander Pol's Oscillator under Combined Periodic and Random Excitation’, IUTAM Symposium, Austria. 20.R.N. Iyengar, J. .S. Goray & H.S. Khushwaha (1989), 'Development of Floor Response spectra using PSD Function’, Trans. 10th Intl. Conf. on SMIRT, Vol. k2, pp 545-550. 21.C.S.Manohar & R.N. Iyengar (1990), ‘Natural Frequencies of Simple Stochastic Structural systems', Proc. Intl. Conf. Str. Testing, Analysis and Design, ICSTAD, Bangalore, India. 22.R.N. Iyengar (1991), 'Approximate Analysis of Nonlinear System under narrow band Random Inputs', Proc. IUTAM Symp. on Nonlinear Stochastic Dynamics, Turin, Italy, (Ed. N. Bellomo), Springer Verlag, pp 309-319. 23.R.N. Iyengar (1993), 'Residence Time characteristics of Strong Motion Accelerograms', Proc. Int. conf. on Continental Collision Zone Earthquakes, Yerevan, Armenia, 1-7 Oct. 24.R.N.I. C.S. Manohar & O.R. Jaiswal (1994), 'Field Study of the Maharashtra Earthquake of 30th September 1993’, Proc.10th Symp on Earthq. Eng., Roorkee, 16-18 November. 25 .R.N. Iyengar (1994), 'Strong Motion Data and Parameters', Proc.10th Symposium on Earthquake Engineering, Roorkee, 26. R.N.Iyengar and K.Popp(1994) ‘Random Vibration of a Galloping Oscillator in Wind’ Proc. ICOSSAR-93, 1711-14, Balkema, Rotterdam. 27. RNI and O.R.Jaiswal (1995) Stochastic Response of Irregular Tracks under Moving Vehicles., Proc.IUTAM Symp.on Nonlinear Stoch. Mechanics, Trondheim, Norway. 213-224., Kluwer Acd.Publ. 28. R.N. Iyengar (1996), ‘Seismic Specification through Spectrum Compatible PSD Functions’, Proc. Symposium on Earthquake Effects on Structures, Plant & Machinery, BHEL, New Delhi. 29.R.N. Iyengar and S. Sarkar (1997), ‘A Statistical Approach to Study of Landslides', Proc. CBRI Golden Jubilee Year Conf., New Delhi, Tata-McGraw-Hill Co. 63-71. 30. R.N. Iyengar, M.P.Jaisingh, (1997), ‘Lateral Strength Masonry Walls’, Proc.CBRI Golden Jubilee Year Conf., New Delhi, Tata-McGraw-Hill Co. 131-142. 31. R.N.Iyengar (1999) ’Seismic Microzonation: A Strategy for Planning Built Environment’ Proc. TCDC Workshop on Natural Disaster Reduction:,Policy Issues and Strategies, Dec.1999, SERC, Madras, III 9-III 18. 32. R.N.Iyengar (1999) ‘Application of Random Vibration in Earthquake Engineering,’ Keynote Address, Proc. of 4^th International Conference on Vibration Problems, (ICOVP) Jadhavpur Univ, Calcutta, Vol A, 1-23. 33. R.N.Iyengar (2000) ‘Probabilistic Approaches in Earthquake Engineering’ Invited Lecture, Proc. ICTAM 2000, (Ed. Aref and Philips).457-471. Kluwer Acad.Publ. Netherlands. 34. R.N.Iyengar (2000), ‘Performance of Indian Rural Stone Masonry during Earthquakes’ Proc. 6^th Intnl. Sem. On Strucutural Masonry for Developing Countries, October 2000, Bangalore, Allied Publ. Ltd., 247-252. 35. R.N.Iyengar (2000), ‘Vibration Problems in Building Structures’ Invited Talk, Proc. VETOMAC-I, October 2000, Bangalore. 36. RNI and S.K.Agarwal (2000) Source Location - an Inverse Structural dynamic Problem, Proc. II Structural Engg.Convention, (SEC-2000), IIT-Bombay, 401-408, Quest Publ. Mumbai. 37. RNI (2001), ‘ Earthquakes in India an Engineering Perspective’ National Sem. on “Design and Construction of Earthquake Resistant Structures” May, 2001,INSTRUCT, A1-A18. 38. RNI (2001) ‘Seismic Microzonation of Vulnerable Cities’ National Sem. on “Design and Construction of Earthquake Resistant Structures” May, 2001,INSTRUCT, Bangalore , B1-B14 39.RNI (2001) ‘Selection of Ground Motion Parameters at Project Sites’ Proc. Workshop on Safety of Nuclear Power Plant Structures, SNPP-2001, July 2001, IISc., Bangalore, 35-52. 40. R.N.Iyengar (2002) Engineering Approaches in Seismic Hazard Estimation, Proc. Advances in Civil Engg., ACE-2002, IIT Kharagpur, pp770-780., Allied Publ. Ltd., N.Delhi. 41. R.N.Iyengar and S.T.G.Raghukanth Earthquake Source Model and Estimation of Strong Ground Motion, Proc. Int. Conf. on Construction Management and Materials in Civil Engineering, CONMAT-2003, IIT Kharagpur. 127-140. Phoenix Publ. House. N.Delhi. 42. R.N.Iyengar and S.T.G.Raghukanth (2003) Attenuation of Strong Ground Motion and Site Specific Seismicity in Peninsular India, Proc. National Seminar on Seismic design of Nuclear Power Plants, 269-291, SERC, Madras. 43.R.N.Iyengar and S.Ghosh (2003) Seismic Hazard Microzonation of Delhi City, Proc. National Sem. on Disaster Management and Mitigation, INAE, 93-108. Phoenix Publ. N.Delhi. 44. RNI and S.Ghosh (2004) Proc. 13^th WCEE, Vancouver, Canada. 45. RNI and S.T.G.Raghukanth (2004) Earthquake source geometry compatible with strong motion data, Proc. 13^th WCEE, Vancouver, Canada. (To be updated) 1. R.N Iyengar, Editor, Natural Hazards in the Urban Habitat , Proc. of CBRI Golden Jubilee Conf. N.Delhi,1997, Tata McGraw-Hill Publ. Co. Ltd. N.Delhi. 2. S.Narayanan and R.N.Iyengar, Editors. IUTAM Symposium on Nonlinearity and Stochastic Dynamics, IIT-Madras, Jan,1999. Kluwer Academic Publishers, Dordrecht. 3. K.S. Jagadish and RNI, Editors. Recent Advances in Structural Engineering, 2005. Universities Press, Hyderabad. 4.RNI: Elements of Mechanical Vibrations, Under Preparation. Technical Reports 1. R.N. Iyengar and K.T.S. Iyengar (1974), 'Seismic Analysis of the Ventilation stack for the Madras Atomic Power Project’, PPED, Dept. of Atomic Energy. 2. R.N. Iyengar, P.K. Dash & C.V. Joga Pao (1974), ‘Random Vibration of Tubular Structures', Ministry of Defence, Govt. of India. 3. R.N. Iyengar, & S.A. Ramu (1975), ‘Random Vibration of a Satellite Structure',, ISRO, Dept. of Space. 4. R.N. Iyengar, S.A. Ramu & K.J. Iyengar (1977), ‘Inelastic Response of Structures under Random and Impulsive Loading’, Aero. Res. and Dev. Board. 5. R.N. Iyengar (1977), 'Prediction of Food Production Including Rainfall Uncertainties', NCST Panel on Futurology, Department of Science and Technology. 6. R.N. Iyengar (1980), 'Vibration of Thin Cylinders', DRDL, Hyderabad. 7. R.N. Iyengar & S.A. Ramu (1982), ‘Dynamic Stresses in Rotating Shells'. DRDL, Hyderabad. 8. R.N. Iyengar & K.P. Rao (1982), ‘Piping Analysis under Seismic Forces'. PPED, Dept. of Atomic Energy. 9. R.N. Iyengar (1983), 'Statistical Analysis of Annual Rainfall in Karnataka', 83, AS4, Centre for Atmospheric Science, IISc., Bangalore. 10. R.N. Iyengar & B.K. Raghuprasad (1983), ‘Spectra Compatible Accelerogram for KAPP’’, PPED, Dept. of Atomic Energy. 11. R.N. Iyengar (1984), 'Track Classification and Track Index', RDSO, Ministry of Railway. 12. R.N. Iyengar (1985), 'Identification of Damping in Railway Vehicles', RDSO, Ministry of Railway. 13. R.N. Iyengar (1986), ' Safety Systems and Coolant Channels of a Nuclear Reactor under Seismic Forces', Nuclear Power Board, Department of Atomic Energy. 14. R.N. Iyengar & K. Meera ( 1986), ‘ Nonlinear Dynamic Analysis of a Railway Freight Car', DST Project Report. 1 5. R.N. Iyengar (1987), 'Uplift and Rocking Motion of a Nuclear Power Plant during Earthquake', Nuclear Power Board, Department of Atomic Energy. 16. R.N. Iyengar (1988), 'Theoretical Slosh Analysis', DRDL, Hyderabad. 17. R.N. Iyengar (1989), 'Spectrum Compatible PSD Function and Floor Response Spectra', NPC, Bombay. 18. R.N. Iyengar & S.A. Ramu (1990), 'Seismic Response of Reactivity Mechanism Assembly', NPC, Bombay. 19. C.S. Manohar & R.N. Iyengar (1990), 'Random Vibration of a Limit Cycle System’, ST/1, Dept. of Civil Engg. IISc. 20. R.N. Iyengar (1990), 'Seismic Response of the NAPP Primary Shutdown System’, NPC, Bombay. 21. R.N. Iyengar (1991), ‘Spectrum Compatible PSD Function and Response Spectra of NAPP', NPC, Bombay. 22. O.R. Jaiswal and R.N. Iyengar (1991), ‘Dynamic Analysis and Random Process Modelling of Railway Tracks' Dept. of Civil Engineering, IISc., Bangalore. 23. S.A. Ramu & R.N. Iyengar (1991), ‘Shock Analysis of Naval Equipment’, Indian Navy, DMDE, Hyderabad. 24. R.N. Iyengar (1992), 'Stochastic Dynamic of a Power Plant Equipments’. BHEL, Hyderabad. 25. R.N. Iyengar (1993), 'Structural Dynamics of a Uniform Missile', DRDL, Hyderabad. 26. R.N. Iyengar (1993), 'Railway Track Tolerances', RDSO, Department of Railway, Lucknow. 27. R.N.Iyengar, O.R. Jaiswal (1995), ‘Passive Devices to control Seismic Response of Structures', 1/95/DRU, CBRI, Roorkee. 28. R.N. Iyengar & D. Sharma (1998), ‘Earthquake History of India in Medieval times’, DRU, CBRI, Roorkee. 29. R.N. Iyengar, Y. Pandey, Dharmaraju (1998), ‘Strong Motion Seismic Instrumentation in and around Delhi Region’, G(S)012, CBRI, Roorkee. 30. RNI (2001) ‘Review of seismic Design Basis for PFBR-Kalpakkam’ For IGCAR, Kalpakkam 31.RNI (2002)‘Seismic Stability of Mullaperiyar Dam’ for CESS, Govt. of Kerala, Trivandrum. 32.RNI(2003) ‘Seismic Stability of Mullaperiyar-Baby Dam’ for IDRB, Govt. of Kerala, Trivandrum. 33. RNI (2004) Seismic hazard at Baku, Azerbaijan, for M/s W. S. Atkins (India) Pvt. Ltd. Bangalore. Courses Developed and Taught at IISc at Masters Level Structural Mechanics. Structural Dynamics Theory of Vibration Engineering Mathematics Introduction to Random Vibration Advanced Random Vibration Design of Structures under Dynamic loading Probabilistic Methods in Structural Engineering Statistical Methods in Meteorology A Short term course entitled 'Probabilistic Approaches in Civil Engineering' for College Teachers was organized during 21 st Feb. to 4th March, 1977. Eighteen lectures were delivered at IIT, Bombay in Dec. 1977, as part of an intensive course on 'Random Vibration'. A Special course on ‘Earthquake Engineering' was developed for staff of M/s. Tata Consulting Engineers, Bangalore, 1987. Special course on 'Structural Dynamics’ was developed for M/s. Kirloskar Electricals Limited, Bangalore, 1989. A Special course on 'Structural Dynamics of Power Plant Equipments' was developed for BHEL, R&D, Hyderabad, 1991. (Courses taught to be updated) (Ph. D) 1. P.K. Das ‘Random Vibration of Nonlinear and Stochastic Systems’. 1978 (Co-guide:C.V.Joga Rao) 2. K.J. Iyengar 'Inelastic Analysis of Structures under Impulsive and Random loads'. 1978 (Co-guide: S.A. Ramu) 3. V.J. Sundaram 'Random Vibration and Fatigue of Viscoelastic Propellants'. 1983 (Co-guide: A.K. Rao) 4 K.C. Prodhan 'Analysis Specification and Synthesis of Strong Motion Earthquakes'. 1987 5. C.S. Manohar 'Random Vibration of limit Cycle Systems and Stochastic Strings’. 1989 6. G.V. Rao 'Vibration on Cables under Deterministic and Random Excitadon 1989 7. O.R. Jaiswal 'Dynamic Analysis of Railway Tracks'. 1995 8. D. Roy 'Chaotic and Random Response in Nolinear Oscillators’. 1995 9. S.K. Agarwal ‘Strong Motion and Seismic Response of Soil Layers’ (From IIT Roorkee) 2000 10 S.T.Raghukanth Engineering Models for Earthquake Sources 2005 11. T.Rehman Seismic hazard in NE India Under progress M.Sc. Engg. 1. K. Meera 'Nonlinear Dynamic Analysis of a Railway Freight Car’. 1986 2. R. Ravi 'Free Vibration of Rectangular Plates with Holes'. 1989 3. O.R. Jaiswal 'Dynamic Analysis and Random Process Modelling of Railway Tracks'. 1991 4. M. Varadarajan 'Spectrum compatible Random Processes as Earthquake Excitations'. 1992 5. S. Basak 'Patterns of Variability of India Monsoon Rainfall’ 1993 (Submitted at Centre for Atmospheric Sciences). 6. S. Raghavachari ‘Vibration of Pipes Resting on Soil Medium’ 1993 7. D.S. Raju ‘Dynamics of Boiler Support Structures’ 1997 8 S.Ghosh Seismic Microzonation of Delhi City 2003 9 Bishaka Basak Critical Excitation and Inverse Approach in Random Vibration 2004 1. R. Muthukrishnan ‘Some Studies in Aiiisotropic Circular Plates’. 1969 2. B.N. Nagaraja ‘Dynamics of Beaiiis under Grinding Forces’ 1974 3. T.K. Saha ‘Response of Caiitilever Structures to Eartliquake Excitation’ 1976 4. G.M. Ammanagi ‘Stochastic Analysis of Monthly River Flows’ 1976 5. M.K. Divekar 'Software for Dynamics for Space Frames’ 1983 6. H.V. Shankar ‘Internal Consistency in Response Spectrum’ 1983 7. M.S. Madangopal ‘Free Vibrational Analysis of Elliptic and Pre forated Plates' 1983 8. H.A. Visweswara ‘Software for Vibration and Hydrostatic Buckling of Ring Stiffened Cylindrical Shelf’ 1983 9. C.S. Manohar ‘Stochastic Critical Seismic Excitations’ 1984 10. N. Vedagiri ‘Application of the Component Element Method’ 1984 11. N. Mahendar ‘Seismic Respoiise of a Power Plant with Base Allowed to Uplift’ 1986 12. K.R. Reddy ‘Non-linear Rocking of a Rigid Block on Rigid Foundation’ 1987 13. D. Roy ‘Structural Dynamics of Spinning Beams’ 1992 14 Raman Kumar Non-Gaussian excitation model in Random Vibration 2004 Industrial Consultancy 'Vibration of Thin Cylinders', Ministry of Defence, 1981. 'Vibration of Multistage Thin Cylinders’, (with S.A. Ramu), Ministry of Defence, 1982-83. 'Structural Dynamic Identification of Railway Vehicles’, Ministry of Railways, 1984. 'Structural Dynamic Identification of Railway Vehicles II phase: Track Index Evaluation'. Ministry of Railway, 1984. 'Piping Analysis under Seismic Forces’, (with K.P. Rao), Department of Atomic Energy, 1982. ‘Spectrum compatible Accelerograms’, Department of Atomic Energy, 1982-83. 'Nonlinear Dynamics of KAPP Raft’, Department of Atomic Energy, 1985-87. ‘NAPP Safety Systems and Coolant Channels under Seismic Forces', Department of Atomic Energy, 1985-86. 'Slosh Frequencies for Cylinders with Inclined Fluid Surface and Ellipsoidal Caps', DRDL, Hyderabad, 1987-88. 'Random Vibration of 500 MW Nuclear Power Plant under SSE Condition', Nuclear Power Corporation. 1988. 'Design of NSTL Shock Tank', MECON, Bangalore, 1985. 'Seismic Analysis of NAPP Reactivity Mechanism', NPC, Bombay, 1989. ‘Classification of Railway Tracks’, RDSO, Lucknow, 1989. 'Floor Response Spectra 235 MW NPP through PSD approach', NPC, Bombay, 1989. 'Shock Analysis of Naval Equipments’, DMDE, Indian Navy, 1988-90. 'Track Geometry Standards', RDSO, Lucknow. 'Dynamics of Power Plant Piping subjected to Stochastic loading', BHEL, Hyderabad. 'Structural Dynamics of a Flight Vehicle’, ARDE, Pune 'Floor Response Spectra of NAPP', NPC, Bombay. 'Dynamics of a Missile', DRDL, Hyderabad. ‘Review of Design Basis seismic Parameters for PFBR Site’ IGCAR, Kalpakkam.2000-2001 ‘Seismic Stability of the Mullaperiyar Dam’ Govt. of Kerala. 2001-2002 ‘Seismic Stability of Mullaperiyar-baby Dam’ govt. of Kerala 2002-2003 About 25 other consultancy projects in which Prof.R.N.Iyengar was personally involved, while he was Director CBRI, are not listed here. Field Experience Junior Engineer, Mysore State PWD, 1962-63. Team Leader: Field Investigations after Khilari (1993) , Chamoli (1999) and Kutch (2001) Earthquakes. Team Leader: Post-disaster Rehabilitation and Capacity Building Tasks at Latur and Chamoli	{"url":"http://civil.iisc.ac.in/~rni/","timestamp":"2024-11-08T02:17:44Z","content_type":"text/html","content_length":"127819","record_id":"<urn:uuid:5af21d86-93e4-4580-9985-21923f0d4b87>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00648.warc.gz"}
5 Islanders - Riddles Guru 5 Islanders Alex came across 5 islanders, all of whom knew one another and would only answer yes-or-no questions. What is the minimum number of questions Alex needs to ask, to successfully determine how many of the islanders are paladins? Alex would need to ask a minimum of 3 questions. Q1: Are there more than 2 paladins? If the answer is no, then there are 0-2 paladins. If the answer is yes, then there are 3-5 paladins. Once the range is known, Alex will need to ask 2 more questions in the form of, “Are there X paladins?”. For example, is the answer to Q1 is yes, then Alex will ask,”Are there 3 paladins?” and “Are there 4 paladins?”. If the answer to both of the questions are no, then we know that there are 5 paladins.	{"url":"https://riddles.guru/riddles/5-islanders/1878/","timestamp":"2024-11-13T18:40:08Z","content_type":"text/html","content_length":"42536","record_id":"<urn:uuid:3e979409-a6e6-4467-af4a-a776f2a04215>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00030.warc.gz"}
Math in Focus Grade 8 Chapter 11 Review Test Answer Key This handy Math in Focus Grade 8 Workbook Answer Key Chapter 11 Review Test detailed solutions for the textbook questions. Math in Focus Grade 8 Course 3 B Chapter 11 Review Test Answer Key Concepts and Skills State whether each event is a simple or compound event. Question 1. Drawing 2 yellow marbles ¡n a row from a bag of yellow and green marbles. Drawing 2 yellow marbles in a row from a bag of yellow and green marbles are compound events. Because the number of marbles is 3 so, the result is the number of outcomes. Question 2. Drawing 1 red pebble and 1 yellow pebble in a row from a bag of red and yellow pebbles. Drawing 1 red pebble and 1 yellow pebble in a row from a bag of red and yellow pebbles is compound. Because the number of pebbles is 2 so, the result is the number of outcomes Question 3. Tossing a coin once. Tossing a coin once is a simple event because it has a result of one outcome. Draw the possibility diagram and state the number of possible outcomes for each compound event. Question 4. From three cards labeled A, B, and C, draw two cards, one at a time with replacement. Question 5. From a pencil case with 1 red pen, 1 green pen, and 1 blue pen, select two pens, one at a time without replacement. Question 6. Toss a fair four-sided number die, labeled 1 to 4, and a coin. Draw the tree diagram for each compound event. Question 7. Spinning a spinner divided into 4 equal areas labeled 1 to 4, and tossing a coin. Question 8. Picking two green apples randomly from a basket of red and green apples. The probability of picking red and green apples is 1/2. State whether each compound event consists of independent events or dependent events. Question 9. From a pencil case, two-color pencils are randomly drawn, one at a time without replacement. Answer: Dependent event Question 10. From two classes of 30 students, one student ¡s selected randomly from each class for a survey. Answer: Independent event Problem Solving Solve. Show your work. Question 11. There are two tables in a room. There are 2 history textbooks and 1 math textbook on the first table. There are 1 history workbook and 1 math workbook on the second table. Use a possibility diagram to find the probability of randomly selecting a history textbook from the first table and a math workbook from the second table. There are two tables in a room. There are 2 history textbooks and 1 math textbook on the first table. 1 + 2 = 3 The probability of randomly selecting a history textbook from the first table and a math workbook from the second table is 1/3. Question 12. A fair four-sided number die is marked 1, 2, 2, and 3. A spinner equally divided into 3 sectors is marked 3, 4, and 7. Jamie tosses the number die and spins the spinner. a) Use a possibility diagram to find the probability that the sum of the two resulting numbers is greater than 5. Note that there are 4 possible outcomes in rolling a die which is 1, 2, 2, and 3, and 3 outcomes for the spinner which are 3, 4 and 7. Add these possible resuLts, then mark the answers that are bigger the 5. Observe that there are 12 possible outcomes and 8 are greater than 5 then Therefore, the probability of obtaining a number that is larger than 5 is $\frac{2}{3}$. b) Use a possibility diagram to find the probability that the product of the two resulting numbers is odd. Notice that there are 4 possible outcomes in rolling a die which is 1, 2, 2. and 3, and 3 outcomes for the spinner which are 3, 4 and 7. Multiply these possible results, then mark the answers that odd numbers. Observe that there are 12 possibLe outcomes and 4 are odd numbers then Therefore, the probability of obtaining an odd number is $\frac{1}{3}$. Question 13. A juggler is giving a performance by juggling a red ball, a yellow ball, and a green ball. All 3 balls have equal chance of dropping. If one ball drops, the juggler will stop and pick up the ball and resume juggling. If another ball drops again, the juggler will stop the performance. a) Draw a tree diagram to represent the possible outcomes and the corresponding probabilities. b) Find the probability of dropping the same colored ball twice. Answer: 1/3 c) Find the probability of dropping one green and one yellow ball. P(G)P(R) = 1/3 × 1/3= 1/9 Question 14. In a marathon, there is a half-marathon and a full-marathon. There are 60 students who participated in the half-marathon and 80 participated in the full-marathon. Half of the students in the half-marathon warm up before the run, while three-quarters of the students in the full-marathon warm up. Assume that warming up and not warming up are mutually exclusive and complementary. a) Draw a tree diagram to represent the possible outcomes and the corresponding probabilities. Since, there are 2 types of marathon, then start the tree diagram with 2 branches. Now, at the end of these branches put the possible outcomes: H for half-marathon, and F for full marathon. Note that the probability of full marathon is $\frac{80}{140}$ and the probability of half-marathon is $\frac{60}{140}$. Next, there are a group who warms up and who does not, then make another 2 branches. At the end of these branches put the possible outcomes: W for warming up and NW for not warming up. Finally, conclude the possible results. Therefore, the tree diagram for the said event is as depicted below. b) What is the probability of randomly picking a marathon participant who warms up before running a full-marathon? Answer: full marathon: 80 runners c) What is the probability of randomly picking a marathon participant who does not warm up before running? 80 + 60 = 140 total: 140 runners; 90 warm-up. The probability that a randomly selected runner warms up is 90/140. Question 15. The probability of Cindy waking up after 8 A.M. on a weekend day is p. Assume the events of Cindy waking up after 8 A.M. and by 8 A.M. are mutually exclusive and complementary. a) If p = 0.3, find the probability that she will wake up after 8 A.M. on two consecutive weekend days. P(waking up after 8 A.M. two weekend days in a row) = p² If p=0.3 p² = (0.3)² p² = 0.09 b) If p = 0.56, find the probability that she will wake up by 8 A.M. on two consecutive weekend days. P(waking up before 8 A.M two weekend days in a row) = (1 – p)² If p = 0.56, 1-p = 0.44 and (1 – p)² = (0.44)² = 0.1936. Question 16. In a jar, there are 2 raisin cookies and 3 oat cookies. Steven takes two cookies one after another without replacement. a) Draw a tree diagram to represent the possible outcomes and the corresponding probabilities. Since there are two different kinds of muffins, which are bran and pumpkin, then start the tree diagram with 2 branches. Now, at the end of these branches put the possible outcomes: P for pumpkin muffin, and B for bran muffins. Note that there are 2 pumpkin muffins out of the total of 5 muffins, and 3 bran muffins out of the 5 total muffins. So the probability for selecting a pumpkin is $\ frac{2}{5}$, and the probability of selecting a bran is $\frac{3}{5}$. Next, there are stilt 2 different kinds of muffins, then again, at the end of these branches put the possible outcomes: P for pumpkin muffin, and B for bran muffins. If a pumpkin muffin is obtained in the first pick, then on the second pick, there are now 1 pumpkin muffin and a totat of 4 muffins left since a muffin is already taken, then the probability of choosing a pumpkin is $\frac{1}{4} $. Also, there are still 3 bran muffin out of 4 muffins left, thus, the probability of selecting a bran muffin is $\frac{3}{4}$. If a bran muffin is obtained in the first pick, then on the second pick, there are still a pumpkin muffin and a total of 4 muffins left since a muffin is already taken, then the probability of choosing a pumpkin is $\frac{2}{4}$. Also, there are only 2 bran muffin left out of 4 muffins, thus, the probability of selecting a bran muffin is $\frac{2}{4}$. Therefore, the tree diagram for the said event is as depicted below. b) Find the probability of Steven randomly getting two of the same type of cookie. The probability of seLecting 2 muffins of a matching type is the probability of obtaining 2 pumpkin muffins, P(P, P), or the probability of choosing 2 bran muffins, P(B, B), then P(P, P) or P(B, B) = P(P, P) + (B, B) = P(P) · P(P) +P(B) · P(B) = $\frac{2}{5}$ · $\frac{1}{4}$ + $\frac{3}{5}$ · $\frac{2}{4}$ = $\frac{2}{20}$ + $\frac{6}{20}$ = $\frac{8}{20}$ Therefore, the probability of picking 2 muffins of a matching type is $\frac{8}{20}$. c) Find the probability of Steven randomly getting at least one raisin cookie. For, the probability of selecting a minimum of one pumpkin muffin, note that the probability of choosing 2 bran muffins, P(B, B), and the probability of picking at least one pumpkin muffin is P(B, B) + P(Minimum of One Pumpkin) = 1 P(Minimiun of One Pumpkin) = 1 – P(B, B) P(Minimum of One Pumpkin) = 1 – P(B) · P(B) P(Minimum of One Pumpkin) = 1 – $\frac{3}{5}$ · $\frac{2}{4}$ P(minimum of One Pumpkin) = 1 – $\frac{6}{20}$ P(Minimum of One Pumpkin) = $\frac{7}{10}$ Therefore, the probabiUty of picking a minimum of one pumpkin is $\frac{7}{10}$. Question 17. Out of 100 raffle tickets, 4 are marked with a prize. Matthew randomly selects two tickets from the box. a) Draw a tree diagram to represent the possible outcomes and the corresponding probabilities. b) What is the probability that Matthew does not win any prizes? Answer: 152/165 c) What is the probability that Matthew gets exactly one of the prizes? Answer: 64/825 Question 18. The tree diagram shows the probability of how Shane spends his day gaming or cycling, depending on the weather. The probability of rain is denoted by a. Assume that gaming and cycling are mutually a) If a = 0.4, find the probability that he will spend his day gaming. Given that a = 0.4 Here the G represents gaming. The probability that he will spend his day on gaming =(1- 0.4) × 1/4 = 0.15. b) If a = 0.75, find the probability that he will spend his day cycling. Given that a = 0.75 Here C represents cycling. The probability that he will spend his day in cycling = (1-0.75) × 3/4 = 0.1875. Leave a Comment	{"url":"https://mathinfocusanswerkey.com/math-in-focus-grade-8-chapter-11-review-test-answer-key/","timestamp":"2024-11-12T00:21:30Z","content_type":"text/html","content_length":"149046","record_id":"<urn:uuid:d22f9b81-ed5a-4f58-9a4c-fc2fc585547d>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00437.warc.gz"}
The Concept of Measures - Basculasbalanzas.com The concept of measures has many applications in mathematics and probability theory. These concepts are the basic premise of measure theory, which studies the properties of s-algebras, measurable functions, and integrals. The main motivation for using measures was to be able to integrate more functions than could be integrated in the Riemann sense. For instance, a measure may assign length to a set of points without a defined area. This is because not all subsets have the same area, and not all can be assigned a length that preserves the measuring process. A measure is a segment of time within a piece of music. It is divided into equal segments by a vertical bar line. Each beat is represented by a particular note value. A bar line indicates the boundary of a measure. This is a useful way to make written music easier to understand because it provides a regular reference point. In addition, the staff symbols represent a batch of data. However, not all measurements have a fixed size. When a business needs to know how many customers it has, it uses a measure. A measure is a percentage of sales. In other words, a measure reflects how many times a customer makes a purchase. This number is called a ‘volume’. If you need to measure the volume of a company, you can use a volume metric. A quantity is a unit of volume. A measurement can also represent the amount of money a business makes. A measure is a generalization of length, area, or volume. It can be used to compare the performance of different elements of a business. For example, a sales representative might want to compare the total sales of a business with the total sales in a given period. If you compare two numbers, you’ll notice that the two measures are different. The difference is in the units of measurement. You can see that a measurement is actually a number. A measure is a generalization of length, area, and volume. It can be interpreted as a mass distribution. A dimension, on the other hand, can be used to represent a logical structure. Its units are different between different systems, but the metric you’re comparing can help you compare metrics from different companies. This is why a measure is a useful measure. The number of measurements that you want to track is very important. A measure is a numerical value. It is the result of a calculation. The values of a measure can be summarized using a number. The same is true for a dimension and a measurement. These two types of numbers can be derived by the same process. Then, you can compare one set against another and use the sum of the two. The results of both can be correlated to each other. Once you have a good understanding of the results, you can use a different type of method for determining the effectiveness of the system. A measure is a category of data. The same metric can be represented by different units. For instance, in a certain music context, a measurement can represent the volume of an instrument, the number of visitors, or the number of sales. If a person is in the same time zone as the instrument, they may have a different concept. If the instruments and objects are similar, the two measures are essentially the same. It is not a matter of what kind of instrument, but rather the number of people who use the software. A measure is a numerical value that is defined by a given period of time. A measure is a number that is calculated for a specific cell or point. It is often used to quantify the results of a system. The values may be expressed as probabilities. In some cases, these numbers are expressed as a percentage. Nevertheless, these concepts can be applied to any kind of system. It may be useful for determining the overall effectiveness of a specific product or service. As you can see, a measure is a unit of time. In contrast, a piece of music can be broken down into multiple measures. The individual beats in a music piece are called measure. This allows for easier understanding. The name of a given metric is also important. For example, a monetary value can be the currency of a nation. A pound of gold is a single unit. In contrast, a unit of gold is measured in terms of time.	{"url":"https://www.basculasbalanzas.com/the-concept-of-measures-3/","timestamp":"2024-11-10T15:39:00Z","content_type":"text/html","content_length":"52987","record_id":"<urn:uuid:36835c90-dc6e-4868-bd29-a0df984838c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00098.warc.gz"}
“Quantum computing is like uncharted territory, with lots of new applications just waiting to be discovered.” What is quantum computing, and what problems will it help us solve? David Sadek, Doctor in Computer Science, and VP, Research, Technology & Innovation at Thales, specifically in charge of Artificial Intelligence & Information Processing, outlines the progress made and the possibilities ahead. Quantum computers exploit the quantum properties of matter to carry out computing tasks, and they're expected to revolutionise the world of computing as we know it. But in terms of practical applications, what impact will these new machines have in the coming decades? Quantum computing is more than just a technological breakthrough – it’s a completely new way of seeing computing in the conventional sense of the term and will give us unprecedented problem-solving and data analysis capabilities. The supercomputers currently used by mathematicians carry out processing tasks on the basis of bits, in other words series of 1s and 0s. A bit has either the value 1 or 0, but cannot have both values at the same time. Quantum computers use quantum bits, or qubits, units of information which, according to the laws of quantum physics, can be in both states – 1 and 0 – simultaneously. Like Schrödinger’s cat, this state of quantum superposition allows computation to be performed simultaneously for all of the "superposed" states of a qubit. Given that the number of computations that can be carried out grows exponentially with the number of qubits used, we can see that a computer capable of handling a large number of qubits would have unprecedented power to address the kind of problems that were previously unsolvable using a conventional approach, or that could only be partly solved after years of computation. Quantum supremacy – the point at which a quantum computer becomes capable of solving problems that no conventional machine could ever solve – will be the next paradigm shift. It will be a real game changer in the world of computing, and like the digital revolution that preceded it, it will open the door to applications that simply didn't exist before. Given that quantum computers capable of handling a large number of qubits are still at the prototype stage, how do we prepare for this transformation? Scientists established the mathematical principles behind this new type of computer long before quantum computers themselves were developed. The first quantum algorithms, designed to leverage the properties of qubits, have been in existence since the 1980s. The Deutsch algorithm, for example, provided the first indication, on a theoretical level, that a quantum computer could one day outperform a conventional computer. Shor's algorithm, which is used for finding the prime factors of an integer, was developed in 1994. Since then, it has come to be seen as a threat – theoretical for the time being, yet nonetheless critical – to public key cryptography systems, which are routinely used in cybersecurity applications, often without users being aware, to protect banking transactions such as credit card payments. To be part of the coming quantum revolution, we first need to understand the fundamental science behind it. This is why Thales is adapting to this new paradigm in computing to produce a new generation of algorithm developers who are capable of comprehending the IT world in quantum terms – something that requires a fundamental change of mindset. It’s also important to understand that quantum computers capable of handling a sufficiently large number of qubits to achieve quantum supremacy – the Holy Grail of the quantum revolution – may only be at the prototype stage today, but machines with the capacity to run quantum algorithms already exist, so the first concrete use cases are well and truly on the table. Which technologies are giving us a foretaste of the new world of quantum computing? They fall into two main categories. First, there are the machines and technologies that exploit the quantum phenomena themselves, such as cold atoms, ion traps, superconducting loops, photon polarisation, etc. Even though the computers which harness these phenomena are currently only capable of handling a small number of qubits, they can be used to run quantum algorithms, so the first proofs-of-concept* (PoC) for real use cases can be developed straight away, and applications could emerge by 2030. For example, a French start-up called Pasqal, which is one of the many companies Thales is working with in this field, is developing a PoC using an actual quantum computer based on cold atom technology, which could soon provide a practical solution for banking security The second category is emulators, which simulate the operation of a quantum computer on a conventional machine so we can implement quantum algorithms and study their performance today. The problem is that emulating a qubit on a conventional machine requires a significant amount of computing power, and the power requirements will rise exponentially as more qubits are added. Even though these two types of technology only provide a foretaste of the capabilities of future quantum computers, they give us important insights into how the machines will work. That means we can already imagine new uses – which will clearly evolve over time – and get to grips with a technology that will go mainstream in the not-too-distant future. We’re a bit like skiers practising on an artificial slope while we're waiting for the start of the winter season so we can get out onto some real snow! How will Thales position itself in the race to develop quantum computers? We’re not aiming to become a manufacturer of quantum computers. But we have significant know-how in key enabling technologies, and this could potentially position us in the supply chain for the future machines. The challenge for a company like ours is to develop practical solutions within the extremely broad field of possibilities opened up by quantum computing. Our objective is clear: we aim to be a pioneer in the development of applications that harness the new possibilities of quantum computing at scale, for both civil and military users, and a leader in the design of the quantum algorithms needed to implement these applications. To achieve this, we will be expanding our capabilities in three key areas simultaneously: the algorithms themselves, by producing a new generation of algorithm developers with expertise in the specific approaches and formalisms used in quantum computing; deployment of actual quantum machines, where Thales is already recognised as one of the only companies in France that can help to benchmark the various quantum computers already in existence; and the software tools that will be needed to take quantum computing into the mainstream and make it accessible to users with less expert knowledge of the underlying theory. By ramping up our capabilities in all these areas and making them available to all the Group's businesses, we will be able to use quantum computing to help solve practical problems in real-world applications that cannot be solved by conventional machines and algorithms. What type of problems could be solved by quantum computing? At Thales, we take a pragmatic approach. Our first step has been to identify the practical challenges to applications in our areas of business that could be overcome by quantum computers when conventional supercomputers reach the limits of their capabilities. Then we classified those practical challenges into six categories of problems that quantum computers could solve: (1) combinatorial optimisation, which involves finding the best option from a large number of possibilities, a classic algorithmic conundrum frequently illustrated by the “travelling salesman problem”*); (2) resolution of linear systems (including differential equations), which could have potentially beneficial applications in the field of electromagnetism; (3) so-called Monte Carlo methods, which are based on the principles of random sampling and can be used for large-scale testing or probabilistic simulations; (4) quantum machine learning, which is at the intersection of artificial intelligence and quantum technology and involves using quantum computers to optimise neural networks; (5) testing cryptographic resistance to decryption by quantum algorithms; and (6) the highly promising field of quantum simulation of matter at molecular level, for example to determine the behaviour of a given chemical or to synthesise new molecules. In all these categories, whenever quantum algorithms and quantum computing could provide new ways of solving problems, we are exploring possible applications in all our areas of business. Can you give us any concrete examples of applications that Thales has started to explore using quantum algorithms and quantum computing technologies? Combinatorial optimisation algorithms are among the most likely to benefit from quantum technology, and we recently launched a PoC relating to a mission planning solution for satellite constellations. To coordinate the movements of satellites, operators have to manage large numbers of parameters and interactions. Even with a small group of satellites, this can quickly create so-called NP-hard problems*, i.e. problems that cannot be solved in a reasonable timeframe by a conventional algorithm, no matter how much computing power is used. In a simulation combining a quantum algorithm with conventional techniques, we have already demonstrated the feasibility of the solution for a small number of satellites, which suggests we could solve the mission planning problem for larger constellations. We are also testing quantum approaches to electromagnetic simulation in radar antenna design, for example using an HHL algorithm (named for its inventors, Harrow, Hassidim and Lloyd) to solve linear equations. Another example is the use of "quantum machine learning" algorithms to expose cyberattacks and for anomaly detection in images. Will quantum computing eventually replace conventional IT as we know it today? It's hard to be sure. What we do know is that a new family of computers will be needed to solve certain problems, such as quantum simulation of matter, which are beyond the computation capabilities of our current machines. We also know for certain that quantum computers will not take over from traditional computers overnight. There will be a long transition, and during this intermediate phase computation tasks will be performed using hybrid solutions that combine high-performance computing (HPC) by conventional supercomputers with the use of quantum processor units (QPUs). In conceptual terms, this hybrid approach is anything but simple, because the tasks to be assigned to the quantum processor and the conventional computer will have to be identified for each operation. It will also be a challenge in terms of the range of skills that the engineers working on these systems will require. But we believe a hybrid approach offers the best opportunities for using quantum algorithms and quantum computing techniques to develop solutions at scale, probably for engineering in the broad sense of the term in the first instance, and then for the development of real-time applications. This is the route that Thales has decided to take. Schrödinger's cat is a thought experiment in quantum mechanics that was proposed by physicist Erwin Schrödinger in 1935. The experiment involves a cat in a sealed box with a vial of poison that may or may not be released depending on the state of a subatomic particle. If the subatomic particle is in a superposition of states, the cat would be both alive and dead until the box is opened and the state of the particle is determined. A Proof of Concept is aimed at demonstrating that a new idea or product is feasible and can be implemented in practice. The travelling salesman problem, a classic mathematical conundrum, has been the focus of extensive research over the years. It continues to be used today, for example as an introduction to computational complexity theory, which focuses on the time and memory required for an algorithm to solve a given problem. The travelling salesman problem can be summarised as follows: given a list of cities to be visited by a salesman, with the distances between each pair of cities being known, what is the shortest possible route that visits each city once and returns to the point of departure? Despite this simple definition, the problem is a complex one: the number of possible routes rises exponentially as the number of cities on the salesman’s route increases. To visit 7 cities, for example, there are a total of 360 possible routes. With 15 cities, the number rises to around 43 billion. If we apply the question to a list of 71 cities, the total would be 51080, roughly equivalent to the number of atoms in the universe. A conventional computer would quickly be overwhelmed by this phenomenon, known as a combinatorial explosion, and would be forced to suggest approximate solutions, due to the vast amount of time that would be required to compute all the different possibilities. A number of algorithms developed in recent years have proposed using quantum computing to solve the problem more efficiently. Running such algorithms on a suitable machine would finally enable our travelling salesman to find his ideal route – but they could also be used to optimise telecom networks, bus routes or logistics operations. ****In computational complexity theory, NP (non-deterministic polynomial time) is a complexity class used to classify decision problems. NP-hard problems are particularly difficult to solve.	{"url":"https://www.thalesgroup.com/en/group/innovation/magazine/quantum-computing-uncharted-territory-lots-new-applications-just-waiting","timestamp":"2024-11-14T04:46:31Z","content_type":"text/html","content_length":"65361","record_id":"<urn:uuid:6da3926a-8093-4b9e-9f38-01c2a96ec753>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00671.warc.gz"}
Nonadiabatic rate constants for proton transfer and proton-coupled electron transfer reactions in solution: Effects of quadratic term in the vibronic coupling expansion Rate constant expressions for vibronically nonadiabatic proton transfer and proton-coupled electron transfer reactions are presented and analyzed. The regimes covered include electronically adiabatic and nonadiabatic reactions, as well as high-frequency and low-frequency proton donor-acceptor vibrational modes. These rate constants differ from previous rate constants derived with the cumulant expansion approach in that the logarithmic expansion of the vibronic coupling in terms of the proton donor-acceptor distance includes a quadratic as well as a linear term. The analysis illustrates that inclusion of this quadratic term in the framework of the cumulant expansion framework may significantly impact the rate constants at high temperatures for proton transfer interfaces with soft proton donor-acceptor modes that are associated with small force constants and weak hydrogen bonds. The effects of the quadratic term may also become significant in these regimes when using the vibronic coupling expansion in conjunction with a thermal averaging procedure for calculating the rate constant. In this case, however, the expansion of the coupling can be avoided entirely by calculating the couplings explicitly for the range of proton donor-acceptor distances sampled. The effects of the quadratic term for weak hydrogen-bonding systems are less significant for more physically realistic models that prevent the sampling of unphysical short proton donor-acceptor distances. Additionally, the rigorous relation between the cumulant expansion and thermal averaging approaches is clarified. In particular, the cumulant expansion rate constant includes effects from dynamical interference between the proton donor-acceptor and solvent motions and becomes equivalent to the thermally averaged rate constant when these dynamical effects are neglected. This analysis identifies the regimes in which each rate constant expression is valid and thus will be important for future applications to proton transfer and proton-coupled electron transfer in chemical and biological processes. All Science Journal Classification (ASJC) codes • General Physics and Astronomy • Physical and Theoretical Chemistry Dive into the research topics of 'Nonadiabatic rate constants for proton transfer and proton-coupled electron transfer reactions in solution: Effects of quadratic term in the vibronic coupling expansion'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/nonadiabatic-rate-constants-for-proton-transfer-and-proton-couple","timestamp":"2024-11-04T19:17:52Z","content_type":"text/html","content_length":"55503","record_id":"<urn:uuid:d0345d72-dd9e-405c-a5a2-c81398eef245>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00170.warc.gz"}
[SOLVED] Spirtes' example of d-separation not leading to independence in a directed cyclic graph with non-linear structural equations ~ Cross Validated ~ TransWikia.com Here is my explanation. I believe the author is right. It comes down to this: for a double arrow relationship $$Wlongleftrightarrow Z,$$ neither $$W$$ nor $$Z$$ is considered a descendant of the other (unless you have other edges relating them). That is, $$W$$ is not a descendant of $$Z,$$ nor is $$Z$$ a descendant of $$W.$$ So let us consider your graph, but only one direction at a time: Here, conditioning on the set $${W,Z}$$ does open up the collider at $$Z$$. However, the path from $$X$$ to $$Y$$ is still blocked by the chain at $$W,$$ since $$W$$ is in the conditioning set. Similarly, if we consider the other "half" of the graph, the same conditioning set opens the collider at $$W$$ but closes the chain at $$Z.$$ In either setting, causal information cannot flow from $$X$$ to $$Y,$$ hence $${W,Z}$$$$d$$-separates $$X$$ and $$Y.$$ References: Causality: Models, Reasoning, and Inference, 2nd Ed., by Judea Pearl, pp. 17-18. Note that in the example of Fig. 1.3(a), Pearl has to resort to the path $$Z_3to Z_2to Z_1$$ to show that $$Z_1$$ is a descendant of $$Z_3;$$ he does not use what would be the obvious $$Z_1longleftrightarrow Z_3$$ relationship. Correct answer by Adrian Keister on December 10, 2020	{"url":"https://transwikia.com/cross-validated/spirtes-example-of-d-separation-not-leading-to-independence-in-a-directed-cyclic-graph-with-non-linear-structural-equations/","timestamp":"2024-11-07T17:03:13Z","content_type":"text/html","content_length":"46428","record_id":"<urn:uuid:bdbfbdd4-ee7f-4aae-b054-76dd03be3b15>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00714.warc.gz"}
Discount factor to interest rate calculator 26 Oct 2010 In a previous post, I described the technique that computer programs like Microsoft Excel use to calculate the XIRR (effective interest rate) as a Interest rates and the time value of money What is the basis of determining discount rate? If so, what other factors besides inflation should be considered? To calculate present value you need a forecast of the future cash flows, and you How to Discount Cash Flow, Calculate PV, FV and Net Present Value How do analysts choose the discount (interest) rate for DCF analysis? divides each FV value by a more substantial discount factor than do mid-period calculations. Discount Factors and Equivalence.. 51-2 the sinking fund factor, you could calculate the neces- The interest rate used in the discount factor formulas is. Table of contents. Chapter 1. Interest rates and factors. 1. 1.1. Interest. 2. 1.2 discounted to a previous point in time, its present value is calculated, taking into So Sam tries once more, but with 7% interest: At 7% Sam gets a Net Present Value of $15. Close enough to zero, Sam doesn't want to calculate any more. 6 Dec 2018 With regard to the discounted rate, this factor is based on how the high-interest loans should be considered when determining the NPV. 23 Aug 2018 Annuity factors are used to calculate present values of annuities, and The Annuity Factor is the sum of the discount factors for maturities 1 Sometimes also known as the Present Value Interest Factor of an Annuity (PVIFA). The discount factor can be calculated based on the discount rate and number of compounding periods. Code to add this calci to your website Expand embed code 8 Apr 2010 annual (p.a.) interest rate given as a per cent value: i = p/100 periods: is calculated by multiplying the principal P by the accumulation factor, 8 Mar 2018 To calculate the discount factor for a cash flow one year from now, divide 1 by the interest rate plus 1. For example, if the interest rate is 5 Some analysts prefer to calculate explicit discount factors in each time period so they can see the effects of compounding more clearly, as well as making the Discount Factor Calculator - calculate the discount factor which is a way of discounting cash flows to get the present value of an investment. Discount factor Formula for the calculation of a discount factor based on the periodic interest rate and the number of interest HOMER calculates the annual real discount rate (also called the real interest rate or HOMER uses the real discount rate to calculate discount factors and The discount rate is the annualized rate of interest and it is denoted by 'i'. Step 2: Now, determine for how long the money is going to remain invested i.e. the tenure If only a nominal interest rate (rate per annum or rate per year) is known, you can calculate the discount rate using the following formula: Simple Amortization 6 Dec 2018 With regard to the discounted rate, this factor is based on how the high-interest loans should be considered when determining the NPV. 10 Apr 2019 This is the interest rate. However, another guy may calculate the return with reference to the future value—his calculation would be ($60,000 - frequencies of compounding, the effective rate of interest and rate of discount, and Solution: We first calculate the discount factors v(4) and v(9). For case (a) 11 May 2017 It is this rate of interest that is known as the discount rate. In a personal injury action, his award will be calculated to ensure that if he Furthermore, if the courts decide that inflation is likely to become an important factor, they Discount Factor Calculation (Step by Step) It can be calculated by using the following steps: Step 1: Firstly, figure out the discount rate for a similar kind of investment based on market information. The discount rate is the annualized rate of interest and it is denoted by ‘i’. The definition of a discount rate depends the context, it's either defined as the interest rate used to calculate net present value or the interest rate charged by the Federal Reserve Bank. There are two discount rate formulas you can use to calculate discount rate, WACC (weighted average cost of capital) and APV (adjusted present value). How to calculate discount rate? The formula used to calculate discount is D=1/(1+P) n where D is discount factor, P = periodic interest rate, n is number of payments. To calculate a discount rate, you first need to know the going interest rate that your business could get from investing capital in an investment with similar risk. You can then calculate the discount rate using the formula 1/(1+i)^n, where i equals the interest rate and n represents how many years until you receive the cash flow. 10 Apr 2019 This is the interest rate. However, another guy may calculate the return with reference to the future value—his calculation would be ($60,000 -	{"url":"https://topoptionskuou.netlify.app/deranick56218do/discount-factor-to-interest-rate-calculator-270","timestamp":"2024-11-02T04:55:53Z","content_type":"text/html","content_length":"32514","record_id":"<urn:uuid:c7806ff8-7f20-49f6-a569-721620bc99e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00135.warc.gz"}
National Population Projections National Population Projections National Population Projections provide projected populations of New Zealand, based on different combinations of fertility, mortality, and migration assumptions. Demographic projections provide an indication of future trends in the size and composition of the population, labour force, families and households. The projections are used for community, business and government planning and policy-making in areas such as health, education, superannuation and transport. The projections, along with the assumptions for fertility, mortality and migration, are typically updated every two to three years. National population projections are produced to assist businesses and government agencies, in planning and policy-making. The projections provide information on the changing characteristics and distribution of the population, which are used to develop social policies in areas such as health and education. For example, the ageing population, population projections can help identify likely future service needs. The projections are neither predictions nor forecasts. They provide an indication of possible future changes in the size and composition of the population. While the projection assumptions are formulated from an assessment of short-term and long-term demographic trends, there is no certainty that any of the assumptions will be realised. Significant events impacting this study series Population concept for all demographic estimates, projections and indices changed from 'de facto' to 'resident'. Population estimates based on the de facto population concept (the estimated de facto population) include visitors from overseas, but made no adjustments for net census undercount or residents temporarily overseas. Population estimates based on the resident population concept (the estimated resident population) include adjustments for net census undercount and residents temporarily overseas, but exclude overseas visitors. The reference date for projections is shifted from 31 March to 30 June. For the first time, Statistics NZ applied a stochastic (probabilistic) approach to producing population projections. Stochastic population projections provide a means of quantifying demographic uncertainty, although it is important to note that estimates of uncertainty are themselves uncertain. By modelling uncertainty in the projection assumptions and deriving simulations, estimates of probability and uncertainty are available for each projection result. No simulation is more likely, or more unlikely, than any other. The simulations provide a probability distribution which can be summarised using percentiles, with the 50th percentile equal to the median. Usage and limitations of the data Nature of Projections These projections are not predictions. The projections should be used as an indication of the overall trend, rather than as exact forecasts. The projections are updated every 2–3 years to maintain their relevance and usefulness, by incorporating new information about demographic trends and developments in methods. The projections are designed to meet both short-term and long-term planning needs, but are not designed to be exact forecasts or to project specific annual variation. These projections are based on assumptions made about future fertility, mortality, and migration patterns of the population. While the assumptions are formulated from an assessment of short-term and long-term demographic trends, there is no certainty that any of the assumptions will be realised. The projections do not take into account non-demographic factors (eg war, catastrophes, major government and business decisions) which may invalidate the projections. Main users of the data Statistics New Zealand, Ministry of Health, Government Planners/Local Body Planners, Ministry of Education, Consultants, Private Businesses. Information Release National Population Projections 2022(base)-2073 Information release National population projections: 2020(base)–2073 How accurate are population estimates and projections? An evaluation of Statistics New Zealand population estimates and projections, 1996–2013. How accurate are population estimates and projections? An evaluation of Statistics New Zealand population estimates and projections, 1996–2013 evaluates the accuracy of recent national and subnational population estimates and projections. The report focuses on estimates and projections of the total population produced and published since 1996, although earlier projections are included where practicable. It is designed to help customers understand the accuracy of Stats NZ’s population estimates and projections relative to observed populations, the reasons for inaccuracies, and discusses current developments that may improve accuracy. Information release National population projections: 2016(base)–2068 Experimental stochastic population projections for New Zealand: 2009 (base) – 2011 Population and migration Data Collection National Population Projections ^en-NZ The 'cohort component' method has been used to derive the population projections. Using this method, the base population is projected forward by calculating the effect of deaths and migration within each age-sex group (or cohort) according to the specified mortality and migration assumptions. New birth cohorts are added to the population by applying the specified fertility assumptions to the female population of childbearing age. The stochastic approach used in the national population projections since the 2011-base projections involves creating 2,000 simulations for the base population, births, deaths, and net migration, and then combining these using the cohort component method. These simulations can be summarised by percentiles, which indicate the probability that the actual result is lower than the percentile. For example, the 25th percentile indicates an estimated 25 percent chance that the actual value will be lower, and a 75 percent chance that the actual result will be higher, than this percentile. Nine alternative percentiles of probability distribution (2.5th, 5th, 10th, 25th, 50th, 75th, 90th, 95th, and 97.5th percentiles) are available in NZ.Stat. Projection Assumptions Projection assumptions are formulated after analysis of short-term and long-term demographic trends, patterns and trends observed in other countries, government policy, information provided by local planners and other relevant information. Assumptions for national projections are derived for each single-year of age to produce projections at one-year intervals. The following describes how assumptions are applied for national Projected (live) births are derived by applying age-specific fertility rates to the mean female population of childbearing age. The mean female population for each age is derived by averaging the population at the start and end of each year. The sum of the number of births derived for each age of mother gives the projected number of births for each year. The female age-specific fertility rates for each year of the projection period represent the number of births to females of each age in each year. The set of age-specific fertility rates for each year is typically summarised by the total fertility rate. For all population projections, a sex ratio at birth of 105.5 males per 100 females is assumed, based on the historical annual average of the total population. The fertility assumptions should not be used as a precise measure of fertility or of fertility differentials between groups. It is important to note that the objective of population projections is not to specifically measure or project the fertility of the population. For projection purposes it is more important to have a realistic yet tractable model for projecting fertility trends (and birth numbers) into the future. Mortality assumptions are formulated in terms of survival rates. This is because in the projection model the base population is survived forward each year. The projected number of deaths is calculated indirectly. Survival rates are applied to births and single years of age. There are different survival rates for each age of life and for males and females. The male and female age-specific survival rates for each year of the projection period represent the proportion of people at each age-sex who will survive for another year. In general, survival rates are highest at ages 5–11 years and then decrease with increasing age. The set of age-sex-specific survival rates for each year is typically summarised by male and female life expectancies at birth. Annual survival rates are applied separately to the population at the start of each year, births and migrants. The mortality assumptions should not be used as a precise measure of mortality or of mortality differentials between groups. It is important to note that the objective of population projections is not to specifically measure or project the life expectancy of the population. For projection purposes it is more important to have a realistic yet tractable model for projecting mortality trends (and death numbers) into the future. Migration assumptions are formulated in terms of a net migration level and an age-sex net migration pattern for each year of the projection period. Where practical, both the level and age-sex pattern are derived from a detailed analysis of net migration, including: • external migration data (from passenger cards): • arrivals and departures by country of citizenship • New Zealand citizen arrivals and departures by country of source/destination • Immigration New Zealand data: • residence applications and approvals • student and work visas 2020-base to 2073 The 2020-base national population projections (released December 2020) have as a base the estimated resident population (ERP) of New Zealand at 30 June 2020, and cover the period to 2073 at one-year intervals. They superseded by the 2016-base national population projections (released October 2016). Detailed information on the 2020-base population, assumptions and 'what if' scenarios used can be found here National Population Projections 2020-base 2016-base to 2068 The 2016-base national population projections (released October 2016) have as a base the provisional estimated resident population (ERP) of New Zealand at 30 June 2016, and cover the period to 2068 at one-year intervals. They supersede the 2014-base national population projections (released November 2014). Detailed information on the base population, assumptions and 'what if' scenarios can be found here: National Population Projections 2016-base 2014-base to 2068 The 2014-base national population projections have been superseded by the 2011-base national population projections (released September 2012). Detailed information on the 2014 base population, assumptions and 'what if' scenarios used can be found here National Population Projections 2014-base	{"url":"https://datainfoplus.stats.govt.nz/item/nz.govt.stats/583ca9da-d6d2-41e0-b626-5743c14deaf5/128","timestamp":"2024-11-08T23:33:08Z","content_type":"text/html","content_length":"58758","record_id":"<urn:uuid:5744626f-5e8c-4074-b398-1126e63d9701>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00002.warc.gz"}
st: Re: testing for cross-sectional independence [Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index] st: Re: testing for cross-sectional independence From Christopher F Baum <[email protected]> To [email protected] Subject st: Re: testing for cross-sectional independence Date Sat, 9 Oct 2004 09:33:30 -0400 On Oct 9, 2004, at 2:33, Levy wrote: I have an unbalanced panel for several hundred firms. the maximium length is 7 whereas the short is 1 for some of the firms. the mean length is 5.4. I want to fit a fixed effect regression while one of my supervisor prefers poolling regression. I don't know what kind of test can i use to choose models between them can u give me some suggestions? i have tried xttest2, but it does not work, i will try xttest0 later, thanks xttest2 is equivalent to the test that could be carried out after a -sureg- estimation. You cannot run -sureg- when N>T, and that is surely the case with your data. xttest2 tries to calculate the residual correlation matrix from your fixed effects model, and that matrix must be singular if there are more firms than observations per firm. I do not see that a fixed effects regression is likely to be very successful in this context when you have so few observations (in some cases only one!) per firm. I would recommend using something like an industry fixed effect if you can argue on economic terms that firms in the same industry might share an 'industry effect', which you imagine will be more important than the within variation in an industry. Kit Baum Faculty Micro Resource Center [ [email protected] ] Boston College Academic Technology Services * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/	{"url":"https://www.stata.com/statalist/archive/2004-10/msg00218.html","timestamp":"2024-11-10T10:59:51Z","content_type":"text/html","content_length":"8966","record_id":"<urn:uuid:bacb3c44-13eb-440e-8b41-7df03d7efed6>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00193.warc.gz"}
ThrustCurve Hobby Rocket Motor Data Flight Simulation It is important to simulate the flight of your rocket before flying for several reasons: • Safety It's critical to choose a motor that is powerful enough for a stable flight of your rocket. • Legality Every flying site has limits on the altitude to which you can fly and simulation will give you an idea of how high your rocket will go. • Recovery Simulation will tell you the optimal delay time for triggering the recovery system. This site provides data files for use with the various flight simulators. On this page is a description of simulation in general. See the simulators page for a list of existing simulator programs. There is also a simple rocket flight simulator available on this site in the motor guide. See also John's flight physics article for more details of these forces and the calculations employed. In-Flight Forces There are three main forces that affect the altitude to which a rocket will fly: ^2 on the rocket. This is the major force that the rocket motor must overcome to lift off. Another force that affect a rocket's flight is wind. However, this is usually left out of simulations and rockets are only flown in low-wind conditions. In the graph above, you can see a rough illustration of how the forces apply to a rocket in flight. Thrust simply follow's the motor's thrust curve. Gravity is a (relatively) constant force. Drag increases sharply with increasing speed (proportional the square of the velocity). Drag is highest near the end of the motor's burn (when all the thrust has been applied and maximum speed achieved). Flight Phases A rocket flight is broken down into several phases: • Liftoff The moment the rocket starts its ascent. • Powered flight The time during which the rocket is being accelerated by the motor. • Burnout The end of the motor's burn. • Coasting The time after motor burnout, while momentum is still causing the rocket to rise. • Apogee The point of maximum altitude (and zero velocity), where the rocket stops rising and starts descending. • Descent The remainder of the flight until the rocket reaches the ground. During powered flight, the motor is providing thrust and the rocket is accelerating upward. Because of this, the velocity and drag are increasing. The thrust applied by the motor varies from moment to moment according to its characteristic burn pattern (graphed by its thrust curve). Near the end of powered fight, hobby rockets reach max Q, which is usually where "shreds" occur. While coasting, the momentum of the rocket is still carrying it upward, but since the motor is no longer providing thrust the speed is decreasing due to gravity and drag. The apogee point is critical in all simulations since it provides the maximum altitude reached by the rocket. The time from burnout to apogee is also important for choosing a delay time when using motor ejection since the recovery system should be triggered when the rocket is moving slowly. Because most flight simulation is concerned with altitude achieved, the descent (or recovery) phase is usually of less interest. The simulations perfomed by the motor guide stop at apogee. Simulation Analysis The graphs below are from a simulation done with a simple rocket flying on an AeroTech M1939 (one of the author's favorite motors). Click on the buttons below the graph to see how the different forces and measurements change during the flight. Thrust is the force provided by the motor. Graphing it over time produces the motor's "thrust curve." This does not include any other factors and comes from an actual static test of the motor. These files, which are specific to each motor, are the purpose of this site. Acceleration is the sum of all forces acting on the rocket. Thrust is pushing the rocket up during the burn, gravity is pulling it down through the entire flight, and drag is slowing its speed. Note that in these graphs, negative values are chopped off at zero. This makes it appear that the acceleration reaches zero and stays there. In reality, the acceleration becomes negative when the rocket starts slowing down. Drag is calculated from the speed of the rocket, since it is proprtional to the square of the velocity. Note how similar the drag and velocity curves are, except that the drag curve is steeper because of the square function. Velocity is the speed the rocket is traveling. This is determined at any given point by taking the velocity at the previous point and applying the acceleration at the current point. (For rockets that stay in the lower atmosphere, max Q occurs at maximum velocity.) Altitude is the height above ground reached by the rocket. This is determined at any given point by taking the altitude at the previous point and applying the velocity at the current point. Apogee is reached when velocity is zero, which defines the highest point reached by the rocket. Note that we stop simulating at apogee; otherwise, the altitude would drop again to zero during the descent For the mathematically minded, the acceleration is the sum of static and dynamic forces, the velocity is an integration of the acceleration, and the altitude is an integration of the velocity. For more information on motor performance, see the motor statistics page. For a list of existing simulator programs, see the simulators page. To find motors for your rocket, try the motor guide.	{"url":"https://www.thrustcurve.org/info/simulation.html","timestamp":"2024-11-04T07:35:44Z","content_type":"text/html","content_length":"17382","record_id":"<urn:uuid:841ee874-e480-4571-ba04-61ff45dd926d>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00190.warc.gz"}
Alexander Hufnagel According to our database , Alexander Hufnagel authored at least 4 papers between 1995 and 1998. Collaborative distances: Book In proceedings Article PhD thesis Dataset Other On csauthors.net: On the Complexity of Computing Mixed Volumes. SIAM J. Comput., 1998 On the Algorithmic Complexity of Minkowski's Reconstruction Theorem Universität Trier, Mathematik/Informatik, Forschungsbericht, 1995 Algorithmic problems in Brunn-Minkowski theory. PhD thesis, 1995 A Polynomial Time Algorithm for Minkowski Reconstruction. Proceedings of the Eleventh Annual Symposium on Computational Geometry, 1995	{"url":"https://www.csauthors.net/alexander-hufnagel/","timestamp":"2024-11-01T23:14:49Z","content_type":"text/html","content_length":"17462","record_id":"<urn:uuid:f0b9ec94-fae7-469f-a835-70d2d531eb8d>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00815.warc.gz"}
Type Promotion Rules Type Promotion Rules¶ Array API specification for type promotion rules. Type promotion rules can be understood at a high level from the following diagram: Type promotion diagram. Promotion between any two types is given by their join on this lattice. Only the types of participating arrays matter, not their values. Dashed lines indicate that behavior for Python scalars is undefined on overflow. Boolean, integer and floating-point dtypes are not connected, indicating mixed-kind promotion is undefined. A conforming implementation of the array API standard must implement the following type promotion rules governing the common result type for two array operands during an arithmetic operation. A conforming implementation of the array API standard may support additional type promotion rules beyond those described in this specification. Type codes are used here to keep tables readable; they are not part of the standard. In code, use the data type objects specified in Data Types (e.g., int16 rather than 'i2'). The following type promotion tables specify the casting behavior for operations involving two array operands. When more than two array operands participate, application of the promotion tables is associative (i.e., the result does not depend on operand order). Signed integer type promotion table¶ i1 i2 i4 i8 i1 i1 i2 i4 i8 i2 i2 i2 i4 i8 i4 i4 i4 i4 i8 i8 i8 i8 i8 i8 • i1: 8-bit signed integer (i.e., int8) • i2: 16-bit signed integer (i.e., int16) • i4: 32-bit signed integer (i.e., int32) • i8: 64-bit signed integer (i.e., int64) Unsigned integer type promotion table¶ u1 u2 u4 u8 u1 u1 u2 u4 u8 u2 u2 u2 u4 u8 u4 u4 u4 u4 u8 u8 u8 u8 u8 u8 • u1: 8-bit unsigned integer (i.e., uint8) • u2: 16-bit unsigned integer (i.e., uint16) • u4: 32-bit unsigned integer (i.e., uint32) • u8: 64-bit unsigned integer (i.e., uint64) Mixed unsigned and signed integer type promotion table¶ u1 u2 u4 i1 i2 i4 i8 i2 i2 i4 i8 i4 i4 i4 i8 i8 i8 i8 i8 Floating-point type promotion table¶ f4 f8 c8 c16 f4 f4 f8 c8 c16 f8 f8 f8 c16 c16 c8 c8 c16 c8 c16 c16 c16 c16 c16 c16 • f4: single-precision (32-bit) floating-point number (i.e., float32) • f8: double-precision (64-bit) floating-point number (i.e., float64) • c8: single-precision complex floating-point number (i.e., complex64) composed of two single-precision (32-bit) floating-point numbers • c16: double-precision complex floating-point number (i.e., complex128) composed of two double-precision (64-bit) floating-point numbers • Type promotion rules must apply when determining the common result type for two array operands during an arithmetic operation, regardless of array dimension. Accordingly, zero-dimensional arrays must be subject to the same type promotion rules as dimensional arrays. • Type promotion of non-numerical data types to numerical data types is unspecified (e.g., bool to intxx or floatxx). Mixed integer and floating-point type promotion rules are not specified because behavior varies between implementations. Mixing arrays with Python scalars¶ Using Python scalars (i.e., instances of bool, int, float, complex) together with arrays must be supported for: • array <op> scalar • scalar <op> array where <op> is a built-in operator (including in-place operators, but excluding the matmul @ operator; see Operators for operators supported by the array object) and scalar has a type and value compatible with the array data type: • a Python bool for a bool array data type. • a Python int within the bounds of the given data type for integer array Data Types. • a Python int or float for real-valued floating-point array data types. • a Python int, float, or complex for complex floating-point array data types. Provided the above requirements are met, the expected behavior is equivalent to: 1. Convert the scalar to zero-dimensional array with the same data type as that of the array used in the expression. 2. Execute the operation for array <op> 0-D array (or 0-D array <op> array if scalar was the left-hand argument). Behavior is not specified when mixing a Python float and an array with an integer data type; this may give float32, float64, or raise an exception. Behavior is implementation-specific. Similarly, behavior is not specified when mixing a Python complex and an array with a real-valued data type; this may give complex64, complex128, or raise an exception. Behavior is Behavior is also not specified for integers outside of the bounds of a given integer data type. Integers outside of bounds may result in overflow or an error.	{"url":"https://data-apis.org/array-api/2022.12/API_specification/type_promotion.html","timestamp":"2024-11-06T19:03:26Z","content_type":"text/html","content_length":"34321","record_id":"<urn:uuid:8693b43f-caf0-4d1f-9c6f-ad11a0d6c8fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00606.warc.gz"}
The Stacks project Lemma 72.10.5. Let $k$ be a field. Let $X$ be a quasi-separated algebraic space over $k$. If there exists a purely transcendental field extension $K/k$ such that $X_ K$ is a scheme, then $X$ is a Proof. Since every algebraic space is the union of its quasi-compact open subspaces, we may assume $X$ is quasi-compact (some details omitted). Recall (Fields, Definition 9.26.1) that the assumption on the extension $K/k$ signifies that $K$ is the fraction field of a polynomial ring (in possibly infinitely many variables) over $k$. Thus $K = \bigcup A$ is the union of subalgebras each of which is a localization of a finite polynomial algebra over $k$. By Limits of Spaces, Lemma 70.5.11 we see that $X_ A$ is a scheme for some $A$. Write for some nonzero $f \in k[x_1, \ldots , x_ n]$. If $k$ is infinite then we can finish the proof as follows: choose $a_1, \ldots , a_ n \in k$ with $f(a_1, \ldots , a_ n) \not= 0$. Then $(a_1, \ldots , a_ n)$ define an $k$-algebra map $A \to k$ mapping $x_ i$ to $a_ i$ and $1/f$ to $1/f(a_1, \ldots , a_ n)$. Thus the base change $X_ A \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(k) \cong X$ is a scheme as desired. In this paragraph we finish the proof in case $k$ is finite. In this case we write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i$ of finite presentation over $k$ and with affine transition morphisms (Limits of Spaces, Lemma 70.10.2). Using Limits of Spaces, Lemma 70.5.11 we see that $X_{i, A}$ is a scheme for some $i$. Thus we may assume $X \to \mathop{\mathrm{Spec}}(k)$ is of finite presentation. Let $x \in \|X\|$ be a closed point. We may represent $x$ by a closed immersion $\mathop{\mathrm{Spec}}(\kappa ) \to X$ (Decent Spaces, Lemma 68.14.6). Then $\mathop{\mathrm{Spec}}(\kappa ) \to \mathop{\mathrm{Spec}}(k)$ is of finite type, hence $\kappa $ is a finite extension of $k$ (by the Hilbert Nullstellensatz, see Algebra, Theorem 10.34.1; some details omitted). Say $[\kappa : k] = d$. Choose an integer $n \gg 0$ prime to $d$ and let $k'/k$ be the extension of degree $n$. Then $k'/k$ is Galois with $G = \text{Aut}(k'/k)$ cyclic of order $n$. If $n$ is large enough there will be $k$-algebra homomorphism $A \to k'$ by the same reason as above. Then $X_{k'}$ is a scheme and $X = X_{k'}/G$ (Lemma 72.10.3). On the other hand, since $n$ and $d$ are relatively prime we see \[ \mathop{\mathrm{Spec}}(\kappa ) \times _{X} X_{k'} = \mathop{\mathrm{Spec}}(\kappa ) \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k') = \mathop{\mathrm{Spec}}(\kappa \otimes _ k k') is the spectrum of a field. In other words, the fibre of $X_{k'} \to X$ over $x$ consists of a single point. Thus by Lemma 72.10.4 we see that $x$ is in the schematic locus of $X$ as desired. $\ Your email address will not be published. Required fields are marked. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). All contributions are licensed under the GNU Free Documentation License. In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B85. Beware of the difference between the letter 'O' and the digit '0'. The tag you filled in for the captcha is wrong. You need to write 0B85, in case you are confused.	{"url":"https://stacks.math.columbia.edu/tag/0B85","timestamp":"2024-11-03T16:11:44Z","content_type":"text/html","content_length":"17101","record_id":"<urn:uuid:81dfbe07-4bde-43ab-a90d-432d8ad38682>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00438.warc.gz"}
Take the circumference of the earth at the equator to be 24,000 miles. An airplane taking off at the equator and flying west at 1000 miles an hour would land at exactly the same time that it started (why?). Moreover, the sun would not move in the plane’s sky during the flight. (Work this out visually in your mind.) Now, at what degree of latitude could a plane flying 500 miles per hour keep up with the sun in this way? Gertrude Goldfish Two young men in tuxedos are walking down the street at about 10 pm. One of them is carrying a round goldfish bowl filled with water. There is a goldfish, named Gertrude, in the bowl. As they pass a round pool in the park, the goldfish gets very excited and jumps out of the bowl into the pool, right at the edge. She starts swimming due north. She hits the wall after she has swum exactly 30 feet. She heads east, and, after going 40 feet, she hits the wall again. Once she regains consciousness after this second collision, she calculates the diameter of the pool. What is it? And what is the story about the two guys in their tuxes? Fences in Circular Region In a circular field, place three fences to make four regions. The fences are all equal in length and their endpoints are on the circular boundary of the field. The four resulting regions have equal area, and the fences don�t intersect within the field. Billy the Goat Billy the goat is tied to the corner of Patty�s barn. The barn is 20 x 40 feet, and the rope is 50 feet long. No trees or other obstructions are in the way. What is the available area of grass that Billy can eat? (Draw a good picture of this area first.) Algebra with Angles If twice ∠A is subtracted from the supplement of ∠A, then the remaining angle exceeds the complement of ∠A by 4°. Find the size of ∠A. Geometric and Arithmetric Sequences There are two positive numbers that may be inserted between 3 and 9 such that the first three numbers are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is: 1. 13½ 2. 11¼ 3. 10½ 4. 109½ Note: There are two other numbers that work, but they’re not both positive. If you go about this problem in a suitably erudite fashion, you’ll turn up this alternative solution too. To make the team, you are going to have to do 89 sit-ups for the coach a week from today. You decide to work up to it. You will start by doing 3 sit-ups today (no sense rushing into things) and end on the 8th day with 89. You don’t know how many you will do tomorrow, but you decide that from the 3rd day on, the number of sit-ups you do will be the sum of what you did on the two preceding days. That is, the number you do on Wednesday will be the sum of the number you did on Monday and the number you did on Tuesday; the number you do on Thursday will be the sum of what you did on Tuesday and Wednesday, and so on. Find out how many sit-ups you should do tomorrow to make this work, so that you come out with 89 a week from today. Hard Functional Equation, f(n) = n The function f satisfies the functional equation f(x) + f(y) = f(x + y) – xy – 1 for every pair x, y of real numbers. If f(1) = 1, then the number of positive integers n for which f(n) = n is: 1. 0 2. 1 3. 2 4. 3 5. infinite Defined Operations 2; Do Composition If 3 h = 10, 7 h = 50, 5 h = 26; and 4 b = 1, 7 b = 2.5, 20 b = 9, then what is n if n hb = 17.5? Garden Crops A large organic nursery has somewhere between 3 and 6 (inclusive) garden plots of herbs when it closes for the season in the fall. In each plot there are between 20 and 30 (inclusive) rosemary plants. If, typically, 10% of those plants don’t winter over successfully until spring, what would be the largest number of plants that could be lost during the winter?	{"url":"https://u.osu.edu/odmp/category/set-8/","timestamp":"2024-11-02T02:32:58Z","content_type":"text/html","content_length":"60830","record_id":"<urn:uuid:e593d98e-afaa-44f4-b25a-7562e72e0af4>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00639.warc.gz"}
ÔÏÍ 14 ÎÏÍÅÒ 1 Physics in Higher Education V. 14, N 1, 2008 The contents 3 Physics and Mathematics "óomplementary" in Education Programs for General Science Subjects N.M. Kozhevnikov 11 Simulation in Training Physicists-Theorists at Classical University M.G. Stulenkov, A.A. Chervova 16 About Teaching of óÏurses "Electrotechnique" and "Radiotechnique" on Physical Faculties of Pedagogical Universities V.D. Semash 20 The Use of Level-Distinguished Problems in Educational Process V.S. Babaev 23 The Study of Some Questions of Polymer Physics in theš Framework of General Physics Course E.V. Prostomolotova 30 System Generating Relations in Physical and Chemical Systems N.E. Deryabina, A.G. Deryabina 35 Studying of a Spectrum of Radiation Heated Tungsten M.B. Shapochkin 42 Theoretical Prediction and Experimental Proof of Audio Frequency when Blowing up Air into a Bottle S.V. Bublikov,š A.E. Boikova, N.A. Bykov 45 The Stude ofš Controlled Physical System Modeling S.V. Borisyonok 54 Combination of Traditional and Modern Computer Technologies in a Laboratory Practical Work N.N. Bezrjadin, T.V. Prokopova, E.M. Agapov, L.V. Vasileva 61 Cycle of Portable Laboratory Works on Disciplines of "Physicist" and "Methods and Devices of the Control of Environment and Ecological Monitoring" N.V.š Kalachev, S.M. Kokin, V.A. Nikitenko, E.K. Silin, M.V. Baharev, A.O. Vorobyov 70 Development of Software for Numerical Modeling Percolation in Bidimentional Regular Structures A.S. Alyoshkin, A.S. Belanov, D.O. Zhukov 77 Numerical Modeling of Percolation Processes in Bidimentional Eegular Structures A.S. Alyoshkin, A.S. Belanov, D.O. Zhukov 89 Studying of Students Knowledge Condition About Natural-Science Cognition Methods E.A. Bershadskaya 104 Exploring Approach in Teaching Physics at the Before-university Stage O.V. Kozachkova, N.K. Shepkina 113 Electronic Studying-methodical Complex and Remote Studying of Courses of "Physicist" and "Concept of Modern Natural Sciences" in a Technical College V.V. Gorbachev, A.F. Smyk 120 Abstracts 125 Information Physics and Mathematics "Complementary" in Education Programs for General Science Subjects N.M. Kozhevnikov St-Petersburg State Polytechnic University General physics and high mathematics mutual influence in contemporary university programs is analyzed. It is shown that the mathematical principles of general physics course could be drawn from the secondary school providing this course inclusion into the first semester schedule. The high mathematics course is oriented to abstract structures investigation and is based on both secondary school and high school physics. Simulation in Training Physicists-Theorists at Classical University M.G. Stulenkov, A.A. Chervova Volzhsky State Engineering and Pedagogical University The article is devoted to a role of simulation in training the students who specialize in the theoretical physics. Advantages of computer models, their pedagogical value and practical application are the basic themes of the article. The model of an actual problem about thin superconducting films is produced here. It is degree work carried out by the student-theorist. The example well illustrates the basic reasons in favor of simulation in training at classical university. About Teaching of Courses "Electrotechnique" and "Radiotechnique" on Physical Faculties of Pedagogical Universities V.D. Semash Moscow State Pedagogical University, Faculty of Physics and Informing Technologies Study of uniform course "Electroradiotechnique and electronics" is proposed on physical faculties of pedagogical universities for training teachers of physics independently of education's trajectories. Principles of selection study's materials, maintenance of course's issues and methodical methods of its teaching are discussed.š The Use of Level-Distinguished Problems in Educational Process V.S. Babaev The St.-Petersburg State Sea Technical University The use of level-distinguished problems in educational process allows realizing an individual approach in teaching pupils and putting into practice monitoring of their knowledge. Analysis of results of this monitoring enables to correct the educational process. The Study of Some Questions of Polymer Physics in the Framework of General Physics Course E.V. Prostomolotova Physical Faculty of the M.V. Lomonosov Moscow State University In the paper we discuss the possibility of the teaching of polymer physics in the framework of general physics course. Many of fundamental results of polymer physics can be understood founding on knowledge of general physics course and higher mathematics. The study of some questions of polymer physics and modern science achievements in the framework of general physics course broad knowledge and range of interests of students and allow involving them into the research work earlier. System Generating Relations in Physical and Chemical Systems N.E. Deryjabina M.V. Lomonosov Moscow State Universityš A.G. Deryabina The Izhevsk State Medical Academy In the article the description of basic kinds of system-generating relations in physical and chemical systems is discussed. It is recommended to use schemes of processes giving visual performance of interacting objects and some kinds of the system-generating relations between them in learning physics and chemistry sciences. Studying of a Spectrum of Radiation Heated Tungsten M.B. Shapochkin Scientific and Technological Centre "LABEKS" It is described researches on measurement of distribution of energy of radiation in a tungsten spectrum. It is shown, that results of experiment well illustrate Planck's distribution for photons of the boze-particles having zero spin. Theoretical Prediction and Experimental Proof of Audio Frequency when Blowing up Air into a Bottle S.V. Bublikov, A.E. Boikova, N.A. Bykov The Herzen State Pedagogical University of Russia, Saint-Petersburg A substantial method of students' cognitive activity intensification consisting in theoretical prediction in combination with experimental proof of audio frequency when blowing up air into a bottle is suggested. The Study of Controlled Physical System Modeling S.V. Borisyonok The Herzen State Pedagogical University of Russia, Saint-Petersburg In the frame of prescriptive approach we discuss the possibilities to study basic applied control theory as an integral part of the modern physics education. Control modeling can be realized effectively in two forms: analytical investigation and simulation. We describe separately the important particular case of dissipative systems. We demonstrate also the application of the traditional control methods to a quantum system as well. Combination of Traditional and Modern Computer Technologies in a Laboratory Practical Work N.N. Bezrjadin, T.V. Prokopova, E.M. Agapov, L.V. Vasileva The Voronezh State Technological Academy The operational experience of physics department of technological academy on formation of the studying-methodical complex combining natural educational experiment and computer variant as much as possible approached to it of a laboratory practical work is presented. Virtual experiment is considered as methodical instructions on the electronic carrier, intended for preparation of students for performance of laboratory work on an instrument breadboard model. The variant of adaptable training restoring school knowledge within the limits of the given studying -methodical complex is offered. Cycle of Portable Laboratory Works on Disciplines of "Physicist" and "Methods and Devices of the Control of Environment and Ecological Monitoring" N.V. Kalachev, S.M. Kokin, V.A. Nikitenko, E.K. Silin*, M.V. Baharev, A.O. Vorobyov* P.N. Lebedev Physical Institute of the Russian Academy of Sciences * Moscow State University of Railway Engineering (MIIT), ** Russian State Open Technical University of Railway Engineering E-mail: kokin1@comtv.ru One more method of the organization of educational process is offered: use of portable complete sets of the equipment, suitable for performance of laboratory works at once on several disciplines as general character (for example, on the physicist), and, on the special. For example, the future engineers-ecologists on older years should perform works on disciplines "Metrology", "Radiating Ecology", "Ecology of Electromagnetic Radiation" "Noise and Vibrations" "Methods and Devices of the Control of Environment" to what practically same devices can be applied. As an example the short description of the several laboratory works created on the basis of computer measuring system and portable devices, used in ecological measurements is resulted. Development of Software for Numerical Modeling Percolation in Bidimentional Regular Structures A.S. Alyoshkin, A.S. Belanov, D.O. Zhukov The Moscow State University of Instrument Making and Computer Science E-mail:š zhukovDm@yandex.ru In this clause considered questions of development of the special software for numerical modeling and visualization of formation of groups clusters the excluded units in the regular bidimentional structures having topology of tree Kaily, square, triangular, six-coal lattices and a lattice with topology 3,122šwhich can be used at preparation of students on a specialty "the Applied mathematics and physics" including as one of techniques computer training and practice of carrying out are considered. Numerical Modeling of Percolation Processes in Bidimentional Eegular Structures A.S. Alyoshkin, A.S. Belanov, D.O. Zhukov The Moscow State University of Instrument Making and Computer Science E-mail:š zhukovDm@yandex.ru In this paper considered questions of numerical modeling of formation of groups clusters the excluded units in the regular bidimentional structures having topology of tree Kaily, square, triangular, six-coal lattices and a lattice with topology 3,122šare considered. Numerical modeling of similar processes can be used for preparation of students on a specialty "the applied mathematics and physics" as a special computer practical work. Studying of Students Knowledge Condition about Natural-Science Cognition Methods E.A. Bershadskaya Military-technical University (Russia) In this paper the theoretical bases are briefly considered, allowing designing system of means for diagnostics of technical college's student's knowledge about natural-science cognition methods and skills to apply them in educational activity. There are given some samples of diagnostic tasks, results of first rate student's inspection in Military-technical University and the analysis of the received data. Exploring Approach in Teaching Physics at the Before-university Stage O.V. Kozachkova, N.K. Shepkina Amur State University The article gives some views on modern trends of exploring approach in studies. The opportunity to use some methods of exploring approach in the process of subject-studies is revealed (on the example of physics). In the article, the value of application of the given method, as dominating at the before-university stage, is declared. Electronic Studying-methodical Complex and Remote Studying of Courses of "Physicist" and "Concept of Modern Natural Sciences" in a Technical College V.V. Gorbachev, A.F. Smyk The Moscow State University of the Press Questions of use of modern remote technologies in educational process at courses of physics and Concept of Modern Natural Sciences are considered.	{"url":"https://pinhe.moomfo.ru/tom14n1.en.htm","timestamp":"2024-11-02T07:33:02Z","content_type":"text/html","content_length":"18295","record_id":"<urn:uuid:e7690a72-262b-44fc-9ac4-29e1bfc17f15>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00717.warc.gz"}
How do you solve 1/2 < n/6? \| HIX Tutor How do you solve #1/2 < n/6#? Answer 1 Using the fact that we can multiply both sides of an inequality by a positive value without changing the validity of the inequality: #1/2 < n/6# #=> 1/2 * 6 < n/6 * 6# #:. 3 < n# Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer from HIX Tutor When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some Not the question you need? HIX Tutor Solve ANY homework problem with a smart AI • 98% accuracy study help • Covers math, physics, chemistry, biology, and more • Step-by-step, in-depth guides • Readily available 24/7	{"url":"https://tutor.hix.ai/question/how-do-you-solve-1-2-n-6-8f9af93594","timestamp":"2024-11-06T20:42:08Z","content_type":"text/html","content_length":"570401","record_id":"<urn:uuid:a6a1d0d7-b95d-4bdd-bdca-055a3a5b5dc1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00239.warc.gz"}
PowerTOST: sampleN.NTID() PowerTOST: sampleN.NTID() [Power / Sample Size] Hi pharm07, ❝ Let me understand again by looking following framework, ❝ Following data is available with me, ❝ Target power : eg, 0.8, .85, 0.9, CwT,CwR,sigWT,SigWR,theta0,theta1,theta2. Although you could give values of , for the FDA keep the defaults of , don’t specify anything): Additionally to passing RSABE and the variance-comparison you must pass conventional ABE. ❝ i want to estimate sample size for low to moderate NTID. More examples are given ❝ Also to estimate sample size by assuming 30 & 40 % dropout rate. See the two supportive functions in the section about dropouts in the named article. ❝ I have gone through the examples. Sometimes error comes as its beyond implied limit. That’s possible if you specify a low or high . Background: $$\eqalign{s_0&=0.1\tag{1}\\ \left\{\theta_{\text{s}_1},\theta_{\text{s}_2}\right\}&=\exp(\mp\theta_\text{s}\cdot s_\text{wR}), }$$ where $\small{s_0}$ is the regulatory switching condition, $\small{\theta_\text{s}}$ the regulatory constant, and finally $\small{\left\{\theta_{\text{s}_1},\theta_{\text{s}_2}\right\}}$ are the implied limits. Say, you assume $\small{CV_\text{wR}=0.1}$. Since $$s_\text{wR}=\sqrt{\log_e(CV_\text{wR}^2+1)}\tag{2}$$ and by using $\small{(1)}$ you end up with $$\left\{\theta_{\text {s}_1},\theta_{\text{s}_2}\right\}=\left\{0.9002,1.1108\right\}.\tag{3}$$ In other words, for any theta0 outside these limits work. That’s by design: sampleN.NTID(CV = 0.1, theta0 = 0.90) +++++++++++ FDA method for NTIDs ++++++++++++ Sample size estimation Study design: 2x2x4 (TRTR\|RTRT) log-transformed data (multiplicative model) 1e+05 studies for each step simulated. alpha = 0.05, target power = 0.8 CVw(T) = 0.1, CVw(R) = 0.1 True ratio = 0.9 ABE limits = 0.8 ... 1.25 Implied scABEL = 0.9002 ... 1.1108 Regulatory settings: FDA - Regulatory const. = 1.053605 - 'CVcap' = 0.2142 Error: theta0 outside implied scABE limits! No sample size estimable. Would you want to estimate a sample size for conventional ABE with a T/R-ratio outside 80–125%? Try: sampleN.TOST(CV = 0.1, theta0 = 0.7999) # any (!) CV, targetpower, design For NTIDs the FDA requires stricter batch-release spec’s (±5% instead of the common ±10%). Therefore, theta0 = 0.975 is the default of this function. I would not go below unless the is relatively high (no scaling if $\small{CV_\text{wR}\geq0.2142}$). ❝ Heteroscedasticity can be challenging for me as T>R. Yes, you are not alone. ❝ Can you check and verify if i am using correct programming. I can’t till you post an example which you consider problematic. Dif-tor heh smusma 🖖🏼 Довге життя Україна! [] Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes Complete thread:	{"url":"https://forum.bebac.at/forum_entry.php?id=22970&order=time","timestamp":"2024-11-04T05:56:07Z","content_type":"text/html","content_length":"18259","record_id":"<urn:uuid:30a4a158-a1f3-46b8-a50e-e60dc2408932>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00663.warc.gz"}
Pi Symbol Copy And Paste Math: Explanations! Pi Symbol Copy and Paste Math: Explanations! The Pi symbol (π) is a crucial mathematical constant that can be easily copied and pasted across various digital platforms. To use the Pi symbol, you can rely on Unicode characters, keyboard shortcuts, or online resources to insert it into documents and communication tools. The Pi symbol represents the ratio of a circle’s circumference to its diameter, approximately equal to 3.14159. Here’s how to copy and paste it: Keyboard Shortcut: On Windows, you can use the Alt code Alt + 227. On Mac, it’s Option + P. Unicode: The Unicode for the Pi symbol is U+03C0. You can insert this in many applications using a Unicode input method. Online Resources: Websites provide the Pi symbol for you to copy and paste directly. Copy from Character Map: Windows users can utilize the Character Map to find and copy the Pi symbol. Copying and pasting the Pi symbol enhances precision in digital mathematical documentation. Key Takeaway The Pi symbol represents the ratio of a circle’s circumference to its diameter. The Pi symbol is widely used in geometry, trigonometry, and physics. Copying and pasting the Pi symbol enhances precision and accuracy in mathematical documentation. Proper usage of the Pi symbol includes considering symbol placement, font style, and labeling units of measurement. Understanding the Pi Symbol Understanding the pi symbol involves recognizing its significance in mathematical calculations and formulas. The pi symbol, denoted by the Greek letter π, represents the ratio of the circumference of a circle to its diameter. It is a fundamental constant in mathematics, with a value approximately equal to This irrational number has widespread applications in various mathematical and scientific fields, such as geometry, trigonometry, and physics. Its importance extends beyond basic geometrical calculations, playing a crucial role in complex equations and mathematical modeling. Additionally, the pi symbol is essential in engineering and technology for accurately determining measurements and designing structures. Overall, comprehending the pi symbol is essential for anyone involved in mathematical or scientific pursuits, as it underpins many fundamental concepts and calculations. Using Pi Symbol in Documents When using the pi symbol in documents, it is essential to consider its proper usage and formatting within mathematical contexts. Copying and pasting the pi symbol accurately is crucial for maintaining the integrity of mathematical expressions. Additionally, ensuring that the pi symbol is correctly displayed in math documents enhances readability and professionalism. Pi Symbol Usage The pi symbol is frequently used in mathematical and scientific documents to represent the constant ratio of a circle’s circumference to its diameter. Its significance in various calculations and formulas makes it an essential element in technical writing. Here is a practical example of how the pi symbol is used in mathematical equations: Equation Description A = πr^2 Calculates the area of a circle. ‘A’ represents the area and ‘r’ is the radius. C = 2πr Computes the circumference of a circle. ‘C’ denotes the circumference and ‘r’ is the radius. V = 4/3πr^3 Determines the volume of a sphere. ‘V’ represents the volume and ‘r’ is the radius. A = 2πrh + 2πr^2 Finds the surface area of a cylinder. ‘A’ denotes the surface area, ‘r’ is the radius, and ‘h’ is the height. Practical example of how the pi symbol is used in mathematical equations Copy and Paste Utilizing the pi symbol when copying and pasting in mathematical and scientific documents ensures accurate representation of mathematical constants and formulas. When copying and pasting, it is crucial to maintain the integrity of mathematical notations, and the pi symbol plays a fundamental role in this regard. Whether it’s in equations, calculations, or statistical analysis, the pi symbol is a cornerstone of mathematical and scientific expressions. By copying and pasting the pi symbol correctly, mathematicians, scientists, and researchers can ensure that their work is accurately presented and easily comprehensible to readers. Additionally, using the pi symbol in documents allows for consistency and precision in conveying complex mathematical concepts. This attention to detail is essential in maintaining the accuracy and clarity of mathematical and scientific discourse. Math Document Formatting Continuing from the previous subtopic, the accurate use of the pi symbol in mathematical and scientific documents is paramount for conveying precise mathematical constants and formulas. When formatting math documents, it’s important to consider the following: 1. Symbol Placement: Ensure the pi symbol is correctly positioned within equations and formulas to maintain accuracy. 2. Font Consistency: Use a consistent font style and size for the pi symbol to uphold uniformity in the document. 3. Clarity in Equations: Clearly denote the pi symbol within equations to avoid confusion and misinterpretation. 4. Units of Measurement: When using the pi symbol in conjunction with measurements, ensure proper unit labeling for comprehensive understanding. Copying Pi Symbol on Different Devices Copying the pi symbol on different devices requires understanding the specific methods for each device’s operating system. On Windows devices, the pi symbol can be copied by using the Alt code method, where the Alt key is held down while entering the Pi symbol’s Alt code (Alt + 227). For macOS, pressing “Option + P” simultaneously will produce the pi symbol. On mobile devices such as iPhones and Android phones, accessing the pi symbol involves a long press on the digit “3” key, which will display the symbol as an option to select. Similarly, on tablets, holding down the specific key associated with the pi symbol will provide access to it. Understanding these device-specific methods ensures efficient copying of the pi symbol across various platforms. Keyboard Shortcuts for Pi Symbol Keyboard shortcuts for the pi symbol provide efficient access to this mathematical symbol on various operating systems and devices. Here are some common keyboard shortcuts to insert the pi symbol: 1. Windows: Press and hold the Alt key, then type 227 on the numeric keypad, and release the Alt key. 2. Mac: Press Option + P. 3. Linux: Press Ctrl + Shift + U, then type 03C0, and press Enter. 4. Microsoft Word: Use the AutoCorrect feature by typing “pi” followed by pressing the spacebar or the Enter key. These shortcuts enable quick and easy insertion of the pi symbol, streamlining mathematical and scientific writing processes. Web-based Pi Symbol Copy and Paste The web-based Pi symbol copy and paste method offers several key points to consider for efficient use. These include Pi symbol shortcuts, tips for cross-browser compatibility, and ensuring accessibility for all users. By understanding and implementing these points, users can effectively navigate the use of Pi symbols across various web platforms. Pi Symbol Shortcuts Web-based Pi symbol copy and paste shortcuts provide a convenient way to incorporate the symbol into mathematical and scientific documents. Here are some web-based shortcuts to easily insert the Pi symbol into your documents: 1. HTML Entity: Use “& pi ;” (remove the spaces) to display the Pi symbol: π 2. Unicode: Press and hold the Alt key, then type “227” using the numeric keypad to produce the Pi symbol: π 3. Keyboard Shortcut (Windows): Press “Alt” and “227” on the numeric keypad to insert the Pi symbol: π 4. Keyboard Shortcut (Mac): Press “Option” and “P” to display the Pi symbol: π These shortcuts can save time and effort, especially when working on mathematical or scientific content that requires frequent use of the Pi symbol. Cross-Browser Compatibility Tips When ensuring cross-browser compatibility for web-based Pi symbol copy and paste methods, it is important to consider the variations in keyboard shortcuts and Unicode rendering across different While some browsers support specific keyboard shortcuts for inserting symbols, others may require alternative methods or may not support them at all. Additionally, the rendering of Unicode characters, including the Pi symbol, can differ slightly across browsers, affecting the appearance and behavior of the copied and pasted symbol. To ensure a consistent user experience, it is essential to test the Pi symbol copy and paste functionality across major browsers and address any compatibility issues that arise. By doing so, users can reliably access the Pi symbol regardless of the browser they use. This emphasizes the importance of accessibility for all. Accessibility for All Ensuring accessibility for all users when copying and pasting the Pi symbol on the web involves addressing cross-browser compatibility and Unicode rendering variations. To achieve this, consider the following: 1. Cross-Browser Compatibility: Test the Pi symbol copy and paste functionality across different web browsers such as Chrome, Firefox, Safari, and Edge to ensure a consistent experience for all 2. Unicode Support: Utilize the appropriate Unicode character for the Pi symbol (π) to ensure proper rendering and compatibility across various devices and platforms. 3. Screen Reader Compatibility: Verify that the Pi symbol is accessible to users of screen reader software by providing alternative text or using ARIA labels where necessary. 4. Keyboard Accessibility: Ensure that users can easily copy and paste the Pi symbol using standard keyboard shortcuts to accommodate individuals with motor disabilities. Additional Tips for Pi Symbol Usage For optimal clarity and precision in mathematical communication, it is essential to understand the various contexts in which the Pi symbol is used. When using the Pi symbol, it’s important to denote whether the value is an approximation or an exact value. In mathematical writing, it’s crucial to follow the standard conventions for formatting the Pi symbol. This includes using the Greek letter π in italicized form when writing by hand or in a document, and using the π symbol in a digital or typeset format for clarity and consistency. Additionally, when utilizing the Pi symbol in equations or formulas, ensure that it is appropriately defined and consistently used throughout the mathematical expression. Forgetting to properly define and use the Pi symbol can lead to confusion and errors in calculations. It is important to double-check its usage to avoid any misunderstandings or inaccuracies in the results. For any further clarification on the usage of Pi in mathematical equations, consider seeking research paper assistance from a knowledgeable source. Adhering to these guidelines will help to maintain accuracy and precision in mathematical communication. The pi symbol is a fundamental mathematical constant that is widely used in various documents and digital platforms. Its significance in mathematics and science cannot be overstated. By understanding how to use and copy the pi symbol, individuals can effectively communicate complex mathematical concepts and equations. Embracing the pi symbol in academic and professional settings can elevate the precision and formality of written communication, adding a touch of irony to the seemingly simple yet profound symbol. Leave a Reply Cancel reply	{"url":"https://symbolismdesk.com/pi-symbol-copy-and-paste-math/","timestamp":"2024-11-07T15:15:33Z","content_type":"text/html","content_length":"139533","record_id":"<urn:uuid:6f75f74a-e396-4d33-bf63-c8b761d11ef7>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00260.warc.gz"}
So SO disappointed by the recent episode of Sell Your Haunted House! 😒😤😭 I haven’t gotten there yet! Just disappointed when dramas lose their steam toward the finale. Yet to watch ep 15… finger crossed it doesn’t disappoint further! saw a bunch of upvotes , thanks Sabu 😘. Welcome back on DB , hoping you and your family are well 🙂 and that you’ll find a better drama next time hehe . Haha! You have found me, Hagsaeng! Me and family are well, thank you!! How’ve you been? What have you been upto! I didn’t know you were on DB until I saw your msg on someone’s post yesterday! Can you believe it’s been over 2 yrs since I came on DB? 😭 Haha Ahhh, no no! The drama I’m watching is great!! I love it to bits…just that ep disappointed me is all. I try to check fanwall once a week or every two week, but only post after I finish a show (kinda a slow watcher recently). I joined discord Beanies and I’m checking it way more often than DB ^^. I’m fine 🙂 , still working from home (which I love hehe). Bring back the fierce Ji-ah please. Raven! How’s Aussi-land? Right? Haha! It’s great Ally! ☺️ I hope you’ve been well! 😌 I’m so impressed with the writer of The Ghost Detective. She should get the writer of the year award. 👏 I still can’t believe how she turned a perfectly good drama into a hot, stinking mess. Only truly talented writers can do that. Definitely checking out her future dramas. And I’m definitely not wearing Esom’s expression right now. She’s actually rocking it better than me. At least her hair is still in place. I might’ve plucked out a few of mine because of how smart, thrilling and suspenseful the latest eps were. The Ghost Detective? More like The Plot Defective. bam! lol @ Plot Defective! Me? No, I’m not upset. Just ignore me Beanies. Just spreading my love for The Plot Defective. 💕💕💕 I totally did not log in just to post this after weeks. I’m glad I’ve dropped it 2 weeks ago. (Thank God for the busy weeks) They still have the last eps left, right? Are you planning to watch it till the end? One more ep gadis. I can do this. I made it this far. Plus it can’t get any [S:worse:S] right? Despite how busy I am with work, I made time to watch this show. And I’m thinking why do I even bother? Same here. I didn’t have time and it didn’t make me excited so stopped. Reading all this am glad :)) Ha! I don’t have time either. I guess I’m finishing it for the sake of finishing it. And here I thought I had gotten better at dropping dramas. Should’ve used this time for 100Million Stars instead:( Have yet to watch that! If it hadn’t started so good, and the actors weren’t so talented, then I wouldn’t be so disappointed. But this “plot” has devolved into a mess of tangled rules, inconsistencies, and the just plain stupid. I do have to say that Park Eun-bin did a good job conveying the subtle mannerisms of Sun-woo even before we supposedly knew she was possessed by her. i’d like to disagree regarding that last point? though I absolutely adore peb and love her with my whole heart, I found her performance as sunwoo-in-yeowool slightly subpar? Before SW’s identity was revealed, visually, I felt that it was still yeowool-in-yeowool, though I knew at the back of my mind that logically, that was unlikely. After SW’s identity was outed it didn’t feel like she was sunwoo-in-yeowool either. She just seemed to be a whole new character / someone trying too hard to be like SW. Then again, it might just be that Lee Ji Ah’s performance as SW was too stellar and distinct that its hard for anyone else to portray in the same manner. Or maybe i’m just being picky and expecting too much THIS. 👏 It can’t get any worse than plain stupid. I’m still so shocked that one writer wrote this. Was there a change of writers midway? Or maybe she is truly talented at creating masterpiece “supposedly knew” haha thanks for the laugh. I knew this was coming a mile away. I was more annoyed that the link I watched on didn’t have a FF option lol. LOL, Raven! I started reading and was like “is there another Ghost Detective around?” Major kudos for bringing us the most hilarious way of nearly killing a detective and sending him to get urgent surgery! 😂😂 Whaddya mean? I’m being honest! I swear! Why do I let stupid dramas make me depressed? 😢 The time I spent on this… Allllll wasted! I’m loving all beanie posts regarding GD 🤣🤣🤣 It need special talent to turn anything to be that level of hot mess Special talent! Exactly what I’m saying… 😂😂 I’m seriously so impressed! How on earth did the writer do it?? Sigh, I lose focus started from ep where Sunwoo was stabbed by Da-il, it went downhill afterwards. I was wondering when exactly it went wrong… This is it!!!! Thank you! I waa actually patient because I thought it couldn’t get any worse that this. Boy was I wrong. When the lawyer and the reporter did that stupid thing, I lost all my cool. That’s it GD, you and I are no longer friends. @vivanesca LS! Stop suffering! You’ve suffered enough! It’s your birthday! Be HAPPY! Happy Birthday chingu! 💖❤🎉💖❤ I fully expected him to say “aniyo” and walk off. But he said “yes”. HE SAID YESSSSSS. AHHHHH. SCREAMING💖💖💖💖💖 HES SO CUTE I NEED TO SQUISH HIM Is the OST that plays right after this out yet? Because it’s bleeping gorgeous. This scene got a new ost (not played yet during the drama) and I’m pretty sure that it’s Jung Joon-Il singing it <3 . So far only 2 ost have been released and I think with that new one there are 3 ost already played but not released yet (I'm waiting for the one in English <3). side note : I replied on another post : that early confession scares me, how are you planning to break our heart writer-nim T_T ost part 2 정세운 (Jeong Sewoon) – 이봐 이봐 이봐 (Told you so) @Kudo Ran – thanks for these links! I found the music on iTunes and promptly downloaded them to my library to listen at work. yeah for Beanies and the unending source of everything! Oh wow – thanks for that! My pleasure 🙂 I think this channel is basically uploading every ost release (or at least for the trending drama) , using it as random playlist is such a delight <3 Yep I saw that comment. I didnt expect him to confess so soon either. And i heard someone say “date” in the preview. I don’t know how to feel lol. I felt giddy watching the scene and now I’m worried where it’s going to take us… Thank you! I did shazam it and it came up with nothing so I figured it wouldn’t be out yet. A little unrelated but the music that plays right before the “yes” is that an instrumental in the show? I mean is it played more often in the drama? That would mean it isn’t just the starting of the ost that plays later. I just saw a kocowa clip of this scene and that music is just stuck with me. It is an instrumental track (and I think it has already been played before in the drama) , then the ost starts when he answered “yes” Oh great. Thanks. Now I’ll have to wait till the end of the show 😂 Yayy! Thank you! Now I’ll just wait till the OwHEN song comes out . You’re welcome ^^ Seen on soompi : the ost part 4 will be I’m OK (괜찮다고 ) by 1415 duo on Oct. 30th So seems like we’ll have to wait November for Owhen ost (=the english ost I’ve been waiting for since ep1 🙁 ) I love Soo-yeon so much for all his honest (sometimes raw) answers. And it actually surprised me that Yeo-reum ask that question so early in the story. “And it actually surprised me that Yeo-reum ask that question so early in the story.” I’m glad you’re bringing that up Gadis. When seeing the scene, my thoughts were : 1) kyah~ he said yes <3 happy dance 2) he said yes but we're just at episode 5 (if we count one-hour ep, gosh I hate the 30min ep count format), only at ep5, not a good sign, writer-nim what are you scheming behind the scene 3) omo omo the ost <3 4) but wait why did she ask THAT ? Wouldn't the usual behaviour to keep wondering why he does things for her and then asks him directly in a few eps ? …or I might have watched too many dramas haha x) 5) ok another day of wait starts ^^' I’m just going to enjoy how refreshingly honest these two can be. It’s a nice change to hear them blurting things out instead of bottling things up like the usual kdrama leads. “refreshingly honest” indeed, I like it a lot, it’s just so unusual in a drama haha ^^’ . Hoping we won’t have a classic noble idiocy down the road (or at least a very short one) if they keep being honest fingers crossed side note : I suspect that he started to fall for her when she told him why she didn’t ask anything about his arm <3 I really loved all their scenes together in this episode. Ahh..I need to do some screenshots, too! Me too!! I wish I could post more but been a bit busy because of work.. I barely had time to watch this. 😩 Drama addict issues! I’m currently watching 6 k-drama plus 1 c-drama. How am I doing this? I don’t know. I’m only one episode behind on one show. I’m kinda proud of myself. Heehee Lol same here! Six dramas and only one ep behind… Until today. Have yet to watch fox and 100 days. So that’s 3 eps. I’m pretty sure they’ll keep piling. Lol HAHAHA. Walked off like a model only to get home, squat outside his door dumbstruck and wonder what the heck did I just do?? 😂 This was cuteeee. But they keep mentioning the cat! why do they keep mentioning the cat?? nervoussss He expected the cat to leave. He is just emotionally unavailable or chooses to not form bonds with another living. Why?? It could be a simple answer that he was an orphan/emotionally unavailable etc etc or he is a man with a tendency to kill and so stays away from the living. If he had a tendency to kill I don’t think he’d stay away from his urges. He’d have to give in eventually. Which makes me hope (since the cat is alive and no new murders) that he’s not a killer but that he may be emotionally stunted which is why he’s so manipulative. He can’t trust people so he uses them instead. And the way he looks for the cat in the terrace. Kanga!! Did he almost worry the cat is gone even though he wanted the cat to leave. I think he’s grown attached to the cat maybe because it reminds him of Jin kang? Because they have to scare us. Animals, especially cats, and potential killers do not mix. We have all heard too many stories of the “He was a quiet child, but the neighborhood cats kept disappearing. “ True, true! I think they are definitely trying to build the tension by showing us the cat in one scene and making it seem like the cat isn’t there anymore (when they panned the camera to show the orange box? outside Moo young’s room. I thought that Moo young had definitely done something to the cat and threw it inside… Or something. I really don’t think she’s in danger. My gut’s telling me he’s got the wrong woman. But who can that card be referring to? More importantly, how is Bon going to get out of this situation with his nosy neighbours witnessing this scene? XD Awwwww. It warms my heart to see Bon notice the little things Ae-ran does for her kids, without thinking, as a mother. It was nice of him to take care of her and make sure she’s eating too! ♡♡♡ I could only think about how SJS had probably suffered when eating his hated carbs lol Say aye if you think he’s enjoying this. Aye♥♥♥♥♥ Omg the 2nd poster freaks me out (but i will still look for this drama) Is it the tiger? 😂 I think it’s awesome XD Yup the tiger bcoz i think its smiling! 😂😂😂 I wonder whats the role of the tiger/pretty girl in this upcoming drama.. havent read much about it. Oh! I didn’t even notice that! It does look like it’s smiling haha. I’m curious too. I’m not going to read about it in case of spoilers so we’ll just have to wait and see. 🙂 Haha the smile is pretty scary 😂😂😭😭 I dont mind spoilers (i do it for the movies coz who wants to to pay for a flop 😝😂) but it will be fine too without spoilers .. i love fairy tales! Me too!! I don’t know much about korean fairy tales though. So this should be interesting! Now that you mentioned it, GF can be very interesting. I only knew western fairytales and fairy godmother now i can expand my fairytale knowledge 😝😝😝 That’s ONE heck of a creative set of posters! This was too cute. 💕💕 Since we’re in an airport- I plane these two. @vivanesca I totally plane those two Looking forward to watching these scenes because the guest stars are Filipino (i.e. girl on the bed). Haha the end of that ep was so dramatic… @egads ‘ alternative title: Circle: when two planes collide takes on a whole different meaning. It’s so funny how KKN is the one helping the lady instead of officer Na, and then he gets emotional after. Lol Lol, yes. I never would’ve thought he was the emotional type! They make a nice pair. Ikr. I thought he’s the tough guy type when he first appeared but nope, he’s a crybaby. 😄 They’d be perfect for each other! ^^ Haha yea I thought that too! can’t decide whether park eun bin or hong shim should get ‘the detective’ xD (btw idk if PEB plays a detective, dont watch the show just guessing) Hmmm. I think Park Eun Bin should be the detective even though she’s not an actual detective lol. She does do more detective work than Hong shim so it fits her character. 😊 Hong Shim = The Fugitive Go Ae-rin = The Nosy Neighbour Ooh I like The Nosy Neighbour! But Hong shim isn’t really a fugitive is she..? I think she technically is. Her father was publicly impugned as a traitor, and both she and her brother were supposed to be killed along with their father. But didn’t her brother beg Master Kim to leave his sister alone? That’s probably why he agreed to work along side Kim. But maybe she might be a fugitive again when Minister Kim finds out she knows the crown prince and so on.. I think the Left State Councillor’s agreement to leave her alone probably isn’t a pardon; it’s just an agreement not to enforce the penalty associated with her supposed crime. She points out to Yul in the last episode that she will have to live in hiding with her brother once he returns (I think either she or her brother said near the border with Great Ming), which suggests to me that she’s still on the run in some way. Right. Yes, that makes sense. 👏 I’m happy to not be The Psychopath. Hahaha no one wants to be the psychopath! 😂 If you had to choose one who would you choose? I think cyborg would be fun for awhile. Which would you choose? Yeah cyborg or I don’t mind beinga spy for a while too. As long as the missions aren’t dangerous hehe I want the spy skills as long as they include martial arts skills. I loved this scene so much! I’m glad Yoo-reum stood up for herself instead of apologising when she wasn’t even in the wrong. I was more mad at the surrounding male supervisors then I was at this woman. I get that Yoo-reum just transferred here and they see this as her creating a problem. But I hated that they put her down in front of everyone and didn’t even bother to listen to her. I’m grateful for her team leader for being proud of her. I didn’t see that coming. And can I just take a moment to swoon at our adorable cyborg?? Swooooon for not only thinking ahead but also for not ignoring our heroine like he used to. I guess she’s growing on him. 😉😁😁 Yes- we are starting to see her character development and growth as a woman. She was just too over the top at the beginning to keep on with how she was. And yeah for her boss who told her “good job”! I would like to say that she was over-eager. Eager to please her bosses, eager to do a great job, to be complimented. This was all good but because she was over eager she didn’t realise that it could get her into trouble. From her perspective it’s all about timing. She was transferred as a trouble-maker and no one’s bothered to give her the benefit of doubt. The male boss had a list of reasons why they shouldn’t keep her in their team when he hadn’t even met her! No wonder she wanted to please everyone from the get-go! He’s so handsome in the last picture. His little smile set my pulse racing. I know! All the small smiles he sent her way today made my heart race! 💕 I teared-up at this scene. Way to go Han Yoo-reom! Thank god she asked for an apology instead. I would’ve been so frustrated otherwise. I did too! She’s such a good actress! Is this supposed to be PPL?? If it is then it’s the best PPL everrr. 😂 The people are pretty, the airport is pretty, the robot is pretty…Shinnamon has competition now! 😂 And it ships Lee Seo-yoon and Han Yoo-reom 😍 Upgrade from that pesky motorbike… The second ‘maybe’ makes all the difference. It’s planting that seed of doubt. Seo In-guk never disappoints, does he? He’s one of the few actors who always comes out with a great role in a great drama. He’s just so immersed in this role and that scares me a bit. Please don’t be evil! Can I even trust you?? I love that Jung So-min came back with another spunky but professional character. I like how she doesn’t sway easily to our hero’s tunes like her naive friend. Her relationship with her brother is already the best thing ever! As expected, Hundred Million Stars had the best opening episodes of the latest batch of dramas. ✴ strong cast ✔ ✴ great female lead ✔ ✴ intriguing plot ✔ ✴ swoony male lead. READ- creepy AND suspicions ✔ ✴ morally grey characters ✔ ✴ great chemistry ✔ ✴ scenes flowed smoothly and set a solid pace ✔ Couldn’t agree more. And it’s always a great thing to find the drama you’re anticipating exceeding your high expectation. I hope the show continue to be this solid. Right? I’m so thankful to all the actors for giving us an amazing performance! I’m pretty sure they’ve got more for us. I trust that the drama will take us in for a solid, captivating ride. SIG and JSM are the only people I can trust in this world. Everyone else disappoints me. Thank God this drama is good. Aw Mindy! hugsss You won’t be disappointed next time. fingers crossed 😊 I know right? I’m grateful that it hooked me from the get-go. I’m sure it’ll only get better! 😁 First fruit and veggies, now check ☑️ marks! Love your usage of emojis! aww haha thanks Jen Jen;D I’ve got 2 wk break from work. Just utilising my creativity… 😂😂 never disappointed me. Always worth the wait and Ican’t seem to pick a favorite among his works. me neither! The only one I haven’t seen is High school king of savvy. But it’s on my list. I LOVE THIS DRAMA. And my husband does too. He just asked me when the next episode airs. I need to start learning to wait until the drama airs to binge, but too much risk being spoiled. Haha yes there are some dramas you need to live watch and endure the challenge of waiting. This is one of them. Especially since we don’t want spoilers. Seo In-gook is just amazingin this role. Gives me the shivers! With the flood of new dramas it was difficult for me to hold on to Life. But hold onto it, I did. Was it the cast? The plot? The characters? Maybe a bit of all three. But mostly because I was curious about the people. How were they going to survive another day at the hospital? I was disappointed that we spent so much time on hospital/business politics yet I was intrigued enough about the characters to keep coming back. There was a steady almost magnetic pull everytime President Gu butted heads with our hospital staff. And I loved every minute of it. I came in think he was THE villain but was pleasantly surprised that this isn’t your typical show with good vs evil. The ending could’ve been better (especially how they resolved Gu’s storyline… And where was his secretary?? She was my fav character!), and I’m glad our Dr Ye got some time off with his brother because boy does he need to cool down a bit! It was funny that he was shooting daggers at the new president but the boy’s gotta chill. Don’t make another enemy! Overall, I’m glad I stuck with the show. 💖💖 Something light to sum up how I felt about the show: Aww, I’m glad you like it. Life is quite unusual for kdrama and it’s one of my fave this year. Also, it doesn’t hurt that I get to see Lee Kyu-hyung playing a rare character like Sun-woo. does he survive the show..without going through some kind of irrevocable pain He survive the show with a happy enough ending. Tbh, I wish the writer didn’t turn him into a punching bag halfway through the show, but since in the end he is content with what he got, it’s good enough for me. Lee Kyu-hyung plays unusual characters very well and I can’t wait to see more of him! Great that you liked it. The characters not being entirely evil was the best part of the show. Well we cannot say that for the chairman but at least President Gu. Plus someone finally posted a photo with Choi Su Hyeon. This was my first time seeing this actress but I was surprised that she has been acting for quite a while in films mostly. She and Won Jin ah have become the actresses I’ll definitely look out for. The chairman is in another world entirely! It’s a pity they didn’t make him balance his bad side with good. It’s like they’re saying there are people who are 100 % bad and we can’t do anything about it. In a way it’s true… I guess? Yes! I really really liked her. I’ve never seen her before but I’ll certainly be keeping an eye out for her. And Woh Ji Nah is a gem! 💎 I enjoyed her performance a lot in Just Between Lovers. I’ve read some spoilers and i think i have the basic ideas about the ending but I’m taking my time watching this in between the other ongoing dramas.. I came for LDW but now start to shift my loyalty to Mr Gu 😂 i really love him being so unsure about himself, unconsiously fidgety when No eul is around That was my plan but with so many new dramas I felt like I’d never get to finish Life. So I sat down yesterday and binged the last 3 eps. Now I don’t feel guilty starting the new shows. 😂 Truthfully I came for everyone hehehe. There’s just so many actors here that I love and I couldn’t miss it for anything. Yes, Gu’s scenes with No eul was adorable. It brought the best out of him! ❤ Good luck finishing it! 😀 LOL thank you, i really need that luck because im juggling in between RL and dramas, and trying to catch up everything but still falling behind 😂😂😂 He smiles! And not just once. This Ep is full of great moments and I can’t believe they ended with another cliffhanger! How am I going to wait a wk!?? 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Translate Each Sentence Into Algebraic Equation - Tessshebaylo Translate Each Sentence Into Algebraic Equation Evaluatea translate each sentence into an algebraic equation 1 a number increased by four is twelve 2 brainly ph solved at tenen oiui ojubiujs 4je o1eisueji ue jequsnu jonbo 01 unao1 decreased pasagot po not activity 3 and 4 please if fou timles added t0 nine the result forty thirty equal to three times eighty five write your answers on answer sheet sentences word problems chilimath how step algebra study com b less than seven five2 difference translating phrases expressions writing equations skills practice you get of following then transtutors Evaluatea Translate Each Sentence Into An Algebraic Equation 1 A Number Increased By Four Is Twelve 2 Brainly Ph Solved Translate Each Sentence Into An Algebraic Equation At Tenen Is Oiui Ojubiujs 4je O1eisueji Ue Jequsnu Increased By Four Twelve Jonbo 01 Unao1 A Number Decreased Solved Pasagot Po Not Activity 3 And 4 Please 1 Translate Each Sentence Into An Algebraic Equation If Fou Timles Number Is Added T0 Nine The Result Forty Thirty Solved Translate Each Sentence Into An Algebraic Equation A Number Increased By Four Is Equal To Twelve Decreased Nine Three Times Eighty Five Translate Each Sentence Into Algebraic Equation Write Your Answers On The Answer Sheet Brainly Ph Algebraic Sentences Word Problems Chilimath How To Translate A Sentence Into 1 Step Equation Algebra Study Com B Translate Each Sentence Into An Algebraic Equation 1 Twelve Less Than Seven Is Five2 The Difference Brainly Ph Translating Phrases Into Algebraic Expressions And Writing Algebraic Equations Skills Practice You Get Answer Translate Each Of The Following Into An Equation And Then Solved Transtutors Ppt Translate Each Phrase Into An Algebraic Expression Powerpoint Presentation Id 2645885 Solved B Translate Each Mathematical Phrase Or Sentence Chegg Com Algebraic Sentences Word Problems Chilimath Basic Algebra Worksheets Digits Guru Solved Activity Translate Each Mathematical Expression Into Algebraic Expressions 1 Thirty Decreased By The Number Three Times Quantity Of Y And 50 Product X D 2 Write Each Sentence As An Algebraic Equation Expressions Writing Solved In Exercises 6 10 Translate Each English Phrase To Chegg Com A Translate The Following Word Phrases Into Algebraic Expressions B Each Sentence An Brainly Ph Translating Algebraic Expressions To Verbal Phrases Math Study Com Algebra I Translating Sentences Into Equations Level 1 Of 2 Examples You Translating Equations Mathbitsnotebook Jrmath Translating Words Into Algebraic Expressions Free Guide Mashup Math Sentence into an algebraic equation oiui ojubiujs 4je o1eisueji ue jequsnu if fou timles a number increased write your answers sentences word problems 1 step translating phrases writing equations skills and then solved Trending Posts This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://www.tessshebaylo.com/translate-each-sentence-into-algebraic-equation/","timestamp":"2024-11-12T16:45:01Z","content_type":"text/html","content_length":"60657","record_id":"<urn:uuid:60da492a-de17-479d-a68c-c3aca0a86b79>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00587.warc.gz"}
Face \| Definition & Meaning Face\|Definition & Meaning To understand the concept of faces we need to understand some related terms as well. These terms include vertices and edges. In the subsequent sections, all three of these terms are explained with the help of suitable examples and diagrams. The following figure shows an example of a cube which serves as a preliminary introduction to these key concepts of geometry when we talk about the three dimensional and two dimensional shapes. Figure 1: Faces, Edges and Vertices of a Cube In this example, the cube has eight vertices, twelve edges, and six faces. The calculations become clearer as we introduce these concepts with the help of examples. What Is a Face? A face in geometry is a three-dimensional geometric shape’s planar (flat) surface. In short, the flat surface of a solid body is known as its face. It’s also important to realise that managing two-dimensional forms is physically impossible because everything around us has three dimensions. Consider the cube given in the figure 1. The smooth surface that makes up this cube’s front is known as its face. A cube has six faces, however, only three are visible in the above figure no. 1. Several faces may be seen on many solid shapes. For instance, a cuboid has six faces as shown in the figure below. Figure 2: Faces, Edges and Vertices of a Cuboid Some geometrical shapes or diagrams are faceless. For instance, a pyramid has faces but a sphere has no face. So a face is made up of one flat surface. In fact, a face is any flat area. The pictures 3 and 4 given below show the faces of a cone and a prism. Figure 3: Faces, Edges and Vertices of a Prism The term “face” refers to any of an object’s distinctive flat surfaces. There are four faces on this tetrahedron as shown in figure 5. Because it could be the face of a polyhedron or the edge of a polygon, the word “side” is not very precise. A solid’s face is any one flat surface. Solids can have several faces. A face is a flat surface (a planar region) that forms part of a solid object’s What Is a Vertex? In any three-dimensional or two dimensional shape, vertices are the locations where two or more line segments or edges cross. They can be thought of like a corner. A single such corner is called a vertex and multiple such corners are termed as vertices. For instance, the cube shown in the figure no. 1 has eight vertices. One of the vertices is hidden in this figure and only seven are displayed. The cone shape shown in the figure below, however, only has one vertex. Figure 4: Faces, Edges and Vertices of a Cone In short, vertex is another name for a corner. Many solids have many vertices or a large number of vertices. What Are Edges? The line segments known as edges act as the meeting places for the faces of a shape and join one vertex to the next. They may be used to describe both two dimensional and three dimensional geometrical shapes. Some shapes, like hemispheres, have curved edges even though many forms have straight edges and lines. A cube, shown in the figure 1, has twelve straight line edges. Nine out of these twelve are visible in the diagram while three are hidden. A quick glance reveals that this cube’s faces intersect one another in a straight line. So we can say that a segment of a line joining two faces or connecting two vertices (corner points) on a polygon’s boundary is referred to as an edge. Many solid forms have several edges. As another example, the tetrahedron shown in the following figure has six edges. Figure 5: Faces, Edges and Vertices of a Tetrahedron Euler’s Theorem The Euler theorem, named after Leonhard Euler, is one of the most important mathematical theorems. The link between a polyhedron’s face, vertex, and edge counts is established by the theorem. To understand Euler’s theorem, we first need to understand the term Polyhedron. Euler’s polyhedron formula is accurate for the majority of polyhedron types. What Is a Polyhedron? A closed-space object formed entirely of polygons is called a polyhedron. The English term “polyhedron” is derived from the Greek words “poly” and “hedron,” which combined denote a base or seat. In other words, a closed solid shape with flat sides and straight edges is known as a polyhedron. There are several varieties of polyhedra. Due of its rounded edges, a cylinder is not a polyhedron; in contrast, a cube is an illustration of one. Polyhedrons have faces, edges, and vertices. The edges of a polyhedron are made up of line segments that span two faces. Vertices are the intersections of three or more edges. In summary, a polyhedron is a three-dimensional solid with faces acting as its only boundary. Mathematical Form of Euler’s Theorem Euler’s Theorem shows how a polyhedron’s faces, vertices, and edges are connected. According to this theorem, for many solid shapes, the total of the number of faces plus the number of vertices minus the number of edges is always equal to two. ( Faces ) + ( Vertices ) – ( Edges ) = 2 Let’s test this condition for our initial example of the cube. We already know that a cube has twelve edges, eight vertices, and six faces. So putting these values in the above formula: ( 6 ) + ( 8 ) – ( 12 ) = 14 – 12 = 2 Hence, the Euler’s theorem holds true for the case of cube. Its important to reiterate that Euler’s Formula only holds true for the Polyhedrons. Numerical Example of Faces Given the following geometrical shape (Octahedron), find the number of faces, edges and vertices. Also check if the Euler’s theorem holds true for this shape or not? Figure 6: Numerical Example for Faces, Edges and Vertices (Octahedron) The given figure has eight faces, twelve edges and six vertices. Putting these values in the Euler’s formula: ( 8 ) + ( 6 ) – ( 12 ) = 14 – 12 = 2 Hence, the Euler’s theorem holds true for the case of cube. All figures and charts have been constructed using GeoGebra.	{"url":"https://www.storyofmathematics.com/glossary/face/","timestamp":"2024-11-08T23:48:36Z","content_type":"text/html","content_length":"174286","record_id":"<urn:uuid:0f3475b1-5e1f-40f4-a799-2ade7479a182>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00556.warc.gz"}
Infinite Dimensional Vector Space Published on Feb 21, 2020 We are familiar with the properties of finite dimensional vector spaces over a field. Many of the results that are valid in finite dimensional vector spaces can very well be extended to infinite dimensional cases sometimes with slight modifications in definitions. But there are certain results that do not hold in infinite dimensional cases. Here we consolidate some of those results and present it in a readable form. We present the whole work in three chapters. All those concepts in vector spaces and linear algebra which we require in the sequel are included in the first chapter. In section I of chapter II we discuss the fundamental concepts and properties of infinite dimensional vector spaces and in section II, the properties of the subspaces of infinite dimensional vector spaces are studied and will find that the chain conditions which hold for finite cases do not hold for infinite cases. The linear transformation on infinite dimensional vector spaces and introduce the concept of infinite matrices. We will show that every linear transformation corresponds to a row finite matrix over the underlying field and vice versa and will prove that the set of all linear transformations of an infinite dimensional vector space in to another is isomorphic to the space of all row finite matrices over the underlying field. In section II we consider the conjugate space of an infinite dimensional vector space and define its dimension and cardinality and will show that the dimension of the conjugate space is greater than the original space. Finally we will show that the conjugate space of the conjugate space of an infinite dimensional vector space cannot be identified with the original space	{"url":"https://www.seminarsonly.com/IT/Infinite%20Dimensional%20Vector%20Space.php","timestamp":"2024-11-04T02:39:37Z","content_type":"text/html","content_length":"14993","record_id":"<urn:uuid:71caf9c5-026c-4d24-9935-d5a2a23b98e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00704.warc.gz"}
February – 2021 challenge Each month, a new set of puzzles will be posted. Come back next month for the solutions and a new set of puzzles, or subscribe to have them sent directly to you. 1. A summer camp counsellor wants to find a length, x, in feet, of the lake as represented in the sketch below. The lengths represented by AB, EB, BD and CD on the sketch were determined to be 1800 feet, 1400 feet, 700 feet, and 800 feet, respectively. Segments AC and DE intersect at B, and angles AEB and CDB have the same measure. What is the value of x? 2. A food truck sells salads for $6.50 each and drinks for $2.00 each. The food truck’s revenue from selling a total of 209 salads and drinks in one day was $836.50. How many salads were sold that 3. How many litres of a 25% saline solution must be added to 3 litres of a 10% saline solution to obtain a 15% saline solution? There are more than one way of doing these puzzles and may well be more than one answer. Please let me and others know what alternatives you find by commenting below. We also welcome general comments on the subject and any feedback you'd like to give. If you have a question that needs a response from me or you would like to contact me privately, please use the contact form. Get more puzzles! If you've enjoyed doing the puzzles, consider ordering the books; • Book One - 150+ of the best puzzles • Book Two - 200+ with new originals and more of your favourites Both in a handy pocket sized format. Click here for full details. Last month's solutions Do you know your Geometry? Can you find the measure of each missing angle (a – m) in the diagram below? (Note: Diagram not drawn to scale.) Some Geometry basic rules: 1. Right angle (L ) = 90º 2. Straight line = 180º (along each side). 3. Triangle = 180º (internal angles). 4. Quadrilateral (or 4-sided polygon) = 360º (internal angles). 5. Irregular Pentagon (or 5-sided polygon) = 540º (internal angles). 6. l, II & lll pairs on a lines represents equal lengths. 1 Comment hemen tıkla on January 9, 2024 at 7:59 pm Also a thing to mention is that an online business administration study course is designed for scholars to be able to effortlessly proceed to bachelor degree programs. The Ninety credit certification meets the other bachelor diploma requirements and once you earn your own associate of arts in BA online, you should have access to the newest technologies in this particular field. Some reasons why students want to get their associate degree in business is because they can be interested in the field and want to have the general knowledge necessary prior to jumping right bachelor education program. Thanks for the tips you provide as part of your blog. Lester Trowers Submit a Comment Cancel reply	{"url":"https://gordonburgin.com/2021/02/february-2021-challenge/","timestamp":"2024-11-08T21:03:17Z","content_type":"text/html","content_length":"255985","record_id":"<urn:uuid:bccb1294-b653-475d-8db5-f82808b30f46>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00382.warc.gz"}
In which quadrant do the lines x=3 and y=-4 intersect? \| HIX Tutor In which quadrant do the lines #x=3# and #y=-4# intersect? Answer 1 Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer from HIX Tutor When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some Not the question you need? HIX Tutor Solve ANY homework problem with a smart AI • 98% accuracy study help • Covers math, physics, chemistry, biology, and more • Step-by-step, in-depth guides • Readily available 24/7	{"url":"https://tutor.hix.ai/question/in-which-quadrant-do-the-lines-x-3-and-y-4-intersect-8f9af913cd","timestamp":"2024-11-03T09:24:40Z","content_type":"text/html","content_length":"565539","record_id":"<urn:uuid:a44c94b4-f893-44b9-9341-367c369bbb25>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00170.warc.gz"}
Members: 3658 Articles: 2'599'751 Articles rated: 2609 05 November 2024 Article overview Measuring inertial mass with Kibble balance Rajendra P. Gupta ; Date: 26 Jun 2022 Abstract: A Kibble balance measures the $gravitational$ mass (weight) of a test mass with extreme precision by balancing the gravitational pull on the test mass against the electromagnetic lift force. The uncertainty in such mass measurement is currently ~$1 imes 10^{-8} $. We show how the same Kibble balance can be used to measure the $inertial$ mass of a test mass, that too with potentially 50% better measurement uncertainty, i.e., ~$5 imes 10^{-9} $. For measuring the inertial mass, the weight of the test mass and the assembly holding it is precisely balanced by a counterweight. The application of the known electromagnetic force accelerates the test mass. Measuring the velocity after a controlled elapsed time provides the acceleration and consequently the inertial mass of the accelerated assembly comprising the Kibble balance coil and the mass holding pan. Repeating the measurement with the test mass added to the assembly and taking the difference between the two measurements yields the inertial mass of the test mass. Thus, the extreme precision inertial and gravitational mass measurement of a test mass with a Kibble balance could provide a test of the equivalence principle. We discuss how the two masses are related to the Planck constant and other coupling constants and how the Kibble balance could be used to test the dynamic constants theories in Dirac cosmology. Source: arXiv, 2207.02680 Services: Forum \| Review \| PDF \| Favorites No review found. Note: answers to reviews or questions about the article must be posted in the forum section. Authors are not allowed to review their own article. They can use the forum section.	{"url":"http://science-advisor.net/article/2207.02680","timestamp":"2024-11-05T20:13:58Z","content_type":"text/html","content_length":"22017","record_id":"<urn:uuid:ed3b7b6e-776b-446c-a65c-296e8b3a98f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00216.warc.gz"}
Checking for prime numbers in Java 1. What is a prime number? A prime number is a positive integer that can only be divided by 1 and itself. For example, 2, 3, 5, 7, 11, 13, 17, 19, 23, etc. are prime numbers. Note: 0 and 1 are not prime numbers. How to check if a positive integer n is a prime number? 1. If n is 0 or 1, it is not a prime number. 2. If n is not divisible by any number from 2 to n-1, then n is a prime number. Otherwise, n is not a prime number. It is easy to see that only 2 is an even prime number, all other even numbers are not prime because they are divisible by 2. Therefore, instead of checking from 2 to n-1, we can only consider from 2 to n/2. In addition, it has been proven that: we only need to check if n is divisible by any number from 2 to the square root of n to determine if n is a prime number or not. 2. How to check prime numbers in Java Using a for loop in Java to check if n is divisible by any number from 2 to n-1 or not? package primenumber; import java.util.Scanner; public class PrimeNumber { public static void main(String[] args) { int num; boolean is_prime = true; System.out.print("Enter a positive integer: "); try (Scanner scanner = new Scanner(System.in)) { num = scanner.nextInt(); // 0 and 1 are not prime numbers if (num == 0 \|\| num == 1) { is_prime = false; // loop to check if n is prime for (int i = 2; i <= num-1; i++) { if (num % i == 0) { is_prime = false; if (is_prime) { System.out.print(num + " is a prime number"); } else { System.out.print(num + " is not a prime number"); In this program, we take a number as input from the user and then check whether it is prime or not by dividing it by all possible divisors from 2 to num-1. If the number is divisible by any of these divisors, then it is not a prime number. If none of these divisors can divide the number, then it is a prime number. We use a boolean variable is_prime to keep track of whether the number is prime or not. We initially assume that the number is prime and set is_prime to true. If we find a divisor that divides the number, then we assigned the value false to is_prime. At the end of the program, we check the value is_prime to determine whether the number is prime or not. We can change the loop condition to only run from 2 to n/2. // loop to check if n is prime for (int i = 2; i <= num/2; i++) { if (num % i == 0) { is_prime = false; You can change the loop condition to only run from 2 to the square root of n (Math.sqrt(n)). // loop to check if n is prime for (int i = 2; i <= Math.sqrt(num); i++) { if (num % i == 0) { is_prime = false; We should only iterate up to the square root of n to make the loop shorter. These are three ways how to check prime numbers in Java. The last way is usually the most optimal one for most cases.	{"url":"https://itlearningcorner.com/programming-language/checking-for-prime-numbers-in-java/","timestamp":"2024-11-13T21:43:50Z","content_type":"text/html","content_length":"67207","record_id":"<urn:uuid:ec64434f-104e-44df-95ec-6673833bd5ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00604.warc.gz"}
98 Meter/Hour Squared to Milligals Meter/Hour Squared [m/h2] Output 98 meter/hour squared in meter/second squared is equal to 0.0000075617283950617 98 meter/hour squared in attometer/second squared is equal to 7561728395061.7 98 meter/hour squared in centimeter/second squared is equal to 0.00075617283950617 98 meter/hour squared in decimeter/second squared is equal to 0.000075617283950617 98 meter/hour squared in dekameter/second squared is equal to 7.5617283950617e-7 98 meter/hour squared in femtometer/second squared is equal to 7561728395.06 98 meter/hour squared in hectometer/second squared is equal to 7.5617283950617e-8 98 meter/hour squared in kilometer/second squared is equal to 7.5617283950617e-9 98 meter/hour squared in micrometer/second squared is equal to 7.56 98 meter/hour squared in millimeter/second squared is equal to 0.0075617283950617 98 meter/hour squared in nanometer/second squared is equal to 7561.73 98 meter/hour squared in picometer/second squared is equal to 7561728.4 98 meter/hour squared in millimeter/hour squared is equal to 98000 98 meter/hour squared in centimeter/hour squared is equal to 9800 98 meter/hour squared in kilometer/hour squared is equal to 0.098 98 meter/hour squared in meter/minute squared is equal to 0.027222222222222 98 meter/hour squared in millimeter/minute squared is equal to 27.22 98 meter/hour squared in centimeter/minute squared is equal to 2.72 98 meter/hour squared in kilometer/minute squared is equal to 0.000027222222222222 98 meter/hour squared in kilometer/hour/second is equal to 0.000027222222222222 98 meter/hour squared in inch/hour/minute is equal to 64.3 98 meter/hour squared in inch/hour/second is equal to 1.07 98 meter/hour squared in inch/minute/second is equal to 0.017862350539516 98 meter/hour squared in inch/hour squared is equal to 3858.27 98 meter/hour squared in inch/minute squared is equal to 1.07 98 meter/hour squared in inch/second squared is equal to 0.00029770584232526 98 meter/hour squared in feet/hour/minute is equal to 5.36 98 meter/hour squared in feet/hour/second is equal to 0.089311752697579 98 meter/hour squared in feet/minute/second is equal to 0.0014885292116263 98 meter/hour squared in feet/hour squared is equal to 321.52 98 meter/hour squared in feet/minute squared is equal to 0.089311752697579 98 meter/hour squared in feet/second squared is equal to 0.000024808820193772 98 meter/hour squared in knot/hour is equal to 0.052915766944444 98 meter/hour squared in knot/minute is equal to 0.00088192944907407 98 meter/hour squared in knot/second is equal to 0.000014698824151235 98 meter/hour squared in knot/millisecond is equal to 1.4698824151235e-8 98 meter/hour squared in mile/hour/minute is equal to 0.0010149062806543 98 meter/hour squared in mile/hour/second is equal to 0.000016915104677572 98 meter/hour squared in mile/hour squared is equal to 0.060894376839259 98 meter/hour squared in mile/minute squared is equal to 0.000016915104677572 98 meter/hour squared in mile/second squared is equal to 4.6986401882144e-9 98 meter/hour squared in yard/second squared is equal to 0.0000082696067312574 98 meter/hour squared in gal is equal to 0.00075617283950617 98 meter/hour squared in galileo is equal to 0.00075617283950617 98 meter/hour squared in centigal is equal to 0.075617283950617 98 meter/hour squared in decigal is equal to 0.0075617283950617 98 meter/hour squared in g-unit is equal to 7.71081704258e-7 98 meter/hour squared in gn is equal to 7.71081704258e-7 98 meter/hour squared in gravity is equal to 7.71081704258e-7 98 meter/hour squared in milligal is equal to 0.75617283950617 98 meter/hour squared in kilogal is equal to 7.5617283950617e-7	{"url":"https://hextobinary.com/unit/acceleration/from/mh2/to/milligal/98","timestamp":"2024-11-11T07:51:22Z","content_type":"text/html","content_length":"96393","record_id":"<urn:uuid:d6c26648-4bcf-4a68-9137-073130ce1d56>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00829.warc.gz"}
Strict Weak Ordering Strict Weak Ordering Category: functors Component type: concept A Strict Weak Ordering is a Binary Predicate that compares two objects, returning true if the first precedes the second. This predicate must satisfy the standard mathematical definition of a strict weak ordering. The precise requirements are stated below, but what they roughly mean is that a Strict Weak Ordering has to behave the way that "less than" behaves: if a is less than b then b is not less than a, if a is less than b and b is less than c then a is less than c, and so on. Refinement of Binary Predicate Associated types │First argument type │The type of the Strict Weak Ordering's first argument. │ │Second argument type│The type of the Strict Weak Ordering's second argument. The first argument type and second argument type must be the same. │ │Result type │The type returned when the Strict Weak Ordering is called. The result type must be convertible to bool. │ F A type that is a model of Strict Weak Ordering X The type of Strict Weak Ordering's arguments. f Object of type F x, y, z Object of type X • Two objects x and y are equivalent if both f(x, y) and f(y, x) are false. Note that an object is always (by the irreflexivity invariant) equivalent to itself. Valid expressions None, except for those defined in the Binary Predicate requirements. Expression semantics │ Name │Expression│ Precondition │ Semantics │ Postcondition │ │Function call│f(x, y) │The ordered pair (x,y) is in the domain of f│Returns true if x precedes y, and false otherwise│The result is either true or false│ Complexity guarantees │Irreflexivity │f(x, x) must be false. │ │Antisymmetry │f(x, y) implies !f(y, x) │ │Transitivity │f(x, y) and f(y, z) imply f(x, z). │ │Transitivity of │Equivalence (as defined above) is transitive: if x is equivalent to y and y is equivalent to z, then x is equivalent to z. (This implies that equivalence does in fact satisfy │ │equivalence │the mathematical definition of an equivalence relation.) [1] │ [1] The first three axioms, irreflexivity, antisymmetry, and transitivity, are the definition of a partial ordering; transitivity of equivalence is required by the definition of a strict weak ordering. A total ordering is one that satisfies an even stronger condition: equivalence must be the same as equality. See also LessThan Comparable, less, Binary Predicate, function objects Copyright © 1999 Silicon Graphics, Inc. All Rights Reserved. TrademarkInformation	{"url":"http://seanborman.com/STL_doc/StrictWeakOrdering.html","timestamp":"2024-11-15T01:29:47Z","content_type":"text/html","content_length":"6952","record_id":"<urn:uuid:fa4ca49c-e9aa-461a-b228-fe3f90e1f50f>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00122.warc.gz"}
Trig Identities Calculator – Simplify Complex Equations Effortlessly Our trig identities calculator will quickly compute values for the fundamental trigonometric functions, making it easier for you to solve trigonometry problems. Trigonometric Identities Calculator Use this calculator to compute the trigonometric functions of an angle. Simply enter the angle in degrees into the input box, select the desired trigonometric function, and click “Calculate”. How to Use the Calculator 1. Enter the angle in degrees in the first input field. 2. Select the trigonometric function you wish to calculate (sin, cos, tan, csc, sec, or cot) from the dropdown menu. 3. Click the “Calculate” button to get the result. How It Calculates The calculator first converts the input angle from degrees to radians by multiplying it by π/180. Then, it calculates the result based on the selected trigonometric function: • Sin – the sine of the angle • Cos – the cosine of the angle • Tan – the tangent of the angle • Csc (cosecant) – the reciprocal of the sine • Sec (secant) – the reciprocal of the cosine • Cot (cotangent) – the reciprocal of the tangent As trigonometric functions like tangent, cosecant, secant, and cotangent can have undefined values, the calculator will display “Undefined” in such cases (e.g., tan(90°) is undefined). Please note that results are rounded to four decimal places. For angles not expressible in finite decimal form, the results can only be approximations of the true values.	{"url":"https://madecalculators.com/trig-identities-calculator/","timestamp":"2024-11-09T16:07:27Z","content_type":"text/html","content_length":"143060","record_id":"<urn:uuid:c95f5e95-25d4-4315-badb-99231a25396a>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00245.warc.gz"}
Singapore Math Grade 1: Online practice Student prior knowledge Prior to starting Singapore Math Grade 1, most young students have already learned to count to 10 and 20, compare groups of objects up to sets of 10 and 20 using proper language: “greater or more than,” “less or fewer than,” and “the same as.” Students also learned to order and compare numbers to 10 and 20. The initial lessons in Singapore Math 1^st grade are both a review and an extension of content covered in Kindergarten.	{"url":"https://esingaporemath.com/program-grade-1","timestamp":"2024-11-06T15:11:31Z","content_type":"text/html","content_length":"253396","record_id":"<urn:uuid:a39016ed-ec40-43fd-a0ca-1b3a6793a836>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00737.warc.gz"}
algebra problem Our users: The best part of The Algebrator is its approach to mathematics. Not only it guides you on the solution but also tells you how to reach that solution. Horace Wagner, MO Wow! A wonderful algebra tutor that has made equation solving easy for me. Rick Parker, MO I was quite frustrated with handling complex numbers. After using this software, I am quite comfortable with it. Complex numbers are no more 'COMPLEX' to me. Perry Huges, KY This version of your algebra software is absolutely great! Thank you so much! It has helped me tremendously. KEEP UP THE GOOD WORK! David Mogorit, DC Students struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. Can you find yours among Search phrases used on 2014-02-24: • algebra 1 online answer key • homework answers glencoe physics • online revision 5grade • freshman algebra tests • free printable 6th grade work • log2 by calculator • prentice hall mathamatics • practise math 9 alberta test final • Heat Transfer equations .ppt • pdf sulla TI-89 • divide dominator • college algebra tips • texas instruments TI-83 Plus program hexadecimal decimal • formulas ratios • practice problems math graphing variables • practice combining terms • free surds worksheets • simplifying radicals to decimals • solve factor equation • what is the standard form of a linear equation in two variables • Cubed Square root of variables • cost accounting online books reading • feet into metres calculater • SAT extended paper KS3 science • high school homework helper • adding, subtracting,multiplying,dividing square numbers • math sheets ks3 free • www.basic concept of algebra.com • free ebook teach math • ged answer sheet free print • Free English Worksheets for 9th Graders • solve a algebra word problem on line • three equations unknowns ti-83 • free printable math worksheets 8th grade • Cookies in ASP Net • printablemathquestions • Pre-algebra by Alan S. Tussy and R. David Gustafson text pages online • divide exponents free worksheet • algebra + 7th grade + integration and equations • Gardening Catalog • Writing Mixed Numbers As Decimals • integers positive and negative words • last 9th grade algebra math regents worksheet • Free, printable sixth grade lesson worksheets • free printouts for first graders • slope intercept form teach • conic graphing calculator online • unknowns and variables in algebra ks3 • fortran solve single non linear equation • Sap Software • 6th Grade Math • rearranging formulae ppt • basic learning for college algebra • College Algebra and Trigonometry 3rd edition homework answers • 10 grade algebra tests online • slope-intercept to standard form worksheet • algebra problem solving worksheets • printable 3rd grade divion problems • grade 11 math cheat sheet • maths work sheet on multiplication for class 6 • maths trivia • coursecompass cheats • exponantial and radicals • converting decimals to halves • ged printable coordinate plane grid • algebra A trig sample questions • smallest common denominator calculator • nine positive integral cubes • front-end estimation, decimals, worksheet • mathematical puzzle,poem or article for class five • Free Family Budget • 1st grade printable homework • free Algebra II workbooks • examples of math trivia with answers mathematics • Los Angeles Cosmetic Eye Surgery • print out algebra regents practice test • ordered pair linear equation calculator • grade nine mathematics • B12 Vitamins • Elephant Insurance • Electronic Catalog • graphing parabola tool • Clipper Cruise Lines • Distance Degrees • Journal Subscribers • Invest Financial Corp • new york 6th grade math tests • "visual basic" +"exponential" • application using vertex of a quadratic • Algebra II question and answer • highest common factor of 26 • maths formulas for yr 10s • Formula to Convert Decimal to Fraction • key algebra concept for solving simultaneous equations • changing numbers from decimals to square roots • Worksheet —expert Arithmetic Radical Problems	{"url":"https://www.algebra-help.com/algebra-help-factor/geometry/free-algebra-problem-solver.html","timestamp":"2024-11-03T15:16:06Z","content_type":"application/xhtml+xml","content_length":"12909","record_id":"<urn:uuid:da564cfa-45d1-42d5-bfe2-eaf308893afa>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00199.warc.gz"}
[EM] Approval Equilibrium Forest W Simmons fsimmons at pcc.edu Wed Jun 13 15:36:45 PDT 2007 Suppose that candidate Y has the greatest pairwise opposition against candidate X. Let the letter n represent the number of ballots on which Y is rated strictly above X, i.e. X's maximum pairwise opposition. If X is an approval equilibrium winner, then the equilibrium approval of Y will be at least n. It may be larger, because there may be some ballots on which X and Y are rated equal top. Setting aside the ballots on which X and Y are rated equal, what is the "cost" of getting an approval of n+1 for X on the remaining ballots. For those remaining ballots on which X is rated top the cost is zero. For those remaining ballots on which X is rated k levels down from the top, the cost is c(k), some increasing function of k. Choose those (n+1) of the (remaining) ballots for which the total cost is a minimum. This total cost T(X) is the cost of making X into an approval The candidate X for which T(X) is minimal is the approval equilibrium What c(k) function should we use? More information about the Election-Methods mailing list	{"url":"http://lists.electorama.com/pipermail/election-methods-electorama.com/2007-June/118442.html","timestamp":"2024-11-03T00:52:49Z","content_type":"text/html","content_length":"3857","record_id":"<urn:uuid:49adc6d6-c358-4626-a04f-9e355050775b>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00565.warc.gz"}
Quantum-Inspired MNIST Posted on December 17, 2021, By Brian N. Siegelwax It doesn’t get easier than this…. What if you could do Machine Learning without building and training a model? What if you could forget about weights and activation functions and optimizers? What if you could exchange normalization and derivatives for addition and subtraction? I’ll use classification as an example, using the MNIST dataset. This should work for clustering, as well, since the quantum algorithm that inspired this experiment works for both. MNIST, for those who don’t know, is a popular dataset of handwritten digits, the numbers 0 through 9. The goal of the MNIST exercise is to take a handwritten single-digit number and accurately identify it as the number it’s intended to be. And, it’s probably impossible to get into Machine Learning without encountering it, because it’s a clean dataset. You can focus on building and training your model, not on cleaning your data. How does classical classification work? At its most fundamental level, classification starts with inputs and outputs and we determine “weights" that, when multiplied by the inputs, produce the known outputs. We then multiply those weights by test inputs and obtain outputs that suggest the most likely classification for each input. inputs * weights = outputs Well, that doesn’t seem so hard, does it? What’s so challenging about that? The challenge is efficiently finding the weights. We start off multiplying the inputs by random weights, determine how far away we are from the known outputs, and adjust the weights in small increments. We do this many, many times, until the small weight adjustments reduce the distances between the weighted inputs and the outputs to their lowest possible values. But, how big should those adjustments be? What if we’re too liberal and we overshoot our target? Or, what if we’re too conservative and training takes forever? Then, we’ll need another model to tune our hyperparameters? So, basic classification is conceptually simple, but it can get complicated. We’re on a quest to optimize both runtime and accuracy. How does quantum classification work? With classical classification, we use weighting to reduce the “distances” between the inputs and the outputs. The word “distances” is important, because there is a quantum algorithm called the SWAP Test which also goes by the name inner products, kernel methods, and, most importantly for this discussion, distance measures. If we map our classical data to quantum states, the SWAP Test determines how close or how distant two quantum states are. The difference between classical classification and quantum classification is that we don’t need weighting, and we don’t need to do any adjusting. We just measure. If our test data measures closest to classification A, then that’s the likely classification for our test data. If, on the other hand, our data point measures closest to classification B, then that’s the likely classification instead. It’s that simple. What is quantum-inspired classification? What if we do the same thing classically that we do quantumly, and just determine the distance between values? distance += absolute_value(MNIST_value - test_data_value) The MNIST dataset contains images that are 28x28 pixels, or 784 total pixels. What if we just loop through, pixel by pixel, and determine the total distance between a test digit and zero, the same test digit and one, the same test digit and two, and so forth? The train dataset contains roughly 6,000 records for each digit, so I like to use mean digits. What are the average intensity values per pixel for all the zeroes, all the ones, and so forth? Then I took the first 100 test digits and did exactly what I just described. I went pixel by pixel and summed the distances between a test digit and the mean zero, then the same digit and the mean one, and so on. The lowest sum represented the shortest distance between the test data and one particular classification, and that’s the likely classification. comparing multi-dimensional data How do the results compare? With a small sample of 100 test digits, the accuracy was 72%. Random guessing would be only 10%, so 72% isn’t terrible. And, considering we’re only doing addition and subtraction, it compares to executing 230,000 operations traditionally. On the low end, I found someone struggling with MNIST accuracy as low as 11%, and I found Kaggle competitions with accuracy above 99%. The reason for both extremes, actually, is the level of complexity we can add. This method beats students who are struggling to learn image classification, on the one hand, while simultaneously retaining quite a bit of room for improvement, on the other. I must also say that quantum-inspired classification is fast. I’ve trained very simple classification models, and the simplest ones still generally take at least a minute. In contrast, 7,840 rounds of addition and subtraction is virtually instantaneous. It actually takes about a second, but that’s probably due to running this in a Jupiter notebook. That’s not optimal. What’s next? My starting point is 72% accuracy. That’s with nothing but addition and subtraction. It couldn’t be simpler. However, I have ideas that would keep this method conceptually simple, while hopefully improving accuracy. First, I have historically been concerned with straying from mean digits, because I’ve been concerned about the train digits overlapping each other. I’m referring to quantum classification here, of course. And, I have had additional concerns regarding dimensionality reduction and limited qubit availability. However, these really aren’t concerns when done classically. Therefore, I do have another “encoding” to try, and we’ll see how that affects accuracy. Second, classical MNIST uses Convolutional Neural Networks (CNN), not the simple approach I described here. We don’t compare single pixel to single pixel, and we don’t do only one comparison. Therefore, I could expand this to compare small groups of pixels to small groups of pixels, like a CNN, and see what happens. Third, I could analyse Kaggle submissions. I’ve never actually done that because I’ve only worked with MNIST quantumly until now. But, maybe there are ideas to be found that can be implemented without adding too much complexity. The point of quantum-inspired MNIST is simplicity, because of quantum MNIST’s simplicity.	{"url":"https://www.quantum-futures.com/post/manage-your-blog-from-your-live-site","timestamp":"2024-11-03T07:13:55Z","content_type":"text/html","content_length":"1050494","record_id":"<urn:uuid:f8f49b17-236d-4c99-9bf7-602da8cb35bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00686.warc.gz"}
M C Squared Project The horizon Designed and produced by the Spanish CoI The c-book unit «The Horizon» was designed by five members of the Spanish CoI. This c-book unit is devoted to the study of the vision of the horizon. Some geometric and algebraic basic concepts provide the necessary tools to understand how the height of the point of vision and of the object one tries to see and the radius act on the vision. Furthermore, refraction makes possible to see things beyond the horizon (under the line of vision). The c-book unit is structured in three phases. The first phase wants to give an answer to the question Q1: What is the line of the horizon? Which factors may affect in setting the horizon? The phase begins with some basic questions about the intuition of the students before starting with the main aims to study in this c-unit: Which is the distance to the horizon from the seashore? How does the use of binoculars or similar tools help to the vision? How would the horizon be in the case that the Earth was a plane?. The second phase focuses on Q2: Which is the distance to the horizon from a certain altitude? Students are asked to do some forecasts and calculations in order to compare the distance to the horizon from different altitudes, as well as the growth of the distance when the altitude increases. Finally, the third phase is devoted to Q3: Is it possible to see beyond the horizon? A picture of watching some mountains in Majorca from a certain point in Barcelona opens this phase, then students are asked about how it is possible to see further the line of the horizon. Then, students are asked to investigate and simulate other possible cases to understand this situation. Available at: http://mc2dme.appspot.com/dwo/dwo.jsp?profile=78&language=en&courseViewNr=5186321119182848	{"url":"http://www.mc2-project.eu/index.php/list-of-c-book-units/146-the-horizon","timestamp":"2024-11-03T13:44:49Z","content_type":"application/xhtml+xml","content_length":"31454","record_id":"<urn:uuid:f371e06c-2521-408b-b511-0b6c5d34a6c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00546.warc.gz"}
Free Group Study Rooms with Timer & Music \| FiveableWhat is a Limit? \| AP Calculus AB/BC Class Notes \| Fiveable A limit denotes theΒ behaviorΒ of a function as itΒ approachesΒ a certain value which is especially important in calculus.Β In mathematical terms, the limit is asking the question "What value is 'y' getting close to as 'x' approaches a number?" and its represented by the expression: Out loud, this would sound something like "the limit of f(x) as x approaches c" • The 'lim' shows that we're finding the limit and not the value of the function • The 'x β c' tells us the value that x is getting closer and closer to but never actually reaching • The 'f(x)' is meant to represent the functions you're working with • Graphically, this would look like following a function to a certain point on the x-axis from the right • Graphically, this would look like following a function to a certain point on the x-axis from the left A limitΒ MUST meet all three criteria in order for it to exist, which is important to remember especially when FRQ's come into play. If the limit from the left and the limit from the right are not equal, then the limit does not exist. An example of this would look something like this graph: Here we can see that as we follow the function from the right hand side, it's going towards y = 2. On the left hand side, we see that the function is approaching y = 1. Since 2 β 1, the limit does not exist at x = 1. β This is when the degree of the polynomial on the top of the function is greater than the degree of the polynomial in the denominator In this case, the answer you are finding is all dependent on whether or not you're being asked to be approaching negative or positive infinity. If you're approaching positive infinity, then the limit of a top-heavy function will also be infinity and vice This is when the degrees of the numerator and denominator are the same. To find the limit here, you'll need to divide the leading coefficients of the numerator and denominator by each other. For example, here we see that the degrees of the top and bottom are 3 and that the leading coefficient of the numerator is 5 and 2 for the denominator.Β Therefore, the limit of f(x) as x approaches positive infinity is 5/2. This is when the degree of the denominator is greater than that of the numerator. If a function is bottom heavy, then the limit is zero, regardless if it is approaching positive or negative infinity π Here are some extra resources to help you out on limits approaching infinity!Β	{"url":"https://hours-zltil9zhf-thinkfiveable.vercel.app/ap-calc/faqs/what-is-a-limit/blog/v17ujCMJV03Ad3iVCcv8","timestamp":"2024-11-11T03:57:39Z","content_type":"text/html","content_length":"240651","record_id":"<urn:uuid:1cc2157d-980f-4158-91ec-f36207d2cc50>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00156.warc.gz"}
Quantile - Sample Quantile Returns the sample p-quantile of the non-missing observations (i.e., divides the sample data into equal parts determined by the percentage p). Quantile(X, p) is the input data sample (one/two-dimensional array of cells (e.g., rows or columns)) is a scalar value between 0 and 1. 1. The time series may include missing values (e.g., #N/A, #VALUE!, #NUM!, empty cell), but they will not be included in the calculations. 2. The Quantile function for any distribution is defined between 0 and 1. Its function is the inverse of the cumulative distribution function (CDF). 3. The Quantile function returns the sample median when $p=0.5$. 4. The Quantile function returns the sample minimum when $p=0$. 5. The Quantile function returns the sample maximum when $p=1$. 6. For any probability distribution, the following holds true for the probability $p$: $$P(X\lt q)\geq p$$ Where: □ $q$ is the sample $p$-quantile. Files Examples Related Links Article is closed for comments.	{"url":"https://support.numxl.com/hc/en-us/articles/215721283-Quantile-Sample-Quantile","timestamp":"2024-11-12T06:19:50Z","content_type":"text/html","content_length":"31717","record_id":"<urn:uuid:36b68009-00aa-46e7-bc7b-d1efa4716b1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00222.warc.gz"}
Erdős Problems If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor \frac{n+1}{2}\rfloor$ distinct distances. Solved by Altman . The stronger variant that says there is one point which determines at least $\lfloor \frac{n+1}{2}\rfloor$ distinct distances is still open. Fishburn in fact conjectures that if $R(x)$ counts the number of distinct distances from $x$ then \[\sum_{x\in A}R(x) \geq \binom{n}{2}.\] Szemerédi conjectured (see [Er97e]) that this stronger variant remains true if we only assume that no three points are on a line, and proved this with the weaker bound of $n/3$. See also [660].	{"url":"https://www.erdosproblems.com/tags/convex","timestamp":"2024-11-07T15:33:14Z","content_type":"text/html","content_length":"11562","record_id":"<urn:uuid:0a499dc2-7c55-4a54-be81-6264c7276756>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00593.warc.gz"}
QR Decomposition with Column Pivoting Go to the first, previous, next, last section, table of contents. The @math{QR} decomposition can be extended to the rank deficient case by introducing a column permutation @math{P}, The first @math{r} columns of this @math{Q} form an orthonormal basis for the range of @math{A} for a matrix with column rank @math{r}. This decomposition can also be used to convert the linear system @math{A x = b} into the triangular system @math{R y = Q^T b, x = P y}, which can be solved by back-substitution and permutation. We denote the @math{QR} decomposition with column pivoting by @math{QRP^T} since @math{A = Q R P^T}. Function: int gsl_linalg_QRPT_decomp (gsl_matrix * A, gsl_vector * tau, gsl_permutation * p, int signum, gsl_vector norm) This function factorizes the @math{M}-by-@math{N} matrix A into the @math{QRP^T} decomposition @math{A = Q R P^T}. On output the diagonal and upper triangular part of the input matrix contain the matrix @math{R}. The permutation matrix @math{P} is stored in the permutation p. The sign of the permutation is given by signum. It has the value @math{(-1)^n}, where @math{n} is the number of interchanges in the permutation. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and vectors which encode the orthogonal matrix Q. The vector tau must be of length @math{k=\min(M,N)}. The matrix @math{Q} is related to these components by, @math{Q = Q_k ... Q_2 Q_1} where @math{Q_i = I - \tau_i v_i v_i^T} and @math{v_i} is the Householder vector @math{v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i))}. This is the same storage scheme as used by LAPACK. On output the norms of each column of R are stored in the vector The algorithm used to perform the decomposition is Householder QR with column pivoting (Golub & Van Loan, Matrix Computations, Algorithm 5.4.1). Function: int gsl_linalg_QRPT_decomp2 (const gsl_matrix * A, gsl_matrix * q, gsl_matrix * r, gsl_vector * tau, gsl_permutation * p, int signum, gsl_vector norm) Function: int gsl_linalg_QRPT_solve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x) Function: int gsl_linalg_QRPT_svx (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, gsl_vector * x) Function: int gsl_linalg_QRPT_QRsolve (const gsl_matrix * Q, const gsl_matrix * R, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x) Function: int gsl_linalg_QRPT_update (gsl_matrix * Q, gsl_matrix * R, const gsl_permutation * p, gsl_vector * u, const gsl_vector * v) This function performs a rank-1 update @math{w v^T} of the @math{QRP^T} decomposition (Q, R,p). The update is given by @math{Q'R' = Q R + w v^T} where the output matrices @math{Q'} and @math{R'} are also orthogonal and right triangular. Note that w is destroyed by the update. The permutation p is not changed. Function: int gsl_linalg_QRPT_Rsolve (const gsl_matrix * QR, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x) Function: int gsl_linalg_QRPT_Rsvx (const gsl_matrix * QR, const gsl_permutation * p, gsl_vector * x) Go to the first, previous, next, last section, table of contents.	{"url":"http://www.math.utah.edu/software/gsl/gsl-ref_205.html","timestamp":"2024-11-09T22:39:32Z","content_type":"text/html","content_length":"6678","record_id":"<urn:uuid:264bd82f-b55c-4beb-b2a4-ee2207b8ba55>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00779.warc.gz"}
TPU from Atmospheric Correction of Landsat 8 OLI Imagery The radiometric conversion of the raw satellite sensor data is performed to calculate the top-of-the-atmosphere radiance and its associated uncertainty is estimated at first step. The next step of the satellite data processing is the atmospheric correction. This study focuses on the TPU created during the atmospheric correction procedure. The atmospheric correction consists of the angular geometry of the sun-sensor orientation and the atmospheric properties. The optically dominant atmospheric components are aerosol, ozone, water vapor. The atmospheric correction introduces additional uncertainties propagated through the atmospheric correction equation. In this study we propose the methodology to estimate the statistical TPU created by the atmospheric correction. The uncertainty of each atmospheric component is estimated first as a standard deviation. Next, the Jacobian matrix as a partial derivative of all surface reflectance bands with respect to several atmospheric parameters is formed and the TPU matrix is calculated using Jacobian matrix and the uncertainty covariance matrxi. The TPU matrix represents the uncertainty of the surface reflectance for each band of each pixel. Any downstream application product will be able to assess its uncertainty based on the associated TPU of the surface reflectance. For example, if a user computes NDVI, the associated uncertainty of the NDVI can be calculated based on the TPU of the surface reflectance. Having the NDVI uncertainty image will allow a user to see the area with larger or smaller NDVI uncertainty and help a decision making in an educated use of the NDVI product. Jun 18th, 5:20 PM TPU from Atmospheric Correction of Landsat 8 OLI Imagery The radiometric conversion of the raw satellite sensor data is performed to calculate the top-of-the-atmosphere radiance and its associated uncertainty is estimated at first step. The next step of the satellite data processing is the atmospheric correction. This study focuses on the TPU created during the atmospheric correction procedure. The atmospheric correction consists of the angular geometry of the sun-sensor orientation and the atmospheric properties. The optically dominant atmospheric components are aerosol, ozone, water vapor. The atmospheric correction introduces additional uncertainties propagated through the atmospheric correction equation. In this study we propose the methodology to estimate the statistical TPU created by the atmospheric correction. The uncertainty of each atmospheric component is estimated first as a standard deviation. Next, the Jacobian matrix as a partial derivative of all surface reflectance bands with respect to several atmospheric parameters is formed and the TPU matrix is calculated using Jacobian matrix and the uncertainty covariance matrxi. The TPU matrix represents the uncertainty of the surface reflectance for each band of each pixel. Any downstream application product will be able to assess its uncertainty based on the associated TPU of the surface reflectance. For example, if a user computes NDVI, the associated uncertainty of the NDVI can be calculated based on the TPU of the surface reflectance. Having the NDVI uncertainty image will allow a user to see the area with larger or smaller NDVI uncertainty and help a decision making in an educated use of the NDVI product.	{"url":"https://digitalcommons.usu.edu/calcon/CALCON2019/all2019content/10/","timestamp":"2024-11-06T16:37:15Z","content_type":"text/html","content_length":"37046","record_id":"<urn:uuid:0f6a196a-cd11-4046-8c84-df5785c22f5b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00175.warc.gz"}
• A new look at subrepresentation formulas, submitted, with C. Hoang and C. Pérez. • Degenerate parabolic p-Laplacian equations: existence, uniqueness and asymptotic behavior of solutions, submitted, with D. Cruz-Uribe and Y. Zhao. • New pointwise bounds by Riesz potential type operators, submitted, with C. Hoang and C. Pérez. • A homage to Guido Weiss and his leadership of the Saint Louis team: Commutators of singular integrals and Sobolev inequalities, to appear in a special volume dedicated to Guido Weiss, Birkhauser/ Springer series, with C. Hoang and C. Pérez. • Bump conditions for general iterated commutators with applications to compactness, to appear Illinois J. Math., with A. Mair and Y. Wen. • Pointwise estimates for rough operators with applications to Sobolev inequalities, to appear J. D'Analyse, with C. Hoang and C. Pérez. • Two weight bump conditions for compactness of commutators, Arch. Math. (Basel) 120 (2023) 47-57, with A. Mair. • New oscillation classes and two weight bump conditions for commutators, Collect. Mat., 74 (2023), 225–246, with D. Cruz-Uribe and Q. Minh Tran. • Weak endpoint bounds for matrix weights, Rev. Mat. Iberoam. 37 (2021), 1513-1538, with D. Cruz-Uribe, J. Isralowitz, S. Pott, and I. Rivera-Rios. • A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities, Georgian Math. J. 27 (2020), 37-42, with D. Cruz-Uribe. • A new approach to norm inequalities on weighted and variable Hardy spaces, Ann. Acad. Sci. Fenn. 45 (2020) 175-198, with D. Cruz-Uribe and H. Nguyen. • Boundedness results for commutators with BMO functions via weighted estimates: a comprehensive approach, Math. Ann. 376 (2020), 853-871, with Á. Bényi, J. M. Martell, E. Stachura, R.H. Torres. • Multilinear fractional Calderon-Zygmund operators on weighted Hardy spaces, Houston Math. J. 45 (2019), 679-713, with D. Cruz-Uribe and H. Nguyen. • Matrix weighted Poincaré inequalities and applications to degenerate elliptic systems, Indiana Univ. Math. J. 68 (2019) 1327-1377, with J. Isralowitz. • The boundedness of multilinear Calderón-Zygmund operators on weighted and variable Hardy spaces, Pub. Mat. 63 (2019) D. Cruz-Uribe and H. Nguyen. • Two weight bump conditions for matrix weights, Integral Equations Operator Theory, 90 (2018), 31 pp., with D. Cruz-Uribe and J. Isralowitz. • Extrapolation in the scale of generalized reverse Hölder weights, Rev. Complut. Mat., 31 (2018), 263–286 with T. Anderson and D. Cruz-Uribe. • Weighted estimates for bilinear fractional integral operators and their commutators Indiana Univ. Math. J., 67 (2018) 397–428, with C. Hoang. • On the harmonic and geometric maximal operators associated to a general basis, Math. Inequal. Appl., 21 (2018) 265-286, with L. Anne Duffee. • Muckenhoupt-Wheeden conjectures for sparse operators, Arch. Math. (Basel), 109 (2017) 49-58, with C. Hoang. • Matrix $A_p$ weights, degenerate Sobolev spaces, and mappings of finite distortion, J. Geom. Anal., 26 (2016) 2797–2830, with D. Cruz-Uribe and S. Rodney. • Unions of Lebesgue spaces and $A_1$ majorants, Pacific J. Math. 280 (2016), 411-432, with G. Knese and J. McCarthy. • Regularity for weak solutions of elliptic PDE below the natural exponent, Ann. Mat. Pura Appl, 195 (2016) 725-740, with D. Cruz-Uribe and S. Rodney. • Compactness properties of commutators of bilinear fractional integrals, Math. Z. 280 (2015) 569-582, with A. Benyi, W. Damien, and R.H. Torres. • Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals, Math. Nachr. 288 (2015) 619-632, with K. Li and W. Sun. • Compact bilinear operators: the weighted case, Michigan Math. J. 64 (2015) 39-51, with A. Benyi, W. Damien, and R.H. Torres. • Logarithmic bump conditions for Calderón-Zygmund operators on spaces of homogeneous type, Publ. Mat. 59 (2015) 17-43, with T. Anderson and D. Cruz-Uribe. • The sharp weighted bound for multilinear maximal functions and Calderón-Zygmund operators, J. Fourier Anal. and Appl. 20 (2014) 751-765, with K. Li and W. Sun. • Bilinear Sobolev-Poincaré inequalities and Leibniz-type rules, J. Geom. Anal. 24 (2014) 1144-1180, with F. Bernicot, D. Maldonado, and V. Naibo. • Higher-order multilinear Poincare and Sobolev inequalities in Carnot groups, Potential Anal. 40 (2014) 231-245, with V. Naibo. • New weighted estimates for bilinear fractional integral operators, Trans. Amer. Math. Soc. 366 (2014) 627-646. • One and two weight norm inequalities for Riesz potentials, Illinois J. Math. 57 (2013) 295-323, with D. Cruz-Uribe. • Mixed $A_p-A_\infty$ estimates with one supremum, Studia Math. 219 (2013) 247-267, with A. Lerner. • A panorama of sampling theory, Excursions in harmonic analysis. Volume 1, 107-127, Appl. Numer. Harmon. Anal. Birkhauser/Springer, New York 2013, with H. Sikic, G. Weiss, and E. Wilson. • A fractional Muckenhoupt-Wheeden theorem and its consequences, Integral Equations Operator Theory 76 (2013) 421-446, with D. Cruz-Uribe. • Regularity of solutions to degenerate $p$-Laplacian equations, J. Math. Anal. Appl. 401 (2013) 458-478, with D. Cruz-Uribe and V. Naibo. • Sharp weighted bounds without testing or extrapolation, Arch. Math. 99 (2012) 457-466. • Sharp norm inequalities for commutators of classical operators, Publ. Mat. 56 (2012) 147-190, with D. Cruz-Uribe. • Weighted multilinear Poincaré inequalities for vector fields of Hormander type, Indiana Univ. Math. J. 60 (2011) 473-506, with D. Maldonado and V. Naibo. • Multiparameter weights with connections to Schauder bases, J. Math. Anal. Appl. 371 (2010) 266-281. • Sharp weighted bounds for fractional integral operators, J. Funct. Anal. 259 (2010) 1073-1097, with M. Lacey, C. Pérez, and R.H. Torres. • Sharp one-weight and two-weight bounds for maximal operators, Studia Math. 194 (2009) 163-180. • Weighted inequalities for multilinear fractional integral operators, Collect. Math. 60 (2009) 213-238.	{"url":"https://kmoen.people.ua.edu/research.html","timestamp":"2024-11-09T13:25:46Z","content_type":"text/html","content_length":"35569","record_id":"<urn:uuid:1a3ad484-da53-48a0-ad52-619339eb5121>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00882.warc.gz"}
Jeopardy Find the Area Missing Length Circumference Perimeter and Area Volume ppt download Presentation is loading. Please wait. To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy , including cookie policy. Ads by Google	{"url":"http://slideplayer.com/slide/4329752/","timestamp":"2024-11-02T05:16:07Z","content_type":"text/html","content_length":"167869","record_id":"<urn:uuid:accf8945-3e6e-413e-91ae-23ed0d140ec3>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00286.warc.gz"}
How do you write a product of a number and 2 as an expression? \| Socratic How do you write a product of a number and 2 as an expression? 1 Answer The product of a given number $x$ and $2$ can be written as $2 x$. $x \cdot 2 = 2 x$ Impact of this question 28261 views around the world	{"url":"https://api-project-1022638073839.appspot.com/questions/how-do-you-write-product-of-a-number-an-2-as-an-expression#112649","timestamp":"2024-11-12T16:42:28Z","content_type":"text/html","content_length":"31917","record_id":"<urn:uuid:14def4f6-3786-4507-be26-638abf04aebd>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00031.warc.gz"}
real part and imaginary part worksheet answers we are given: f(t)=4 e^(jWt) + 3 e^(2jWt) in volts I have the answer for this question: [Real part of f(t)]^2 =25 [Imaginary part of f(t)]^2 =24cos(Wt) how did we get these 2 values? Complex numbers are written in the form a + bi, where a is called the real term and the coefficient of i is the imaginary part. endobj endobj 'Positive' and 'Negative' are defined only on the real number line, which is part of the system of complex numbers. 11 0 obj<> Real part 2 , imaginary part -5i Complex numbers written like this are in the rectangular form. Complex number = (Real Part) + (Imaginary Part) i The set of complex numbers is denoted by C. endobj Learn more about matlab, symbolic Symbolic Math Toolbox, MATLAB Fetal Pig Dissection Thoracic Cavity The Circulatory System. Approach: A complex number can be represented as Z = x + yi, where x is real part and y is imaginary. Answer with 5 significant digits. Printable Worksheets @ www.mathworksheets4kids.com Name : Answer key Real Part and Imaginary Part Sheet 1 A) Complete the table. But I don't know how to do it s logarithm 12 0 obj<> Hence the set of real numbers, denoted R, is a subset of the set of complex numbers, denoted C. Adding and subtracting complex numbers is similar to adding and subtracting like terms. Complex Numbers. To download/print, click on pop-out icon or print icon to worksheet to print or download. Response times vary by subject and question complexity. The numbers that have parts in them an imaginary part and a real part are what we term as complex numbers. 18 0 obj<> This worksheet is designed to give students practice at imaginary number operations, specifically adding, subtracting and multiplying complex numbers. endobj x�}T}PSW 1�˳�@y�Bݒ��H��X�Y[E[��֖� $`�G�%&Z ��Mи��Pi5T+b�-�8u:�mwv��q�;{^�;{_��ٿv�̙{ι�w��޹J��H$ �x�N_ěVm4U��',� \�j�(N��d��L�� p6Q��u��m�V��diD�&f�j��̘]�mЛ��}f��m�-7�u&^g1�3� ��\��r8��3-� �� 5:��3��q+��U��# X܎Z��m��&(��-�=��h0gP��Vo(�+_��T�j8��Rۨ��jj;�G=C��T���$T"a�,� I�K�X��U��tF��8Y��\|�\|�Ёƅ��b��K�L��u�Љ'Ux=�A�4��U��a�][�zL\|Z��WX��1�n��'�U�7��&Or� gG�V��4Y�-{ۇ��'z�\�� _& �l��նV:�R�i��s��Z�xHpؒ�@k@��Z�� 17 0 obj<> 29 scaffolded questions that start relatively easy and end with some real challenges. A complex number is a number with a Real part, a, and an imaginary part, bi written in the form I. Complex numbers is vital in high school math. -7+8i the real part : : - 7 (my answer) the imaginary part : ...” in Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. What is the real part and imaginary part of the original load impedance (Zl)? Free worksheet(pdf) and answer key on Complex Numbers. We will follow the below steps to separate out real and imaginary part. Imaginary And Complex Numbers - Displaying top 8 worksheets found for this concept.. ]:��,�=}�c��_�. I need the real and imaginary part of $\log \sin (x+iy)$. Median response time is 34 minutes and may be longer for new subjects. With this quiz, you can test your knowledge of imaginary numbers. Plus model problems explained step by step Worksheet will open in a new window. ... Identifying Real And Imaginary Part Complex Numbers Number Worksheets Algebra . 1 0 obj<> A complex number can be divided into two parts, the imaginary part, and the real part. In the complex number {eq}z = a + bi {/eq}, a is the real part, and b is the imaginary part. %PDF-1.3 stream x�c`� 13 0 obj<> You can & download or print using the browser document reader options. 28 Operations With Complex Numbers Worksheet In 2020 Motivational Interviewing Word Problem Worksheets Number Worksheets . 45 question end of unit review sheet on adding, subtracting, mulitplying, dividing and simplify complex and imaginary numbers. Although arbitrary, there is also some sense of a positive and negative imaginary numbers. %�� stream Find an answer to your question “Check my answer? Answer with 5 significant digits. This frees you up to go around and tuto stream View worksheet. There are scrambled answers at the bottom so students can check their work as they go through this worksheet. Free worksheet(pdf) and answer key on Simplifying Imaginary numbers (radicals) and powers of i. B) Form the complex numbers with the given real parts and imaginary parts. x�� 1 Answers for math worksheets, quiz, homework, and lessons. Actually, imaginary numbers are used quite frequently in engineering and physics, such as an alternating current in electrical engineering, whic… ��9��8E̵㻶�� mT˞�I�іT�R;hv��Or�L{wz�Q!�f�e��`�% A�� ժp�K�g��/D��=8�a�X3��[� \|:��I� �MrXB��\��#L)�fΡbè��r�쪬ى��eW��\|x��!c��9P+��d3�g ��d��R��U^ g?:�@�6H�Yx�z�8Nī~J�u�S]��T��캶�Q�&�u��Uu�S7��T��WA+��H�!�! For example, the real number 5 is also a complex number because it can be written as 5 + 0 i with a real part of 5 and an imaginary part of 0. Think of imaginary numbers as numbers that are typically used in mathematical computations to get to/from “real” numbers (because they are more easily used in advanced computations), but really don’t exist in life as we know it. Learn more about complex number, real part, imaginary part, matlab I expand $\sin(x+iy)=\ sin x \cosh y+i \cos x \sinh y$. ... Answer Key. 16 0 obj<> endobj If we have 0 + bi we have a pure imaginary number. Real and Imaginary part of this function please. Yet they are real in the sense that they do exist and can be explained quite easily in terms of math as the square root of a negative number. For my system of equations, the procedure described in Solving complex equations of using Reduce works no more. 20 0 obj<> Some of the worksheets for this concept are Operations with complex numbers, Complex numbers and powers of i, Dividing complex numbers, Adding and subtracting complex numbers, Real part and imaginary part 1 a complete the, Complex numbers, Complex numbers, Properties of complex numbers. Given the setup, calculate the magnitude of the equivalent source voltage (Vs,eq) seen by Zm? endstream 5 (c) The real part of is and the imaginary part is I. (b) The real part of (8 + 8i)(8 - 81) is 128 and the imaginary part is o (Type integers or simplified fractions.) This one-page worksheet contains 12 multi-step problems. endobj endobj Imaginary And Complex Numbers - Displaying top 8 worksheets found for this concept. (d) The real part of 1-si is and the imaginary part is (Type integers or simplified fractions.) Free worksheet pdf and answer key on simplifying imaginary numbers radicals and powers of i. endobj If we have a + bi a != 0, b != 0 then we have a complex number with a real part and an imaginary part. endobj Since the real part, the imaginary part, and the indeterminate i in a complex number are all considered as numbers in themselves, two complex numbers, given as z = x + yi and w = u + vi are multiplied under the rules of the distributive property, the commutative properties and the defining property i 2 = … endstream Because then I could use Solve[equations, vars, Reals].Nevertheless I hope for a simpler way to overcome this issue. This Worksheet Section 1.5 Analysis: Real and Imaginary Numbers Worksheet is suitable for 10th - 12th Grade. Some of the worksheets for this concept are Operations with complex numbers, Complex numbers and powers of i, Dividing complex numbers, Adding and subtracting complex numbers, Real part and imaginary part 1 a complete the, Complex numbers, Complex numbers, Properties of complex numbers. endobj Answer with 5 significant digits. (Part (2+3 i) + (-4+5i — —49 ((5+14i) -(10- -25 (5+4i)- (-1-21 -3i (-5i ) 2i (3i2) C-2)C3) 3i(2i) 64 Start Here 2(3+2i) 2i-(3+2j 3i (2+3i) 18 3 0 obj<> endobj what are the real and imaginary parts of the complex number? real and imaginary part of complex number . Solution for i) Find real and imaginary part of complex number (V3 + i)° by using De Moivre's Theorem. Perform operations like addition, subtraction and multiplication on complex numbers, write the complex numbers in standard form, identify the real and imaginary parts, find the conjugate, graph complex numbers, rationalize the denominator, find the absolute value, modulus, and argument in this collection of printable complex number worksheets. Output: Real part: 6, Imaginary part: 8 Recommended: Please try your approach on first, before moving on to the solution. Found worksheet you are looking for? 14 0 obj<> The general definition is a + bi Where a and b are real numbers and i is the imaginary unit i = sqrt(-1) If we have a + 0i we have a real number. 4- i (Type integers or simplified fractions.) In this real and imaginary numbers worksheet, students solve algebraic expressions containing real and imaginary numbers. Real and imaginary part of (x+iy)e^{ix−y} ? Any number that is written with ‘iota’ is an imaginary number, these are negative numbers in a radical. 15 0 obj<> Real (Part 81 25 (6+2 (1-2i) End Here Complete the maze by simplifying each expression, shade the squares that Imaginary contain imaginary numbers, and following the path of complex numbers. Model Problems In this example we will simplifying imaginary numbers. or what is the real and imaginary part of f(t) if you can get it in another form but it should be simplified as the above answers please show me your way thank you endobj 2 0 obj<> Let’s explore this topic with our easy-to-use complex number worksheets that are tailor-made for students in high school and is the perfect resource to introduce this new concept. Imaginary numbers of the form bi are numbers that when squared result in a negative number. Some examples are 3 +4i, 2— 5i, —6 +0i, 0— i. We can also graph these numbers. Dec 13, 2018 - Complex number worksheets feature standard form, identifying real and imaginary part, rationalize the denominator, graphing, conjugate, modulus and more! 19 0 obj<> How can I separate the real and imaginary part of the equations? Consider the complex number 4 + 3i: 4 is called the real part, 3 is called the imaginary part. So it makes sense to say, for example $1 -100i$ is positive and $-1 + 100i$ is negative, based upon their real number values. Learn more about simplify, complex function, real and imaginary parts … Note: 3i is not the imaginary part. A complex number has two parts, a real part and an imaginary part. Download/Print, click on pop-out icon or print using the browser document reader options 4 is called the part... Download/Print, click on pop-out icon or print using the browser document reader options De Moivre 's Theorem is. Is real part of 1-si is and the imaginary part, a, an! 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Randomized algorithm A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. One has to distinguish between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort^[1]), and algorithms which have a chance of producing an incorrect result (Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem^[2]) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem.^[3] In common practice, randomized algorithms are approximated using a pseudorandom number generator in place of a true source of random bits; such an implementation may deviate from the expected theoretical behavior. As a motivating example, consider the problem of finding an ‘a’ in an array of n elements. Input: An array of n≥2 elements, in which half are ‘a’s and the other half are ‘b’s. Output: Find an ‘a’ in the array. We give two versions of the algorithm, one Las Vegas algorithm and one Monte Carlo algorithm. Las Vegas algorithm: findingA_LV(array A, n) Randomly select one element out of n elements. until 'a' is found This algorithm succeeds with probability 1. The number of iterations varies and can be arbitrarily large, but the expected number of iterations is ${\displaystyle \lim _{n\to \infty }\sum _{i=1}^{n}{\frac {i}{2^{i}}}=2}$ Since it is constant the expected run time over many calls is ${\displaystyle \Theta (1)}$. (See Big O notation) Monte Carlo algorithm: findingA_MC(array A, n, k) Randomly select one element out of n elements. i = i + 1 until i=k or 'a' is found If an ‘a’ is found, the algorithm succeeds, else the algorithm fails. After k iterations, the probability of finding an ‘a’ is: ${\displaystyle \Pr[\mathrm {find~a} ]=1-(1/2)^{k}}$ This algorithm does not guarantee success, but the run time is bounded. The number of iterations is always less than or equal to k. Taking k to be constant the run time (expected and absolute) is ${\ displaystyle \Theta (1)}$. Randomized algorithms are particularly useful when faced with a malicious "adversary" or attacker who deliberately tries to feed a bad input to the algorithm (see worst-case complexity and competitive analysis (online algorithm)) such as in the Prisoner's dilemma. It is for this reason that randomness is ubiquitous in cryptography. In cryptographic applications, pseudo-random numbers cannot be used, since the adversary can predict them, making the algorithm effectively deterministic. Therefore, either a source of truly random numbers or a cryptographically secure pseudo-random number generator is required. Another area in which randomness is inherent is quantum computing. In the example above, the Las Vegas algorithm always outputs the correct answer, but its running time is a random variable. The Monte Carlo algorithm (related to the Monte Carlo method for simulation) is guaranteed to complete in an amount of time that can be bounded by a function the input size and its parameter k, but allows a small probability of error. Observe that any Las Vegas algorithm can be converted into a Monte Carlo algorithm (via Markov's inequality), by having it output an arbitrary, possibly incorrect answer if it fails to complete within a specified time. Conversely, if an efficient verification procedure exists to check whether an answer is correct, then a Monte Carlo algorithm can be converted into a Las Vegas algorithm by running the Monte Carlo algorithm repeatedly till a correct answer is obtained. Computational complexity Computational complexity theory models randomized algorithms as probabilistic Turing machines. Both Las Vegas and Monte Carlo algorithms are considered, and several complexity classes are studied. The most basic randomized complexity class is RP, which is the class of decision problems for which there is an efficient (polynomial time) randomized algorithm (or probabilistic Turing machine) which recognizes NO-instances with absolute certainty and recognizes YES-instances with a probability of at least 1/2. The complement class for RP is co-RP. Problem classes having (possibly nonterminating) algorithms with polynomial time average case running time whose output is always correct are said to be in ZPP. The class of problems for which both YES and NO-instances are allowed to be identified with some error is called BPP. This class acts as the randomized equivalent of P, i.e. BPP represents the class of efficient randomized algorithms. Historically, the first randomized algorithm was a method developed by Michael O. Rabin for the closest pair problem in computational geometry.^[4] The study of randomized algorithms was spurred by the 1977 discovery of a randomized primality test (i.e., determining the primality of a number) by Robert M. Solovay and Volker Strassen. Soon afterwards Michael O. Rabin demonstrated that the 1976 Miller's primality test can be turned into a randomized algorithm. At that time, no practical deterministic algorithm for primality was known. The Miller–Rabin primality test relies on a binary relation between two positive integers k and n that can be expressed by saying that k "is a witness to the compositeness of" n. It can be shown that • If there is a witness to the compositeness of n, then n is composite (i.e., n is not prime), and • If n is composite then at least three-fourths of the natural numbers less than n are witnesses to its compositeness, and • There is a fast algorithm that, given k and n, ascertains whether k is a witness to the compositeness of n. Observe that this implies that the primality problem is in Co-RP. If one randomly chooses 100 numbers less than a composite number n, then the probability of failing to find such a "witness" is (1/4)^100 so that for most practical purposes, this is a good primality test. If n is big, there may be no other test that is practical. The probability of error can be reduced to an arbitrary degree by performing enough independent tests. Therefore, in practice, there is no penalty associated with accepting a small probability of error, since with a little care the probability of error can be made astronomically small. Indeed, even though a deterministic polynomial-time primality test has since been found (see AKS primality test), it has not replaced the older probabilistic tests in cryptographic software nor is it expected to do so for the foreseeable future. Quicksort is a familiar, commonly used algorithm in which randomness can be useful. Many deterministic versions of this algorithm require O(n^2) time to sort n numbers for some well-defined class of degenerate inputs (such as an already sorted array), with the specific class of inputs that generate this behavior defined by the protocol for pivot selection. However, if the algorithm selects pivot elements uniformly at random, it has a provably high probability of finishing in O(n log n) time regardless of the characteristics of the input. Randomized incremental constructions in geometry In computational geometry, a standard technique to build a structure like a convex hull or Delaunay triangulation is to randomly permute the input points and then insert them one by one into the existing structure. The randomization ensures that the expected number of changes to the structure caused by an insertion is small, and so the expected running time of the algorithm can be bounded from above. This technique is known as randomized incremental construction.^[5] Min cut Input: A graph G(V,E) Output: A cut partitioning the vertices into L and R, with the minimum number of edges between L and R. Recall that the contraction of two nodes, u and v, in a (multi-)graph yields a new node u ' with edges that are the union of the edges incident on either u or v, except from any edge(s) connecting u and v. Figure 1 gives an example of contraction of vertex A and B. After contraction, the resulting graph may have parallel edges, but contains no self loops. Karger's^[6] basic algorithm: i = 1 Take a random edge (u,v) ∈ E in G replace u and v with the contraction u' until only 2 nodes remain obtain the corresponding cut result C[i] i = i + 1 until i = m output the minimum cut among C[1], C[2], ..., C[m]. In each execution of the outer loop, the algorithm repeats the inner loop until only 2 nodes remain, the corresponding cut is obtained. The run time of one execution is ${\displaystyle O(n)}$, and n denotes the number of vertices. After m times executions of the outer loop, we output the minimum cut among all the results. The figure 2 gives an example of one execution of the algorithm. After execution, we get a cut of size 3. Lemma 1: Let k be the min cut size, and let C = {e[1], e[2], ..., e[k]} be the min cut. If, during iteration i, no edge e ∈ C is selected for contraction, then C[i] = C. Proof: If G is not connected, then G can be partitioned into L and R without any edge between them. So the min cut in a disconnected graph is 0. Now, assume G is connected. Let V=L∪R be the partition of V induced by C : C = { {u,v} ∈ E : u ∈ L,v ∈ R } (well-defined since G is connected). Consider an edge {u,v} of C. Initially, u,v are distinct vertices. As long as we pick an edge ${\displaystyle feq e}$, u and v do not get merged. Thus, at the end of the algorithm, we have two compound nodes covering the entire graph, one consisting of the vertices of L and the other consisting of the vertices of R. As in figure 2, the size of min cut is 1, and C = {(A,B)}. If we don't select (A,B) for contraction, we can get the min cut. Lemma 2: If G is a multigraph with p vertices and whose min cut has size k, then G has at least pk/2 edges. Proof: Because the min cut is k, every vertex v must satisfy degree(v) ≥ k. Therefore, the sum of the degree is at least pk. But it is well known that the sum of vertex degrees equals 2\|E\|. The lemma Analysis of algorithm The probability that the algorithm succeeds is 1 − the probability that all attempts fail. By independence, the probability that all attempts fail is By lemma 1, the probability that C[i] = C is the probability that no edge of C is selected during iteration i. Consider the inner loop and let G[j] denote the graph after j edge contractions, where j ∈ {0, 1, …, n − 3}. G[j] has n − j vertices. We use the chain rule of conditional possibilities. The probability that the edge chosen at iteration j is not in C, given that no edge of C has been chosen before, is ${\displaystyle 1-{\frac {k}{\|E(G_{j})\|}}}$. Note that G[j] still has min cut of size k, so by Lemma 2, it still has at least ${\displaystyle {\frac {(n-j)k}{2}}}$ edges. Thus, ${\displaystyle 1-{\frac {k}{\|E(G_{j})\|}}\geq 1-{\frac {2}{n-j}}={\frac {n-j-2}{n-j}}}$. So by the chain rule, the probability of finding the min cut C is Cancellation gives ${\displaystyle \Pr[C_{i}=C]\geq {\frac {2}{n(n-1)}}}$. Thus the probability that the algorithm succeeds is at least ${\displaystyle 1-\left(1-{\frac {2}{n(n-1)}}\right)^{m}}$. For ${\displaystyle m={\frac {n(n-1)}{2}}\ln n}$, this is equivalent to ${\displaystyle 1-{\frac {1}{n}}}$. The algorithm finds the min cut with probability ${\displaystyle 1-{\frac {1}{n}}}$, in time $ {\displaystyle O(mn)=O(n^{3}\log n)}$. Randomness can be viewed as a resource, like space and time. Derandomization is then the process of removing randomness (or using as little of it as possible). It is not currently known if all algorithms can be derandomized without significantly increasing their running time. For instance, in computational complexity, it is unknown whether P = BPP, i.e., we do not know whether we can take an arbitrary randomized algorithm that runs in polynomial time with a small error probability and derandomize it to run in polynomial time without using randomness. There are specific methods that can be employed to derandomize particular randomized algorithms: • the method of conditional probabilities, and its generalization, pessimistic estimators • discrepancy theory (which is used to derandomize geometric algorithms) • the exploitation of limited independence in the random variables used by the algorithm, such as the pairwise independence used in universal hashing • the use of expander graphs (or dispersers in general) to amplify a limited amount of initial randomness (this last approach is also referred to as generating pseudorandom bits from a random source, and leads to the related topic of pseudorandomness) Where randomness helps When the model of computation is restricted to Turing machines, it is currently an open question whether the ability to make random choices allows some problems to be solved in polynomial time that cannot be solved in polynomial time without this ability; this is the question of whether P = BPP. However, in other contexts, there are specific examples of problems where randomization yields strict improvements. • Based on the initial motivating example: given an exponentially long string of 2^k characters, half a's and half b's, a random-access machine requires 2^k−1 lookups in the worst-case to find the index of an a; if it is permitted to make random choices, it can solve this problem in an expected polynomial number of lookups. • The natural way of carrying out a numerical computation in embedded systems or cyber-physical systems is to provide a result that approximates the correct one with high probability (or Probably Approximately Correct Computation (PACC)). The hard problem associated with the evaluation of the discrepancy loss between the approximated and the correct computation can be effectively addressed by resorting to randomization^[7] • In communication complexity, the equality of two strings can be verified to some reliability using ${\displaystyle \log n}$ bits of communication with a randomized protocol. Any deterministic protocol requires ${\displaystyle \Theta (n)}$ bits if defending against a strong opponent.^[8] • The volume of a convex body can be estimated by a randomized algorithm to arbitrary precision in polynomial time.^[9]Bárány and Füredi showed that no deterministic algorithm can do the same.^[10] This is true unconditionally, i.e. without relying on any complexity-theoretic assumptions, assuming the convex body can be queried only as a black box. • A more complexity-theoretic example of a place where randomness appears to help is the class IP. IP consists of all languages that can be accepted (with high probability) by a polynomially long interaction between an all-powerful prover and a verifier that implements a BPP algorithm. IP = PSPACE.^[11] However, if it is required that the verifier be deterministic, then IP = NP. • In a chemical reaction network (a finite set of reactions like A+B → 2C + D operating on a finite number of molecules), the ability to ever reach a given target state from an initial state is decidable, while even approximating the probability of ever reaching a given target state (using the standard concentration-based probability for which reaction will occur next) is undecidable. More specifically, a limited Turing machine can be simulated with arbitrarily high probability of running correctly for all time, only if a random chemical reaction network is used. With a simple nondeterministic chemical reaction network (any possible reaction can happen next), the computational power is limited to primitive recursive functions.^[12] See also • Probabilistic analysis of algorithms • Atlantic City algorithm • Bogosort • Principle of deferred decision • Randomized algorithms as zero-sum games • Probabilistic roadmap • HyperLogLog • count–min sketch • approximate counting algorithm 1. ^ Hoare, C. A. R. (July 1961). "Algorithm 64: Quicksort". Commun. ACM. 4 (7): 321–. doi:10.1145/366622.366644. ISSN 0001-0782. 2. ^ Kudelić, Robert (2016-04-01). "Monte-Carlo randomized algorithm for minimal feedback arc set problem". Applied Soft Computing. 41: 235–246. doi:10.1016/j.asoc.2015.12.018. 3. ^ "In testing primality of very large numbers chosen at random, the chance of stumbling upon a value that fools the Fermat test is less than the chance that cosmic radiation will cause the computer to make an error in carrying out a 'correct' algorithm. Considering an algorithm to be inadequate for the first reason but not for the second illustrates the difference between mathematics and engineering." Hal Abelson and Gerald J. Sussman (1996). Structure and Interpretation of Computer Programs. MIT Press, section 1.2. 4. ^ Smid, Michiel. Closest point problems in computational geometry. Max-Planck-Institut für Informatik\|year=1995 5. ^ A. A. Tsay, W. S. Lovejoy, David R. Karger, Random Sampling in Cut, Flow, and Network Design Problems, Mathematics of Operations Research, 24(2):383–413, 1999. 6. ^ Alippi, Cesare (2014), Intelligence for Embedded Systems, Springer, ISBN 978-3-319-05278-6. 7. ^ Kushilevitz, Eyal; Nisan, Noam (2006), Communication Complexity, Cambridge University Press, ISBN 9780521029834. For the deterministic lower bound see p. 11; for the logarithmic randomized upper bound see pp. 31–32. 8. ^ Dyer, M.; Frieze, A.; Kannan, R. (1991), "A random polynomial-time algorithm for approximating the volume of convex bodies" (PDF), Journal of the ACM, 38 (1): 1–17, doi:10.1145/102782.102783 9. ^ Füredi, Z.; Bárány, I. (1986), "Computing the volume is difficult", Proc. 18th ACM Symposium on Theory of Computing (Berkeley, California, May 28–30, 1986) (PDF), New York, NY: ACM, pp. 442–447, CiteSeerX , doi:10.1145/12130.12176, ISBN 0-89791-193-8 10. ^ Shamir, A. (1992), "IP = PSPACE", Journal of the ACM, 39 (4): 869–877, doi:10.1145/146585.146609 11. ^ Cook, Matthew; Soloveichik, David; Winfree, Erik; Bruck, Jehoshua (2009), "Programmability of chemical reaction networks", in Condon, Anne; Harel, David; Kok, Joost N.; Salomaa, Arto; Winfree, Erik (eds.), Algorithmic Bioprocesses (PDF), Natural Computing Series, Springer-Verlag, pp. 543–584, doi:10.1007/978-3-540-88869-7_27. • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw–Hill, 1990. ISBN 0-262-03293-7. Chapter 5: Probabilistic Analysis and Randomized Algorithms, pp. 91–122. • Dirk Draheim. "Semantics of the Probabilistic Typed Lambda Calculus (Markov Chain Semantics, Termination Behavior, and Denotational Semantics)." Springer, 2017. • Jon Kleinberg and Éva Tardos. Algorithm Design. Chapter 13: "Randomized algorithms". • Fallis, D. (2000). "The reliability of randomized algorithms". The British Journal for the Philosophy of Science. 51 (2): 255–271. doi:10.1093/bjps/51.2.255. • M. Mitzenmacher and E. Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, New York (NY), 2005. • Rajeev Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, New York (NY), 1995. • Rajeev Motwani and P. Raghavan. Randomized Algorithms. A survey on Randomized Algorithms. • Christos Papadimitriou (1993), Computational Complexity (1st ed.), Addison Wesley, ISBN 978-0-201-53082-7 Chapter 11: Randomized computation, pp. 241–278. • Rabin, Michael O. (1980). "Probabilistic algorithm for testing primality". Journal of Number Theory. 12: 128–138. doi:. • A. A. Tsay, W. S. Lovejoy, David R. Karger, Random Sampling in Cut, Flow, and Network Design Problems, Mathematics of Operations Research, 24(2):383–413, 1999.	{"url":"https://codedocs.org/what-is/randomized-algorithm","timestamp":"2024-11-08T15:42:27Z","content_type":"text/html","content_length":"82351","record_id":"<urn:uuid:ccc7a5e9-4788-4038-89f0-d8c534bdac9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00082.warc.gz"}
First Grade Fraction Worksheets 📆 Updated: 20 Aug 2024 🔖 Category: 1st Grade First-grade Fraction Worksheets provide a comprehensive and engaging way for young learners to develop a solid understanding of fractions. Designed specifically for first-grade students, these worksheets include various exercises and activities that introduce the concept of fractions in a clear and concise manner. With good visuals and age-appropriate content, these worksheets make learning about fractions enjoyable and accessible for first graders. By using these worksheets, 1st grade students will understand about main concepts of fractions such as fraction calculation, simplifying fractions, and equal fractions or equivalent fractions. Whether it's identifying parts of a whole or comparing fractions, these worksheets offer an effective tool to help young learners develop their understanding of fractions. Enhancing early math skills can be fun with our first grade fraction worksheets, providing engaging ways to teach fractions. Explore the mathematical world with these First Grade Fraction Worksheets! What is Fraction in Math? Fraction is one of the elements that students need to understand while learning math. It is a rational number in the form of in-line or ancient notation (a/b). The number a at the beginning (or top) is a numerator. Meanwhile, the b (or bottom) is a denominator. According to the GCF (Goodwill Community Foundation) Global, the definition of fraction is a part of something whole and complete. It is less than the entire one thing; but more than zero (0). Fraction might look intimidating. However, we use this mathematical element daily. Easy visualization of fractions would be using a pizza. If you cut an entire pizza into eight (8) pieces and took one (1) slice, it would be 1/8 of the pizza. When the number of numerator and denominator are the same, it means the whole, or the same as one (1) (e.g. 3/3, 7/7). What are The Types of Fraction? According to the Central Bucks School District, there are three types of fractions in mathematics, proper, improper, and mixed. • A proper fraction is a fraction with a numerator more than the denominator (e.g. 3/7, 6/10). • An Improper fraction is a fraction with the numerator lesser than the denominator (e,g. 6/3, 8/2). • An amixed fraction is a fraction that consists of one whole number and a proper fraction (e.g. 1 4/6, 7 5/9). The mixed reaction is always more than number one (1). We could change the improper to be a mixed fraction with the following steps: 1. Break the numerator with the denominator. 2. Take note of the complete number. 3. Write any residue number on the top of the denominator. We could also convert the opposites (mixed fractions to improper fractions) with these steps: 1. Multiply the whole number by the denominator number. 2. Add the result to the numerator. 3. Write the final answer above the denominator. Why Should Students Learn Fractions? As one of the field topics in mathematics, fractions are essential subjects that students should master. Fraction is the basis of a more complicated mathematical formula, such as algebra. According to Francis Fennel from the National Council of Teachers of Mathematics, many teachers wish their students to be skillful in fractions before the algebra class. Hence, the teacher needs to build a solid fraction base. The ability to work with fractions is the essential foundation for students to learn more breakthrough mathematic topics. Francis said that fractions introduce abstract concepts to the students in the sense of mathematics. How to Apply Fractions in The Real Life? "Why do we learn this topic, though? We will never use this in real life." These types of responses might not be strange for mathematics teachers. It is not a surprise many students are struggling with mathematics, which is understandable. For some of those students, mathematics is abstract knowledge. The teachers should help students notice how close mathematics in human life. Almost everything around us is related to math. Below are the real-life applications of fractions: 1. Splitting the bill with friends. 2. Divide meals with siblings. 3. Calculate the price of discounted stuff. 4. Recreate a food recipe. 5. Manage to take a well-balanced picture with a camera. 6. Count the time using a clock. 7. A Doctor makes prescriptions for medicine. What are The Challenges in Teaching Fraction? Every teacher might meet with challenges in their classroom. These disputes could come from the students, topic, strategy, mediums, and teacher themselves. In learning fractions, the students might struggle because they are not familiar with the form of the portion and a bunch of strange terms such as numerators, denominators, proper, improper, and more. The teachers should find the roots of the problems and find the solution. The solution could be a change in learning strategy and medium, a four-eye discussion with students, and a talk with other mathematics teachers. How to Teach Fraction? Learning about fractions might be challenging for some students. Hence, the teachers should make a proper and suitable learning strategy to help the students understand the topic. These are some tips to teach fractions to students. • Some education researchers recommend that teachers use real-life objects familiar to the students in teaching fractions. Use solid items such as stationery, snacks, toys, and more. • You can ask the children to draw pictures of the item on the board to involve students in learning. • When the students grasp the concept of fractions, the teacher can continue learning with rational numbers. • In teaching fractions, the teacher should remember it takes a long time for the students to master this. • Let the students absorb the concept and know all the elements. • The teachers should find many teaching mediums to help the students understand fractions. The teacher can use visual helpers such as posters, flashcards, videos, or these First Grade Fraction Worksheets to aid the students. What is included in the First Grade Fraction Worksheet? First Grade Fraction Worksheet is a worksheet designed for 1st grade students to help them understand about the main concept of fractions. These worksheets provide various simple shapes to learn An example is a square divided into four parts. So, students will understand how to write and read a fraction using that shape. Teachers and parents can use First Grade Fraction Worksheets as a learning tool to teach fractions to 1st grade students in a fun way. These worksheets are easy to use. Teachers and parents just need to download and print our worksheets. Then, introduce and explain it to children. Fraction is one of basic math concepts that should be learned by students, especially 1st grade students. Because fraction is important lesson, teachers and parents can use our First Grade Fraction Worksheets as the best tool to teach fraction to 1st grade students. The ability to work with fractions (fraction addition, fraction subtraction, fraction multiplication, and fraction division) is the essential foundation for students to learn more breakthrough mathematic topics, such as algebra. Some of informations, names, images and video detail mentioned are the property of their respective owners & source. Who is Worksheeto? At Worksheeto, we are committed to delivering an extensive and varied portfolio of superior quality worksheets, designed to address the educational demands of students, educators, and parents. Popular Categories	{"url":"https://www.worksheeto.com/post_first-grade-fraction-worksheets_144692/","timestamp":"2024-11-14T06:50:17Z","content_type":"text/html","content_length":"79610","record_id":"<urn:uuid:bda17161-8aa2-42d7-9b71-5a1bb0340869>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00784.warc.gz"}
Python practice Remove K digits Given a non-negative integer represented as a string num and an integer k, remove k digits from the number so that the new number is the smallest possible. The remaining digits should maintain their original order in the string. Return the result as a string. Input Format: • The first line contains the string num, representing the non-negative integer. • The second line contains the integer k. Output Format: • Print the smallest possible integer as a string after removing k digits. Stack based solution: Coach Manish Manish is the coach of a high school football team. He has come up with a training regime of difficulty ‘d’ but he is afraid that it might result in his team getting exhausted quickly and prone to So he has decided to break the training in ’n’ number of days where every day has a certain amount of difficulty and the difficulty of each day adds up to ‘d’ the original difficulty. But he also wants to make sure that his training results in a greater output so he cannot spread the difficulty of training without planning. The result can be determined by calculating the GCD of the difficulty of all the days. Return the greatest result that can be achieved from the training. Input Format: You are given two space-separated integers d and n, denoting the total difficulty and number of days it is divided into. Output Format: You have to return a single integer denoting the maximum result that can be achieved from the training. Sample Input: Sample Output: We can break the difficulty in 3 and 6 whose GCD results in 3 which is the greatest output possible. GCD based solution: Duplicate Letters Dholu is very creative and wants to play with strings and he has a very good idea. You have given a string s, remove duplicate letters so that every letter appears once and only once. You must make sure your result is the smallest in lexicographical order Input Format: • first line consists of a string Output Format: Sample Input: Sample Output: Explanation: Since if we only collect unique letters, they come out to be ‘c’, ‘b’, ‘a’ and ‘d’ and when they are arranged lexicographically the output is abcd. 1<= s.length<= 10000	{"url":"https://jyos-sw.medium.com/python-practice-7d00a01b8db2?source=user_profile_page---------0-------------c6b4914c32ee---------------","timestamp":"2024-11-09T03:55:13Z","content_type":"text/html","content_length":"109350","record_id":"<urn:uuid:e0676a27-a314-4756-98f4-677389648230>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00197.warc.gz"}
[...]: The values in the data set can be measured. [...]: The values in the data set can be measured. [...]: The values in the data set can be measured. If you want to change selection, open document below and click on "Move attachment" Subject 3. Frequency Distributions ed into two types: discrete and continuous. Discrete: The values in the data set can be counted. There are distinct spaces between the values, such as the number of children in a family or the number of shares comprising an index. <span> Continuous: The values in the data set can be measured. There are normally lots of decimal places involved and (theoretically, at least) there are no gaps between permissible values (i.e., all values can be included in the data set). Example status not learned measured difficulty 37% [default] last interval [days] repetition number in this series 0 memorised on scheduled repetition scheduled repetition interval last repetition or drill No repetitions	{"url":"https://buboflash.eu/bubo5/show-dao2?d=1636464987404","timestamp":"2024-11-05T06:19:05Z","content_type":"text/html","content_length":"36282","record_id":"<urn:uuid:ef4c1950-3810-4e4b-8a8b-0890d7bc25ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00611.warc.gz"}
One-Variable Equations (Employment Themed) Worksheets Download One-Variable Equations (Employment Themed) Worksheets Click the button below to get instant access to these premium worksheets for use in the classroom or at a home. This worksheet can be edited by Premium members using the free Google Slides online software. Click the Edit button above to get started. • In algebra, an equation can be defined as a mathematical sentence composed of an equal symbol between two algebraic expressions that have the same value. • One-variable equation is a mathematical statement that is consist of a variable and whose left-hand side is exactly the same as what’s on the right-hand side. It is made up of two expressions connected by an equal sign. Equation represents balance. With equal sign 3x = 2x + x Equation Expression It is made up of two expressions connected by an equal sign. Equation represents balance. It may be a number, a variable, or a combination of numbers and variables and operation symbols. With equal sign Without equal sign 3x = 2x + x 3x, 2x, x Examples of One-Variable Equation Non-examples of One-Variable Equations 2x = 4 -m = 2½ r -13 = 55(w – 7) = 26 2x – 3r11c – 525m – 4n + 7100 Here are the commonly used properties of equality when solving one-variable equations. If a, b, and c are real numbers, then… Reflexive Property Symmetric Property Transitive Property a = a If a = b , then b=a If a = b and b = c, then a = c Distributive Property Addition Property of Equality (APE) Multiplication Property of Equality (MPE) a(b+c) = ab + ac If a = b, then a + c = b + c If a = b, then a x c = b x c Please take note of following steps in solving one-variable equations. 1. Simplify both sides of the equation. 2. Use the addition property of equality to put the terms with variable on one side of the equation and the constant terms on the other. 3. Use the multiplication property of equality to make the coefficient of the variable term equal to 1. 4. Check your answer by substituting your solution into the original equation. One-Variable Equations (Employment Themed) Worksheets This is a fantastic bundle which includes everything you need to know about One-Variable Equations across 21 in-depth pages. Each ready to use worksheet collection includes 10 activities and an answer guide. Not teaching common core standards? Don’t worry! All our worksheets are completely editable so can be tailored for your curriculum and target audience. Resource Examples Click any of the example images below to view a larger version. Worksheets Activities Included Ages 10-11 (Basic) • The Curriculum Vitae Problem • Job Interview Day • On-the-Job Training • Day 1 of Work • My Day: Works and Math Ages 11-12 (Advanced) • My First Salary • My Productivity Rate • Did My Co-Worker Do it Right? • Yes! It’s a Double-Pay Day • It’s Weekend Rest Day!	{"url":"https://helpingwithmath.com/worksheet/one-variable-equations-employment-themed-worksheets/","timestamp":"2024-11-09T04:34:02Z","content_type":"text/html","content_length":"147792","record_id":"<urn:uuid:570b4313-c8a7-4e84-909d-b7345c77570e>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00719.warc.gz"}
Cycle Benchmarking (CB) Cycle Benchmarking (CB) See make_cb() for API documentation. CB is a fully scalable protocol for assessing the performance of a specified clock cycle containing any combination of single gates, parallelized gates, and idle qubits, across an entire $n$-qubit device [9, 10]. The performance is measured in one of two ways: the first and most common is the total probability of error of the cycle ($e_F$), and the second is the probability of some specified Pauli error ocurring during the cycle ($e_P$ for some Pauli $P$). Examples of terms that may be returned from the analysis are as follows. These descriptions are also available via mouse-overs when running True-Q™ in a Jupyter or Colab notebook. Estimated Parameters ${e}_{F}$ - The probability of any error occurring during a dressed cycle, where a dressed cycle is the cycle of interest composed with a random cycle of local gates. This is also known as the process infidelity of the dressed cycle. By default, the process infidelity is reported for the entire cycle and includes, for example, errors that occur on qubits that are idle during the cycle of interest. However this quantity can optionally be computed for any subset of qubits, corresponding to a marginal process infidelity. If $\mathcal{C}$ is superoperator of the cycle of interest and $\mathcal{T}$ is the superoperator of a twirling operation, then the process infidelity of the dressed cycle is defined as \[e_F = 1 - \mathbb{E}\left[ \operatorname{Tr} \left( \mathcal{T}^\dagger \mathcal{C}^\dagger \tilde{\mathcal{C}} \tilde{\mathcal{T}} \right)\right]\] where a tilde denotes the noisy implementation of a superoperator, and the expectation value is taken over randomizations of the twirling group. Note that if the twirling group is perfectly implemented, then we have the simpler formula \[e_F = 1 - \frac{\operatorname{Tr}(\mathcal{C}^\dagger \tilde{\mathcal{C}})}{d^2}\] which is the process infidelity of the cycle of interest alone. The process infidelity is closely related to the process fidelity $F_E$ (also known as the entanglement fidelity), and the average gate infidelity $r = 1 - \int d\psi \operatorname{Tr}(\psi, E (\psi))$, via the formula $e_F = 1 - F_E = (1 + 1/d) r$. Here, $d$ is the total dimension of the system across all qubits. Note that the process fidelity is stable under tensor products, whereas the average gate infidelity must be converted to the process fidelity to estimate the fidelity of a tensor product of processes. Because this protocol uses the same twirling group as randomized compiling (RC), estimates of this parameter accurately predict the in-situ characteristics of this cycle when present in a circuit that is run using RC. ${e}_{ZXZX}$ - The marginal error rate of the subscripted Pauli error occuring on the qubits in question. In other words, the sum of the probabilities of all global errors that act as the subscripted error on the specified qubit labels. For example, if an error rate $e_X=0.1$ is measured for the qubit label (0, ), then \[\sum_Q e_{X \otimes Q} = 0.1\] where $Q$ is summed over all Paulis acting on different qubits. Examples: $e_{X}$, $e_{XYYI}$, $e_{ZZ}$, $e_{XYYI}$, $e_{ZZ}$. ${A}_{YZXX}$ - SPAM parameter of the exponential decay $Ap^m$ for the given Pauli term. Examples: $A_X$, $A_{XYYI}$, $A_{ZZ}$, describing the $A$ parameter for each of the respective Paulis. ${p}_{YZXX}$ - Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. Examples: $p_X$, $p_{XYYI}$, $p_{ZZ}$, describing the $p$ parameter for each of the respective Paulis.	{"url":"https://trueq.quantumbenchmark.com/guides/error_diagnostics/cb.html","timestamp":"2024-11-11T04:34:25Z","content_type":"text/html","content_length":"29274","record_id":"<urn:uuid:fec2f950-66d0-4835-b50e-b61e45710566>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00779.warc.gz"}
sfp 1.0.0 Software floating point package Software floating point package with decimal number base Dr. Heinrich Hohl <hohl@isartext.de> Version 1.0.0 - 2016-10-03 This package provides a compact software floating point package. The floating point numbers used in this package are represented by a composite single length number. 24 bits are used for the significand, and the top 8 bits for the exponent. Both numbers are based on the decimal system. The significand must be normalized in order to obtain a valid floating point number. It contains an implied decimal point five digits from the right. Valid significands are: Negative: -9.99999 ... -1.00000 Zero: 0.00000 Positive: +1.00000 ... +9.99999 This means that floating point numbers have a resolution of 6 decimal digits in total. Zero is always given by 0-sig 0-exp. The exponent has a range of -128 ... +127. Floating point numbers are recognized if their syntax is conform with the ANS Forth standard. See The optional Floating-Point word set, in particular section "12.3.7 Text interpreter input number conversion". Examples of valid floating point numbers are: 1E 1.E 1.E0 +1.23E-1 -1.23E+1 Decimal base is mandatory for input and output of floating point numbers. Note that a capital E is mandatory for the input number conversion. The string conversion routine >FLOAT is more tolerant and will recognize D, d, E, e as exponent identifiers. The unified stack model is used, i.e. floating point numbers are put on the data stack. Stack manipulation and storage is the same as for single length numbers. Synonyms for the related operators have been provided for clarity. Decimal number base Floating point packages are usually based on the binary numbers. This package uses the decimal number base for several reasons: • I read Martin Tracy's article "Zen Floating Point" in Dr Dobb's Toolbook of Forth II. A floating point package in just one screen! I was impressed by this compact and simple solution. The file CHAP18.LST contains the original listing that came with the book. • I needed a small package. Four decimal digits for the significand were sufficient, and only basic arithmetic operations were required. This should be possible without going to binary. • A small floating point package is easier to develop, test and understand if the used number base is decimal instead of binary. • Converting a number from decimal to binary and back may introduce small rounding errors (e.g. 0.100 ends up as 0.099). Because my application was intended to acquire and save data, rounding errors were not acceptable. So I had to stay in the decimal number base. For an excellent discussion on binary vs. decimal floating point, read the following articles: The package is available in versions for the following FORTH systems: The package was developed under LMI PC/FORTH in 1992. The first version SFP16 used a 16-bit significand and a 16-bit exponent. It had a resolution of 4 decimal digits and was used to acquire, save and plot data in automated measurement systems. The second version SFP24 was written in 1998 and used a 24-bit significand and an 8-bit exponent. This increased the resolution to 6 decimal digits but required the introduction of triple length number operators. The LMI version is included for completeness and because someone might still want to use the package on a 16-bit system. It is also instructive to see how much simpler the SFP24 code became after rewriting the package for 32-bit systems. The following description will concentrate on the 32-bit versions (SwiftForth, VFX Forth) of the package. Use the following commands to install the package: • SwiftForth: INCLUDE sfp_sf.f • VFX Forth: INCLUDE sfp_vfx.fth This makes the fixed point words available. In addition, the number conversion routine of the Forth system is extended: If integer conversion has failed, the system attempts conversion to a floating point number. The package adds the following words to the system: F@ F! D>F S>F >FLOAT F>D F>S PLACES (F.) F. F.R (FS.) FS. FS.R FDROP FDUP FOVER FSWAP FROT F2DROP F2DUP F2OVER F2SWAP F0= F0<> F= F0< F0> FABS FNEGATE F+ F- F* F/ F10* F10/ F< F> FMAX FMIN See glossary.md for stack comments and descriptions of the defined words. DECIMAL number base is mandatory for input and output of floating point numbers. Any base may be used for input and output of integers. However, this may cause confusion if numbers contain an "E". Use PLACES to specify the number of places behind the decimal point. DECIMAL 5 PLACES 834 . --> 834 ok 22.537 D. --> 22537 ok 17.483E F. --> 17.48300 ok 17.483E FS. --> 1.74830E1 ok 0.1E3 FS. --> 1.00000E2 3.84E4 227.37E-6 F* FS. --> 8.73100E0 3.84E4 227.37E-6 F/ FS. --> 1.68887E8 947.84E-6 FLOG F. --> -3.02326 ok Important rules 1. On 32-bit systems, fixed point numbers are represented by single length numbers. Although standard words could be used for storage and stack manipulation, you should always use the provided synonyms. This is important for clarity of the code. 2. This package is not intended for extensive number crunching. For these tasks you should rely on a package that utilizes the NDP (Numerical Data Processor) and provides a separate FP stack. This package largely conforms to Forth-83 (LMI) and ANS Forth (SwiftForth, VFX). Nonstandard words that should be mentioned: • DPL contains the decimal point location after an integer number conversion has been performed • { ... } marks block comments in SwiftForth • (* ... *) marks block comments in VFX Forth • PACKAGE PRIVATE PUBLIC END-PACKAGE are used for information hiding in SwiftForth • MODULE EXPORT END-MODULE are used for information hiding in VFX Forth • UPPER converts a character to upper case in SwiftForth • UPC converts a character to upper case in VFX Forth • AKA old new creates a synonym in SwiftForth • SYNONYM new old creates a synonym in VFX Forth I want to thank Stephen Pelc from MPE who informed me how to patch the floating point number conversion routine into VFX Forth.	{"url":"https://theforth.net/package/sfp/1.0.0","timestamp":"2024-11-14T21:30:40Z","content_type":"text/html","content_length":"14790","record_id":"<urn:uuid:4c377a12-4afc-4c3a-bd66-7dbf08b75f63>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00466.warc.gz"}
Notational Conventions • To: mathgroup at smc.vnet.net • Subject: [mg129974] Notational Conventions • From: Marsh <marshfeldman at gmail.com> • Date: Thu, 28 Feb 2013 21:28:22 -0500 (EST) • Delivered-to: l-mathgroup@mail-archive0.wolfram.com • Delivered-to: l-mathgroup@wolfram.com • Delivered-to: mathgroup-newout@smc.vnet.net • Delivered-to: mathgroup-newsend@smc.vnet.net I am very new to Mathematica but see its potential. I'm unsure of how to combine text and Mathematica computations in a notebook so that the final document reads properly, and I haven't found a tutorial or any other help that covers this. A near as I can make out, one writes the document as if it were an ordinary word processing document, interspersing Mathematica calculations and commands, and then one collapses the cells that one does not want to display in the final document. Is this correct? What I'd like to do is to adopt certain notational conventions and have mathematica deal with them correctly. For example, one set of conventions italicizes names of all variables, uses uppercase letters to name random variables and matrices, but distinguishes random variables from matrices by displaying all matrix names in boldface. A different set of conventions will use italicized uppercase letters to designate points in a diagram or constant parameters in an equation, with matrices now designated as uppercase letters with the dimensions of the matrix as a subscript (e.g., n x m). Ideally, one would have a mathematical style sheet that makes changing the conventions used in a document. But even if one were to be willing to use a fixed notation in a document (notebook), I'm not sure how it would work. How does one use X to designate a random variable and bold X to designate a matrix? How does one use the Greek letter pi as a variable and have it show up in the document as the Greek letter rather than as the word "pi"? So what's the best way to deal with these issues? Next by Date: Re: World map as Texture Previous by thread: An unknown Greek matrix	{"url":"https://forums.wolfram.com/mathgroup/archive/2013/Feb/msg00338.html","timestamp":"2024-11-10T08:33:03Z","content_type":"text/html","content_length":"31539","record_id":"<urn:uuid:8c5842a9-8737-469d-a5ba-9bbd37893213>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00780.warc.gz"}
The 2D Binning Dialog Box 17.1.7.1 The 2D Binning Dialog Box Supporting Information For help with range controls, see: Specifying Your Input Data Specify the data range for the X values. Specify the data range for the Y values. Choose the row ranges on which the 2D Frequency Counts/Binning is performed. The row to begin 2D Frequency Counts/Binning. The row to end 2D Frequency Counts/Binning. Use this branch to specify the X binning range. The minimum and maximum X values are displayed to the right of the node. Specify the way to define the X binning range. • Bin Centers Specify Binning Range by X binning range is indicated by bin centers. • Bin Ends X binning range is indicated by bin ends. Minimum Bin Beginning Specify the minimum X bin starting value. Maximum Bin End Specify the maximum X bin ending value. Step by Specify bin construction, including Bin Size and Number of Bins. Bin Size Specify the fixed step size for X bins. Number of Bins Generated automatically by the bin range and increment. Periodical Specify whether or not the data is periodic. Period Specify a period for the data. Control available when Periodical is checked. Include Outliers Add outliers to lowest and highest bins. Only available when Periodical is selected. Include Outliers < Minimum Include outliers that are less than bin minimum of the lowest bin. Available when Periodical is cleared. Include Outliers >= Maximum Border Options Include outliers that are greater than or equal to the bin maximum of the highest bin. Available when Periodical is cleared. Separately Count Minimum Bin the minimum value separately. Separately Count Maximum Bin the maximum value separately. Specify the sort order of the bin data output. • Ascending Output Binning Order Output bin data in ascending order. • Descending Output bin data in descending order. Settings for Y binning range. Specify the parameter that defines the Y binning range. • Bin Centers Specify Y Binning Range by Y binning range is indicated by bin centers. • Bin Ends Y binning range is indicated by bin ends. Minimum Bin Beginning The minimum value for the lowest Y bin. Maximum Bin End The maximum value for the lowest Y bin. Step by Specify bin construction, including Bin Size and Number of Bins. Bin Size Specify the fixed step size for Y bins. Number of Bins Generated automatically by the bin range and increment. Periodical Specify whether or not the data is periodic. Period Specify a period for the data. It is available when Periodical is checked. Include Outliers Add outliers to lowest and highest bins. Only available when Periodical is selected. Include Outliers < Minimum Include outliers that are less than bin minimum of the lowest bin. Available when Periodical is cleared. Include Outliers>=Maximum Border Options Include outliers that are greater than or equal to the bin maximum of the highest bin. Available when Periodical is cleared.. Separately Count Minimum Bin the minimum value separately. Separately Count Maximum Bin the maximum value separately. Specify the sort order of the bin data output. • Ascending Output Binning Order Output bin data in ascending order. • Descending Output bin data in descending order. Quantity to Compute Compute selected bin statistics. Minimum Output minimum for each bin. Maximum Output maximum for each bin. Mean Compute and output mean for each bin. Median Output median for each bin. Sum Compute and output sum for each bin. Count Compute and output count for each bin. Percent Frequency Compute and output percent frequency for each bin. Column to Compute Quantity Available when an option other than Count is chosen from the Quantity to Compute drop-down list. Select a column to compute the Quantity to Compute. The Customized option allows choosing an arbitrary column in the current worksheet. Base Column Available only when Column to Compute Quantity = Customized. Allows you choose an arbitrary column in the current worksheet. Bin Output Specify whether to output the Bin, Bin Begin and Bin Center/Bin End for specified X Binning Ranges. By default, these three boxes are not checked. Output Worksheet Output results to a worksheet. Subtotal Count for Each Binned Y Available when two conditions are met: • The Output Worksheet check box is selected. • The Quantity to Compute is Sum, Count or Percent Frequency. Output the subtotal count for each binned Y to the Output Worksheet. Output Matrix Output the result to a matrix. Matrix Plots Available when Output Matrix is selected. 3D Bars Plot a 3D bar graph from the output matrix. Image Plot Plot an image plot from the output matrix.	{"url":"https://d2mvzyuse3lwjc.cloudfront.net/doc/en/Origin-Help/2D-Binning-Dialog","timestamp":"2024-11-03T06:07:04Z","content_type":"text/html","content_length":"153396","record_id":"<urn:uuid:bbd06326-14b1-43a7-b7c4-1e0730cd087a>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00519.warc.gz"}
Arithmetic without Calculators Lesson Plans Mental Math Arithmetic without Calculators My Algebra students have spent most of their school careers being told that they can't use calculators. Thus, they are stunned the day they walk into my Algebra One class and are told they're free to use them. I become their instant hero. But I do give them a couple caveats. First, of course, is that I might, at some point, give them a test or quiz that they're not allowed to use them, but that I'll always warn them of that in advance. (The reason I do that is that occasionally we do things like order of operations practice, or fractions problems, that I need to know they have the process down so they'll be able to handle it once variables are inserted in the The second thing I tell them is, even though they are free to use their calculators, I find that, in general, students who rely on their calculators do the work about a quarter the speed as students who don't rely on them. "What happens," I say, "if I ask you what 6 times 8 is? Those of you who don't know that multiplication fact will pull your calculator out..." here I, with exaggerated slowness, pull out an imaginary calculator and begin punching imaginary buttons painfully slow, "...and punch in SIX...TIMES...EIGHT...EQUALS...and while you've done that, your classmate who knows that 6x 8 is 48 will have already gone through the next three steps, and maybe even have moved on to the next problem." So even though I tell them they can use the calculators, I encourage them to think first, to decide if they really need it. Occasionally when we're all working a problem together on the board, I'll have a very simple math fact I need, like "What's 5 x 10?" and after a dramatic pause, I'll say with mock horror, "Did I really just hear someone reaching for their calculator?" I also, when we're working problems together, try to surprise them with math facts that I know. For example, we might be trying to factor the quadratic x^2 - 2x - 168, and I'll say without hesitation (even though I don't have a calculator) that must be (x - 14)(x + 12). They want to know how I'm able to do that so quickly in my head, and instead of answering them, I'll say, "Someday I'll show you; you don't know enough algebra yet." This puzzles them, because they don't see how algebra can help you do arithmetic. Later (usually when they're taking Algebra Two), there will come a day when we have some time to kill in class, and I'll say to them, "Would you like to know how I use algebra to help me do arithmetic?" Of course, it gets them out of doing something else, so they're agreeable. I write on the board something like 168 = 14 x 12. "Would you like to know how I did that in my head?" "You memorized the answer!" someone always says. "No...actually, I didn't. I knew the answer because I knew that 169 is 13^2." I write on the board: 168 = 169 - 1 = 13^2 - 1^2. "That," I say, "is a difference of squares." And I finish by writing: 168 = (13 - 1)(13 + 1) That takes a moment to sink in. Then I say, "I also know that 165 = 11 times 15. Why? Because 165 = 13^2 - 2^2 = (13 - 2)(13 + 2)." Then I let them try. Most of my students know that 144 is 12^2, so I write "143 =" and let them finish by concluding that this is 12^2 - 1^2, which means it's (12 - 1)(12 + 1) = 11 x 13. Of course, this is a very special case requiring that the number be a difference of squares and that the students have a lot of perfect squares memorized. The reverse is also true; if a student has to multiply 19 x 21, it's easiest to think of that as (20 - 1)(20 + 1) = 20^2 - 1 = 400 - 1 = 399. Of if they need to multiply 14 x 18, that's (16 - 2)(16 + 2) = 16^2 - 2^2 = 256 - 4 = 252. Other factoring rules can come into play, too. For example: 1001 = 10^3 + 1^3 = (10 + 1)(10^2 - 10 + 1) = 11(91) 27008 = 30^3 + 2^3 = (30 + 2)(30^2 - 30x2 + 4) = 32 · 844 = 2^5 · (2^2 · 211) = 2^7 · 211 Furthermore, the "sum and product" rule of factoring can come into play sometimes. When I see the number 198, the first thing that pops into my mind is "what two numbers add to 9 and multiply to 8?" If I can find an answer to that question, it might help me factor the number. In this case, the numbers are 1 and 8. 198 = (10 + 1)(10 + 8) = 11 · 18 = 2 · 3^2 · 11. If I see the number 1224, I think: 1224 = 12 x 100 + 12 x 2 = 12(100 + 2) = 12(102) = (2^2 · 3)(2 · 3 · 17) = 2^3 · 3^2 · 17 These kinds of math tricks are not often useful, but if you're watching for them, it might surprise you how often you can use algebraic methods to factor or multiply numbers. Many students will roll their eyes and turn back to their calculators. But other students will start looking for these techniques whenever they have numbers they need to manipulate. For that handful of students, arithmetic ceases to be a boring chore, and becomes a possibility of discovering something fun and different. In the meantime, they keep those arithmetic facts fresh in their minds, and don't become too dependent on their calculators. Lesson by Mr. Twitchell Blogs on This Site Reviews and book lists - books we love! The site administrator fields questions from visitors.	{"url":"https://www.theproblemsite.com/lesson-plans/math/mental-math/arithmetic-without-calculators","timestamp":"2024-11-03T22:02:18Z","content_type":"text/html","content_length":"25068","record_id":"<urn:uuid:3a74b3b1-4610-42dd-9203-5eaefc14075b>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00125.warc.gz"}
Excel Chart Multiple Values Per Day 2024 - Multiplication Chart Printable Excel Chart Multiple Values Per Day Excel Chart Multiple Values Per Day – You could make a multiplication chart in Excel by using a format. You will discover numerous types of web templates and learn how to formatting your multiplication graph utilizing them. Here are some tips and tricks to make a multiplication graph or chart. When you have a template, all you need to do is copy the solution and mixture it in a new cellular. Then you can use this method to increase a series of phone numbers by another established. Excel Chart Multiple Values Per Day. Multiplication kitchen table web template You may want to learn how to write a simple formula if you are in the need to create a multiplication table. Very first, you need to locking mechanism row one of many header column, then flourish the number on row A by mobile phone B. Another way to develop a multiplication desk is by using combined references. In this case, you would key in $A2 into line A and B$1 into row B. The effect is actually a multiplication desk having a formula that works both for columns and rows. You can use the multiplication table template to create your table if you are using an Excel program. Just open the spreadsheet together with your multiplication dinner table template and change the brand for the student’s label. You can even alter the sheet to fit your personal needs. There is an solution to affect the color of the tissues to change the appearance of the multiplication table, way too. Then, it is possible to modify all the different multiples to meet your requirements. Building a multiplication graph in Stand out When you’re utilizing multiplication kitchen table application, you can easily develop a straightforward multiplication table in Excel. Basically create a sheet with rows and columns numbered from a to 40. Where columns and rows intersect may be the response. For example, if a row has a digit of three, and a column has a digit of five, then the answer is three times five. The same thing goes for the opposite. First, it is possible to go into the amounts that you have to flourish. If you need to multiply two digits by three, you can type a formula for each number in cell A1, for example. To make the figures greater, choose the cells at A1 and A8, then click the correct arrow to choose an array of tissue. You may then variety the multiplication formula within the cellular material from the other columns and rows. Gallery of Excel Chart Multiple Values Per Day Excel Chart Of Values From E g All Mondays In A List Of Dates Working With Multiple Data Series In Excel Pryor Learning Solutions How To Calculate Daily Averages With A Pivot Table Excel Campus Leave a Comment	{"url":"https://www.multiplicationchartprintable.com/excel-chart-multiple-values-per-day/","timestamp":"2024-11-06T11:21:51Z","content_type":"text/html","content_length":"52076","record_id":"<urn:uuid:472ca73a-fe05-4a72-b45a-941341692596>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00828.warc.gz"}
Space Turtle 2004 Canadian Computing Competition, Stage 1 Problem S4: Space Turtle Space Turtle is a fearless space adventurer. His spaceship, the Tortoise, is a little outdated, but still gets him where he needs to go. The Tortoise can do only two things - move forward an integer number of light-years, and turn in one of four directions (relative to the current orientation): right, left, up and down. In fact, strangely enough, we can even think of the Tortoise as a ship which travels along a 3-dimensional co-ordinate grid, measured in light-years. In today's adventure, Space Turtle is searching for the fabled Golden Shell, which lies on a deserted planet somewhere in uncharted space. Space Turtle plans to fly around randomly looking for the planet, hoping that his turtle instincts will lead him to the treasure. You have the lonely job of being the keeper of the fabled Golden Shell. Being lonely, your only hobby is to observe and record how close various treasure seekers come to finding the deserted planet and its hidden treasure. Given your observations of Space Turtle's movements, determine the closest distance Space Turtle comes to reaching the Golden Shell. The first line consists of three integers sx, sy, and sz, which give the coordinates of Space Turtle's starting point. Space Turtle is originally oriented in the positive x direction, with the top of his spaceship pointing in the positive z direction, and with the positive y direction to his left. Each of these integers are between -100 and 100. The second line consists of three integers tx, ty, and tz, which give the coordinates of the deserted planet. Each of these integers are between -10000 and 10000. The rest of the lines describe Space Turtle's flight plan in his search for the Golden Shell. Each line consists of an integer, d, 0 ≤ d ≤ 100, and a letter c, separated by a space. The integer indicates the distance in light-years that the Tortoise moves forward, and the letter indicates the direction the ship turns after having moved forward. `L', `R', `U', and `D' stand for left, right, up and down, respectively. There will be no more than 100 such lines. On the last line of input, instead of one of the four direction letters, the letter `E' is given instead, indicating the end of today's adventure. Output the closest distance that Space Turtle gets to the hidden planet, rounded to 2 decimal places. If Space Turtle's coordinates coincide with the planet's coordinates during his flight indicate that with a distance of 0.00. He safely lands on the planet and finds the Golden Shell. Sample Input 2 L 2 L 2 U 2 U 2 L 2 L 2 U 2 E Sample Output All Submissions Best Solutions Point Value: 12 Time Limit: 2.00s Memory Limit: 16M Added: Sep 28, 2008 Problem Types: [Show] [Hide] Languages Allowed: C++03, PAS, C, HASK, ASM, RUBY, PYTH2, JAVA, PHP, SCM, CAML, PERL, C#, C++11, PYTH3 Does the turtle stay facing -> and goes up in this position or does it turn upward so the front is facing up and goes upward? Does the turtle stay facing -> and goes up in this position or does it turn upward so the front is facing up and goes upward? The U instruction means that the Space Turtle adjusts himself so that whatever direction currently seems to him to be UP is now his FRONT, and LEFT and RIGHT stay the same as they were before. The U instruction means that the Space Turtle adjusts himself so that whatever direction currently seems to him to be UP is now his FRONT, and LEFT and RIGHT stay the same as they were before. Can someone please explain how the input works. I know the problem statement tells me how to do it but I'm still quite confused. You start facing a positive x direction. The you move in that direction and afterwards you change direction? So for the sample, you start off with 0 0 0 Then you move to 2 0 0? Then you move to 2 2 0? Can someone please explain how the input works. I know the problem statement tells me how to do it but I'm still quite confused. You start facing a positive x direction. The you move in that direction and afterwards you change direction? So for the sample, you start off with 0 0 0 Then you move to 2 0 0? Then you move to 2 2 0?	{"url":"https://wcipeg.com/problem/ccc04s4","timestamp":"2024-11-13T21:04:18Z","content_type":"text/html","content_length":"16678","record_id":"<urn:uuid:c8310f7f-fe71-49d3-8fed-eef81bc38df3>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00333.warc.gz"}
Handbook of Homotopy Theory Added final table of contents and publication date. We should see if we can track down copies of the missing chapters, for instance on peoples’ websites. diff, v13, current Added today’s □ Gijs Heuts, Lie algebra models for unstable homotopy theory, (arXiv:1907.13055) diff, v11, current Added todays’ □ David Gepner, An Introduction to Higher Categorical Algebra, (arXiv:1907.02904) diff, v9, current Added today’s • Lars Hesselholt, Thomas Nikolaus, Topological cyclic homology, (arXiv:1905.08984) diff, v7, current Added another chapter, by Wickelgren and Williams on Unstable Motivic Homotopy Theory. diff, v5, current Added Arone and Ching’s contribution. diff, v4, current Added the contribution of Barthel and Beaudry diff, v3, current Added Mark Behrens’s contribution diff, v2, current Page created, but author did not leave any comments. v1, current Reordered list of contributions to match the contents page, and added a few more arXiv links I found in the process. Only three of the chapters are not available online, and only two of those online are non-arXiv. All the chapters are listed here, whether or not they are electronically available. diff, v14, current Added new arXiv link Paul Balmer, A guide to tensor-triangular classification, (arXiv:1912.08963) diff, v15, current Added today’s arXiv link for Tyler Lawson, $E_n$-ring spectra and Dyer-Lashof operations. It’s not clear how different this is to the version available on Lawson’s webpage, so I left that link too. diff, v16, current Added new arXiv link for Gunnar Carlsson, Persistent homology and applied homotopy theory (arXiv:2004.00738) diff, v17, current added to most items pointer to the nLab entry concerned with their topic BTW, It looks like most items added to the list here have not been added yet to the references in the respective entry. diff, v18, current Added bibliographic data. diff, v19, current A curious fact: the publication of this handbook could be considered the “official point” at which (most of) algebraic topology got renamed as homotopy theory. Haynes Miller writes about this in the This volume may be regarded as a successor to the “Handbook of Algebraic Topology,” edited by Ioan James and published a quarter of a century ago. In calling it the “Handbook of Homotopy Theory,” I am recognizing that the discipline has expanded and deepened, and traditional questions of topology, as classically understood, are now only one of many distinct mathematical disciplines in which it has had a profound impact and which serve as sources of motivation for research directions within homotopy theory proper. Although one could also point to Clark Barwick’s August 2017 manifesto “The future of homotopy theory”: https://ncatlab.org/nlab/files/BarwickFutureOfHomotopyTheory.pdf as another “official renaming Hi Dmitri, this comment is worthwhile to include in the entry itself! Added a citation from the introduction. diff, v21, current have added the publication data (CRC Press 2019, ISBN:9780815369707) diff, v23, current oh, now I see such data was there on another line, away from the author/title block. Have re-organized now. diff, v23, current	{"url":"https://nforum.ncatlab.org/discussion/9356/handbook-of-homotopy-theory/?Focus=88513","timestamp":"2024-11-14T11:35:16Z","content_type":"application/xhtml+xml","content_length":"72388","record_id":"<urn:uuid:b8aeddf7-d951-4608-a07b-9759dec13f30>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00278.warc.gz"}
Formaldehyde consists of 40.0% carbon, 6.7% hydrogen, and 53.3% oxygen. What is its empirical formula? \| Socratic Formaldehyde consists of 40.0% carbon, 6.7% hydrogen, and 53.3% oxygen. What is its empirical formula? 1 Answer The empirical formula is $\text{CH"_2"O}$. Assume that you have 100 g of formaldehyde. Then you have 40.0 g of $\text{C}$, 6.7 g of $\text{H}$, and 53.3 g of $\text{O}$. Our job is to calculate the ratio of the moles of each element. $\text{Moles of C" =40.0 color(red)(cancel(color(black)("g C"))) × "1 mol C"/(12.01 color(red)(cancel(color(black)("g C")))) = "3.331 mol C}$ $\text{Moles of H" = 6.7 color(red)(cancel(color(black)("g H"))) × "1 mol H"/(1.008color(red)(cancel(color(black)("g H")))) = "6.65 mol H}$ $\text{Moles of O" = 53.3 color(red)(cancel(color(black)("g O"))) × "1 mol O"/(16.00 color(red)(cancel(color(black)("g O")))) = "3.331 mol O}$ To get the molar ratio, we divide each number of moles by the smallest number ($3.331$). From here on, I like to summarize the calculations in a table. $\text{Element"color(white)(X) "Mass/g"color(white)(X) "Moles"color(white)(X) "Ratio" color(white)(X)"Integers}$ #stackrel(—————————————————-—)(color(white)(l)"C" color(white)(XXXX)40.0 color(white)(Xlll)3.331 color(white)(Xll)1 color(white)(XXXlll)1)# $\textcolor{w h i t e}{X} \text{H} \textcolor{w h i t e}{X X X X l} 6.7 \textcolor{w h i t e}{X X l} 6.65 \textcolor{w h i t e}{X X l} 2.00 \textcolor{w h i t e}{X X l} 2$ $\textcolor{w h i t e}{X} \text{O} \textcolor{w h i t e}{X X X l l} 53.3 \textcolor{w h i t e}{X X l} 3.331 \textcolor{w h i t e}{X l l} 1.00 \textcolor{w h i t e}{X X l} 1$ The ratio comes out as $\text{C:H:O} = 1 : 2 : 1$. Thus, the empirical formula is $\text{CH"_2"O}$. Impact of this question 56913 views around the world	{"url":"https://socratic.org/questions/8-formaldehyde-consists-of-40-0-carbon-6-7-hydrogen-and-53-3-oxygen-what-is-its-","timestamp":"2024-11-14T08:37:14Z","content_type":"text/html","content_length":"36872","record_id":"<urn:uuid:f35c1fc2-ac17-4c58-b87a-2a1a3352e95f>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00313.warc.gz"}
[Solved] Design a combinational logic circuit that \| SolutionInn Answered step by step Verified Expert Solution Design a combinational logic circuit that displays the hexadecimal value of a Balanced Gray code input according to the specifications given in the assignment Debug Design a combinational logic circuit that displays the hexadecimal value of a Balanced Gray code input according to the specifications given in the assignment Debug and test your design by simulating it using the Logisim simulator 3 Document your work in a short report This text was automatically generated from the attachment Please refer to the attachment to view this question There are 3 Steps involved in it Step: 1 1 Introduction The objective of this project is to reinforce your understanding of binary codes combinational logic design and logic simulation You must i design a combinational logic circuit that Get Instant Access to Expert-Tailored Solutions See step-by-step solutions with expert insights and AI powered tools for academic success Ace Your Homework with AI Get the answers you need in no time with our AI-driven, step-by-step assistance Get Started Recommended Textbook for Authors: M. Rafiquzzaman 6th edition 1-118-85579-9, 1118855795, 9781118969304, 978-1118855799 More Books Students also viewed these Algorithms questions View Answer in SolutionInn App	{"url":"https://www.solutioninn.com/study-help/questions/design-a-combinational-logic-circuit-that-displays-the-hexadecimal-value-of-a-164911","timestamp":"2024-11-09T04:30:48Z","content_type":"text/html","content_length":"111427","record_id":"<urn:uuid:84a6e158-6502-43e3-9fdf-35a295915e7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00684.warc.gz"}
API Reference# Module Configuration# Output Data Type Configuration# Method to set cuML’s single GPU estimators global output type. It will be used by all estimators unless overridden in their initialization with their own output_type parameter. Can also be overridden by the context manager method using_output_type(). output_type{‘input’, ‘cudf’, ‘cupy’, ‘numpy’} (default = ‘input’) Desired output type of results and attributes of the estimators. ■ 'input' will mean that the parameters and methods will mirror the format of the data sent to the estimators/methods as much as possible. Specifically: Input type Output type cuDF DataFrame or Series cuDF DataFrame or Series NumPy arrays NumPy arrays Pandas DataFrame or Series NumPy arrays Numba device arrays Numba device arrays CuPy arrays CuPy arrays Other __cuda_array_interface__ objs CuPy arrays ■ 'cudf' will return cuDF Series for single dimensional results and DataFrames for the rest. ■ 'cupy' will return CuPy arrays. ■ 'numpy' will return NumPy arrays. 'cupy' and 'numba' options (as well as 'input' when using Numba and CuPy ndarrays for input) have the least overhead. cuDF add memory consumption and processing time needed to build the Series and DataFrames. 'numpy' has the biggest overhead due to the need to transfer data to CPU memory. >>> import cuml >>> import cupy as cp >>> ary = [[1.0, 4.0, 4.0], [2.0, 2.0, 2.0], [5.0, 1.0, 1.0]] >>> ary = cp.asarray(ary) >>> prev_output_type = cuml.global_settings.output_type >>> cuml.set_global_output_type('cudf') >>> dbscan_float = cuml.DBSCAN(eps=1.0, min_samples=1) >>> dbscan_float.fit(ary) >>> # cuML output type >>> dbscan_float.labels_ dtype: int32 >>> type(dbscan_float.labels_) <class 'cudf.core.series.Series'> >>> cuml.set_global_output_type(prev_output_type) Context manager method to set cuML’s global output type inside a with statement. It gets reset to the prior value it had once the with code block is executer. output_type{‘input’, ‘cudf’, ‘cupy’, ‘numpy’} (default = ‘input’) Desired output type of results and attributes of the estimators. ■ 'input' will mean that the parameters and methods will mirror the format of the data sent to the estimators/methods as much as possible. Specifically: Input type Output type cuDF DataFrame or Series cuDF DataFrame or Series NumPy arrays NumPy arrays Pandas DataFrame or Series NumPy arrays Numba device arrays Numba device arrays CuPy arrays CuPy arrays Other __cuda_array_interface__ objs CuPy arrays ■ 'cudf' will return cuDF Series for single dimensional results and DataFrames for the rest. ■ 'cupy' will return CuPy arrays. ■ 'numpy' will return NumPy arrays. >>> import cuml >>> import cupy as cp >>> ary = [[1.0, 4.0, 4.0], [2.0, 2.0, 2.0], [5.0, 1.0, 1.0]] >>> ary = cp.asarray(ary) >>> with cuml.using_output_type('cudf'): ... dbscan_float = cuml.DBSCAN(eps=1.0, min_samples=1) ... dbscan_float.fit(ary) ... print("cuML output inside 'with' context") ... print(dbscan_float.labels_) ... print(type(dbscan_float.labels_)) cuML output inside 'with' context dtype: int32 <class 'cudf.core.series.Series'> >>> # use cuml again outside the context manager >>> dbscan_float2 = cuml.DBSCAN(eps=1.0, min_samples=1) >>> dbscan_float2.fit(ary) >>> # cuML default output >>> dbscan_float2.labels_ array([0, 1, 2], dtype=int32) >>> isinstance(dbscan_float2.labels_, cp.ndarray) CPU / GPU Device Selection (Experimental)# cuML provides experimental support for running selected estimators and operators on either the GPU or CPU. This document covers the set of operators for which CPU/GPU device selection capabilities are supported as of the current nightly packages. If an operator isn’t listed here, it can only be run on the GPU. Prior versions of cuML may have reduced support compared to the following table. Category Operator Clustering HDBSCAN Dimensionality Reduction and Manifold Learning PCA Dimensionality Reduction and Manifold Learning TruncatedSVD Dimensionality Reduction and Manifold Learning UMAP Neighbors NearestNeighbors Regression and Classification ElasticNet Regression and Classification Lasso Regression and Classification LinearRegression Regression and Classification LogisticRegression Regression and Classification Ridge If a CUDA-enabled GPU is available on the system, cuML will default to using it. Users can configure CPU or GPU execution for supported operators via context managers or global configuration. from cuml.linear_model import Lasso from cuml.common.device_selection import using_device_type, set_global_device_type with using_device_type("CPU"): # Alternatively, using_device_type("GPU") model = Lasso() model.fit(X_train, y_train) predictions = model.predict(X_test) # All operators supporting CPU execution will run on the CPU after this configuration model = Lasso() model.fit(X_train, y_train) predictions = model.predict(X_test) For more detailed examples, please see the Execution Device Interoperability Notebook in the User Guide. Verbosity Levels# cuML follows a verbosity model similar to Scikit-learn’s: The verbose parameter can be a boolean, or a numeric value, and higher numeric values mean more verbosity. The exact values can be set directly, or through the cuml.common.logger module, and they are: Numeric value cuml.common.logger value Verbosity level 0 cuml.common.logger.level_off Disables all log messages 1 cuml.common.logger.level_critical Enables only critical messages 2 cuml.common.logger.level_error Enables all messages up to and including errors. 3 cuml.common.logger.level_warn Enables all messages up to and including warnings. 4 or False cuml.common.logger.level_info Enables all messages up to and including information messages. 5 or True cuml.common.logger.level_debug Enables all messages up to and including debug messages. 6 cuml.common.logger.level_trace Enables all messages up to and including trace messages. Preprocessing, Metrics, and Utilities# Model Selection and Data Splitting# cuml.model_selection.train_test_split(X, y=None, test_size: Optional[Union[float, int]] = None, train_size: Optional[Union[float, int]] = None, shuffle: bool = True, random_state: Optional[Union[ int, RandomState, RandomState]] = None, stratify=None)[source]# Partitions device data into four collated objects, mimicking Scikit-learn’s train_test_split. Xcudf.DataFrame or cuda_array_interface compliant device array Data to split, has shape (n_samples, n_features) ystr, cudf.Series or cuda_array_interface compliant device array Set of labels for the data, either a series of shape (n_samples) or the string label of a column in X (if it is a cuDF DataFrame) containing the labels train_sizefloat or int, optional If float, represents the proportion [0, 1] of the data to be assigned to the training set. If an int, represents the number of instances to be assigned to the training set. Defaults to 0.8 shufflebool, optional Whether or not to shuffle inputs before splitting random_stateint, CuPy RandomState or NumPy RandomState optional If shuffle is true, seeds the generator. Unseeded by default stratify: cudf.Series or cuda_array_interface compliant device array, optional parameter. When passed, the input is split using this as column to startify on. Default=None X_train, X_test, y_train, y_testcudf.DataFrame or array-like objects Partitioned dataframes if X and y were cuDF objects. If y was provided as a column name, the column was dropped from X. Partitioned numba device arrays if X and y were Numba device arrays. Partitioned CuPy arrays for any other input. >>> import cudf >>> from cuml.model_selection import train_test_split >>> # Generate some sample data >>> df = cudf.DataFrame({'x': range(10), ... 'y': [0, 1] * 5}) >>> print(f'Original data: {df.shape[0]} elements') Original data: 10 elements >>> # Suppose we want an 80/20 split >>> X_train, X_test, y_train, y_test = train_test_split(df, 'y', ... train_size=0.8) >>> print(f'X_train: {X_train.shape[0]} elements') X_train: 8 elements >>> print(f'X_test: {X_test.shape[0]} elements') X_test: 2 elements >>> print(f'y_train: {y_train.shape[0]} elements') y_train: 8 elements >>> print(f'y_test: {y_test.shape[0]} elements') y_test: 2 elements >>> # Alternatively, if our labels are stored separately >>> labels = df['y'] >>> df = df.drop(['y'], axis=1) >>> # we can also do >>> X_train, X_test, y_train, y_test = train_test_split(df, labels, ... train_size=0.8) Feature and Label Encoding (Single-GPU)# class cuml.preprocessing.LabelEncoder.LabelEncoder(, handle_unknown='error', handle=None, verbose=False, output_type=None)[source]# An nvcategory based implementation of ordinal label encoding handle_unknown{‘error’, ‘ignore’}, default=’error’ Whether to raise an error or ignore if an unknown categorical feature is present during transform (default is to raise). When this parameter is set to ‘ignore’ and an unknown category is encountered during transform or inverse transform, the resulting encoding will be null. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. Converting a categorical implementation to a numerical one >>> from cudf import DataFrame, Series >>> from cuml.preprocessing import LabelEncoder >>> data = DataFrame({'category': ['a', 'b', 'c', 'd']}) >>> # There are two functionally equivalent ways to do this >>> le = LabelEncoder() >>> le.fit(data.category) # le = le.fit(data.category) also works >>> encoded = le.transform(data.category) >>> print(encoded) dtype: uint8 >>> # This method is preferred >>> le = LabelEncoder() >>> encoded = le.fit_transform(data.category) >>> print(encoded) dtype: uint8 >>> # We can assign this to a new column >>> data = data.assign(encoded=encoded) >>> print(data.head()) category encoded 0 a 0 1 b 1 2 c 2 3 d 3 >>> # We can also encode more data >>> test_data = Series(['c', 'a']) >>> encoded = le.transform(test_data) >>> print(encoded) dtype: uint8 >>> # After train, ordinal label can be inverse_transform() back to >>> # string labels >>> ord_label = cudf.Series([0, 0, 1, 2, 1]) >>> str_label = le.inverse_transform(ord_label) >>> print(str_label) 0 a 1 a 2 b 3 c 4 b dtype: object fit(y[, _classes]) Fit a LabelEncoder (nvcategory) instance to a set of categories fit_transform(y[, z]) Simultaneously fit and transform an input get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(y) Revert ordinal label to original label transform(y) Transform an input into its categorical keys. fit(y, _classes=None)[source]# Fit a LabelEncoder (nvcategory) instance to a set of categories ycudf.Series, pandas.Series, cupy.ndarray or numpy.ndarray Series containing the categories to be encoded. It’s elements may or may not be unique _classes: int or None. Passed by the dask client when dask LabelEncoder is used. A fitted instance of itself to allow method chaining fit_transform(y, z=None) Series[source]# Simultaneously fit and transform an input This is functionally equivalent to (but faster than) LabelEncoder().fit(y).transform(y) Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. inverse_transform(y: Series) Series[source]# Revert ordinal label to original label ycudf.Series, pandas.Series, cupy.ndarray or numpy.ndarray dtype=int32 Ordinal labels to be reverted revertedthe same type as y Reverted labels transform(y) Series[source]# Transform an input into its categorical keys. This is intended for use with small inputs relative to the size of the dataset. For fitting and transforming an entire dataset, prefer fit_transform. ycudf.Series, pandas.Series, cupy.ndarray or numpy.ndarray Input keys to be transformed. Its values should match the categories given to fit The ordinally encoded input series if a category appears that was not seen in fit class cuml.preprocessing.LabelBinarizer(, neg_label=0, pos_label=1, sparse_output=False, handle=None, verbose=False, output_type=None)[source]# A multi-class dummy encoder for labels. neg_labelinteger (default=0) label to be used as the negative binary label pos_labelinteger (default=1) label to be used as the positive binary label sparse_outputbool (default=False) whether to return sparse arrays for transformed output Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. Create an array with labels and dummy encode them >>> import cupy as cp >>> import cupyx >>> from cuml.preprocessing import LabelBinarizer >>> labels = cp.asarray([0, 5, 10, 7, 2, 4, 1, 0, 0, 4, 3, 2, 1], ... dtype=cp.int32) >>> lb = LabelBinarizer() >>> encoded = lb.fit_transform(labels) >>> print(str(encoded)) [[1 0 0 0 0 0 0 0] [0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 1] [0 0 0 0 0 0 1 0] [0 0 1 0 0 0 0 0] [0 0 0 0 1 0 0 0] [0 1 0 0 0 0 0 0] [1 0 0 0 0 0 0 0] [1 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0] [0 0 0 1 0 0 0 0] [0 0 1 0 0 0 0 0] [0 1 0 0 0 0 0 0]] >>> decoded = lb.inverse_transform(encoded) >>> print(str(decoded)) [ 0 5 10 7 2 4 1 0 0 4 3 2 1] fit(y) Fit label binarizer fit_transform(y) Fit label binarizer and transform multi-class labels to their dummy-encoded representation. get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(y[, threshold]) Transform binary labels back to original multi-class labels transform(y) Transform multi-class labels to their dummy-encoded representation labels. fit(y) LabelBinarizer[source]# Fit label binarizer yarray of shape [n_samples,] or [n_samples, n_classes] Target values. The 2-d matrix should only contain 0 and 1, represents multilabel classification. selfreturns an instance of self. fit_transform(y) SparseCumlArray[source]# Fit label binarizer and transform multi-class labels to their dummy-encoded representation. yarray of shape [n_samples,] or [n_samples, n_classes] arrarray with encoded labels Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. inverse_transform(y, threshold=None) CumlArray[source]# Transform binary labels back to original multi-class labels yarray of shape [n_samples, n_classes] thresholdfloat this value is currently ignored arrarray with original labels transform(y) SparseCumlArray[source]# Transform multi-class labels to their dummy-encoded representation labels. yarray of shape [n_samples,] or [n_samples, n_classes] arrarray with encoded labels cuml.preprocessing.label_binarize(y, classes, neg_label=0, pos_label=1, sparse_output=False) SparseCumlArray[source]# A stateless helper function to dummy encode multi-class labels. yarray-like of size [n_samples,] or [n_samples, n_classes] classesthe set of unique classes in the input neg_labelinteger the negative value for transformed output pos_labelinteger the positive value for transformed output sparse_outputbool whether to return sparse array class cuml.preprocessing.OneHotEncoder(, categories='auto', drop=None, sparse='deprecated', sparse_output=True, dtype=<class 'numpy.float32'>, handle_unknown='error', handle=None, verbose=False, Encode categorical features as a one-hot numeric array. The input to this estimator should be a cuDF.DataFrame or a cupy.ndarray, denoting the unique values taken on by categorical (discrete) features. The features are encoded using a one-hot (aka ‘one-of-K’ or ‘dummy’) encoding scheme. This creates a binary column for each category and returns a sparse matrix or dense array (depending on the sparse parameter). By default, the encoder derives the categories based on the unique values in each feature. Alternatively, you can also specify the categories manually. a one-hot encoding of y labels should use a LabelBinarizer instead. categories‘auto’ an cupy.ndarray or a cudf.DataFrame, default=’auto’ Categories (unique values) per feature: ■ ‘auto’ : Determine categories automatically from the training data. ■ DataFrame/ndarray : categories[col] holds the categories expected in the feature col. drop‘first’, None, a dict or a list, default=None Specifies a methodology to use to drop one of the categories per feature. This is useful in situations where perfectly collinear features cause problems, such as when feeding the resulting data into a neural network or an unregularized regression. ■ None : retain all features (the default). ■ ‘first’ : drop the first category in each feature. If only one category is present, the feature will be dropped entirely. ■ dict/list : drop[col] is the category in feature col that should be dropped. sparse_outputbool, default=True This feature is not fully supported by cupy yet, causing incorrect values when computing one hot encodings. See cupy/cupy#3223 New in version 24.06: sparse was renamed to sparse_output sparsebool, default=True Will return sparse matrix if set True else will return an array. Deprecated since version 24.06: sparse is deprecated in 24.06 and will be removed in 25.08. Use sparse_output instead. dtypenumber type, default=np.float Desired datatype of transform’s output. handle_unknown{‘error’, ‘ignore’}, default=’error’ Whether to raise an error or ignore if an unknown categorical feature is present during transform (default is to raise). When this parameter is set to ‘ignore’ and an unknown category is encountered during transform, the resulting one-hot encoded columns for this feature will be all zeros. In the inverse transform, an unknown category will be denoted as None. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. drop_idx_array of shape (n_features,) drop_idx_[i] is the index in categories_[i] of the category to be dropped for each feature. None if all the transformed features will be retained. fit(X[, y]) Fit OneHotEncoder to X. fit_transform(X[, y]) Fit OneHotEncoder to X, then transform X. Equivalent to fit(X).transform(X). get_feature_names([input_features]) Return feature names for output features. get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(X) Convert the data back to the original representation. transform(X) Transform X using one-hot encoding. fit(X, y=None)[source]# Fit OneHotEncoder to X. Parameters ———- Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas Ignored. This parameter exists for compatibility only. fit_transform(X, y=None)[source]# Fit OneHotEncoder to X, then transform X. Equivalent to fit(X).transform(X). Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas Ignored. This parameter exists for compatibility only. X_outsparse matrix if sparse=True else a 2-d array Transformed input. Return feature names for output features. input_featureslist of str of shape (n_features,) String names for input features if available. By default, “x0”, “x1”, … “xn_features” is used. output_feature_namesndarray of shape (n_output_features,) Array of feature names. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. Convert the data back to the original representation. In case unknown categories are encountered (all zeros in the one-hot encoding), None is used to represent this category. The return type is the same as the type of the input used by the first call to fit on this estimator instance. Xarray-like or sparse matrix, shape [n_samples, n_encoded_features] The transformed data. X_trcudf.DataFrame or cupy.ndarray Inverse transformed array. Transform X using one-hot encoding. Parameters ———- Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas X_outsparse matrix if sparse=True else a 2-d array Transformed input. class cuml.preprocessing.TargetEncoder.TargetEncoder(n_folds=4, smooth=0, seed=42, split_method='interleaved', output_type='auto', stat='mean')[source]# A cudf based implementation of target encoding [1], which converts one or multiple categorical variables, ‘Xs’, with the average of corresponding values of the target variable, ‘Y’. The input data is grouped by the columns Xs and the aggregated mean value of Y of each group is calculated to replace each value of Xs. Several optimizations are applied to prevent label leakage and parallelize the execution. n_foldsint (default=4) Default number of folds for fitting training data. To prevent label leakage in fit, we split data into n_folds and encode one fold using the target variables of the remaining folds. smoothint or float (default=0) Count of samples to smooth the encoding. 0 means no smoothing. seedint (default=42) Random seed split_method{‘random’, ‘continuous’, ‘interleaved’}, (default=’interleaved’) Method to split train data into n_folds. ‘random’: random split. ‘continuous’: consecutive samples are grouped into one folds. ‘interleaved’: samples are assign to each fold in a round robin way. ‘customize’: customize splitting by providing a fold_ids array in fit() or fit_transform() functions. output_type{‘cupy’, ‘numpy’, ‘auto’}, default = ‘auto’ The data type of output. If ‘auto’, it matches input data. stat{‘mean’,’var’,’median’}, default = ‘mean’ The statistic used in encoding, mean, variance or median of the target. Converting a categorical implementation to a numerical one >>> from cudf import DataFrame, Series >>> from cuml.preprocessing import TargetEncoder >>> train = DataFrame({'category': ['a', 'b', 'b', 'a'], ... 'label': [1, 0, 1, 1]}) >>> test = DataFrame({'category': ['a', 'c', 'b', 'a']}) >>> encoder = TargetEncoder() >>> train_encoded = encoder.fit_transform(train.category, train.label) >>> test_encoded = encoder.transform(test.category) >>> print(train_encoded) [1. 1. 0. 1.] >>> print(test_encoded) [1. 0.75 0.5 1. ] fit(x, y[, fold_ids]) Fit a TargetEncoder instance to a set of categories fit_transform(x, y[, fold_ids]) Simultaneously fit and transform an input get_params([deep]) Returns a dict of all params owned by this class. transform(x) Transform an input into its categorical keys. fit(x, y, fold_ids=None)[source]# Fit a TargetEncoder instance to a set of categories xcudf.Series or cudf.DataFrame or cupy.ndarray categories to be encoded. It’s elements may or may not be unique ycudf.Series or cupy.ndarray Series containing the target variable. fold_idscudf.Series or cupy.ndarray Series containing the indices of the customized folds. Its values should be integers in range [0, N-1] to split data into N folds. If None, fold_ids is generated based on A fitted instance of itself to allow method chaining fit_transform(x, y, fold_ids=None)[source]# Simultaneously fit and transform an input This is functionally equivalent to (but faster than) TargetEncoder().fit(y).transform(y) xcudf.Series or cudf.DataFrame or cupy.ndarray categories to be encoded. It’s elements may or may not be unique ycudf.Series or cupy.ndarray Series containing the target variable. fold_idscudf.Series or cupy.ndarray Series containing the indices of the customized folds. Its values should be integers in range [0, N-1] to split data into N folds. If None, fold_ids is generated based on The ordinally encoded input series Returns a dict of all params owned by this class. Transform an input into its categorical keys. This is intended for test data. For fitting and transforming the training data, prefer fit_transform. Input keys to be transformed. Its values doesn’t have to match the categories given to fit The ordinally encoded input series Feature Scaling and Normalization (Single-GPU)# class cuml.preprocessing.MaxAbsScaler(args, kwargs)[source]# Scale each feature by its maximum absolute value. This estimator scales and translates each feature individually such that the maximal absolute value of each feature in the training set will be 1.0. It does not shift/center the data, and thus does not destroy any sparsity. This scaler can also be applied to sparse CSR or CSC matrices. copyboolean, optional, default is True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Equivalent function without the estimator API. NaNs are treated as missing values: disregarded in fit, and maintained in transform. >>> from cuml.preprocessing import MaxAbsScaler >>> import cupy as cp >>> X = [[ 1., -1., 2.], ... [ 2., 0., 0.], ... [ 0., 1., -1.]] >>> X = cp.array(X) >>> transformer = MaxAbsScaler().fit(X) >>> transformer >>> transformer.transform(X) array([[ 0.5, -1. , 1. ], [ 1. , 0. , 0. ], [ 0. , 1. , -0.5]]) scale_ndarray, shape (n_features,) Per feature relative scaling of the data. max_abs_ndarray, shape (n_features,) Per feature maximum absolute value. The number of samples processed by the estimator. Will be reset on new calls to fit, but increments across partial_fit calls. fit(X[, y]) Compute the maximum absolute value to be used for later scaling. get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(X) Scale back the data to the original representation partial_fit(X[, y]) Online computation of max absolute value of X for later scaling. transform(X) Scale the data fit(X, y=None) MaxAbsScaler[source]# Compute the maximum absolute value to be used for later scaling. X{array-like, sparse matrix}, shape [n_samples, n_features] The data used to compute the per-feature minimum and maximum used for later scaling along the features axis. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. inverse_transform(X) SparseCumlArray[source]# Scale back the data to the original representation X{array-like, sparse matrix} The data that should be transformed back. partial_fit(X, y=None) MaxAbsScaler[source]# Online computation of max absolute value of X for later scaling. All of X is processed as a single batch. This is intended for cases when fit() is not feasible due to very large number of n_samples or because X is read from a continuous stream. X{array-like, sparse matrix}, shape [n_samples, n_features] The data used to compute the mean and standard deviation used for later scaling along the features axis. Transformer instance. transform(X) SparseCumlArray[source]# Scale the data X{array-like, sparse matrix} The data that should be scaled. class cuml.preprocessing.MinMaxScaler(args, *kwargs)[source]# Transform features by scaling each feature to a given range. This estimator scales and translates each feature individually such that it is in the given range on the training set, e.g. between zero and one. The transformation is given by: X_std = (X - X.min(axis=0)) / (X.max(axis=0) - X.min(axis=0)) X_scaled = X_std (max - min) + min where min, max = feature_range. This transformation is often used as an alternative to zero mean, unit variance scaling. feature_rangetuple (min, max), default=(0, 1) Desired range of transformed data. copybool, default=True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Equivalent function without the estimator API. NaNs are treated as missing values: disregarded in fit, and maintained in transform. >>> from cuml.preprocessing import MinMaxScaler >>> import cupy as cp >>> data = [[-1, 2], [-0.5, 6], [0, 10], [1, 18]] >>> data = cp.array(data) >>> scaler = MinMaxScaler() >>> print(scaler.fit(data)) >>> print(scaler.data_max_) [ 1. 18.] >>> print(scaler.transform(data)) [[0. 0. ] [0.25 0.25] [0.5 0.5 ] [1. 1. ]] >>> print(scaler.transform(cp.array([[2, 2]]))) [[1.5 0. ]] min_ndarray of shape (n_features,) Per feature adjustment for minimum. Equivalent to min - X.min(axis=0) * self.scale_ scale_ndarray of shape (n_features,) Per feature relative scaling of the data. Equivalent to (max - min) / (X.max(axis=0) - X.min(axis=0)) data_min_ndarray of shape (n_features,) Per feature minimum seen in the data data_max_ndarray of shape (n_features,) Per feature maximum seen in the data data_range_ndarray of shape (n_features,) Per feature range (data_max_ - data_min_) seen in the data The number of samples processed by the estimator. It will be reset on new calls to fit, but increments across partial_fit calls. fit(X[, y]) Compute the minimum and maximum to be used for later scaling. get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(X) Undo the scaling of X according to feature_range. partial_fit(X[, y]) Online computation of min and max on X for later scaling. transform(X) Scale features of X according to feature_range. fit(X, y=None) MinMaxScaler[source]# Compute the minimum and maximum to be used for later scaling. Xarray-like of shape (n_samples, n_features) The data used to compute the per-feature minimum and maximum used for later scaling along the features axis. Fitted scaler. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. inverse_transform(X) CumlArray[source]# Undo the scaling of X according to feature_range. Xarray-like of shape (n_samples, n_features) Input data that will be transformed. It cannot be sparse. Xtarray-like of shape (n_samples, n_features) Transformed data. partial_fit(X, y=None) MinMaxScaler[source]# Online computation of min and max on X for later scaling. All of X is processed as a single batch. This is intended for cases when fit() is not feasible due to very large number of n_samples or because X is read from a continuous stream. Xarray-like of shape (n_samples, n_features) The data used to compute the mean and standard deviation used for later scaling along the features axis. Transformer instance. transform(X) CumlArray[source]# Scale features of X according to feature_range. Xarray-like of shape (n_samples, n_features) Input data that will be transformed. Xtarray-like of shape (n_samples, n_features) Transformed data. class cuml.preprocessing.Normalizer(args, kwargs)[source]# Normalize samples individually to unit norm. Each sample (i.e. each row of the data matrix) with at least one non zero component is rescaled independently of other samples so that its norm (l1, l2 or inf) equals one. This transformer is able to work both with dense numpy arrays and sparse matrix Scaling inputs to unit norms is a common operation for text classification or clustering for instance. For instance the dot product of two l2-normalized TF-IDF vectors is the cosine similarity of the vectors and is the base similarity metric for the Vector Space Model commonly used by the Information Retrieval community. norm‘l1’, ‘l2’, or ‘max’, optional (‘l2’ by default) The norm to use to normalize each non zero sample. If norm=’max’ is used, values will be rescaled by the maximum of the absolute values. copyboolean, optional, default True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Equivalent function without the estimator API. This estimator is stateless (besides constructor parameters), the fit method does nothing but is useful when used in a pipeline. >>> from cuml.preprocessing import Normalizer >>> import cupy as cp >>> X = [[4, 1, 2, 2], ... [1, 3, 9, 3], ... [5, 7, 5, 1]] >>> X = cp.array(X) >>> transformer = Normalizer().fit(X) # fit does nothing. >>> transformer >>> transformer.transform(X) array([[0.8, 0.2, 0.4, 0.4], [0.1, 0.3, 0.9, 0.3], [0.5, 0.7, 0.5, 0.1]]) fit(X[, y]) Do nothing and return the estimator unchanged transform(X[, copy]) Scale each non zero row of X to unit norm fit(X, y=None) Normalizer[source]# Do nothing and return the estimator unchanged This method is just there to implement the usual API and hence work in pipelines. X{array-like, CSR matrix} transform(X, copy=None) SparseCumlArray[source]# Scale each non zero row of X to unit norm X{array-like, CSR matrix}, shape [n_samples, n_features] The data to normalize, row by row. copybool, optional (default: None) Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. class cuml.preprocessing.RobustScaler(args, *kwargs)[source]# Scale features using statistics that are robust to outliers. This Scaler removes the median and scales the data according to the quantile range (defaults to IQR: Interquartile Range). The IQR is the range between the 1st quartile (25th quantile) and the 3rd quartile (75th quantile). Centering and scaling happen independently on each feature by computing the relevant statistics on the samples in the training set. Median and interquartile range are then stored to be used on later data using the transform method. Standardization of a dataset is a common requirement for many machine learning estimators. Typically this is done by removing the mean and scaling to unit variance. However, outliers can often influence the sample mean / variance in a negative way. In such cases, the median and the interquartile range often give better results. with_centeringboolean, default=True If True, center the data before scaling. This will cause transform to raise an exception when attempted on sparse matrices, because centering them entails building a dense matrix which in common use cases is likely to be too large to fit in memory. with_scalingboolean, default=True If True, scale the data to interquartile range. quantile_rangetuple (q_min, q_max), 0.0 < q_min < q_max < 100.0 Default: (25.0, 75.0) = (1st quantile, 3rd quantile) = IQR Quantile range used to calculate scale_. copyboolean, optional, default=True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Equivalent function without the estimator API. Further removes the linear correlation across features with whiten=True. >>> from cuml.preprocessing import RobustScaler >>> import cupy as cp >>> X = [[ 1., -2., 2.], ... [ -2., 1., 3.], ... [ 4., 1., -2.]] >>> X = cp.array(X) >>> transformer = RobustScaler().fit(X) >>> transformer >>> transformer.transform(X) array([[ 0. , -2. , 0. ], [-1. , 0. , 0.4], [ 1. , 0. , -1.6]]) center_array of floats The median value for each feature in the training set. scale_array of floats The (scaled) interquartile range for each feature in the training set. fit(X[, y]) Compute the median and quantiles to be used for scaling. get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(X) Scale back the data to the original representation transform(X) Center and scale the data. fit(X, y=None) RobustScaler[source]# Compute the median and quantiles to be used for scaling. X{array-like, CSC matrix}, shape [n_samples, n_features] The data used to compute the median and quantiles used for later scaling along the features axis. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. inverse_transform(X) SparseCumlArray[source]# Scale back the data to the original representation X{array-like, sparse matrix} The data used to scale along the specified axis. transform(X) SparseCumlArray[source]# Center and scale the data. X{array-like, sparse matrix} The data used to scale along the specified axis. class cuml.preprocessing.StandardScaler(args, *kwargs)[source]# Standardize features by removing the mean and scaling to unit variance The standard score of a sample x is calculated as: where u is the mean of the training samples or zero if with_mean=False, and s is the standard deviation of the training samples or one if with_std=False. Centering and scaling happen independently on each feature by computing the relevant statistics on the samples in the training set. Mean and standard deviation are then stored to be used on later data using transform(). Standardization of a dataset is a common requirement for many machine learning estimators: they might behave badly if the individual features do not more or less look like standard normally distributed data (e.g. Gaussian with 0 mean and unit variance). For instance many elements used in the objective function of a learning algorithm (such as the RBF kernel of Support Vector Machines or the L1 and L2 regularizers of linear models) assume that all features are centered around 0 and have variance in the same order. If a feature has a variance that is orders of magnitude larger that others, it might dominate the objective function and make the estimator unable to learn from other features correctly as expected. This scaler can also be applied to sparse CSR or CSC matrices by passing with_mean=False to avoid breaking the sparsity structure of the data. copyboolean, optional, default True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. with_meanboolean, True by default If True, center the data before scaling. This does not work (and will raise an exception) when attempted on sparse matrices, because centering them entails building a dense matrix which in common use cases is likely to be too large to fit in memory. with_stdboolean, True by default If True, scale the data to unit variance (or equivalently, unit standard deviation). See also Equivalent function without the estimator API. Further removes the linear correlation across features with ‘whiten=True’. NaNs are treated as missing values: disregarded in fit, and maintained in transform. We use a biased estimator for the standard deviation, equivalent to numpy.std(x, ddof=0). Note that the choice of ddof is unlikely to affect model performance. >>> from cuml.preprocessing import StandardScaler >>> import cupy as cp >>> data = [[0, 0], [0, 0], [1, 1], [1, 1]] >>> data = cp.array(data) >>> scaler = StandardScaler() >>> print(scaler.fit(data)) >>> print(scaler.mean_) [0.5 0.5] >>> print(scaler.transform(data)) [[-1. -1.] [-1. -1.] [ 1. 1.] [ 1. 1.]] >>> print(scaler.transform(cp.array([[2, 2]]))) [[3. 3.]] scale_ndarray or None, shape (n_features,) Per feature relative scaling of the data. This is calculated using sqrt(var_). Equal to None when with_std=False. mean_ndarray or None, shape (n_features,) The mean value for each feature in the training set. Equal to None when with_mean=False. var_ndarray or None, shape (n_features,) The variance for each feature in the training set. Used to compute scale_. Equal to None when with_std=False. n_samples_seen_int or array, shape (n_features,) The number of samples processed by the estimator for each feature. If there are not missing samples, the n_samples_seen will be an integer, otherwise it will be an array. Will be reset on new calls to fit, but increments across partial_fit calls. fit(X[, y]) Compute the mean and std to be used for later scaling. get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(X[, copy]) Scale back the data to the original representation partial_fit(X[, y]) Online computation of mean and std on X for later scaling. transform(X[, copy]) Perform standardization by centering and scaling fit(X, y=None) StandardScaler[source]# Compute the mean and std to be used for later scaling. X{array-like, sparse matrix}, shape [n_samples, n_features] The data used to compute the mean and standard deviation used for later scaling along the features axis. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. inverse_transform(X, copy=None) SparseCumlArray[source]# Scale back the data to the original representation X{array-like, sparse matrix}, shape [n_samples, n_features] The data used to scale along the features axis. copybool, optional (default: None) Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. X_tr{array-like, sparse matrix}, shape [n_samples, n_features] Transformed array. partial_fit(X, y=None) StandardScaler[source]# Online computation of mean and std on X for later scaling. All of X is processed as a single batch. This is intended for cases when fit() is not feasible due to very large number of n_samples or because X is read from a continuous stream. The algorithm for incremental mean and std is given in Equation 1.5a,b in Chan, Tony F., Gene H. Golub, and Randall J. LeVeque. “Algorithms for computing the sample variance: Analysis and recommendations.” The American Statistician 37.3 (1983): 242-247: X{array-like, sparse matrix}, shape [n_samples, n_features] The data used to compute the mean and standard deviation used for later scaling along the features axis. Transformer instance. transform(X, copy=None) SparseCumlArray[source]# Perform standardization by centering and scaling X{array-like, sparse matrix}, shape [n_samples, n_features] The data used to scale along the features axis. copybool, optional (default: None) Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. cuml.preprocessing.maxabs_scale(X, , axis=0, copy=True)[source]# Scale each feature to the [-1, 1] range without breaking the sparsity. This estimator scales each feature individually such that the maximal absolute value of each feature in the training set will be 1.0. This scaler can also be applied to sparse CSR or CSC matrices. X{array-like, sparse matrix}, shape (n_samples, n_features) The data. axisint (0 by default) axis used to scale along. If 0, independently scale each feature, otherwise (if 1) scale each sample. copyboolean, optional, default is True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Performs scaling to the [-1, 1] range using the``Transformer`` API NaNs are treated as missing values: disregarded to compute the statistics, and maintained during the data transformation. cuml.preprocessing.minmax_scale(X, feature_range=(0, 1), , axis=0, copy=True)[source]# Transform features by scaling each feature to a given range. This estimator scales and translates each feature individually such that it is in the given range on the training set, i.e. between zero and one. The transformation is given by (when axis=0): X_std = (X - X.min(axis=0)) / (X.max(axis=0) - X.min(axis=0)) X_scaled = X_std (max - min) + min where min, max = feature_range. The transformation is calculated as (when axis=0): X_scaled = scale * X + min - X.min(axis=0) * scale where scale = (max - min) / (X.max(axis=0) - X.min(axis=0)) This transformation is often used as an alternative to zero mean, unit variance scaling. Xarray-like of shape (n_samples, n_features) The data. feature_rangetuple (min, max), default=(0, 1) Desired range of transformed data. axisint, default=0 Axis used to scale along. If 0, independently scale each feature, otherwise (if 1) scale each sample. copybool, default=True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Performs scaling to a given range using the``Transformer`` API cuml.preprocessing.normalize(X, norm='l2', , axis=1, copy=True, return_norm=False)[source]# Scale input vectors individually to unit norm (vector length). X{array-like, sparse matrix}, shape [n_samples, n_features] The data to normalize, element by element. Please provide CSC matrix to normalize on axis 0, conversely provide CSR matrix to normalize on axis 1 norm‘l1’, ‘l2’, or ‘max’, optional (‘l2’ by default) The norm to use to normalize each non zero sample (or each non-zero feature if axis is 0). axis0 or 1, optional (1 by default) axis used to normalize the data along. If 1, independently normalize each sample, otherwise (if 0) normalize each feature. copyboolean, optional, default True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. return_normboolean, default False whether to return the computed norms X{array-like, sparse matrix}, shape [n_samples, n_features] Normalized input X. normsarray, shape [n_samples] if axis=1 else [n_features] An array of norms along given axis for X. When X is sparse, a NotImplementedError will be raised for norm ‘l1’ or ‘l2’. See also Performs normalization using the Transformer API cuml.preprocessing.robust_scale(X, , axis=0, with_centering=True, with_scaling=True, quantile_range=(25.0, 75.0), copy=True)[source]# Standardize a dataset along any axis Center to the median and component wise scale according to the interquartile range. X{array-like, sparse matrix} The data to center and scale. axisint (0 by default) axis used to compute the medians and IQR along. If 0, independently scale each feature, otherwise (if 1) scale each sample. with_centeringboolean, True by default If True, center the data before scaling. with_scalingboolean, True by default If True, scale the data to unit variance (or equivalently, unit standard deviation). quantile_rangetuple (q_min, q_max), 0.0 < q_min < q_max < 100.0 Default: (25.0, 75.0) = (1st quantile, 3rd quantile) = IQR Quantile range used to calculate scale_. copyboolean, optional, default is True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Performs centering and scaling using the Transformer API This implementation will refuse to center sparse matrices since it would make them non-sparse and would potentially crash the program with memory exhaustion problems. Instead the caller is expected to either set explicitly with_centering=False (in that case, only variance scaling will be performed on the features of the CSR matrix) or to densify the matrix if he/she expects the materialized dense array to fit in memory. To avoid memory copy the caller should pass a CSR matrix. cuml.preprocessing.scale(X, , axis=0, with_mean=True, with_std=True, copy=True)[source]# Standardize a dataset along any axis Center to the mean and component wise scale to unit variance. X{array-like, sparse matrix} The data to center and scale. axisint (0 by default) axis used to compute the means and standard deviations along. If 0, independently standardize each feature, otherwise (if 1) standardize each sample. with_meanboolean, True by default If True, center the data before scaling. with_stdboolean, True by default If True, scale the data to unit variance (or equivalently, unit standard deviation). copyboolean, optional, default True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Performs scaling to unit variance using the``Transformer`` API This implementation will refuse to center sparse matrices since it would make them non-sparse and would potentially crash the program with memory exhaustion problems. Instead the caller is expected to either set explicitly with_mean=False (in that case, only variance scaling will be performed on the features of the sparse matrix) or to densify the matrix if he /she expects the materialized dense array to fit in memory. For optimal processing the caller should pass a CSC matrix. NaNs are treated as missing values: disregarded to compute the statistics, and maintained during the data transformation. We use a biased estimator for the standard deviation, equivalent to numpy.std(x, ddof=0). Note that the choice of ddof is unlikely to affect model performance. Other preprocessing methods (Single-GPU)# class cuml.preprocessing.Binarizer(args, *kwargs)[source]# Binarize data (set feature values to 0 or 1) according to a threshold Values greater than the threshold map to 1, while values less than or equal to the threshold map to 0. With the default threshold of 0, only positive values map to 1. Binarization is a common operation on text count data where the analyst can decide to only consider the presence or absence of a feature rather than a quantified number of occurrences for It can also be used as a pre-processing step for estimators that consider boolean random variables (e.g. modelled using the Bernoulli distribution in a Bayesian setting). thresholdfloat, optional (0.0 by default) Feature values below or equal to this are replaced by 0, above it by 1. Threshold may not be less than 0 for operations on sparse matrices. copyboolean, optional, default True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Equivalent function without the estimator API. If the input is a sparse matrix, only the non-zero values are subject to update by the Binarizer class. This estimator is stateless (besides constructor parameters), the fit method does nothing but is useful when used in a pipeline. >>> from cuml.preprocessing import Binarizer >>> import cupy as cp >>> X = [[ 1., -1., 2.], ... [ 2., 0., 0.], ... [ 0., 1., -1.]] >>> X = cp.array(X) >>> transformer = Binarizer().fit(X) # fit does nothing. >>> transformer >>> transformer.transform(X) array([[1., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) fit(X[, y]) Do nothing and return the estimator unchanged transform(X[, copy]) Binarize each element of X fit(X, y=None) Binarizer[source]# Do nothing and return the estimator unchanged This method is just there to implement the usual API and hence work in pipelines. X{array-like, sparse matrix} transform(X, copy=None) SparseCumlArray[source]# Binarize each element of X X{array-like, sparse matrix}, shape [n_samples, n_features] The data to binarize, element by element. Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. class cuml.preprocessing.FunctionTransformer(args, *kwargs)[source]# Constructs a transformer from an arbitrary callable. A FunctionTransformer forwards its X (and optionally y) arguments to a user-defined function or function object and returns the result of this function. This is useful for stateless transformations such as taking the log of frequencies, doing custom scaling, etc. Note: If a lambda is used as the function, then the resulting transformer will not be pickleable. funccallable, default=None The callable to use for the transformation. This will be passed the same arguments as transform, with args and kwargs forwarded. If func is None, then func will be the identity function. inverse_funccallable, default=None The callable to use for the inverse transformation. This will be passed the same arguments as inverse transform, with args and kwargs forwarded. If inverse_func is None, then inverse_func will be the identity function. accept_sparsebool, default=False Indicate that func accepts a sparse matrix as input. Otherwise, if accept_sparse is false, sparse matrix inputs will cause an exception to be raised. check_inversebool, default=True Whether to check that or func followed by inverse_func leads to the original inputs. It can be used for a sanity check, raising a warning when the condition is not fulfilled. kw_argsdict, default=None Dictionary of additional keyword arguments to pass to func. inv_kw_argsdict, default=None Dictionary of additional keyword arguments to pass to inverse_func. >>> import cupy as cp >>> from cuml.preprocessing import FunctionTransformer >>> transformer = FunctionTransformer(func=cp.log1p) >>> X = cp.array([[0, 1], [2, 3]]) >>> transformer.transform(X) array([[0. , 0.6931...], [1.0986..., 1.3862...]]) fit(X[, y]) Fit transformer by checking X. inverse_transform(X) Transform X using the inverse function. transform(X) Transform X using the forward function. fit(X, y=None) FunctionTransformer[source]# Fit transformer by checking X. X{array-like, sparse matrix}, shape (n_samples, n_features) Input array. inverse_transform(X) SparseCumlArray[source]# Transform X using the inverse function. X{array-like, sparse matrix}, shape (n_samples, n_features) Input array. X_out{array-like, sparse matrix}, shape (n_samples, n_features) Transformed input. transform(X) SparseCumlArray[source]# Transform X using the forward function. X{array-like, sparse matrix}, shape (n_samples, n_features) Input array. X_out{array-like, sparse matrix}, shape (n_samples, n_features) Transformed input. class cuml.preprocessing.KBinsDiscretizer(args, *kwargs)[source]# Bin continuous data into intervals. n_binsint or array-like, shape (n_features,) (default=5) The number of bins to produce. Raises ValueError if n_bins < 2. encode{‘onehot’, ‘onehot-dense’, ‘ordinal’}, (default=’onehot’) Method used to encode the transformed result. Encode the transformed result with one-hot encoding and return a sparse matrix. Ignored features are always stacked to the right. Encode the transformed result with one-hot encoding and return a dense array. Ignored features are always stacked to the right. Return the bin identifier encoded as an integer value. strategy{‘uniform’, ‘quantile’, ‘kmeans’}, (default=’quantile’) Strategy used to define the widths of the bins. All bins in each feature have identical widths. All bins in each feature have the same number of points. Values in each bin have the same nearest center of a 1D k-means cluster. See also Class used to bin values as 0 or 1 based on a parameter threshold. In bin edges for feature i, the first and last values are used only for inverse_transform. During transform, bin edges are extended to: np.concatenate([-np.inf, bin_edges_[i][1:-1], np.inf]) You can combine KBinsDiscretizer with cuml.compose.ColumnTransformer if you only want to preprocess part of the features. KBinsDiscretizer might produce constant features (e.g., when encode = 'onehot' and certain bins do not contain any data). These features can be removed with feature selection algorithms (e.g., >>> from cuml.preprocessing import KBinsDiscretizer >>> import cupy as cp >>> X = [[-2, 1, -4, -1], ... [-1, 2, -3, -0.5], ... [ 0, 3, -2, 0.5], ... [ 1, 4, -1, 2]] >>> X = cp.array(X) >>> est = KBinsDiscretizer(n_bins=3, encode='ordinal', strategy='uniform') >>> est.fit(X) >>> Xt = est.transform(X) >>> Xt array([[0, 0, 0, 0], [1, 1, 1, 0], [2, 2, 2, 1], [2, 2, 2, 2]], dtype=int32) Sometimes it may be useful to convert the data back into the original feature space. The inverse_transform function converts the binned data into the original feature space. Each value will be equal to the mean of the two bin edges. >>> est.bin_edges_[0] array([-2., -1., 0., 1.]) >>> est.inverse_transform(Xt) array([[-1.5, 1.5, -3.5, -0.5], [-0.5, 2.5, -2.5, -0.5], [ 0.5, 3.5, -1.5, 0.5], [ 0.5, 3.5, -1.5, 1.5]]) n_bins_int array, shape (n_features,) Number of bins per feature. Bins whose width are too small (i.e., <= 1e-8) are removed with a warning. bin_edges_array of arrays, shape (n_features, ) The edges of each bin. Contain arrays of varying shapes (n_bins_, ) Ignored features will have empty arrays. fit(X[, y]) Fit the estimator. get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(Xt) Transform discretized data back to original feature space. transform(X) Discretize the data. fit(X, y=None) KBinsDiscretizer[source]# Fit the estimator. Xnumeric array-like, shape (n_samples, n_features) Data to be discretized. Ignored. This parameter exists only for compatibility with sklearn.pipeline.Pipeline. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. inverse_transform(Xt) SparseCumlArray[source]# Transform discretized data back to original feature space. Note that this function does not regenerate the original data due to discretization rounding. Xtnumeric array-like, shape (n_sample, n_features) Transformed data in the binned space. Xinvnumeric array-like Data in the original feature space. transform(X) SparseCumlArray[source]# Discretize the data. Xnumeric array-like, shape (n_samples, n_features) Data to be discretized. Xtnumeric array-like or sparse matrix Data in the binned space. class cuml.preprocessing.KernelCenterer(args, *kwargs)[source]# Center a kernel matrix Let K(x, z) be a kernel defined by phi(x)^T phi(z), where phi is a function mapping x to a Hilbert space. KernelCenterer centers (i.e., normalize to have zero mean) the data without explicitly computing phi(x). It is equivalent to centering phi(x) with cuml.preprocessing.StandardScaler(with_std=False). >>> import cupy as cp >>> from cuml.preprocessing import KernelCenterer >>> from cuml.metrics import pairwise_kernels >>> X = cp.array([[ 1., -2., 2.], ... [ -2., 1., 3.], ... [ 4., 1., -2.]]) >>> K = pairwise_kernels(X, metric='linear') >>> K array([[ 9., 2., -2.], [ 2., 14., -13.], [ -2., -13., 21.]]) >>> transformer = KernelCenterer().fit(K) >>> transformer >>> transformer.transform(K) array([[ 5., 0., -5.], [ 0., 14., -14.], [ -5., -14., 19.]]) K_fit_rows_array, shape (n_samples,) Average of each column of kernel matrix Average of kernel matrix fit(K[, y]) Fit KernelCenterer transform(K[, copy]) Center kernel matrix. fit(K, y=None) KernelCenterer[source]# Fit KernelCenterer Knumpy array of shape [n_samples, n_samples] Kernel matrix. selfreturns an instance of self. transform(K, copy=True) CumlArray[source]# Center kernel matrix. Knumpy array of shape [n_samples1, n_samples2] Kernel matrix. copyboolean, optional, default True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. K_newnumpy array of shape [n_samples1, n_samples2] class cuml.preprocessing.MissingIndicator(args, *kwargs)[source]# Binary indicators for missing values. Note that this component typically should not be used in a vanilla Pipeline consisting of transformers and a classifier, but rather could be added using a FeatureUnion or ColumnTransformer. missing_valuesnumber, string, np.nan (default) or None The placeholder for the missing values. All occurrences of missing_values will be imputed. For pandas’ dataframes with nullable integer dtypes with missing values, missing_values should be set to np.nan, since pd.NA will be converted to np.nan. featuresstr, default=None Whether the imputer mask should represent all or a subset of features. ○ If “missing-only” (default), the imputer mask will only represent features containing missing values during fit time. ○ If “all”, the imputer mask will represent all features. sparseboolean or “auto”, default=None Whether the imputer mask format should be sparse or dense. ○ If “auto” (default), the imputer mask will be of same type as input. ○ If True, the imputer mask will be a sparse matrix. ○ If False, the imputer mask will be a numpy array. error_on_newboolean, default=None If True (default), transform will raise an error when there are features with missing values in transform that have no missing values in fit. This is applicable only when features= >>> import numpy as np >>> from sklearn.impute import MissingIndicator >>> X1 = np.array([[np.nan, 1, 3], ... [4, 0, np.nan], ... [8, 1, 0]]) >>> X2 = np.array([[5, 1, np.nan], ... [np.nan, 2, 3], ... [2, 4, 0]]) >>> indicator = MissingIndicator() >>> indicator.fit(X1) >>> X2_tr = indicator.transform(X2) >>> X2_tr array([[False, True], [ True, False], [False, False]]) features_ndarray, shape (n_missing_features,) or (n_features,) The features indices which will be returned when calling transform. They are computed during fit. For features='all', it is to range(n_features). fit(X[, y]) Fit the transformer on X. fit_transform(X[, y]) Generate missing values indicator for X. get_param_names() Returns a list of hyperparameter names owned by this class. transform(X) Generate missing values indicator for X. fit(X, y=None) MissingIndicator[source]# Fit the transformer on X. X{array-like, sparse matrix}, shape (n_samples, n_features) Input data, where n_samples is the number of samples and n_features is the number of features. Returns self. fit_transform(X, y=None) SparseCumlArray[source]# Generate missing values indicator for X. X{array-like, sparse matrix}, shape (n_samples, n_features) The input data to complete. Xt{ndarray or sparse matrix}, shape (n_samples, n_features) or (n_samples, n_features_with_missing) The missing indicator for input data. The data type of Xt will be boolean. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. transform(X) SparseCumlArray[source]# Generate missing values indicator for X. X{array-like, sparse matrix}, shape (n_samples, n_features) The input data to complete. Xt{ndarray or sparse matrix}, shape (n_samples, n_features) or (n_samples, n_features_with_missing) The missing indicator for input data. The data type of Xt will be boolean. class cuml.preprocessing.PolynomialFeatures(args, kwargs)[source]# Generate polynomial and interaction features. Generate a new feature matrix consisting of all polynomial combinations of the features with degree less than or equal to the specified degree. For example, if an input sample is two dimensional and of the form [a, b], the degree-2 polynomial features are [1, a, b, a^2, ab, b^2]. The degree of the polynomial features. Default = 2. interaction_onlyboolean, default = False If true, only interaction features are produced: features that are products of at most degree distinct input features (so not x[1] 2, x[0] * x[2] ** 3, etc.). If True (default), then include a bias column, the feature in which all polynomial powers are zero (i.e. a column of ones - acts as an intercept term in a linear model). orderstr in {‘C’, ‘F’}, default ‘C’ Order of output array in the dense case. ‘F’ order is faster to compute, but may slow down subsequent estimators. Be aware that the number of features in the output array scales polynomially in the number of features of the input array, and exponentially in the degree. High degrees can cause overfitting. >>> import numpy as np >>> from cuml.preprocessing import PolynomialFeatures >>> X = np.arange(6).reshape(3, 2) >>> X array([[0, 1], [2, 3], [4, 5]]) >>> poly = PolynomialFeatures(2) >>> poly.fit_transform(X) array([[ 1., 0., 1., 0., 0., 1.], [ 1., 2., 3., 4., 6., 9.], [ 1., 4., 5., 16., 20., 25.]]) >>> poly = PolynomialFeatures(interaction_only=True) >>> poly.fit_transform(X) array([[ 1., 0., 1., 0.], [ 1., 2., 3., 6.], [ 1., 4., 5., 20.]]) powers_array, shape (n_output_features, n_input_features) powers_[i, j] is the exponent of the jth input in the ith output. The total number of input features. The total number of polynomial output features. The number of output features is computed by iterating over all suitably sized combinations of input features. fit(X[, y]) Compute number of output features. get_feature_names([input_features]) Return feature names for output features get_param_names() Returns a list of hyperparameter names owned by this class. transform(X) Transform data to polynomial features fit(X, y=None) PolynomialFeatures[source]# Compute number of output features. Xarray-like, shape (n_samples, n_features) The data. Return feature names for output features input_featureslist of string, length n_features, optional String names for input features if available. By default, “x0”, “x1”, … “xn_features” is used. output_feature_nameslist of string, length n_output_features Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. transform(X) SparseCumlArray[source]# Transform data to polynomial features X{array-like, sparse matrix}, shape [n_samples, n_features] The data to transform, row by row. Prefer CSR over CSC for sparse input (for speed), but CSC is required if the degree is 4 or higher. If the degree is less than 4 and the input format is CSC, it will be converted to CSR, have its polynomial features generated, then converted back to CSC. If the degree is 2 or 3, the method described in “Leveraging Sparsity to Speed Up Polynomial Feature Expansions of CSR Matrices Using K-Simplex Numbers” by Andrew Nystrom and John Hughes is used, which is much faster than the method used on CSC input. For this reason, a CSC input will be converted to CSR, and the output will be converted back to CSC prior to being returned, hence the preference of CSR. XP{array-like, sparse matrix}, shape [n_samples, NP] The matrix of features, where NP is the number of polynomial features generated from the combination of inputs. class cuml.preprocessing.PowerTransformer(args, kwargs)[source]# Apply a power transform featurewise to make data more Gaussian-like. Power transforms are a family of parametric, monotonic transformations that are applied to make data more Gaussian-like. This is useful for modeling issues related to heteroscedasticity (non-constant variance), or other situations where normality is desired. Currently, PowerTransformer supports the Box-Cox transform and the Yeo-Johnson transform. The optimal parameter for stabilizing variance and minimizing skewness is estimated through maximum Box-Cox requires input data to be strictly positive, while Yeo-Johnson supports both positive or negative data. By default, zero-mean, unit-variance normalization is applied to the transformed data. methodstr, (default=’yeo-johnson’) The power transform method. Available methods are: ○ ‘yeo-johnson’ [1], works with positive and negative values ○ ‘box-cox’ [2], only works with strictly positive values standardizeboolean, default=True Set to True to apply zero-mean, unit-variance normalization to the transformed output. copyboolean, optional, default=True Set to False to perform inplace computation during transformation. See also Equivalent function without the estimator API. Maps data to a standard normal distribution with the parameter output_distribution='normal'. NaNs are treated as missing values: disregarded in fit, and maintained in transform. I.K. Yeo and R.A. Johnson, “A new family of power transformations to improve normality or symmetry.” Biometrika, 87(4), pp.954-959, (2000). G.E.P. Box and D.R. Cox, “An Analysis of Transformations”, Journal of the Royal Statistical Society B, 26, 211-252 (1964). >>> import cupy as cp >>> from cuml.preprocessing import PowerTransformer >>> pt = PowerTransformer() >>> data = cp.array([[1, 2], [3, 2], [4, 5]]) >>> print(pt.fit(data)) >>> print(pt.lambdas_) [ 1.386... -3.100...] >>> print(pt.transform(data)) [[-1.316... -0.707...] [ 0.209... -0.707...] [ 1.106... 1.414...]] lambdas_array of float, shape (n_features,) The parameters of the power transformation for the selected features. fit(X[, y]) Estimate the optimal parameter lambda for each feature. fit_transform(X[, y]) Fit to data, then transform it. get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(X) Apply the inverse power transformation using the fitted lambdas. transform(X) Apply the power transform to each feature using the fitted lambdas. fit(X, y=None) PowerTransformer[source]# Estimate the optimal parameter lambda for each feature. The optimal lambda parameter for minimizing skewness is estimated on each feature independently using maximum likelihood. Xarray-like, shape (n_samples, n_features) The data used to estimate the optimal transformation parameters. fit_transform(X, y=None) CumlArray[source]# Fit to data, then transform it. Fits transformer to X and y with optional parameters fit_params and returns a transformed version of X. X{array-like, sparse matrix, dataframe} of shape (n_samples, n_features) yndarray of shape (n_samples,), default=None Target values. Additional fit parameters. X_newndarray array of shape (n_samples, n_features_new) Transformed array. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. inverse_transform(X) CumlArray[source]# Apply the inverse power transformation using the fitted lambdas. The inverse of the Box-Cox transformation is given by: if lambda_ == 0: X = exp(X_trans) X = (X_trans lambda_ + 1) ** (1 / lambda_) The inverse of the Yeo-Johnson transformation is given by: if X >= 0 and lambda_ == 0: X = exp(X_trans) - 1 elif X >= 0 and lambda_ != 0: X = (X_trans * lambda_ + 1) ** (1 / lambda_) - 1 elif X < 0 and lambda_ != 2: X = 1 - (-(2 - lambda_) * X_trans + 1) ** (1 / (2 - lambda_)) elif X < 0 and lambda_ == 2: X = 1 - exp(-X_trans) Xarray-like, shape (n_samples, n_features) The transformed data. Xarray-like, shape (n_samples, n_features) The original data transform(X) CumlArray[source]# Apply the power transform to each feature using the fitted lambdas. Xarray-like, shape (n_samples, n_features) The data to be transformed using a power transformation. X_transarray-like, shape (n_samples, n_features) The transformed data. class cuml.preprocessing.QuantileTransformer(args, kwargs)[source]# Transform features using quantiles information. This method transforms the features to follow a uniform or a normal distribution. Therefore, for a given feature, this transformation tends to spread out the most frequent values. It also reduces the impact of (marginal) outliers: this is therefore a robust preprocessing scheme. The transformation is applied on each feature independently. First an estimate of the cumulative distribution function of a feature is used to map the original values to a uniform distribution. The obtained values are then mapped to the desired output distribution using the associated quantile function. Features values of new/unseen data that fall below or above the fitted range will be mapped to the bounds of the output distribution. Note that this transform is non-linear. It may distort linear correlations between variables measured at the same scale but renders variables measured at different scales more directly comparable. n_quantilesint, optional (default=1000 or n_samples) Number of quantiles to be computed. It corresponds to the number of landmarks used to discretize the cumulative distribution function. If n_quantiles is larger than the number of samples, n_quantiles is set to the number of samples as a larger number of quantiles does not give a better approximation of the cumulative distribution function estimator. output_distributionstr, optional (default=’uniform’) Marginal distribution for the transformed data. The choices are ‘uniform’ (default) or ‘normal’. ignore_implicit_zerosbool, optional (default=False) Only applies to sparse matrices. If True, the sparse entries of the matrix are discarded to compute the quantile statistics. If False, these entries are treated as zeros. subsampleint, optional (default=1e5) Maximum number of samples used to estimate the quantiles for computational efficiency. Note that the subsampling procedure may differ for value-identical sparse and dense matrices. random_stateint, RandomState instance or None, optional (default=None) Determines random number generation for subsampling and smoothing noise. Please see subsample for more details. Pass an int for reproducible results across multiple function calls. See copyboolean, optional, (default=True) Set to False to perform inplace transformation and avoid a copy (if the input is already a numpy array). See also Equivalent function without the estimator API. Perform mapping to a normal distribution using a power transform. Perform standardization that is faster, but less robust to outliers. Perform robust standardization that removes the influence of outliers but does not put outliers and inliers on the same scale. NaNs are treated as missing values: disregarded in fit, and maintained in transform. >>> import cupy as cp >>> from cuml.preprocessing import QuantileTransformer >>> rng = cp.random.RandomState(0) >>> X = cp.sort(rng.normal(loc=0.5, scale=0.25, size=(25, 1)), axis=0) >>> qt = QuantileTransformer(n_quantiles=10, random_state=0) >>> qt.fit_transform(X) The actual number of quantiles used to discretize the cumulative distribution function. quantiles_ndarray, shape (n_quantiles, n_features) The values corresponding the quantiles of reference. references_ndarray, shape(n_quantiles, ) Quantiles of references. fit(X[, y]) Compute the quantiles used for transforming. get_param_names() Returns a list of hyperparameter names owned by this class. inverse_transform(X) Back-projection to the original space. transform(X) Feature-wise transformation of the data. fit(X, y=None) QuantileTransformer[source]# Compute the quantiles used for transforming. Xndarray or sparse matrix, shape (n_samples, n_features) The data used to scale along the features axis. If a sparse matrix is provided, it will be converted into a sparse csc_matrix. Additionally, the sparse matrix needs to be nonnegative if ignore_implicit_zeros is False. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. inverse_transform(X) SparseCumlArray[source]# Back-projection to the original space. Xndarray or sparse matrix, shape (n_samples, n_features) The data used to scale along the features axis. If a sparse matrix is provided, it will be converted into a sparse csc_matrix. Additionally, the sparse matrix needs to be nonnegative if ignore_implicit_zeros is False. Xtndarray or sparse matrix, shape (n_samples, n_features) The projected data. transform(X) SparseCumlArray[source]# Feature-wise transformation of the data. Xndarray or sparse matrix, shape (n_samples, n_features) The data used to scale along the features axis. If a sparse matrix is provided, it will be converted into a sparse csc_matrix. Additionally, the sparse matrix needs to be nonnegative if ignore_implicit_zeros is False. Xtndarray or sparse matrix, shape (n_samples, n_features) The projected data. class cuml.preprocessing.SimpleImputer(args, *kwargs)[source]# Imputation transformer for completing missing values. missing_valuesnumber, string, np.nan (default) or None The placeholder for the missing values. All occurrences of missing_values will be imputed. For pandas’ dataframes with nullable integer dtypes with missing values, missing_values should be set to np.nan, since pd.NA will be converted to np.nan. strategystring, default=’mean’ The imputation strategy. ○ If “mean”, then replace missing values using the mean along each column. Can only be used with numeric data. ○ If “median”, then replace missing values using the median along each column. Can only be used with numeric data. ○ If “most_frequent”, then replace missing using the most frequent value along each column. Can be used with strings or numeric data. ○ If “constant”, then replace missing values with fill_value. Can be used with strings or numeric data. strategy=”constant” for fixed value imputation. fill_valuestring or numerical value, default=None When strategy == “constant”, fill_value is used to replace all occurrences of missing_values. If left to the default, fill_value will be 0 when imputing numerical data and “missing_value” for strings or object data types. verboseinteger, default=0 Controls the verbosity of the imputer. copyboolean, default=True If True, a copy of X will be created. If False, imputation will be done in-place whenever possible. Note that, in the following cases, a new copy will always be made, even if copy=False: ○ If X is not an array of floating values; ○ If X is encoded as a CSR matrix; ○ If add_indicator=True. add_indicatorboolean, default=False If True, a MissingIndicator transform will stack onto output of the imputer’s transform. This allows a predictive estimator to account for missingness despite imputation. If a feature has no missing values at fit/train time, the feature won’t appear on the missing indicator even if there are missing values at transform/test time. See also Multivariate imputation of missing values. Columns which only contained missing values at fit() are discarded upon transform() if strategy is not “constant”. >>> import cupy as cp >>> from cuml.preprocessing import SimpleImputer >>> imp_mean = SimpleImputer(missing_values=cp.nan, strategy='mean') >>> imp_mean.fit(cp.asarray([[7, 2, 3], [4, cp.nan, 6], [10, 5, 9]])) >>> X = [[cp.nan, 2, 3], [4, cp.nan, 6], [10, cp.nan, 9]] >>> print(imp_mean.transform(cp.asarray(X))) [[ 7. 2. 3. ] [ 4. 3.5 6. ] [10. 3.5 9. ]] statistics_array of shape (n_features,) The imputation fill value for each feature. Computing statistics can result in np.nan values. During transform(), features corresponding to np.nan statistics will be discarded. fit(X[, y]) Fit the imputer on X. get_param_names() Returns a list of hyperparameter names owned by this class. transform(X) Impute all missing values in X. fit(X, y=None) SimpleImputer[source]# Fit the imputer on X. X{array-like, sparse matrix}, shape (n_samples, n_features) Input data, where n_samples is the number of samples and n_features is the number of features. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. transform(X) SparseCumlArray[source]# Impute all missing values in X. X{array-like, sparse matrix}, shape (n_samples, n_features) The input data to complete. cuml.preprocessing.add_dummy_feature(X, value=1.0)[source]# Augment dataset with an additional dummy feature. This is useful for fitting an intercept term with implementations which cannot otherwise fit it directly. X{array-like, sparse matrix}, shape [n_samples, n_features] Value to use for the dummy feature. X{array, sparse matrix}, shape [n_samples, n_features + 1] Same data with dummy feature added as first column. >>> from cuml.preprocessing import add_dummy_feature >>> import cupy as cp >>> add_dummy_feature(cp.array([[0, 1], [1, 0]])) array([[1., 0., 1.], [1., 1., 0.]]) cuml.preprocessing.binarize(X, , threshold=0.0, copy=True)[source]# Boolean thresholding of array-like or sparse matrix X{array-like, sparse matrix}, shape [n_samples, n_features] The data to binarize, element by element. thresholdfloat, optional (0.0 by default) Feature values below or equal to this are replaced by 0, above it by 1. Threshold may not be less than 0 for operations on sparse matrices. copyboolean, optional, default True Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. See also Performs binarization using the Transformer API class cuml.compose.ColumnTransformer(args, kwargs)[source]# Applies transformers to columns of an array or dataframe. This estimator allows different columns or column subsets of the input to be transformed separately and the features generated by each transformer will be concatenated to form a single feature space. This is useful for heterogeneous or columnar data, to combine several feature extraction mechanisms or transformations into a single transformer. transformerslist of tuples List of (name, transformer, columns) tuples specifying the transformer objects to be applied to subsets of the data: Like in Pipeline and FeatureUnion, this allows the transformer and its parameters to be set using set_params and searched in grid search. transformer{‘drop’, ‘passthrough’} or estimator Estimator must support fit and transform. Special-cased strings ‘drop’ and ‘passthrough’ are accepted as well, to indicate to drop the columns or to pass them through untransformed, respectively. columnsstr, array-like of str, int, array-like of int, array-like of bool, slice or callable Indexes the data on its second axis. Integers are interpreted as positional columns, while strings can reference DataFrame columns by name. A scalar string or int should be used where transformer expects X to be a 1d array-like (vector), otherwise a 2d array will be passed to the transformer. A callable is passed the input data X and can return any of the above. To select multiple columns by name or dtype, you can use make_column_selector. remainder{‘drop’, ‘passthrough’} or estimator, default=’drop’ By default, only the specified columns in transformers are transformed and combined in the output, and the non-specified columns are dropped. (default of 'drop'). By specifying remainder= 'passthrough', all remaining columns that were not specified in transformers will be automatically passed through. This subset of columns is concatenated with the output of the transformers. By setting remainder to be an estimator, the remaining non-specified columns will use the remainder estimator. The estimator must support fit and transform. Note that using this feature requires that the DataFrame columns input at fit and transform have identical order. sparse_thresholdfloat, default=0.3 If the output of the different transformers contains sparse matrices, these will be stacked as a sparse matrix if the overall density is lower than this value. Use sparse_threshold=0 to always return dense. When the transformed output consists of all dense data, the stacked result will be dense, and this keyword will be ignored. n_jobsint, default=None Number of jobs to run in parallel. None means 1 unless in a joblib.parallel_backend context. -1 means using all processors. for more details. transformer_weightsdict, default=None Multiplicative weights for features per transformer. The output of the transformer is multiplied by these weights. Keys are transformer names, values the weights. verbosebool, default=False If True, the time elapsed while fitting each transformer will be printed as it is completed. See also Convenience function for combining the outputs of multiple transformer objects applied to column subsets of the original feature space. Convenience function for selecting columns based on datatype or the columns name with a regex pattern. The order of the columns in the transformed feature matrix follows the order of how the columns are specified in the transformers list. Columns of the original feature matrix that are not specified are dropped from the resulting transformed feature matrix, unless specified in the passthrough keyword. Those columns specified with passthrough are added at the right to the output of the transformers. >>> import cupy as cp >>> from cuml.compose import ColumnTransformer >>> from cuml.preprocessing import Normalizer >>> ct = ColumnTransformer( ... [("norm1", Normalizer(norm='l1'), [0, 1]), ... ("norm2", Normalizer(norm='l1'), slice(2, 4))]) >>> X = cp.array([[0., 1., 2., 2.], ... [1., 1., 0., 1.]]) >>> # Normalizer scales each row of X to unit norm. A separate scaling >>> # is applied for the two first and two last elements of each >>> # row independently. >>> ct.fit_transform(X) array([[0. , 1. , 0.5, 0.5], [0.5, 0.5, 0. , 1. ]]) The collection of fitted transformers as tuples of (name, fitted_transformer, column). fitted_transformer can be an estimator, ‘drop’, or ‘passthrough’. In case there were no columns selected, this will be the unfitted transformer. If there are remaining columns, the final element is a tuple of the form: (‘remainder’, transformer, remaining_columns) corresponding to the remainder parameter. If there are remaining columns, then len(transformers_)==len(transformers)+1, otherwise len(transformers_)==len(transformers). Access the fitted transformer by name. Boolean flag indicating whether the output of transform is a sparse matrix or a dense numpy array, which depends on the output of the individual transformers and the sparse_threshold fit(X, y=None) ColumnTransformer[source]# Fit all transformers using X. X{array-like, dataframe} of shape (n_samples, n_features) Input data, of which specified subsets are used to fit the transformers. yarray-like of shape (n_samples,…), default=None Targets for supervised learning. This estimator fit_transform(X, y=None) SparseCumlArray[source]# Fit all transformers, transform the data and concatenate results. X{array-like, dataframe} of shape (n_samples, n_features) Input data, of which specified subsets are used to fit the transformers. yarray-like of shape (n_samples,), default=None Targets for supervised learning. X_t{array-like, sparse matrix} of shape (n_samples, sum_n_components) hstack of results of transformers. sum_n_components is the sum of n_components (output dimension) over transformers. If any result is a sparse matrix, everything will be converted to sparse matrices. Get feature names from all transformers. feature_nameslist of strings Names of the features produced by transform. Get parameters for this estimator. Returns the parameters given in the constructor as well as the estimators contained within the transformers of the ColumnTransformer. deepbool, default=True If True, will return the parameters for this estimator and contained subobjects that are estimators. Parameter names mapped to their values. property named_transformers_# Access the fitted transformer by name. Read-only attribute to access any transformer by given name. Keys are transformer names and values are the fitted transformer objects. Set the parameters of this estimator. Valid parameter keys can be listed with get_params(). Note that you can directly set the parameters of the estimators contained in transformers of ColumnTransformer. transform(X) SparseCumlArray[source]# Transform X separately by each transformer, concatenate results. X{array-like, dataframe} of shape (n_samples, n_features) The data to be transformed by subset. X_t{array-like, sparse matrix} of shape (n_samples, sum_n_components) hstack of results of transformers. sum_n_components is the sum of n_components (output dimension) over transformers. If any result is a sparse matrix, everything will be converted to sparse matrices. class cuml.compose.make_column_selector(pattern=None, , dtype_include=None, dtype_exclude=None)[source]# Create a callable to select columns to be used with ColumnTransformer. make_column_selector() can select columns based on datatype or the columns name with a regex. When using multiple selection criteria, all criteria must match for a column to be selected. patternstr, default=None Name of columns containing this regex pattern will be included. If None, column selection will not be selected based on pattern. dtype_includecolumn dtype or list of column dtypes, default=None A selection of dtypes to include. For more details, see pandas.DataFrame.select_dtypes(). dtype_excludecolumn dtype or list of column dtypes, default=None A selection of dtypes to exclude. For more details, see pandas.DataFrame.select_dtypes(). Callable for column selection to be used by a ColumnTransformer. See also Class that allows combining the outputs of multiple transformer objects used on column subsets of the data into a single feature space. >>> from cuml.preprocessing import StandardScaler, OneHotEncoder >>> from cuml.compose import make_column_transformer >>> from cuml.compose import make_column_selector >>> import cupy as cp >>> import cudf >>> X = cudf.DataFrame({'city': ['London', 'London', 'Paris', 'Sallisaw'], ... 'rating': [5, 3, 4, 5]}) >>> ct = make_column_transformer( ... (StandardScaler(), ... make_column_selector(dtype_include=cp.number)), # rating ... (OneHotEncoder(), ... make_column_selector(dtype_include=object))) # city >>> ct.fit_transform(X) array([[ 0.90453403, 1. , 0. , 0. ], [-1.50755672, 1. , 0. , 0. ], [-0.30151134, 0. , 1. , 0. ], [ 0.90453403, 0. , 0. , 1. ]]) cuml.compose.make_column_transformer(transformers, remainder='drop', sparse_threshold=0.3, n_jobs=None, verbose=False)[source]# Construct a ColumnTransformer from the given transformers. This is a shorthand for the ColumnTransformer constructor; it does not require, and does not permit, naming the transformers. Instead, they will be given names automatically based on their types. It also does not allow weighting with transformer_weights. Tuples of the form (transformer, columns) specifying the transformer objects to be applied to subsets of the data: transformer{‘drop’, ‘passthrough’} or estimator Estimator must support fit and transform. Special-cased strings ‘drop’ and ‘passthrough’ are accepted as well, to indicate to drop the columns or to pass them through untransformed, respectively. columnsstr, array-like of str, int, array-like of int, slice, array-like of bool or callable Indexes the data on its second axis. Integers are interpreted as positional columns, while strings can reference DataFrame columns by name. A scalar string or int should be used where transformer expects X to be a 1d array-like (vector), otherwise a 2d array will be passed to the transformer. A callable is passed the input data X and can return any of the above. To select multiple columns by name or dtype, you can use make_column_selector. remainder{‘drop’, ‘passthrough’} or estimator, default=’drop’ By default, only the specified columns in transformers are transformed and combined in the output, and the non-specified columns are dropped. (default of 'drop'). By specifying remainder= 'passthrough', all remaining columns that were not specified in transformers will be automatically passed through. This subset of columns is concatenated with the output of the transformers. By setting remainder to be an estimator, the remaining non-specified columns will use the remainder estimator. The estimator must support fit and transform. sparse_thresholdfloat, default=0.3 If the transformed output consists of a mix of sparse and dense data, it will be stacked as a sparse matrix if the density is lower than this value. Use sparse_threshold=0 to always return dense. When the transformed output consists of all sparse or all dense data, the stacked result will be sparse or dense, respectively, and this keyword will be ignored. n_jobsint, default=None Number of jobs to run in parallel. None means 1 unless in a joblib.parallel_backend context. -1 means using all processors. See Glossary for more details. verbosebool, default=False If True, the time elapsed while fitting each transformer will be printed as it is completed. See also Class that allows combining the outputs of multiple transformer objects used on column subsets of the data into a single feature space. >>> from cuml.preprocessing import StandardScaler, OneHotEncoder >>> from cuml.compose import make_column_transformer >>> make_column_transformer( ... (StandardScaler(), ['numerical_column']), ... (OneHotEncoder(), ['categorical_column'])) ColumnTransformer(transformers=[('standardscaler', StandardScaler(...), ('onehotencoder', OneHotEncoder(...), Text Preprocessing (Single-GPU)# class cuml.preprocessing.text.stem.PorterStemmer(mode='NLTK_EXTENSIONS')[source]# A word stemmer based on the Porter stemming algorithm. Porter, M. “An algorithm for suffix stripping.” Program 14.3 (1980): 130-137. See http://www.tartarus.org/~martin/PorterStemmer/ for the homepage of the algorithm. Martin Porter has endorsed several modifications to the Porter algorithm since writing his original paper, and those extensions are included in the implementations on his website. Additionally, others have proposed further improvements to the algorithm, including NLTK contributors. Only below mode is supported currently PorterStemmer.NLTK_EXTENSIONS ☆ Implementation that includes further improvements devised by NLTK contributors or taken from other modified implementations found on the web. mode: Modes of stemming (Only supports (NLTK_EXTENSIONS) currently) >>> import cudf >>> from cuml.preprocessing.text.stem import PorterStemmer >>> stemmer = PorterStemmer() >>> word_str_ser = cudf.Series(['revival','singing','adjustable']) >>> print(stemmer.stem(word_str_ser)) 0 reviv 1 sing 2 adjust dtype: object stem(word_str_ser) Stem Words using Porter stemmer Stem Words using Porter stemmer A string series of words to stem Stemmed words strings series Feature and Label Encoding (Dask-based Multi-GPU)# class cuml.dask.preprocessing.LabelBinarizer(, client=None, *kwargs)[source]# A distributed version of LabelBinarizer for one-hot encoding a collection of labels. Create an array with labels and dummy encode them >>> import cupy as cp >>> import cupyx >>> from cuml.dask.preprocessing import LabelBinarizer >>> from dask_cuda import LocalCUDACluster >>> from dask.distributed import Client >>> import dask >>> cluster = LocalCUDACluster() >>> client = Client(cluster) >>> labels = cp.asarray([0, 5, 10, 7, 2, 4, 1, 0, 0, 4, 3, 2, 1], ... dtype=cp.int32) >>> labels = dask.array.from_array(labels) >>> lb = LabelBinarizer() >>> encoded = lb.fit_transform(labels) >>> print(encoded.compute()) [[1 0 0 0 0 0 0 0] [0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 1] [0 0 0 0 0 0 1 0] [0 0 1 0 0 0 0 0] [0 0 0 0 1 0 0 0] [0 1 0 0 0 0 0 0] [1 0 0 0 0 0 0 0] [1 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0] [0 0 0 1 0 0 0 0] [0 0 1 0 0 0 0 0] [0 1 0 0 0 0 0 0]] >>> decoded = lb.inverse_transform(encoded) >>> print(decoded.compute()) [ 0 5 10 7 2 4 1 0 0 4 3 2 1] >>> client.close() >>> cluster.close() fit(y) Fit label binarizer fit_transform(y) Fit the label encoder and return transformed labels inverse_transform(y[, threshold]) Invert a set of encoded labels back to original labels transform(y) Transform and return encoded labels class cuml.dask.preprocessing.LabelEncoder.LabelEncoder(, client=None, verbose=False, *kwargs)[source]# A cuDF-based implementation of ordinal label encoding handle_unknown{‘error’, ‘ignore’}, default=’error’ Whether to raise an error or ignore if an unknown categorical feature is present during transform (default is to raise). When this parameter is set to ‘ignore’ and an unknown category is encountered during transform or inverse transform, the resulting encoding will be null. Converting a categorical implementation to a numerical one >>> from dask_cuda import LocalCUDACluster >>> from dask.distributed import Client >>> import cudf >>> import dask_cudf >>> from cuml.dask.preprocessing import LabelEncoder >>> import pandas as pd >>> pd.set_option('display.max_colwidth', 2000) >>> cluster = LocalCUDACluster(threads_per_worker=1) >>> client = Client(cluster) >>> df = cudf.DataFrame({'num_col':[10, 20, 30, 30, 30], ... 'cat_col':['a','b','c','a','a']}) >>> ddf = dask_cudf.from_cudf(df, npartitions=2) >>> # There are two functionally equivalent ways to do this >>> le = LabelEncoder() >>> le.fit(ddf.cat_col) # le = le.fit(data.category) also works <cuml.dask.preprocessing.LabelEncoder.LabelEncoder object at 0x...> >>> encoded = le.transform(ddf.cat_col) >>> print(encoded.compute()) dtype: uint8 >>> # This method is preferred >>> le = LabelEncoder() >>> encoded = le.fit_transform(ddf.cat_col) >>> print(encoded.compute()) dtype: uint8 >>> # We can assign this to a new column >>> ddf = ddf.assign(encoded=encoded.values) >>> print(ddf.compute()) num_col cat_col encoded 0 10 a 0 1 20 b 1 2 30 c 2 3 30 a 0 4 30 a 0 >>> # We can also encode more data >>> test_data = cudf.Series(['c', 'a']) >>> encoded = le.transform(dask_cudf.from_cudf(test_data, ... npartitions=2)) >>> print(encoded.compute()) dtype: uint8 >>> # After train, ordinal label can be inverse_transform() back to >>> # string labels >>> ord_label = cudf.Series([0, 0, 1, 2, 1]) >>> ord_label = le.inverse_transform( ... dask_cudf.from_cudf(ord_label,npartitions=2)) >>> print(ord_label.compute()) 0 a 1 a 2 b 3 c 4 b dtype: object >>> client.close() >>> cluster.close() fit(y) Fit a LabelEncoder instance to a set of categories fit_transform(y[, delayed]) Simultaneously fit and transform an input inverse_transform(y[, delayed]) Convert the data back to the original representation. transform(y[, delayed]) Transform an input into its categorical keys. Fit a LabelEncoder instance to a set of categories Series containing the categories to be encoded. Its elements may or may not be unique A fitted instance of itself to allow method chaining Number of unique classes will be collected at the client. It’ll consume memory proportional to the number of unique classes. fit_transform(y, delayed=True)[source]# Simultaneously fit and transform an input This is functionally equivalent to (but faster than) LabelEncoder().fit(y).transform(y) inverse_transform(y, delayed=True)[source]# Convert the data back to the original representation. In case unknown categories are encountered (all zeros in the one-hot encoding), None is used to represent this category. Xdask_cudf Series The string representation of the categories. delayedbool (default = True) Whether to execute as a delayed task or eager. Distributed object containing the inverse transformed array. transform(y, delayed=True)[source]# Transform an input into its categorical keys. This is intended for use with small inputs relative to the size of the dataset. For fitting and transforming an entire dataset, prefer fit_transform. Input keys to be transformed. Its values should match the categories given to fit The ordinally encoded input series if a category appears that was not seen in fit class cuml.dask.preprocessing.OneHotEncoder(, client=None, verbose=False, kwargs)[source]# Encode categorical features as a one-hot numeric array. The input to this transformer should be a dask_cuDF.DataFrame or cupy dask.Array, denoting the values taken on by categorical features. The features are encoded using a one-hot (aka ‘one-of-K’ or ‘dummy’) encoding scheme. This creates a binary column for each category and returns a sparse matrix or dense array (depending on the sparse parameter). By default, the encoder derives the categories based on the unique values in each feature. Alternatively, you can also specify the categories manually. categories‘auto’, cupy.ndarray or cudf.DataFrame, default=’auto’ Categories (unique values) per feature. All categories are expected to fit on one GPU. ■ ‘auto’ : Determine categories automatically from the training data. ■ DataFrame/ndarray : categories[col] holds the categories expected in the feature col. drop‘first’, None or a dict, default=None Specifies a methodology to use to drop one of the categories per feature. This is useful in situations where perfectly collinear features cause problems, such as when feeding the resulting data into a neural network or an unregularized regression. ■ None : retain all features (the default). ■ ‘first’ : drop the first category in each feature. If only one category is present, the feature will be dropped entirely. ■ Dict : drop[col] is the category in feature col that should be dropped. sparsebool, default=False This feature was deactivated and will give an exception when True. The reason is because sparse matrix are not fully supported by cupy yet, causing incorrect values when computing one hot encodings. See cupy/cupy#3223 dtypenumber type, default=np.float Desired datatype of transform’s output. handle_unknown{‘error’, ‘ignore’}, default=’error’ Whether to raise an error or ignore if an unknown categorical feature is present during transform (default is to raise). When this parameter is set to ‘ignore’ and an unknown category is encountered during transform, the resulting one-hot encoded columns for this feature will be all zeros. In the inverse transform, an unknown category will be denoted as None. fit(X) Fit a multi-node multi-gpu OneHotEncoder to X. inverse_transform(X[, delayed]) Convert the data back to the original representation. transform(X[, delayed]) Transform X using one-hot encoding. Fit a multi-node multi-gpu OneHotEncoder to X. XDask cuDF DataFrame or CuPy backed Dask Array The data to determine the categories of each feature. inverse_transform(X, delayed=True)[source]# Convert the data back to the original representation. In case unknown categories are encountered (all zeros in the one-hot encoding), None is used to represent this category. XCuPy backed Dask Array, shape [n_samples, n_encoded_features] The transformed data. delayedbool (default = True) Whether to execute as a delayed task or eager. X_trDask cuDF DataFrame or CuPy backed Dask Array Distributed object containing the inverse transformed array. transform(X, delayed=True)[source]# Transform X using one-hot encoding. XDask cuDF DataFrame or CuPy backed Dask Array The data to encode. delayedbool (default = True) Whether to execute as a delayed task or eager. outDask cuDF DataFrame or CuPy backed Dask Array Distributed object containing the transformed input. Feature Extraction (Single-GPU)# class cuml.feature_extraction.text.CountVectorizer(input=None, encoding=None, decode_error=None, strip_accents=None, lowercase=True, preprocessor=None, tokenizer=None, stop_words=None, token_pattern=None, ngram_range=(1, 1), analyzer='word', max_df=1.0, min_df=1, max_features=None, vocabulary=None, binary=False, dtype=<class 'numpy.float32'>, delimiter=' ')[source]# Convert a collection of text documents to a matrix of token counts If you do not provide an a-priori dictionary then the number of features will be equal to the vocabulary size found by analyzing the data. lowercaseboolean, True by default Convert all characters to lowercase before tokenizing. preprocessorcallable or None (default) Override the preprocessing (string transformation) stage while preserving the tokenizing and n-grams generation steps. stop_wordsstring {‘english’}, list, or None (default) If ‘english’, a built-in stop word list for English is used. If a list, that list is assumed to contain stop words, all of which will be removed from the input documents. If None, no stop words will be used. max_df can be set to a value to automatically detect and filter stop words based on intra corpus document frequency of terms. ngram_rangetuple (min_n, max_n), default=(1, 1) The lower and upper boundary of the range of n-values for different word n-grams or char n-grams to be extracted. All values of n such such that min_n <= n <= max_n will be used. For example an ngram_range of (1, 1) means only unigrams, (1, 2) means unigrams and bigrams, and (2, 2) means only bigrams. analyzerstring, {‘word’, ‘char’, ‘char_wb’} Whether the feature should be made of word n-gram or character n-grams. Option ‘char_wb’ creates character n-grams only from text inside word boundaries; n-grams at the edges of words are padded with space. max_dffloat in range [0.0, 1.0] or int, default=1.0 When building the vocabulary ignore terms that have a document frequency strictly higher than the given threshold (corpus-specific stop words). If float, the parameter represents a proportion of documents, integer absolute counts. This parameter is ignored if vocabulary is not None. min_dffloat in range [0.0, 1.0] or int, default=1 When building the vocabulary ignore terms that have a document frequency strictly lower than the given threshold. This value is also called cut-off in the literature. If float, the parameter represents a proportion of documents, integer absolute counts. This parameter is ignored if vocabulary is not None. max_featuresint or None, default=None If not None, build a vocabulary that only consider the top max_features ordered by term frequency across the corpus. This parameter is ignored if vocabulary is not None. vocabularycudf.Series, optional If not given, a vocabulary is determined from the input documents. binaryboolean, default=False If True, all non zero counts are set to 1. This is useful for discrete probabilistic models that model binary events rather than integer counts. dtypetype, optional Type of the matrix returned by fit_transform() or transform(). delimiterstr, whitespace by default String used as a replacement for stop words if stop_words is not None. Typically the delimiting character between words is a good choice. Array mapping from feature integer indices to feature name. Terms that were ignored because they either: ★ occurred in too many documents (max_df) ★ occurred in too few documents (min_df) ★ were cut off by feature selection (max_features). This is only available if no vocabulary was given. fit(raw_documents[, y]) Build a vocabulary of all tokens in the raw documents. fit_transform(raw_documents[, y]) Build the vocabulary and return document-term matrix. get_feature_names() Array mapping from feature integer indices to feature name. inverse_transform(X) Return terms per document with nonzero entries in X. transform(raw_documents) Transform documents to document-term matrix. fit(raw_documents, y=None)[source]# Build a vocabulary of all tokens in the raw documents. raw_documentscudf.Series or pd.Series A Series of string documents fit_transform(raw_documents, y=None)[source]# Build the vocabulary and return document-term matrix. Equivalent to self.fit(X).transform(X) but preprocess X only once. raw_documentscudf.Series or pd.Series A Series of string documents Xcupy csr array of shape (n_samples, n_features) Document-term matrix. Array mapping from feature integer indices to feature name. A list of feature names. Return terms per document with nonzero entries in X. Xarray-like of shape (n_samples, n_features) Document-term matrix. X_invlist of cudf.Series of shape (n_samples,) List of Series of terms. Transform documents to document-term matrix. Extract token counts out of raw text documents using the vocabulary fitted with fit or the one provided to the constructor. raw_documentscudf.Series or pd.Series A Series of string documents Xcupy csr array of shape (n_samples, n_features) Document-term matrix. class cuml.feature_extraction.text.HashingVectorizer(input=None, encoding=None, decode_error=None, strip_accents=None, lowercase=True, preprocessor=None, tokenizer=None, stop_words=None, token_pattern=None, ngram_range=(1, 1), analyzer='word', n_features=1048576, binary=False, norm='l2', alternate_sign=True, dtype=<class 'numpy.float32'>, delimiter=' ')[source]# Convert a collection of text documents to a matrix of token occurrences It turns a collection of text documents into a cupyx.scipy.sparse matrix holding token occurrence counts (or binary occurrence information), possibly normalized as token frequencies if norm= ’l1’ or projected on the euclidean unit sphere if norm=’l2’. This text vectorizer implementation uses the hashing trick to find the token string name to feature integer index mapping. This strategy has several advantages: ○ it is very low memory scalable to large datasets as there is no need to store a vocabulary dictionary in memory which is even more important as GPU’s that are often memory constrained ○ it is fast to pickle and un-pickle as it holds no state besides the constructor parameters ○ it can be used in a streaming (partial fit) or parallel pipeline as there is no state computed during fit. There are also a couple of cons (vs using a CountVectorizer with an in-memory vocabulary): ○ there is no way to compute the inverse transform (from feature indices to string feature names) which can be a problem when trying to introspect which features are most important to a ○ there can be collisions: distinct tokens can be mapped to the same feature index. However in practice this is rarely an issue if n_features is large enough (e.g. 2 18 for text classification problems). ○ no IDF weighting as this would render the transformer stateful. The hash function employed is the signed 32-bit version of Murmurhash3. lowercasebool, default=True Convert all characters to lowercase before tokenizing. preprocessorcallable or None (default) Override the preprocessing (string transformation) stage while preserving the tokenizing and n-grams generation steps. stop_wordsstring {‘english’}, list, default=None If ‘english’, a built-in stop word list for English is used. There are several known issues with ‘english’ and you should consider an alternative. If a list, that list is assumed to contain stop words, all of which will be removed from the resulting tokens. Only applies if analyzer == 'word'. ngram_rangetuple (min_n, max_n), default=(1, 1) The lower and upper boundary of the range of n-values for different word n-grams or char n-grams to be extracted. All values of n such such that min_n <= n <= max_n will be used. For example an ngram_range of (1, 1) means only unigrams, (1, 2) means unigrams and bigrams, and (2, 2) means only bigrams. analyzerstring, {‘word’, ‘char’, ‘char_wb’} Whether the feature should be made of word n-gram or character n-grams. Option ‘char_wb’ creates character n-grams only from text inside word boundaries; n-grams at the edges of words are padded with space. n_featuresint, default=(2 20) The number of features (columns) in the output matrices. Small numbers of features are likely to cause hash collisions, but large numbers will cause larger coefficient dimensions in linear learners. binarybool, default=False. If True, all non zero counts are set to 1. This is useful for discrete probabilistic models that model binary events rather than integer counts. norm{‘l1’, ‘l2’}, default=’l2’ Norm used to normalize term vectors. None for no normalization. alternate_signbool, default=True When True, an alternating sign is added to the features as to approximately conserve the inner product in the hashed space even for small n_features. This approach is similar to sparse random projection. dtypetype, optional Type of the matrix returned by fit_transform() or transform(). delimiterstr, whitespace by default String used as a replacement for stop words if stop_words is not None. Typically the delimiting character between words is a good choice. >>> from cuml.feature_extraction.text import HashingVectorizer >>> import pandas as pd >>> corpus = [ ... 'This is the first document.', ... 'This document is the second document.', ... 'And this is the third one.', ... 'Is this the first document?', ... ] >>> vectorizer = HashingVectorizer(n_features=24) >>> X = vectorizer.fit_transform(pd.Series(corpus)) >>> print(X.shape) (4, 16) fit(X[, y]) This method only checks the input type and the model parameter. fit_transform(X[, y]) Transform a sequence of documents to a document-term matrix. partial_fit(X[, y]) Does nothing: This transformer is stateless This method is just there to mark the fact that this transformer can work in a streaming setup. transform(raw_documents) Transform documents to document-term matrix. fit(X, y=None)[source]# This method only checks the input type and the model parameter. It does not do anything meaningful as this transformer is stateless Xcudf.Series or pd.Series A Series of string documents fit_transform(X, y=None)[source]# Transform a sequence of documents to a document-term matrix. Xiterable over raw text documents, length = n_samples Samples. Each sample must be a text document (either bytes or unicode strings, file name or file object depending on the constructor argument) which will be tokenized and hashed. Ignored. This parameter exists only for compatibility with sklearn.pipeline.Pipeline. Xsparse CuPy CSR matrix of shape (n_samples, n_features) Document-term matrix. partial_fit(X, y=None)[source]# Does nothing: This transformer is stateless This method is just there to mark the fact that this transformer can work in a streaming setup. Xcudf.Series(A Series of string documents). Transform documents to document-term matrix. Extract token counts out of raw text documents using the vocabulary fitted with fit or the one provided to the constructor. raw_documentscudf.Series or pd.Series A Series of string documents Xsparse CuPy CSR matrix of shape (n_samples, n_features) Document-term matrix. class cuml.feature_extraction.text.TfidfVectorizer(input=None, encoding=None, decode_error=None, strip_accents=None, lowercase=True, preprocessor=None, tokenizer=None, stop_words=None, token_pattern=None, ngram_range=(1, 1), analyzer='word', max_df=1.0, min_df=1, max_features=None, vocabulary=None, binary=False, dtype=<class 'numpy.float32'>, delimiter=' ', norm='l2', use_idf=True, smooth_idf=True, sublinear_tf=False)[source]# Convert a collection of raw documents to a matrix of TF-IDF features. Equivalent to CountVectorizer followed by TfidfTransformer. lowercaseboolean, True by default Convert all characters to lowercase before tokenizing. preprocessorcallable or None (default) Override the preprocessing (string transformation) stage while preserving the tokenizing and n-grams generation steps. stop_wordsstring {‘english’}, list, or None (default) If ‘english’, a built-in stop word list for English is used. If a list, that list is assumed to contain stop words, all of which will be removed from the input documents. If None, no stop words will be used. max_df can be set to a value to automatically detect and filter stop words based on intra corpus document frequency of terms. ngram_rangetuple (min_n, max_n), default=(1, 1) The lower and upper boundary of the range of n-values for different word n-grams or char n-grams to be extracted. All values of n such such that min_n <= n <= max_n will be used. For example an ngram_range of (1, 1) means only unigrams, (1, 2) means unigrams and bigrams, and (2, 2) means only bigrams. analyzerstring, {‘word’, ‘char’, ‘char_wb’}, default=’word’ Whether the feature should be made of word n-gram or character n-grams. Option ‘char_wb’ creates character n-grams only from text inside word boundaries; n-grams at the edges of words are padded with space. max_dffloat in range [0.0, 1.0] or int, default=1.0 When building the vocabulary ignore terms that have a document frequency strictly higher than the given threshold (corpus-specific stop words). If float, the parameter represents a proportion of documents, integer absolute counts. This parameter is ignored if vocabulary is not None. min_dffloat in range [0.0, 1.0] or int, default=1 When building the vocabulary ignore terms that have a document frequency strictly lower than the given threshold. This value is also called cut-off in the literature. If float, the parameter represents a proportion of documents, integer absolute counts. This parameter is ignored if vocabulary is not None. max_featuresint or None, default=None If not None, build a vocabulary that only consider the top max_features ordered by term frequency across the corpus. This parameter is ignored if vocabulary is not None. vocabularycudf.Series, optional If not given, a vocabulary is determined from the input documents. binaryboolean, default=False If True, all non zero counts are set to 1. This is useful for discrete probabilistic models that model binary events rather than integer counts. dtypetype, optional Type of the matrix returned by fit_transform() or transform(). delimiterstr, whitespace by default String used as a replacement for stop words if stop_words is not None. Typically the delimiting character between words is a good choice. norm{‘l1’, ‘l2’}, default=’l2’ Each output row will have unit norm, either: ★ ‘l2’: Sum of squares of vector elements is 1. The cosine similarity between two vectors is their dot product when l2 norm has been applied. ★ ‘l1’: Sum of absolute values of vector elements is 1. use_idfbool, default=True Enable inverse-document-frequency reweighting. smooth_idfbool, default=True Smooth idf weights by adding one to document frequencies, as if an extra document was seen containing every term in the collection exactly once. Prevents zero divisions. sublinear_tfbool, default=False Apply sublinear tf scaling, i.e. replace tf with 1 + log(tf). The stop_words_ attribute can get large and increase the model size when pickling. This attribute is provided only for introspection and can be safely removed using delattr or set to None before pickling. This class is largely based on scikit-learn 0.23.1’s TfIdfVectorizer code, which is provided under the BSD-3 license. idf_array of shape (n_features) The inverse document frequency (IDF) vector; only defined if use_idf is True. Array mapping from feature integer indices to feature name. Terms that were ignored because they either: ★ occurred in too many documents (max_df) ★ occurred in too few documents (min_df) ★ were cut off by feature selection (max_features). This is only available if no vocabulary was given. fit(raw_documents) Learn vocabulary and idf from training set. fit_transform(raw_documents[, y]) Learn vocabulary and idf, return document-term matrix. get_feature_names() Array mapping from feature integer indices to feature name. transform(raw_documents) Transform documents to document-term matrix. Learn vocabulary and idf from training set. raw_documentscudf.Series or pd.Series A Series of string documents Fitted vectorizer. fit_transform(raw_documents, y=None)[source]# Learn vocabulary and idf, return document-term matrix. This is equivalent to fit followed by transform, but more efficiently implemented. raw_documentscudf.Series or pd.Series A Series of string documents Xcupy csr array of shape (n_samples, n_features) Tf-idf-weighted document-term matrix. Array mapping from feature integer indices to feature name. A list of feature names. Transform documents to document-term matrix. Uses the vocabulary and document frequencies (df) learned by fit (or fit_transform). raw_documentscudf.Series or pd.Series A Series of string documents Xcupy csr array of shape (n_samples, n_features) Tf-idf-weighted document-term matrix. Feature Extraction (Dask-based Multi-GPU)# class cuml.dask.feature_extraction.text.TfidfTransformer(, client=None, verbose=False, kwargs)[source]# Distributed TF-IDF transformer >>> import cupy as cp >>> from sklearn.datasets import fetch_20newsgroups >>> from sklearn.feature_extraction.text import CountVectorizer >>> from dask_cuda import LocalCUDACluster >>> from dask.distributed import Client >>> from cuml.dask.common import to_sparse_dask_array >>> from cuml.dask.naive_bayes import MultinomialNB >>> import dask >>> from cuml.dask.feature_extraction.text import TfidfTransformer >>> # Create a local CUDA cluster >>> cluster = LocalCUDACluster() >>> client = Client(cluster) >>> # Load corpus >>> twenty_train = fetch_20newsgroups(subset='train', ... shuffle=True, random_state=42) >>> cv = CountVectorizer() >>> xformed = cv.fit_transform(twenty_train.data).astype(cp.float32) >>> X = to_sparse_dask_array(xformed, client) >>> y = dask.array.from_array(twenty_train.target, asarray=False, ... fancy=False).astype(cp.int32) >>> multi_gpu_transformer = TfidfTransformer() >>> X_transformed = multi_gpu_transformer.fit_transform(X) >>> X_transformed.compute_chunk_sizes() >>> model = MultinomialNB() >>> model.fit(X_transformed, y) <cuml.dask.naive_bayes.naive_bayes.MultinomialNB object at 0x...> >>> result = model.score(X_transformed, y) >>> print(result) >>> client.close() >>> cluster.close() fit(X[, y]) Fit distributed TFIDF Transformer fit_transform(X[, y]) Fit distributed TFIDFTransformer and then transform the given set of data samples. transform(X[, y]) Use distributed TFIDFTransformer to transform the given set of data samples. fit(X, y=None)[source]# Fit distributed TFIDF Transformer Xdask.Array with blocks containing dense or sparse cupy arrays cuml.dask.feature_extraction.text.TfidfTransformer instance fit_transform(X, y=None)[source]# Fit distributed TFIDFTransformer and then transform the given set of data samples. Xdask.Array with blocks containing dense or sparse cupy arrays dask.Array with blocks containing transformed sparse cupy arrays transform(X, y=None)[source]# Use distributed TFIDFTransformer to transform the given set of data samples. Xdask.Array with blocks containing dense or sparse cupy arrays dask.Array with blocks containing transformed sparse cupy arrays Dataset Generation (Single-GPU)# Determines random number generation for dataset creation. Pass an int for reproducible output across multiple function calls. cuml.datasets.make_blobs(n_samples=100, n_features=2, centers=None, cluster_std=1.0, center_box=(-10.0, 10.0), shuffle=True, random_state=None, return_centers=False, order='F', dtype='float32') Generate isotropic Gaussian blobs for clustering. n_samplesint or array-like, optional (default=100) If int, it is the total number of points equally divided among clusters. If array-like, each element of the sequence indicates the number of samples per cluster. n_featuresint, optional (default=2) The number of features for each sample. centersint or array of shape [n_centers, n_features], optional (default=None) The number of centers to generate, or the fixed center locations. If n_samples is an int and centers is None, 3 centers are generated. If n_samples is array-like, centers must be either None or an array of length equal to the length of n_samples. cluster_stdfloat or sequence of floats, optional (default=1.0) The standard deviation of the clusters. center_boxpair of floats (min, max), optional (default=(-10.0, 10.0)) The bounding box for each cluster center when centers are generated at random. shuffleboolean, optional (default=True) Shuffle the samples. random_stateint, RandomState instance, default=None Determines random number generation for dataset creation. Pass an int for reproducible output across multiple function calls. return_centersbool, optional (default=False) If True, then return the centers of each cluster order: str, optional (default=’F’) The order of the generated samples dtypestr, optional (default=’float32’) Dtype of the generated samples Xdevice array of shape [n_samples, n_features] The generated samples. ydevice array of shape [n_samples] The integer labels for cluster membership of each sample. centersdevice array, shape [n_centers, n_features] The centers of each cluster. Only returned if return_centers=True. See also a more intricate variant >>> from sklearn.datasets import make_blobs >>> X, y = make_blobs(n_samples=10, centers=3, n_features=2, ... random_state=0) >>> print(X.shape) (10, 2) >>> y array([0, 0, 1, 0, 2, 2, 2, 1, 1, 0]) >>> X, y = make_blobs(n_samples=[3, 3, 4], centers=None, n_features=2, ... random_state=0) >>> print(X.shape) (10, 2) >>> y array([0, 1, 2, 0, 2, 2, 2, 1, 1, 0]) cuml.datasets.make_classification(n_samples=100, n_features=20, n_informative=2, n_redundant=2, n_repeated=0, n_classes=2, n_clusters_per_class=2, weights=None, flip_y=0.01, class_sep=1.0, hypercube=True, shift=0.0, scale=1.0, shuffle=True, random_state=None, order='F', dtype='float32', _centroids=None, _informative_covariance=None, _redundant_covariance=None, _repeated_indices Generate a random n-class classification problem. This initially creates clusters of points normally distributed (std=1) about vertices of an n_informative-dimensional hypercube with sides of length 2class_sep and assigns an equal number of clusters to each class. It introduces interdependence between these features and adds various types of further noise to the data. Without shuffling, X horizontally stacks features in the following order: the primary n_informative features, followed by n_redundant linear combinations of the informative features, followed by n_repeated duplicates, drawn randomly with replacement from the informative and redundant features. The remaining features are filled with random noise. Thus, without shuffling, all useful features are contained in the columns X[:, :n_informative + n_redundant + n_repeated]. n_samplesint, optional (default=100) The number of samples. n_featuresint, optional (default=20) The total number of features. These comprise n_informative informative features, n_redundant redundant features, n_repeated duplicated features and n_features-n_informative- n_redundant-n_repeated useless features drawn at random. n_informativeint, optional (default=2) The number of informative features. Each class is composed of a number of gaussian clusters each located around the vertices of a hypercube in a subspace of dimension n_informative. For each cluster, informative features are drawn independently from N(0, 1) and then randomly linearly combined within each cluster in order to add covariance. The clusters are then placed on the vertices of the hypercube. n_redundantint, optional (default=2) The number of redundant features. These features are generated as random linear combinations of the informative features. n_repeatedint, optional (default=0) The number of duplicated features, drawn randomly from the informative and the redundant features. n_classesint, optional (default=2) The number of classes (or labels) of the classification problem. n_clusters_per_classint, optional (default=2) The number of clusters per class. weightsarray-like of shape (n_classes,) or (n_classes - 1,), (default=None) The proportions of samples assigned to each class. If None, then classes are balanced. Note that if len(weights) == n_classes - 1, then the last class weight is automatically inferred. More than n_samples samples may be returned if the sum of weights exceeds 1. flip_yfloat, optional (default=0.01) The fraction of samples whose class is assigned randomly. Larger values introduce noise in the labels and make the classification task harder. class_sepfloat, optional (default=1.0) The factor multiplying the hypercube size. Larger values spread out the clusters/classes and make the classification task easier. hypercubeboolean, optional (default=True) If True, the clusters are put on the vertices of a hypercube. If False, the clusters are put on the vertices of a random polytope. shiftfloat, array of shape [n_features] or None, optional (default=0.0) Shift features by the specified value. If None, then features are shifted by a random value drawn in [-class_sep, class_sep]. scalefloat, array of shape [n_features] or None, optional (default=1.0) Multiply features by the specified value. If None, then features are scaled by a random value drawn in [1, 100]. Note that scaling happens after shifting. shuffleboolean, optional (default=True) Shuffle the samples and the features. random_stateint, RandomState instance or None (default) Determines random number generation for dataset creation. Pass an int for reproducible output across multiple function calls. See Glossary. order: str, optional (default=’F’) The order of the generated samples dtypestr, optional (default=’float32’) Dtype of the generated samples _centroids: array of centroids of shape (n_clusters, n_informative) _informative_covariance: array for covariance between informative features of shape (n_clusters, n_informative, n_informative) _redundant_covariance: array for covariance between redundant features of shape (n_informative, n_redundant) _repeated_indices: array of indices for the repeated features of shape (n_repeated, ) Xdevice array of shape [n_samples, n_features] The generated samples. ydevice array of shape [n_samples] The integer labels for class membership of each sample. The algorithm is adapted from Guyon [1] and was designed to generate the “Madelon” dataset. How we optimized for GPUs: 1. Firstly, we generate X from a standard univariate instead of zeros. This saves memory as we don’t need to generate univariates each time for each feature class (informative, repeated, etc.) while also providing the added speedup of generating a big matrix on GPU 2. We generate order=F construction. We exploit the fact that X is a generated from a univariate normal, and covariance is introduced with matrix multiplications. Which means, we can generate X as a 1D array and just reshape it to the desired order, which only updates the metadata and eliminates copies 3. Lastly, we also shuffle by construction. Centroid indices are permuted for each sample, and then we construct the data for each centroid. This shuffle works for both order=C and order =F and eliminates any need for secondary copies I. Guyon, “Design of experiments for the NIPS 2003 variable selection benchmark”, 2003. >>> from cuml.datasets.classification import make_classification >>> X, y = make_classification(n_samples=10, n_features=4, ... n_informative=2, n_classes=2, ... random_state=10) >>> print(X) [[-1.7974224 0.24425316 0.39062843 -0.38293394] [ 0.6358963 1.4161923 0.06970507 -0.16085647] [-0.22802866 -1.1827322 0.3525861 0.276615 ] [ 1.7308872 0.43080002 0.05048406 0.29837844] [-1.9465544 0.5704457 -0.8997551 -0.27898186] [ 1.0575483 -0.9171263 0.09529338 0.01173469] [ 0.7917619 -1.0638094 -0.17599393 -0.06420116] [-0.6686142 -0.13951421 -0.6074711 0.21645583] [-0.88968956 -0.914443 0.1302423 0.02924336] [-0.8817671 -0.84549576 0.1845096 0.02556021]] >>> print(y) [1 0 1 1 1 1 1 1 1 0] cuml.datasets.make_regression(n_samples=100, n_features=2, n_informative=2, n_targets=1, bias=0.0, effective_rank=None, tail_strength=0.5, noise=0.0, shuffle=True, coef=False, random_state=None, dtype='single', handle=None) Union[Tuple[CumlArray, CumlArray], Tuple[CumlArray, CumlArray, CumlArray]][source]# Generate a random regression problem. See https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_regression.html n_samplesint, optional (default=100) The number of samples. n_featuresint, optional (default=2) The number of features. n_informativeint, optional (default=2) The number of informative features, i.e., the number of features used to build the linear model used to generate the output. n_targetsint, optional (default=1) The number of regression targets, i.e., the dimension of the y output vector associated with a sample. By default, the output is a scalar. biasfloat, optional (default=0.0) The bias term in the underlying linear model. effective_rankint or None, optional (default=None) if not None: The approximate number of singular vectors required to explain most of the input data by linear combinations. Using this kind of singular spectrum in the input allows the generator to reproduce the correlations often observed in practice. if None: The input set is well conditioned, centered and gaussian with unit variance. tail_strengthfloat between 0.0 and 1.0, optional (default=0.5) The relative importance of the fat noisy tail of the singular values profile if effective_rank is not None. noisefloat, optional (default=0.0) The standard deviation of the gaussian noise applied to the output. shuffleboolean, optional (default=True) Shuffle the samples and the features. coefboolean, optional (default=False) If True, the coefficients of the underlying linear model are returned. random_stateint, RandomState instance or None (default) Seed for the random number generator for dataset creation. dtype: string or numpy dtype (default: ‘single’) Type of the data. Possible values: float32, float64, ‘single’, ‘float’ or ‘double’. handle: cuml.Handle If it is None, a new one is created just for this function call outdevice array of shape [n_samples, n_features] The input samples. valuesdevice array of shape [n_samples, n_targets] The output values. coefdevice array of shape [n_features, n_targets], optional The coefficient of the underlying linear model. It is returned only if coef is True. >>> from cuml.datasets.regression import make_regression >>> from cuml.linear_model import LinearRegression >>> # Create regression problem >>> data, values = make_regression(n_samples=200, n_features=12, ... n_informative=7, bias=-4.2, ... noise=0.3, random_state=10) >>> # Perform a linear regression on this problem >>> lr = LinearRegression(fit_intercept = True, normalize = False, ... algorithm = "eig") >>> reg = lr.fit(data, values) >>> print(reg.coef_) [-2.6980877e-02 7.7027252e+01 1.1498465e+01 8.5468025e+00 5.8548538e+01 6.0772545e+01 3.6876743e+01 4.0023815e+01 4.3908358e-03 -2.0275116e-02 3.5066366e-02 -3.4512520e-02] cuml.datasets.make_arima(batch_size=1000, n_obs=100, order=(1, 1, 1), seasonal_order=(0, 0, 0, 0), intercept=False, random_state=None, dtype='double', handle=None)[source]# Generates a dataset of time series by simulating an ARIMA process of a given order. batch_size: int Number of time series to generate n_obs: int Number of observations per series orderTuple[int, int, int] Order (p, d, q) of the simulated ARIMA process seasonal_order: Tuple[int, int, int, int] Seasonal ARIMA order (P, D, Q, s) of the simulated ARIMA process intercept: bool or int Whether to include a constant trend mu in the simulated ARIMA process random_state: int, RandomState instance or None (default) Seed for the random number generator for dataset creation. dtype: string or numpy dtype (default: ‘single’) Type of the data. Possible values: float32, float64, ‘single’, ‘float’ or ‘double’ handle: cuml.Handle If it is None, a new one is created just for this function call out: array-like, shape (n_obs, batch_size) Array of the requested type containing the generated dataset from cuml.datasets import make_arima y = make_arima(1000, 100, (2,1,2), (0,1,2,12), 0) Dataset Generation (Dask-based Multi-GPU)# cuml.dask.datasets.blobs.make_blobs(n_samples=100, n_features=2, centers=None, cluster_std=1.0, n_parts=None, center_box=(-10, 10), shuffle=True, random_state=None, return_centers=False, verbose= False, order='F', dtype='float32', client=None, workers=None)[source]# Makes labeled Dask-Cupy arrays containing blobs for a randomly generated set of centroids. This function calls make_blobs from cuml.datasets on each Dask worker and aggregates them into a single Dask Dataframe. For more information on Scikit-learn’s make_blobs. number of rows number of features centersint or array of shape [n_centers, n_features], optional (default=None) The number of centers to generate, or the fixed center locations. If n_samples is an int and centers is None, 3 centers are generated. If n_samples is array-like, centers must be either None or an array of length equal to the length of n_samples. cluster_stdfloat (default = 1.0) standard deviation of points around centroid n_partsint (default = None) number of partitions to generate (this can be greater than the number of workers) center_boxtuple (int, int) (default = (-10, 10)) the bounding box which constrains all the centroids random_stateint (default = None) sets random seed (or use None to reinitialize each time) return_centersbool, optional (default=False) If True, then return the centers of each cluster verboseint or boolean (default = False) Logging level. shufflebool (default=False) Shuffles the samples on each worker. order: str, optional (default=’F’) The order of the generated samples dtypestr, optional (default=’float32’) Dtype of the generated samples clientdask.distributed.Client (optional) Dask client to use workersoptional, list of strings Dask addresses of workers to use for computation. If None, all available Dask workers will be used. (e.g. : workers = list(client.scheduler_info()['workers'].keys())) Xdask.array backed by CuPy array of shape [n_samples, n_features] The input samples. ydask.array backed by CuPy array of shape [n_samples] The output values. centersdask.array backed by CuPy array of shape [n_centers, n_features], optional The centers of the underlying blobs. It is returned only if return_centers is True. >>> from dask_cuda import LocalCUDACluster >>> from dask.distributed import Client >>> from cuml.dask.datasets import make_blobs >>> cluster = LocalCUDACluster(threads_per_worker=1) >>> client = Client(cluster) >>> workers = list(client.scheduler_info()['workers'].keys()) >>> X, y = make_blobs(1000, 10, centers=42, cluster_std=0.1, ... workers=workers) >>> client.close() >>> cluster.close() cuml.dask.datasets.classification.make_classification(n_samples=100, n_features=20, n_informative=2, n_redundant=2, n_repeated=0, n_classes=2, n_clusters_per_class=2, weights=None, flip_y=0.01, class_sep=1.0, hypercube=True, shift=0.0, scale=1.0, shuffle=True, random_state=None, order='F', dtype='float32', n_parts=None, client=None)[source]# Generate a random n-class classification problem. This initially creates clusters of points normally distributed (std=1) about vertices of an n_informative-dimensional hypercube with sides of length 2 * class_sep and assigns an equal number of clusters to each class. It introduces interdependence between these features and adds various types of further noise to the data. Without shuffling, X horizontally stacks features in the following order: the primary n_informative features, followed by n_redundant linear combinations of the informative features, followed by n_repeated duplicates, drawn randomly with replacement from the informative and redundant features. The remaining features are filled with random noise. Thus, without shuffling, all useful features are contained in the columns X[:, :n_informative + n_redundant + n_repeated]. n_samplesint, optional (default=100) The number of samples. n_featuresint, optional (default=20) The total number of features. These comprise n_informative informative features, n_redundant redundant features, n_repeated duplicated features and n_features-n_informative- n_redundant-n_repeated useless features drawn at random. n_informativeint, optional (default=2) The number of informative features. Each class is composed of a number of gaussian clusters each located around the vertices of a hypercube in a subspace of dimension n_informative. For each cluster, informative features are drawn independently from N(0, 1) and then randomly linearly combined within each cluster in order to add covariance. The clusters are then placed on the vertices of the hypercube. n_redundantint, optional (default=2) The number of redundant features. These features are generated as random linear combinations of the informative features. n_repeatedint, optional (default=0) The number of duplicated features, drawn randomly from the informative and the redundant features. n_classesint, optional (default=2) The number of classes (or labels) of the classification problem. n_clusters_per_classint, optional (default=2) The number of clusters per class. weightsarray-like of shape (n_classes,) or (n_classes - 1,) , (default=None) The proportions of samples assigned to each class. If None, then classes are balanced. Note that if len(weights) == n_classes - 1, then the last class weight is automatically inferred. More than n_samples samples may be returned if the sum of weights exceeds 1. flip_yfloat, optional (default=0.01) The fraction of samples whose class is assigned randomly. Larger values introduce noise in the labels and make the classification task harder. class_sepfloat, optional (default=1.0) The factor multiplying the hypercube size. Larger values spread out the clusters/classes and make the classification task easier. hypercubeboolean, optional (default=True) If True, the clusters are put on the vertices of a hypercube. If False, the clusters are put on the vertices of a random polytope. shiftfloat, array of shape [n_features] or None, optional (default=0.0) Shift features by the specified value. If None, then features are shifted by a random value drawn in [-class_sep, class_sep]. scalefloat, array of shape [n_features] or None, optional (default=1.0) Multiply features by the specified value. If None, then features are scaled by a random value drawn in [1, 100]. Note that scaling happens after shifting. shuffleboolean, optional (default=True) Shuffle the samples and the features. random_stateint, RandomState instance or None (default) Determines random number generation for dataset creation. Pass an int for reproducible output across multiple function calls. See Glossary. order: str, optional (default=’F’) The order of the generated samples dtypestr, optional (default=’float32’) Dtype of the generated samples n_partsint (default = None) number of partitions to generate (this can be greater than the number of workers) Xdask.array backed by CuPy array of shape [n_samples, n_features] The generated samples. ydask.array backed by CuPy array of shape [n_samples] The integer labels for class membership of each sample. How we extended the dask MNMG version from the single GPU version: 1. We generate centroids of shape (n_centroids, n_informative) 2. We generate an informative covariance of shape (n_centroids, n_informative, n_informative) 3. We generate a redundant covariance of shape (n_informative, n_redundant) 4. We generate the indices for the repeated features We pass along the references to the futures of the above arrays with each part to the single GPU cuml.datasets.classification.make_classification so that each part (and worker) has access to the correct values to generate data from the same covariances >>> from dask.distributed import Client >>> from dask_cuda import LocalCUDACluster >>> from cuml.dask.datasets.classification import make_classification >>> cluster = LocalCUDACluster() >>> client = Client(cluster) >>> X, y = make_classification(n_samples=10, n_features=4, ... random_state=1, n_informative=2, ... n_classes=2) >>> print(X.compute()) [[-1.1273878 1.2844919 -0.32349187 0.1595734 ] [ 0.80521786 -0.65946865 -0.40753683 0.15538901] [ 1.0404129 -1.481386 1.4241115 1.2664981 ] [-0.92821544 -0.6805706 -0.26001272 0.36004275] [-1.0392245 -1.1977317 0.16345565 -0.21848428] [ 1.2273135 -0.529214 2.4799604 0.44108105] [-1.9163864 -0.39505136 -1.9588828 -1.8881643 ] [-0.9788184 -0.89851004 -0.08339313 0.1130247 ] [-1.0549078 -0.8993015 -0.11921967 0.04821599] [-1.8388828 -1.4063598 -0.02838472 -1.0874642 ]] >>> print(y.compute()) [1 0 0 0 0 1 0 0 0 0] >>> client.close() >>> cluster.close() cuml.dask.datasets.regression.make_low_rank_matrix(n_samples=100, n_features=100, effective_rank=10, tail_strength=0.5, random_state=None, n_parts=1, n_samples_per_part=None, dtype='float32') Generate a mostly low rank matrix with bell-shaped singular values n_samplesint, optional (default=100) The number of samples. n_featuresint, optional (default=100) The number of features. effective_rankint, optional (default=10) The approximate number of singular vectors required to explain most of the data by linear combinations. tail_strengthfloat between 0.0 and 1.0, optional (default=0.5) The relative importance of the fat noisy tail of the singular values profile. random_stateint, CuPy RandomState instance, Dask RandomState instance or None (default) Determines random number generation for dataset creation. Pass an int for reproducible output across multiple function calls. n_partsint, optional (default=1) The number of parts of work. dtype: str, optional (default=’float32’) dtype of generated data XDask-CuPy array of shape [n_samples, n_features] The matrix. cuml.dask.datasets.regression.make_regression(n_samples=100, n_features=100, n_informative=10, n_targets=1, bias=0.0, effective_rank=None, tail_strength=0.5, noise=0.0, shuffle=False, coef=False, random_state=None, n_parts=1, n_samples_per_part=None, order='F', dtype='float32', client=None, use_full_low_rank=True)[source]# Generate a random regression problem. The input set can either be well conditioned (by default) or have a low rank-fat tail singular profile. The output is generated by applying a (potentially biased) random linear regression model with “n_informative” nonzero regressors to the previously generated input and some gaussian centered noise with some adjustable scale. n_samplesint, optional (default=100) The number of samples. n_featuresint, optional (default=100) The number of features. n_informativeint, optional (default=10) The number of informative features, i.e., the number of features used to build the linear model used to generate the output. n_targetsint, optional (default=1) The number of regression targets, i.e., the dimension of the y output vector associated with a sample. By default, the output is a scalar. biasfloat, optional (default=0.0) The bias term in the underlying linear model. effective_rankint or None, optional (default=None) if not None: The approximate number of singular vectors required to explain most of the input data by linear combinations. Using this kind of singular spectrum in the input allows the generator to reproduce the correlations often observed in practice. if None: The input set is well conditioned, centered and gaussian with unit variance. tail_strengthfloat between 0.0 and 1.0, optional (default=0.5) The relative importance of the fat noisy tail of the singular values profile if “effective_rank” is not None. noisefloat, optional (default=0.0) The standard deviation of the gaussian noise applied to the output. shuffleboolean, optional (default=False) Shuffle the samples and the features. coefboolean, optional (default=False) If True, the coefficients of the underlying linear model are returned. random_stateint, CuPy RandomState instance, Dask RandomState instance or None (default) Determines random number generation for dataset creation. Pass an int for reproducible output across multiple function calls. n_partsint, optional (default=1) The number of parts of work. orderstr, optional (default=’F’) Row-major or Col-major dtype: str, optional (default=’float32’) dtype of generated data use_full_low_rankboolean (default=True) Whether to use the entire dataset to generate the low rank matrix. If False, it creates a low rank covariance and uses the corresponding covariance to generate a multivariate normal distribution on the remaining chunks XDask-CuPy array of shape [n_samples, n_features] The input samples. yDask-CuPy array of shape [n_samples] or [n_samples, n_targets] The output values. coefDask-CuPy array of shape [n_features] or [n_features, n_targets], optional The coefficient of the underlying linear model. It is returned only if coef is True. Known Performance Limitations: 1. When effective_rank is set and use_full_low_rank is True, we cannot generate order F by construction, and an explicit transpose is performed on each part. This may cause memory to spike (other parameters make order F by construction) 2. When n_targets > 1 and order = 'F' as above, we have to explicitly transpose the y array. If coef = True, then we also explicitly transpose the ground_truth array 3. When shuffle = True and order = F, there are memory spikes to shuffle the F order arrays If out-of-memory errors are encountered in any of the above configurations, try increasing the n_parts parameter. Metrics (regression, classification, and distance)# Metrics (clustering and manifold learning)# V-measure metric of a cluster labeling given a ground truth. The V-measure is the harmonic mean between homogeneity and completeness: v = (1 + beta) * homogeneity * completeness / (beta * homogeneity + completeness) This metric is independent of the absolute values of the labels: a permutation of the class or cluster label values won’t change the score value in any way. This metric is furthermore symmetric: switching label_true with label_pred will return the same score value. This can be useful to measure the agreement of two independent label assignments strategies on the same dataset when the real ground truth is not known. labels_predarray-like (device or host) shape = (n_samples,) The labels predicted by the model for the test dataset. Acceptable formats: cuDF DataFrame, NumPy ndarray, Numba device ndarray, cuda array interface compliant array like CuPy labels_truearray-like (device or host) shape = (n_samples,) The ground truth labels (ints) of the test dataset. Acceptable formats: cuDF DataFrame, NumPy ndarray, Numba device ndarray, cuda array interface compliant array like CuPy betafloat, default=1.0 Ratio of weight attributed to homogeneity vs completeness. If beta is greater than 1, completeness is weighted more strongly in the calculation. If beta is less than 1, homogeneity is weighted more strongly. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. score between 0.0 and 1.0. 1.0 stands for perfectly complete labeling class cuml.benchmark.algorithms.AlgorithmPair(cpu_class, cuml_class, shared_args, cuml_args={}, cpu_args={}, name=None, accepts_labels=True, cpu_data_prep_hook=None, cuml_data_prep_hook=None, accuracy_function=None, bench_func=<function fit>, setup_cpu_func=None, setup_cuml_func=None)[source]# Wraps a cuML algorithm and (optionally) a cpu-based algorithm (typically scikit-learn, but does not need to be as long as it offers fit and predict or transform methods). Provides mechanisms to run each version with default arguments. If no CPU-based version of the algorithm is available, pass None for the cpu_class when instantiating Class for CPU version of algorithm. Set to None if not available. Class for cuML algorithm Arguments passed to both implementations’s initializer Arguments only passed to cuml’s initializer cpu_args dict Arguments only passed to sklearn’s initializer If True, the fit methods expects both X and y inputs. Otherwise, it expects only an X input. data_prep_hookfunction (data -> data) Optional function to run on input data before passing to fit accuracy_functionfunction (y_test, y_pred) Function that returns a scalar representing accuracy bench_funccustom function to perform fit/predict/transform run_cpu(data[, bench_args]) Runs the cpu-based algorithm's fit method on specified data run_cuml(data[, bench_args]) Runs the cuml-based algorithm's fit method on specified data run_cpu(data, bench_args={}, override_setup_args)[source]# Runs the cpu-based algorithm’s fit method on specified data run_cuml(data, bench_args={}, override_setup_args)[source]# Runs the cuml-based algorithm’s fit method on specified data Returns the algorithm pair with the name ‘name’ (case-insensitive) Returns all defined AlgorithmPair objects Wrappers to run ML benchmarks class cuml.benchmark.runners.AccuracyComparisonRunner(bench_rows, bench_dims, dataset_name='blobs', input_type='numpy', test_fraction=0.1, n_reps=1)[source]# Wrapper to run an algorithm with multiple dataset sizes and compute accuracy and speedup of cuml relative to sklearn baseline. class cuml.benchmark.runners.BenchmarkTimer(reps=1)[source]# Provides a context manager that runs a code block reps times and records results to the instance variable timings. Use like: timer = BenchmarkTimer(rep=5) for _ in timer.benchmark_runs(): ... do something ... class cuml.benchmark.runners.SpeedupComparisonRunner(bench_rows, bench_dims, dataset_name='blobs', input_type='numpy', n_reps=1)[source]# Wrapper to run an algorithm with multiple dataset sizes and compute speedup of cuml relative to sklearn baseline. cuml.benchmark.runners.run_variations(algos, dataset_name, bench_rows, bench_dims, param_override_list=[{}], cuml_param_override_list=[{}], cpu_param_override_list=[{}], dataset_param_override_list=[{}], dtype=<class 'numpy.float32'>, input_type='numpy', test_fraction=0.1, run_cpu=True, device_list=('gpu',), raise_on_error=False, n_reps=1)[source]# Runs each algo in algos once per bench_rows X bench_dims X params_override_list X cuml_param_override_list combination and returns a dataframe containing timing and accuracy data. algosstr or list Name of algorithms to run and evaluate Name of dataset to use bench_rowslist of int Dataset row counts to test bench_dimslist of int Dataset column counts to test param_override_listlist of dict Dicts containing parameters to pass to __init__. Each dict specifies parameters to override in one run of the algorithm. cuml_param_override_listlist of dict Dicts containing parameters to pass to __init__ of the cuml algo only. cpu_param_override_listlist of dict Dicts containing parameters to pass to __init__ of the cpu algo only. Dicts containing parameters to pass to dataset generator function dtype: [np.float32\|np.float64] Specifies the dataset precision to be used for benchmarking. The fraction of data to use for testing. If True, run the cpu-based algorithm for comparison Data generators for cuML benchmarks The main entry point for consumers is gen_data, which wraps the underlying data generators. Notes when writing new generators: Each generator is a function that accepts: ☆ n_samples (set to 0 for ‘default’) ☆ n_features (set to 0 for ‘default’) ☆ random_state ☆ (and optional generator-specific parameters) The function should return a 2-tuple (X, y), where X is a Pandas dataframe and y is a Pandas series. If the generator does not produce labels, it can return (X, None) A set of helper functions (convert_) can convert these to alternative formats. Future revisions may support generating cudf dataframes or GPU arrays directly instead. cuml.benchmark.datagen.gen_data(dataset_name, dataset_format, n_samples=0, n_features=0, test_fraction=0.0, datasets_root_dir='.', dtype=<class 'numpy.float32'>, kwargs)[source]# Returns a tuple of data from the specified generator. Dataset to use. Can be a synthetic generator (blobs or regression) or a specified dataset (higgs currently, others coming soon) Type of data to return. (One of cudf, numpy, pandas, gpuarray) Total number of samples to loaded including training and testing samples Fraction of the dataset to partition randomly into the test set. If this is 0.0, no test set will be created. (train_features, train_labels, test_features, test_labels) tuple containing matrices or dataframes of the requested format. test_features and test_labels may be None if no splitting was done. Regression and Classification# Linear Regression# class cuml.LinearRegression(, algorithm='eig', fit_intercept=True, copy_X=None, normalize=False, handle=None, verbose=False, output_type=None)# LinearRegression is a simple machine learning model where the response y is modelled by a linear combination of the predictors in X. cuML’s LinearRegression expects either a cuDF DataFrame or a NumPy matrix and provides 2 algorithms SVD and Eig to fit a linear model. SVD is more stable, but Eig (default) is much faster. algorithm{‘svd’, ‘eig’, ‘qr’, ‘svd-qr’, ‘svd-jacobi’}, (default = ‘eig’) Choose an algorithm: ■ ‘svd’ - alias for svd-jacobi; ■ ‘eig’ - use an eigendecomposition of the covariance matrix; ■ ‘qr’ - use QR decomposition algorithm and solve Rx = Q^T y ■ ‘svd-qr’ - compute SVD decomposition using QR algorithm ■ ‘svd-jacobi’ - compute SVD decomposition using Jacobi iterations. Among these algorithms, only ‘svd-jacobi’ supports the case when the number of features is larger than the sample size; this algorithm is force-selected automatically in such a case. For the broad range of inputs, ‘eig’ and ‘qr’ are usually the fastest, followed by ‘svd-jacobi’ and then ‘svd-qr’. In theory, SVD-based algorithms are more stable. fit_interceptboolean (default = True) If True, LinearRegression tries to correct for the global mean of y. If False, the model expects that you have centered the data. copy_Xbool, default=True If True, it is guaranteed that a copy of X is created, leaving the original X unchanged. However, if set to False, X may be modified directly, which would reduce the memory usage of the normalizeboolean (default = False) This parameter is ignored when fit_intercept is set to False. If True, the predictors in X will be normalized by dividing by the column-wise standard deviation. If False, no scaling will be done. Note: this is in contrast to sklearn’s deprecated normalize flag, which divides by the column-wise L2 norm; but this is the same as if using sklearn’s StandardScaler. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. LinearRegression suffers from multicollinearity (when columns are correlated with each other), and variance explosions from outliers. Consider using Ridge Regression to fix the multicollinearity problem, and consider maybe first DBSCAN to remove the outliers, or statistical analysis to filter possible outliers. Applications of LinearRegression LinearRegression is used in regression tasks where one wants to predict say sales or house prices. It is also used in extrapolation or time series tasks, dynamic systems modelling and many other machine learning tasks. This model should be first tried if the machine learning problem is a regression task (predicting a continuous variable). For additional information, see scikitlearn’s OLS documentation. For an additional example see the OLS notebook. Starting from version 23.08, the new ‘copy_X’ parameter defaults to ‘True’, ensuring a copy of X is created after passing it to fit(), preventing any changes to the input, but with increased memory usage. This represents a change in behavior from previous versions. With copy_X=False a copy might still be created if necessary. >>> import cupy as cp >>> import cudf >>> # Both import methods supported >>> from cuml import LinearRegression >>> from cuml.linear_model import LinearRegression >>> lr = LinearRegression(fit_intercept = True, normalize = False, ... algorithm = "eig") >>> X = cudf.DataFrame() >>> X['col1'] = cp.array([1,1,2,2], dtype=cp.float32) >>> X['col2'] = cp.array([1,2,2,3], dtype=cp.float32) >>> y = cudf.Series(cp.array([6.0, 8.0, 9.0, 11.0], dtype=cp.float32)) >>> reg = lr.fit(X,y) >>> print(reg.coef_) 0 1.0 1 2.0 dtype: float32 >>> print(reg.intercept_) >>> X_new = cudf.DataFrame() >>> X_new['col1'] = cp.array([3,2], dtype=cp.float32) >>> X_new['col2'] = cp.array([5,5], dtype=cp.float32) >>> preds = lr.predict(X_new) >>> print(preds) 0 15.999... 1 14.999... dtype: float32 coef_array, shape (n_features) The estimated coefficients for the linear regression model. The independent term. If fit_intercept is False, will be 0. fit(X, y[, convert_dtype, sample_weight]) Fit the model with X and y. get_param_names() Returns a list of hyperparameter names owned by this class. fit(X, y, convert_dtype=True, sample_weight=None) LinearRegression[source]# Fit the model with X and y. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas yarray-like (device or host) shape = (n_samples, 1) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the train method will, when necessary, convert y to be the same data type as X if they differ. This will increase memory used for the method. sample_weightarray-like (device or host) shape = (n_samples,), default=None The weights for each observation in X. If None, all observations are assigned equal weight. Acceptable dense formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/ Series, NumPy ndarray and Pandas DataFrame/Series. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. Logistic Regression# class cuml.LogisticRegression(, penalty='l2', tol=0.0001, C=1.0, fit_intercept=True, class_weight=None, max_iter=1000, linesearch_max_iter=50, verbose=False, l1_ratio=None, solver='qn', handle=None, LogisticRegression is a linear model that is used to model probability of occurrence of certain events, for example probability of success or fail of an event. cuML’s LogisticRegression can take array-like objects, either in host as NumPy arrays or in device (as Numba or __cuda_array_interface__ compliant), in addition to cuDF objects. It provides both single-class (using sigmoid loss) and multiple-class (using softmax loss) variants, depending on the input variables Only one solver option is currently available: Quasi-Newton (QN) algorithms. Even though it is presented as a single option, this solver resolves to two different algorithms underneath: ☆ Orthant-Wise Limited Memory Quasi-Newton (OWL-QN) if there is l1 regularization ☆ Limited Memory BFGS (L-BFGS) otherwise. Note that, just like in Scikit-learn, the bias will not be regularized. penalty‘none’, ‘l1’, ‘l2’, ‘elasticnet’ (default = ‘l2’) Used to specify the norm used in the penalization. If ‘none’ or ‘l2’ are selected, then L-BFGS solver will be used. If ‘l1’ is selected, solver OWL-QN will be used. If ‘elasticnet’ is selected, OWL-QN will be used if l1_ratio > 0, otherwise L-BFGS will be used. tolfloat (default = 1e-4) Tolerance for stopping criteria. The exact stopping conditions depend on the chosen solver. Check the solver’s documentation for more details: Cfloat (default = 1.0) Inverse of regularization strength; must be a positive float. fit_interceptboolean (default = True) If True, the model tries to correct for the global mean of y. If False, the model expects that you have centered the data. class_weightdict or ‘balanced’, default=None By default all classes have a weight one. However, a dictionary can be provided with weights associated with classes in the form {class_label: weight}. The “balanced” mode uses the values of y to automatically adjust weights inversely proportional to class frequencies in the input data as n_samples / (n_classes * np.bincount(y)). Note that these weights will be multiplied with sample_weight (passed through the fit method) if sample_weight is specified. max_iterint (default = 1000) Maximum number of iterations taken for the solvers to converge. linesearch_max_iterint (default = 50) Max number of linesearch iterations per outer iteration used in the lbfgs and owl QN solvers. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. l1_ratiofloat or None, optional (default=None) The Elastic-Net mixing parameter, with 0 <= l1_ratio <= 1 solver‘qn’ (default=’qn’) Algorithm to use in the optimization problem. Currently only qn is supported, which automatically selects either L-BFGS or OWL-QN depending on the conditions of the l1 regularization described above. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. cuML’s LogisticRegression uses a different solver that the equivalent Scikit-learn, except when there is no penalty and solver=lbfgs is used in Scikit-learn. This can cause (smaller) differences in the coefficients and predictions of the model, similar to using different solvers in Scikit-learn. For additional information, see Scikit-learn’s LogisticRegression. >>> import cudf >>> import numpy as np >>> # Both import methods supported >>> # from cuml import LogisticRegression >>> from cuml.linear_model import LogisticRegression >>> X = cudf.DataFrame() >>> X['col1'] = np.array([1,1,2,2], dtype = np.float32) >>> X['col2'] = np.array([1,2,2,3], dtype = np.float32) >>> y = cudf.Series(np.array([0.0, 0.0, 1.0, 1.0], dtype=np.float32)) >>> reg = LogisticRegression() >>> reg.fit(X,y) >>> print(reg.coef_) 0 0.69861 0.570058 >>> print(reg.intercept_) 0 -2.188... dtype: float32 >>> X_new = cudf.DataFrame() >>> X_new['col1'] = np.array([1,5], dtype = np.float32) >>> X_new['col2'] = np.array([2,5], dtype = np.float32) >>> preds = reg.predict(X_new) >>> print(preds) 0 0.0 1 1.0 dtype: float32 coef_: dev array, dim (n_classes, n_features) or (n_classes, n_features+1) The estimated coefficients for the logistic regression model. intercept_: device array (n_classes, 1) The independent term. If fit_intercept is False, will be 0. decision_function(X[, convert_dtype]) Gives confidence score for X fit(X, y[, sample_weight, convert_dtype]) Fit the model with X and y. get_param_names() Returns a list of hyperparameter names owned by this class. predict(X[, convert_dtype]) Predicts the y for X. predict_log_proba(X[, convert_dtype]) Predicts the log class probabilities for each class in X predict_proba(X[, convert_dtype]) Predicts the class probabilities for each class in X set_params(params) Accepts a dict of params and updates the corresponding ones owned by this class. decision_function(X, convert_dtype=True) CumlArray[source]# Gives confidence score for X Xarray-like (device or host) shape = (n_samples, n_features) Dense or sparse matrix containing floats or doubles. Acceptable dense formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the decision_function method will, when necessary, convert the input to the data type which was used to train the model. This will increase memory used for the scorecuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, n_classes) Confidence score For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. fit(X, y, sample_weight=None, convert_dtype=True) LogisticRegression[source]# Fit the model with X and y. Xarray-like (device or host) shape = (n_samples, n_features) Dense or sparse matrix containing floats or doubles. Acceptable dense formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas yarray-like (device or host) shape = (n_samples, 1) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas sample_weightarray-like (device or host) shape = (n_samples,), default=None The weights for each observation in X. If None, all observations are assigned equal weight. Acceptable dense formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/ Series, NumPy ndarray and Pandas DataFrame/Series. convert_dtypebool, optional (default = True) When set to True, the train method will, when necessary, convert y to be the same data type as X if they differ. This will increase memory used for the method. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. predict(X, convert_dtype=True) CumlArray[source]# Xarray-like (device or host) shape = (n_samples, n_features) Dense or sparse matrix containing floats or doubles. Acceptable dense formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the predict method will, when necessary, convert the input to the data type which was used to train the model. This will increase memory used for the method. predscuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, 1) Predicted values For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. predict_log_proba(X, convert_dtype=True) CumlArray[source]# Predicts the log class probabilities for each class in X Xarray-like (device or host) shape = (n_samples, n_features) Dense or sparse matrix containing floats or doubles. Acceptable dense formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the predict_log_proba method will, when necessary, convert the input to the data type which was used to train the model. This will increase memory used for the predscuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, n_classes) Logaright of predicted class probabilities For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. predict_proba(X, convert_dtype=True) CumlArray[source]# Predicts the class probabilities for each class in X Xarray-like (device or host) shape = (n_samples, n_features) Dense or sparse matrix containing floats or doubles. Acceptable dense formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the predict_proba method will, when necessary, convert the input to the data type which was used to train the model. This will increase memory used for the method. predscuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, n_classes) Predicted class probabilities For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. Accepts a dict of params and updates the corresponding ones owned by this class. If the child class has appropriately overridden the get_param_names method and does not need anything other than what is, there in this method, then it doesn’t have to override this method Ridge Regression# class cuml.Ridge(, alpha=1.0, solver='eig', fit_intercept=True, normalize=False, handle=None, output_type=None, verbose=False)# Ridge extends LinearRegression by providing L2 regularization on the coefficients when predicting response y with a linear combination of the predictors in X. It can reduce the variance of the predictors, and improves the conditioning of the problem. cuML’s Ridge can take array-like objects, either in host as NumPy arrays or in device (as Numba or __cuda_array_interface__ compliant), in addition to cuDF objects. It provides 3 algorithms: SVD, Eig and CD to fit a linear model. In general SVD uses significantly more memory and is slower than Eig. If using CUDA 10.1, the memory difference is even bigger than in the other supported CUDA versions. However, SVD is more stable than Eig (default). CD uses Coordinate Descent and can be faster when data is large. alphafloat (default = 1.0) Regularization strength - must be a positive float. Larger values specify stronger regularization. Array input will be supported later. solver{‘eig’, ‘svd’, ‘cd’} (default = ‘eig’) Eig uses a eigendecomposition of the covariance matrix, and is much faster. SVD is slower, but guaranteed to be stable. CD or Coordinate Descent is very fast and is suitable for large fit_interceptboolean (default = True) If True, Ridge tries to correct for the global mean of y. If False, the model expects that you have centered the data. normalizeboolean (default = False) If True, the predictors in X will be normalized by dividing by the column-wise standard deviation. If False, no scaling will be done. Note: this is in contrast to sklearn’s deprecated normalize flag, which divides by the column-wise L2 norm; but this is the same as if using sklearn’s StandardScaler. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. Ridge provides L2 regularization. This means that the coefficients can shrink to become very small, but not zero. This can cause issues of interpretability on the coefficients. Consider using Lasso, or thresholding small coefficients to zero. Applications of Ridge Ridge Regression is used in the same way as LinearRegression, but does not suffer from multicollinearity issues. Ridge is used in insurance premium prediction, stock market analysis and much For additional docs, see Scikit-learn’s Ridge Regression. >>> import cupy as cp >>> import cudf >>> # Both import methods supported >>> from cuml import Ridge >>> from cuml.linear_model import Ridge >>> alpha = 1e-5 >>> ridge = Ridge(alpha=alpha, fit_intercept=True, normalize=False, ... solver="eig") >>> X = cudf.DataFrame() >>> X['col1'] = cp.array([1,1,2,2], dtype = cp.float32) >>> X['col2'] = cp.array([1,2,2,3], dtype = cp.float32) >>> y = cudf.Series(cp.array([6.0, 8.0, 9.0, 11.0], dtype=cp.float32)) >>> result_ridge = ridge.fit(X, y) >>> print(result_ridge.coef_) 0 1.000... 1 1.999... >>> print(result_ridge.intercept_) >>> X_new = cudf.DataFrame() >>> X_new['col1'] = cp.array([3,2], dtype=cp.float32) >>> X_new['col2'] = cp.array([5,5], dtype=cp.float32) >>> preds = result_ridge.predict(X_new) >>> print(preds) 0 15.999... 1 14.999... coef_array, shape (n_features) The estimated coefficients for the linear regression model. The independent term. If fit_intercept is False, will be 0. fit(X, y[, convert_dtype, sample_weight]) Fit the model with X and y. get_param_names() Returns a list of hyperparameter names owned by this class. set_params(params) Accepts a dict of params and updates the corresponding ones owned by this class. fit(X, y, convert_dtype=True, sample_weight=None) Ridge[source]# Fit the model with X and y. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas yarray-like (device or host) shape = (n_samples, 1) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the train method will, when necessary, convert y to be the same data type as X if they differ. This will increase memory used for the method. sample_weightarray-like (device or host) shape = (n_samples,), default=None The weights for each observation in X. If None, all observations are assigned equal weight. Acceptable dense formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/ Series, NumPy ndarray and Pandas DataFrame/Series. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. Accepts a dict of params and updates the corresponding ones owned by this class. If the child class has appropriately overridden the get_param_names method and does not need anything other than what is, there in this method, then it doesn’t have to override this method Lasso Regression# class cuml.Lasso(, alpha=1.0, fit_intercept=True, normalize=False, max_iter=1000, tol=0.001, solver='cd', selection='cyclic', handle=None, output_type=None, verbose=False)[source]# Lasso extends LinearRegression by providing L1 regularization on the coefficients when predicting response y with a linear combination of the predictors in X. It can zero some of the coefficients for feature selection and improves the conditioning of the problem. cuML’s Lasso can take array-like objects, either in host as NumPy arrays or in device (as Numba or __cuda_array_interface__ compliant), in addition to cuDF objects. It uses coordinate descent to fit a linear model. This estimator supports cuML’s experimental device selection capabilities. It can be configured to run on either the CPU or the GPU. To learn more, please see CPU / GPU Device Selection alphafloat (default = 1.0) Constant that multiplies the L1 term. alpha = 0 is equivalent to an ordinary least square, solved by the LinearRegression object. For numerical reasons, using alpha = 0 with the Lasso object is not advised. Given this, you should use the LinearRegression object. fit_interceptboolean (default = True) If True, Lasso tries to correct for the global mean of y. If False, the model expects that you have centered the data. normalizeboolean (default = False) If True, the predictors in X will be normalized by dividing by the column-wise standard deviation. If False, no scaling will be done. Note: this is in contrast to sklearn’s deprecated normalize flag, which divides by the column-wise L2 norm; but this is the same as if using sklearn’s StandardScaler. max_iterint (default = 1000) The maximum number of iterations tolfloat (default = 1e-3) The tolerance for the optimization: if the updates are smaller than tol, the optimization code checks the dual gap for optimality and continues until it is smaller than tol. solver{‘cd’, ‘qn’} (default=’cd’) Choose an algorithm: ■ ‘cd’ - coordinate descent ■ ‘qn’ - quasi-newton You may find the alternative ‘qn’ algorithm is faster when the number of features is sufficiently large, but the sample size is small. selection{‘cyclic’, ‘random’} (default=’cyclic’) If set to ‘random’, a random coefficient is updated every iteration rather than looping over features sequentially by default. This (setting to ‘random’) often leads to significantly faster convergence especially when tol is higher than 1e-4. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. For additional docs, see scikitlearn’s Lasso. >>> import numpy as np >>> import cudf >>> from cuml.linear_model import Lasso >>> ls = Lasso(alpha = 0.1, solver='qn') >>> X = cudf.DataFrame() >>> X['col1'] = np.array([0, 1, 2], dtype = np.float32) >>> X['col2'] = np.array([0, 1, 2], dtype = np.float32) >>> y = cudf.Series( np.array([0.0, 1.0, 2.0], dtype = np.float32) ) >>> result_lasso = ls.fit(X, y) >>> print(result_lasso.coef_) 0 0.425 1 0.425 dtype: float32 >>> print(result_lasso.intercept_) >>> X_new = cudf.DataFrame() >>> X_new['col1'] = np.array([3,2], dtype = np.float32) >>> X_new['col2'] = np.array([5,5], dtype = np.float32) >>> preds = result_lasso.predict(X_new) >>> print(preds) 0 3.549997 1 3.124997 dtype: float32 coef_array, shape (n_features) The estimated coefficients for the linear regression model. The independent term. If fit_intercept is False, will be 0. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. ElasticNet Regression# class cuml.ElasticNet(, alpha=1.0, l1_ratio=0.5, fit_intercept=True, normalize=False, max_iter=1000, tol=0.001, solver='cd', selection='cyclic', handle=None, output_type=None, verbose=False)# ElasticNet extends LinearRegression with combined L1 and L2 regularizations on the coefficients when predicting response y with a linear combination of the predictors in X. It can reduce the variance of the predictors, force some coefficients to be small, and improves the conditioning of the problem. cuML’s ElasticNet an array-like object or cuDF DataFrame, uses coordinate descent to fit a linear model. alphafloat (default = 1.0) Constant that multiplies the L1 term. alpha = 0 is equivalent to an ordinary least square, solved by the LinearRegression object. For numerical reasons, using alpha = 0 with the Lasso object is not advised. Given this, you should use the LinearRegression object. l1_ratiofloat (default = 0.5) The ElasticNet mixing parameter, with 0 <= l1_ratio <= 1. For l1_ratio = 0 the penalty is an L2 penalty. For l1_ratio = 1 it is an L1 penalty. For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2. fit_interceptboolean (default = True) If True, Lasso tries to correct for the global mean of y. If False, the model expects that you have centered the data. normalizeboolean (default = False) If True, the predictors in X will be normalized by dividing by the column-wise standard deviation. If False, no scaling will be done. Note: this is in contrast to sklearn’s deprecated normalize flag, which divides by the column-wise L2 norm; but this is the same as if using sklearn’s StandardScaler. max_iterint (default = 1000) The maximum number of iterations tolfloat (default = 1e-3) The tolerance for the optimization: if the updates are smaller than tol, the optimization code checks the dual gap for optimality and continues until it is smaller than tol. solver{‘cd’, ‘qn’} (default=’cd’) Choose an algorithm: ■ ‘cd’ - coordinate descent ■ ‘qn’ - quasi-newton You may find the alternative ‘qn’ algorithm is faster when the number of features is sufficiently large, but the sample size is small. selection{‘cyclic’, ‘random’} (default=’cyclic’) If set to ‘random’, a random coefficient is updated every iteration rather than looping over features sequentially by default. This (setting to ‘random’) often leads to significantly faster convergence especially when tol is higher than 1e-4. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. For additional docs, see scikitlearn’s ElasticNet. >>> import cupy as cp >>> import cudf >>> from cuml.linear_model import ElasticNet >>> enet = ElasticNet(alpha = 0.1, l1_ratio=0.5, solver='qn') >>> X = cudf.DataFrame() >>> X['col1'] = cp.array([0, 1, 2], dtype = cp.float32) >>> X['col2'] = cp.array([0, 1, 2], dtype = cp.float32) >>> y = cudf.Series(cp.array([0.0, 1.0, 2.0], dtype = cp.float32) ) >>> result_enet = enet.fit(X, y) >>> print(result_enet.coef_) 0 0.445... 1 0.445... dtype: float32 >>> print(result_enet.intercept_) >>> X_new = cudf.DataFrame() >>> X_new['col1'] = cp.array([3,2], dtype = cp.float32) >>> X_new['col2'] = cp.array([5,5], dtype = cp.float32) >>> preds = result_enet.predict(X_new) >>> print(preds) 0 3.674... 1 3.228... dtype: float32 coef_array, shape (n_features) The estimated coefficients for the linear regression model. The independent term. If fit_intercept is False, will be 0. fit(X, y[, convert_dtype, sample_weight]) Fit the model with X and y. get_param_names() Returns a list of hyperparameter names owned by this class. set_params(params) Accepts a dict of params and updates the corresponding ones owned by this class. fit(X, y, convert_dtype=True, sample_weight=None) ElasticNet[source]# Fit the model with X and y. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas yarray-like (device or host) shape = (n_samples, 1) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the train method will, when necessary, convert y to be the same data type as X if they differ. This will increase memory used for the method. sample_weightarray-like (device or host) shape = (n_samples,), default=None The weights for each observation in X. If None, all observations are assigned equal weight. Acceptable dense formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/ Series, NumPy ndarray and Pandas DataFrame/Series. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. Accepts a dict of params and updates the corresponding ones owned by this class. If the child class has appropriately overridden the get_param_names method and does not need anything other than what is, there in this method, then it doesn’t have to override this method Mini Batch SGD Classifier# class cuml.MBSGDClassifier(, loss='hinge', penalty='l2', alpha=0.0001, l1_ratio=0.15, fit_intercept=True, epochs=1000, tol=0.001, shuffle=True, learning_rate='constant', eta0=0.001, power_t=0.5, batch_size=32, n_iter_no_change=5, handle=None, verbose=False, output_type=None)# Linear models (linear SVM, logistic regression, or linear regression) fitted by minimizing a regularized empirical loss with mini-batch SGD. The MBSGD Classifier implementation is experimental and and it uses a different algorithm than sklearn’s SGDClassifier. In order to improve the results obtained from cuML’s MBSGDClassifier: □ Reduce the batch size □ Increase the eta0 □ Increase the number of iterations Since cuML is analyzing the data in batches using a small eta0 might not let the model learn as much as scikit learn does. Furthermore, decreasing the batch size might seen an increase in the time required to fit the model. loss{‘hinge’, ‘log’, ‘squared_loss’} (default = ‘hinge’) ‘hinge’ uses linear SVM ‘log’ uses logistic regression ‘squared_loss’ uses linear regression penalty{‘none’, ‘l1’, ‘l2’, ‘elasticnet’} (default = ‘l2’) ‘none’ does not perform any regularization ‘l1’ performs L1 norm (Lasso) which minimizes the sum of the abs value of coefficients ‘l2’ performs L2 norm (Ridge) which minimizes the sum of the square of the coefficients ‘elasticnet’ performs Elastic Net regularization which is a weighted average of L1 and L2 norms alphafloat (default = 0.0001) The constant value which decides the degree of regularization l1_ratiofloat (default=0.15) The l1_ratio is used only when penalty = elasticnet. The value for l1_ratio should be 0 <= l1_ratio <= 1. When l1_ratio = 0 then the penalty = 'l2' and if l1_ratio = 1 then penalty = 'l1' batch_sizeint (default = 32) It sets the number of samples that will be included in each batch. fit_interceptboolean (default = True) If True, the model tries to correct for the global mean of y. If False, the model expects that you have centered the data. epochsint (default = 1000) The number of times the model should iterate through the entire dataset during training (default = 1000) tolfloat (default = 1e-3) The training process will stop if current_loss > previous_loss - tol shuffleboolean (default = True) True, shuffles the training data after each epoch False, does not shuffle the training data after each epoch eta0float (default = 0.001) Initial learning rate power_tfloat (default = 0.5) The exponent used for calculating the invscaling learning rate learning_rate{‘optimal’, ‘constant’, ‘invscaling’, ‘adaptive’} (default = ‘constant’) optimal option will be supported in a future version constant keeps the learning rate constant adaptive changes the learning rate if the training loss or the validation accuracy does not improve for n_iter_no_change epochs. The old learning rate is generally divided by 5 n_iter_no_changeint (default = 5) the number of epochs to train without any improvement in the model Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. For additional docs, see scikitlearn’s SGDClassifier. >>> import cupy as cp >>> import cudf >>> from cuml.linear_model import MBSGDClassifier >>> X = cudf.DataFrame() >>> X['col1'] = cp.array([1,1,2,2], dtype = cp.float32) >>> X['col2'] = cp.array([1,2,2,3], dtype = cp.float32) >>> y = cudf.Series(cp.array([1, 1, 2, 2], dtype=cp.float32)) >>> pred_data = cudf.DataFrame() >>> pred_data['col1'] = cp.asarray([3, 2], dtype=cp.float32) >>> pred_data['col2'] = cp.asarray([5, 5], dtype=cp.float32) >>> cu_mbsgd_classifier = MBSGDClassifier(learning_rate='constant', ... eta0=0.05, epochs=2000, ... fit_intercept=True, ... batch_size=1, tol=0.0, ... penalty='l2', ... loss='squared_loss', ... alpha=0.5) >>> cu_mbsgd_classifier.fit(X, y) >>> print("cuML intercept : ", cu_mbsgd_classifier.intercept_) cuML intercept : 0.725... >>> print("cuML coef : ", cu_mbsgd_classifier.coef_) cuML coef : 0 0.273... 1 0.182... dtype: float32 >>> cu_pred = cu_mbsgd_classifier.predict(pred_data) >>> print("cuML predictions : ", cu_pred) cuML predictions : 0 1.0 1 1.0 dtype: float32 fit(X, y[, convert_dtype]) Fit the model with X and y. get_param_names() Returns a list of hyperparameter names owned by this class. predict(X[, convert_dtype]) Predicts the y for X. set_params(params) Accepts a dict of params and updates the corresponding ones owned by this class. fit(X, y, convert_dtype=True) MBSGDClassifier[source]# Fit the model with X and y. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas yarray-like (device or host) shape = (n_samples, 1) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the train method will, when necessary, convert y to be the same data type as X if they differ. This will increase memory used for the method. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. predict(X, convert_dtype=True) CumlArray[source]# Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the predict method will, when necessary, convert the input to the data type which was used to train the model. This will increase memory used for the method. predscuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, 1) Predicted values For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. Accepts a dict of params and updates the corresponding ones owned by this class. If the child class has appropriately overridden the get_param_names method and does not need anything other than what is, there in this method, then it doesn’t have to override this method Mini Batch SGD Regressor# class cuml.MBSGDRegressor(, loss='squared_loss', penalty='l2', alpha=0.0001, l1_ratio=0.15, fit_intercept=True, epochs=1000, tol=0.001, shuffle=True, learning_rate='constant', eta0=0.001, power_t= 0.5, batch_size=32, n_iter_no_change=5, handle=None, verbose=False, output_type=None)# Linear regression model fitted by minimizing a regularized empirical loss with mini-batch SGD. The MBSGD Regressor implementation is experimental and and it uses a different algorithm than sklearn’s SGDClassifier. In order to improve the results obtained from cuML’s MBSGD Regressor: □ Reduce the batch size □ Increase the eta0 □ Increase the number of iterations Since cuML is analyzing the data in batches using a small eta0 might not let the model learn as much as scikit learn does. Furthermore, decreasing the batch size might seen an increase in the time required to fit the model. loss‘squared_loss’ (default = ‘squared_loss’) ‘squared_loss’ uses linear regression penalty‘none’, ‘l1’, ‘l2’, ‘elasticnet’ (default = ‘l2’) ‘none’ does not perform any regularization ‘l1’ performs L1 norm (Lasso) which minimizes the sum of the abs value of coefficients ‘l2’ performs L2 norm (Ridge) which minimizes the sum of the square of the coefficients ‘elasticnet’ performs Elastic Net regularization which is a weighted average of L1 and L2 norms alphafloat (default = 0.0001) The constant value which decides the degree of regularization fit_interceptboolean (default = True) If True, the model tries to correct for the global mean of y. If False, the model expects that you have centered the data. l1_ratiofloat (default=0.15) The l1_ratio is used only when penalty = elasticnet. The value for l1_ratio should be 0 <= l1_ratio <= 1. When l1_ratio = 0 then the penalty = 'l2' and if l1_ratio = 1 then penalty = 'l1' batch_sizeint (default = 32) It sets the number of samples that will be included in each batch. epochsint (default = 1000) The number of times the model should iterate through the entire dataset during training (default = 1000) tolfloat (default = 1e-3) The training process will stop if current_loss > previous_loss - tol shuffleboolean (default = True) True, shuffles the training data after each epoch False, does not shuffle the training data after each epoch eta0float (default = 0.001) Initial learning rate power_tfloat (default = 0.5) The exponent used for calculating the invscaling learning rate learning_rate{‘optimal’, ‘constant’, ‘invscaling’, ‘adaptive’} (default = ‘constant’) optimal option will be supported in a future version constant keeps the learning rate constant adaptive changes the learning rate if the training loss or the validation accuracy does not improve for n_iter_no_change epochs. The old learning rate is generally divided by 5 n_iter_no_changeint (default = 5) the number of epochs to train without any improvement in the model Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. For additional docs, see scikitlearn’s SGDRegressor. >>> import cupy as cp >>> import cudf >>> from cuml.linear_model import MBSGDRegressor as cumlMBSGDRegressor >>> X = cudf.DataFrame() >>> X['col1'] = cp.array([1,1,2,2], dtype = cp.float32) >>> X['col2'] = cp.array([1,2,2,3], dtype = cp.float32) >>> y = cudf.Series(cp.array([1, 1, 2, 2], dtype=cp.float32)) >>> pred_data = cudf.DataFrame() >>> pred_data['col1'] = cp.asarray([3, 2], dtype=cp.float32) >>> pred_data['col2'] = cp.asarray([5, 5], dtype=cp.float32) >>> cu_mbsgd_regressor = cumlMBSGDRegressor(learning_rate='constant', ... eta0=0.05, epochs=2000, ... fit_intercept=True, ... batch_size=1, tol=0.0, ... penalty='l2', ... loss='squared_loss', ... alpha=0.5) >>> cu_mbsgd_regressor.fit(X, y) >>> print("cuML intercept : ", cu_mbsgd_regressor.intercept_) cuML intercept : 0.725... >>> print("cuML coef : ", cu_mbsgd_regressor.coef_) cuML coef : 0 0.273... 1 0.182... dtype: float32 >>> cu_pred = cu_mbsgd_regressor.predict(pred_data) >>> print("cuML predictions : ", cu_pred) cuML predictions : 0 2.456... 1 2.183... dtype: float32 fit(X, y[, convert_dtype]) Fit the model with X and y. get_param_names() Returns a list of hyperparameter names owned by this class. predict(X[, convert_dtype]) Predicts the y for X. set_params(params) Accepts a dict of params and updates the corresponding ones owned by this class. fit(X, y, convert_dtype=True) MBSGDRegressor[source]# Fit the model with X and y. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas yarray-like (device or host) shape = (n_samples, 1) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the train method will, when necessary, convert y to be the same data type as X if they differ. This will increase memory used for the method. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. predict(X, convert_dtype=True) CumlArray[source]# Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the predict method will, when necessary, convert the input to the data type which was used to train the model. This will increase memory used for the method. predscuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, 1) Predicted values For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. Accepts a dict of params and updates the corresponding ones owned by this class. If the child class has appropriately overridden the get_param_names method and does not need anything other than what is, there in this method, then it doesn’t have to override this method Multiclass Classification# class cuml.multiclass.MulticlassClassifier(estimator, , handle=None, verbose=False, output_type=None, strategy='ovr')[source]# Wrapper around scikit-learn multiclass classifiers that allows to choose different multiclass strategies. The input can be any kind of cuML compatible array, and the output type follows cuML’s output type configuration rules. Berofe passing the data to scikit-learn, it is converted to host (numpy) array. Under the hood the data is partitioned for binary classification, and it is transformed back to the device by the cuML estimator. These copies back and forth the device and the host have some overhead. For more details see issue rapidsai/cuml#2876. estimatorcuML estimator Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. strategy: string {‘ovr’, ‘ovo’}, default=’ovr’ Multiclass classification strategy: ‘ovr’: one vs. rest or ‘ovo’: one vs. one >>> from cuml.linear_model import LogisticRegression >>> from cuml.multiclass import MulticlassClassifier >>> from cuml.datasets.classification import make_classification >>> X, y = make_classification(n_samples=10, n_features=6, ... n_informative=4, n_classes=3, ... random_state=137) >>> cls = MulticlassClassifier(LogisticRegression(), strategy='ovo') >>> cls.fit(X,y) >>> cls.predict(X) array([1, 1, 1, 1, 1, 1, 2, 1, 1, 2]) classes_float, shape (n_classes_) Array of class labels. Number of classes. decision_function(X) Calculate the decision function. fit(X, y) Fit a multiclass classifier. get_param_names() Returns a list of hyperparameter names owned by this class. predict(X) Predict using multi class classifier. decision_function(X) CumlArray[source]# Calculate the decision function. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas resultscuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, 1) Decision function values For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. fit(X, y) MulticlassClassifier[source]# Fit a multiclass classifier. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas yarray-like (device or host) shape = (n_samples, 1) Dense matrix of any dtype. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas DataFrame/Series. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. predict(X) CumlArray[source]# Predict using multi class classifier. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas predscuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, 1) Predicted values For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. class cuml.multiclass.OneVsOneClassifier(estimator, args, handle=None, verbose=False, output_type=None)[source]# Wrapper around Sckit-learn’s class with the same name. The input can be any kind of cuML compatible array, and the output type follows cuML’s output type configuration rules. Berofe passing the data to scikit-learn, it is converted to host (numpy) array. Under the hood the data is partitioned for binary classification, and it is transformed back to the device by the cuML estimator. These copies back and forth the device and the host have some overhead. For more details see issue rapidsai/cuml#2876. For documentation see scikit-learn’s OneVsOneClassifier. estimatorcuML estimator Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. >>> from cuml.linear_model import LogisticRegression >>> from cuml.multiclass import OneVsOneClassifier >>> from cuml.datasets.classification import make_classification >>> X, y = make_classification(n_samples=10, n_features=6, ... n_informative=4, n_classes=3, ... random_state=137) >>> cls = OneVsOneClassifier(LogisticRegression()) >>> cls.fit(X,y) >>> cls.predict(X) array([1, 1, 1, 1, 1, 1, 2, 1, 1, 2]) Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. class cuml.multiclass.OneVsRestClassifier(estimator, args, handle=None, verbose=False, output_type=None)[source]# Wrapper around Sckit-learn’s class with the same name. The input can be any kind of cuML compatible array, and the output type follows cuML’s output type configuration rules. Berofe passing the data to scikit-learn, it is converted to host (numpy) array. Under the hood the data is partitioned for binary classification, and it is transformed back to the device by the cuML estimator. These copies back and forth the device and the host have some overhead. For more details see issue rapidsai/cuml#2876. For documentation see scikit-learn’s OneVsRestClassifier. estimatorcuML estimator Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. >>> from cuml.linear_model import LogisticRegression >>> from cuml.multiclass import OneVsRestClassifier >>> from cuml.datasets.classification import make_classification >>> X, y = make_classification(n_samples=10, n_features=6, ... n_informative=4, n_classes=3, ... random_state=137) >>> cls = OneVsRestClassifier(LogisticRegression()) >>> cls.fit(X,y) >>> cls.predict(X) array([1, 1, 1, 1, 1, 1, 2, 1, 1, 2]) Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. Naive Bayes# class cuml.naive_bayes.MultinomialNB(, alpha=1.0, fit_prior=True, class_prior=None, output_type=None, handle=None, verbose=False)[source]# Naive Bayes classifier for multinomial models The multinomial Naive Bayes classifier is suitable for classification with discrete features (e.g., word counts for text classification). The multinomial distribution normally requires integer feature counts. However, in practice, fractional counts such as tf-idf may also work. alphafloat (default=1.0) Additive (Laplace/Lidstone) smoothing parameter (0 for no smoothing). fit_priorboolean (default=True) Whether to learn class prior probabilities or no. If false, a uniform prior will be used. class_priorarray-like, size (n_classes) (default=None) Prior probabilities of the classes. If specified, the priors are not adjusted according to the data. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. Load the 20 newsgroups dataset from Scikit-learn and train a Naive Bayes classifier. >>> import cupy as cp >>> import cupyx >>> from sklearn.datasets import fetch_20newsgroups >>> from sklearn.feature_extraction.text import CountVectorizer >>> from cuml.naive_bayes import MultinomialNB >>> # Load corpus >>> twenty_train = fetch_20newsgroups(subset='train', shuffle=True, ... random_state=42) >>> # Turn documents into term frequency vectors >>> count_vect = CountVectorizer() >>> features = count_vect.fit_transform(twenty_train.data) >>> # Put feature vectors and labels on the GPU >>> X = cupyx.scipy.sparse.csr_matrix(features.tocsr(), ... dtype=cp.float32) >>> y = cp.asarray(twenty_train.target, dtype=cp.int32) >>> # Train model >>> model = MultinomialNB() >>> model.fit(X, y) >>> # Compute accuracy on training set >>> model.score(X, y) class_count_ndarray of shape (n_classes) Number of samples encountered for each class during fitting. class_log_prior_ndarray of shape (n_classes) Log probability of each class (smoothed). classes_ndarray of shape (n_classes,) Class labels known to the classifier feature_count_ndarray of shape (n_classes, n_features) Number of samples encountered for each (class, feature) during fitting. feature_log_prob_ndarray of shape (n_classes, n_features) Empirical log probability of features given a class, P(x_i\|y). Number of features of each sample. class cuml.naive_bayes.BernoulliNB(, alpha=1.0, binarize=0.0, fit_prior=True, class_prior=None, output_type=None, handle=None, verbose=False)[source]# Naive Bayes classifier for multivariate Bernoulli models. Like MultinomialNB, this classifier is suitable for discrete data. The difference is that while MultinomialNB works with occurrence counts, BernoulliNB is designed for binary/boolean features. alphafloat, default=1.0 Additive (Laplace/Lidstone) smoothing parameter (0 for no smoothing). binarizefloat or None, default=0.0 Threshold for binarizing (mapping to booleans) of sample features. If None, input is presumed to already consist of binary vectors. fit_priorbool, default=True Whether to learn class prior probabilities or not. If false, a uniform prior will be used. class_priorarray-like of shape (n_classes,), default=None Prior probabilities of the classes. If specified the priors are not adjusted according to the data. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. C.D. Manning, P. Raghavan and H. Schuetze (2008). Introduction to Information Retrieval. Cambridge University Press, pp. 234-265. https://nlp.stanford.edu/IR-book/html/htmledition/ the-bernoulli-model-1.html A. McCallum and K. Nigam (1998). A comparison of event models for naive Bayes text classification. Proc. AAAI/ICML-98 Workshop on Learning for Text Categorization, pp. 41-48. V. Metsis, I. Androutsopoulos and G. Paliouras (2006). Spam filtering with naive Bayes – Which naive Bayes? 3rd Conf. on Email and Anti-Spam (CEAS). >>> import cupy as cp >>> rng = cp.random.RandomState(1) >>> X = rng.randint(5, size=(6, 100), dtype=cp.int32) >>> Y = cp.array([1, 2, 3, 4, 4, 5]) >>> from cuml.naive_bayes import BernoulliNB >>> clf = BernoulliNB() >>> clf.fit(X, Y) >>> print(clf.predict(X[2:3])) class_count_ndarray of shape (n_classes) Number of samples encountered for each class during fitting. class_log_prior_ndarray of shape (n_classes) Log probability of each class (smoothed). classes_ndarray of shape (n_classes,) Class labels known to the classifier feature_count_ndarray of shape (n_classes, n_features) Number of samples encountered for each (class, feature) during fitting. feature_log_prob_ndarray of shape (n_classes, n_features) Empirical log probability of features given a class, P(x_i\|y). Number of features of each sample. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. class cuml.naive_bayes.ComplementNB(, alpha=1.0, fit_prior=True, class_prior=None, norm=False, output_type=None, handle=None, verbose=False)[source]# The Complement Naive Bayes classifier described in Rennie et al. (2003). The Complement Naive Bayes classifier was designed to correct the “severe assumptions” made by the standard Multinomial Naive Bayes classifier. It is particularly suited for imbalanced data sets. alphafloat, default=1.0 Additive (Laplace/Lidstone) smoothing parameter (0 for no smoothing). fit_priorbool, default=True Whether to learn class prior probabilities or not. If false, a uniform prior will be used. class_priorarray-like of shape (n_classes,), default=None Prior probabilities of the classes. If specified the priors are not adjusted according to the data. normbool, default=False Whether or not a second normalization of the weights is performed. The default behavior mirrors the implementation found in Mahout and Weka, which do not follow the full algorithm described in Table 9 of the paper. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. Rennie, J. D., Shih, L., Teevan, J., & Karger, D. R. (2003). Tackling the poor assumptions of naive bayes text classifiers. In ICML (Vol. 3, pp. 616-623). https://people.csail.mit.edu/jrennie/ >>> import cupy as cp >>> rng = cp.random.RandomState(1) >>> X = rng.randint(5, size=(6, 100), dtype=cp.int32) >>> Y = cp.array([1, 2, 3, 4, 4, 5]) >>> from cuml.naive_bayes import ComplementNB >>> clf = ComplementNB() >>> clf.fit(X, Y) >>> print(clf.predict(X[2:3])) class_count_ndarray of shape (n_classes) Number of samples encountered for each class during fitting. class_log_prior_ndarray of shape (n_classes) Log probability of each class (smoothed). classes_ndarray of shape (n_classes,) Class labels known to the classifier feature_count_ndarray of shape (n_classes, n_features) Number of samples encountered for each (class, feature) during fitting. feature_log_prob_ndarray of shape (n_classes, n_features) Empirical log probability of features given a class, P(x_i\|y). Number of features of each sample. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. class cuml.naive_bayes.GaussianNB(, priors=None, var_smoothing=1e-09, output_type=None, handle=None, verbose=False)[source]# Gaussian Naive Bayes (GaussianNB) Can perform online updates to model parameters via partial_fit(). For details on algorithm used to update feature means and variance online, see Stanford CS tech report STAN-CS-79-773 by Chan, Golub, and LeVeque: priorsarray-like of shape (n_classes,) Prior probabilities of the classes. If specified the priors are not adjusted according to the data. var_smoothingfloat, default=1e-9 Portion of the largest variance of all features that is added to variances for calculation stability. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. >>> import cupy as cp >>> X = cp.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], ... [3, 2]], cp.float32) >>> Y = cp.array([1, 1, 1, 2, 2, 2], cp.float32) >>> from cuml.naive_bayes import GaussianNB >>> clf = GaussianNB() >>> clf.fit(X, Y) >>> print(clf.predict(cp.array([[-0.8, -1]], cp.float32))) >>> clf_pf = GaussianNB() >>> clf_pf.partial_fit(X, Y, cp.unique(Y)) >>> print(clf_pf.predict(cp.array([[-0.8, -1]], cp.float32))) fit(X, y[, sample_weight]) Fit Gaussian Naive Bayes classifier according to X, y get_param_names() Returns a list of hyperparameter names owned by this class. partial_fit(X, y[, classes, sample_weight]) Incremental fit on a batch of samples. fit(X, y, sample_weight=None) GaussianNB[source]# Fit Gaussian Naive Bayes classifier according to X, y X{array-like, cupy sparse matrix} of shape (n_samples, n_features) Training vectors, where n_samples is the number of samples and n_features is the number of features. yarray-like shape (n_samples) Target values. sample_weightarray-like of shape (n_samples) Weights applied to individual samples (1. for unweighted). Currently sample weight is ignored. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. partial_fit(X, y, classes=None, sample_weight=None) GaussianNB[source]# Incremental fit on a batch of samples. This method is expected to be called several times consecutively on different chunks of a dataset so as to implement out-of-core or online learning. This is especially useful when the whole dataset is too big to fit in memory at once. This method has some performance overhead hence it is better to call partial_fit on chunks of data that are as large as possible (as long as fitting in the memory budget) to hide the overhead. X{array-like, cupy sparse matrix} of shape (n_samples, n_features) Training vectors, where n_samples is the number of samples and n_features is the number of features. A sparse matrix in COO format is preferred, other formats will go through a conversion to COO. yarray-like of shape (n_samples) Target values. classesarray-like of shape (n_classes) List of all the classes that can possibly appear in the y vector. Must be provided at the first call to partial_fit, can be omitted in subsequent calls. sample_weightarray-like of shape (n_samples) Weights applied to individual samples (1. for unweighted). Currently sample weight is ignored. class cuml.naive_bayes.CategoricalNB(, alpha=1.0, fit_prior=True, class_prior=None, output_type=None, handle=None, verbose=False)[source]# Naive Bayes classifier for categorical features The categorical Naive Bayes classifier is suitable for classification with discrete features that are categorically distributed. The categories of each feature are drawn from a categorical distribution. alphafloat, default=1.0 Additive (Laplace/Lidstone) smoothing parameter (0 for no smoothing). fit_priorbool, default=True Whether to learn class prior probabilities or not. If false, a uniform prior will be used. class_priorarray-like of shape (n_classes,), default=None Prior probabilities of the classes. If specified the priors are not adjusted according to the data. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. >>> import cupy as cp >>> rng = cp.random.RandomState(1) >>> X = rng.randint(5, size=(6, 100), dtype=cp.int32) >>> y = cp.array([1, 2, 3, 4, 5, 6]) >>> from cuml.naive_bayes import CategoricalNB >>> clf = CategoricalNB() >>> clf.fit(X, y) >>> print(clf.predict(X[2:3])) category_count_ndarray of shape (n_features, n_classes, n_categories) With n_categories being the highest category of all the features. This array provides the number of samples encountered for each feature, class and category of the specific feature. class_count_ndarray of shape (n_classes,) Number of samples encountered for each class during fitting. class_log_prior_ndarray of shape (n_classes,) Smoothed empirical log probability for each class. classes_ndarray of shape (n_classes,) Class labels known to the classifier feature_log_prob_ndarray of shape (n_features, n_classes, n_categories) With n_categories being the highest category of all the features. Each array of shape (n_classes, n_categories) provides the empirical log probability of categories given the respective feature and class, P(x_i\|y). This attribute is not available when the model has been trained with sparse data. Number of features of each sample. fit(X, y[, sample_weight]) Fit Naive Bayes classifier according to X, y partial_fit(X, y[, classes, sample_weight]) Incremental fit on a batch of samples. fit(X, y, sample_weight=None) CategoricalNB[source]# Fit Naive Bayes classifier according to X, y Xarray-like of shape (n_samples, n_features) Training vectors, where n_samples is the number of samples and n_features is the number of features. Here, each feature of X is assumed to be from a different categorical distribution. It is further assumed that all categories of each feature are represented by the numbers 0, …, n - 1, where n refers to the total number of categories for the given feature. This can, for instance, be achieved with the help of OrdinalEncoder. yarray-like of shape (n_samples,) Target values. sample_weightarray-like of shape (n_samples), default=None Weights applied to individual samples (1. for unweighted). Currently sample weight is ignored. partial_fit(X, y, classes=None, sample_weight=None) CategoricalNB[source]# Incremental fit on a batch of samples. This method is expected to be called several times consecutively on different chunks of a dataset so as to implement out-of-core or online learning. This is especially useful when the whole dataset is too big to fit in memory at once. This method has some performance overhead hence it is better to call partial_fit on chunks of data that are as large as possible (as long as fitting in the memory budget) to hide the overhead. Xarray-like of shape (n_samples, n_features) Training vectors, where n_samples is the number of samples and n_features is the number of features. Here, each feature of X is assumed to be from a different categorical distribution. It is further assumed that all categories of each feature are represented by the numbers 0, …, n - 1, where n refers to the total number of categories for the given feature. This can, for instance, be achieved with the help of OrdinalEncoder. yarray-like of shape (n_samples) Target values. classesarray-like of shape (n_classes), default=None List of all the classes that can possibly appear in the y vector. Must be provided at the first call to partial_fit, can be omitted in subsequent calls. sample_weightarray-like of shape (n_samples), default=None Weights applied to individual samples (1. for unweighted). Currently sample weight is ignored. Stochastic Gradient Descent# class cuml.SGD(, loss='squared_loss', penalty='none', alpha=0.0001, l1_ratio=0.15, fit_intercept=True, epochs=1000, tol=0.001, shuffle=True, learning_rate='constant', eta0=0.001, power_t=0.5, batch_size=32, n_iter_no_change=5, handle=None, output_type=None, verbose=False)# Stochastic Gradient Descent is a very common machine learning algorithm where one optimizes some cost function via gradient steps. This makes SGD very attractive for large problems when the exact solution is hard or even impossible to find. cuML’s SGD algorithm accepts a numpy matrix or a cuDF DataFrame as the input dataset. The SGD algorithm currently works with linear regression, ridge regression and SVM models. loss‘hinge’, ‘log’, ‘squared_loss’ (default = ‘squared_loss’) ‘hinge’ uses linear SVM ‘log’ uses logistic regression ‘squared_loss’ uses linear regression penalty‘none’, ‘l1’, ‘l2’, ‘elasticnet’ (default = ‘none’) ‘none’ does not perform any regularization ‘l1’ performs L1 norm (Lasso) which minimizes the sum of the abs value of coefficients ‘l2’ performs L2 norm (Ridge) which minimizes the sum of the square of the coefficients ‘elasticnet’ performs Elastic Net regularization which is a weighted average of L1 and L2 norms alphafloat (default = 0.0001) The constant value which decides the degree of regularization fit_interceptboolean (default = True) If True, the model tries to correct for the global mean of y. If False, the model expects that you have centered the data. epochsint (default = 1000) The number of times the model should iterate through the entire dataset during training (default = 1000) tolfloat (default = 1e-3) The training process will stop if current_loss > previous_loss - tol shuffleboolean (default = True) True, shuffles the training data after each epoch False, does not shuffle the training data after each epoch eta0float (default = 0.001) Initial learning rate power_tfloat (default = 0.5) The exponent used for calculating the invscaling learning rate batch_sizeint (default=32) The number of samples to use for each batch. learning_rate‘optimal’, ‘constant’, ‘invscaling’, ‘adaptive’ (default = ‘constant’) Optimal option supported in the next version constant keeps the learning rate constant adaptive changes the learning rate if the training loss or the validation accuracy does not improve for n_iter_no_change epochs. The old learning rate is generally divide by 5 n_iter_no_changeint (default = 5) The number of epochs to train without any improvement in the model Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. >>> import numpy as np >>> import cudf >>> from cuml.solvers import SGD as cumlSGD >>> X = cudf.DataFrame() >>> X['col1'] = np.array([1,1,2,2], dtype=np.float32) >>> X['col2'] = np.array([1,2,2,3], dtype=np.float32) >>> y = cudf.Series(np.array([1, 1, 2, 2], dtype=np.float32)) >>> pred_data = cudf.DataFrame() >>> pred_data['col1'] = np.asarray([3, 2], dtype=np.float32) >>> pred_data['col2'] = np.asarray([5, 5], dtype=np.float32) >>> cu_sgd = cumlSGD(learning_rate='constant', eta0=0.005, epochs=2000, ... fit_intercept=True, batch_size=2, ... tol=0.0, penalty='none', loss='squared_loss') >>> cu_sgd.fit(X, y) >>> cu_pred = cu_sgd.predict(pred_data).to_numpy() >>> print(" cuML intercept : ", cu_sgd.intercept_) cuML intercept : 0.00418... >>> print(" cuML coef : ", cu_sgd.coef_) cuML coef : 0 0.9841... 1 0.0097... dtype: float32 >>> print("cuML predictions : ", cu_pred) cuML predictions : [3.0055... 2.0214...] fit(X, y[, convert_dtype]) Fit the model with X and y. get_param_names() Returns a list of hyperparameter names owned by this class. predict(X[, convert_dtype]) Predicts the y for X. predictClass(X[, convert_dtype]) Predicts the y for X. fit(X, y, convert_dtype=True) SGD[source]# Fit the model with X and y. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas yarray-like (device or host) shape = (n_samples, 1) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the train method will, when necessary, convert y to be the same data type as X if they differ. This will increase memory used for the method. Returns a list of hyperparameter names owned by this class. It is expected that every child class overrides this method and appends its extra set of parameters that it in-turn owns. This is to simplify the implementation of get_params and set_params methods. predict(X, convert_dtype=True) CumlArray[source]# Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the predict method will, when necessary, convert the input to the data type which was used to train the model. This will increase memory used for the method. predscuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, 1) Predicted values For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. predictClass(X, convert_dtype=True) CumlArray[source]# Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas convert_dtypebool, optional (default = True) When set to True, the predictClass method will, when necessary, convert the input to the data type which was used to train the model. This will increase memory used for the method. predscuDF, CuPy or NumPy object depending on cuML’s output type configuration, shape = (n_samples, 1) Predicted values For more information on how to configure cuML’s output type, refer to: Output Data Type Configuration. Random Forest# class cuml.ensemble.RandomForestClassifier(, split_criterion=0, handle=None, verbose=False, output_type=None, *kwargs)# Implements a Random Forest classifier model which fits multiple decision tree classifiers in an ensemble. Note that the underlying algorithm for tree node splits differs from that used in scikit-learn. By default, the cuML Random Forest uses a quantile-based algorithm to determine splits, rather than an exact count. You can tune the size of the quantiles with the n_bins parameter. n_estimatorsint (default = 100) Number of trees in the forest. (Default changed to 100 in cuML 0.11) split_criterionint or string (default = 0 ('gini')) The criterion used to split nodes. ■ 0 or 'gini' for gini impurity ■ 1 or 'entropy' for information gain (entropy) ■ 2 or 'mse' for mean squared error ■ 4 or 'poisson' for poisson half deviance ■ 5 or 'gamma' for gamma half deviance ■ 6 or 'inverse_gaussian' for inverse gaussian deviance only 0/'gini' and 1/'entropy' valid for classification bootstrapboolean (default = True) Control bootstrapping. ■ If True, eachtree in the forest is built on a bootstrapped sample with replacement. ■ If False, the whole dataset is used to build each tree. max_samplesfloat (default = 1.0) Ratio of dataset rows used while fitting each tree. max_depthint (default = 16) Maximum tree depth. Must be greater than 0. Unlimited depth (i.e, until leaves are pure) is not supported. This default differs from scikit-learn’s random forest, which defaults to unlimited depth. max_leavesint (default = -1) Maximum leaf nodes per tree. Soft constraint. Unlimited, If -1. max_featuresint, float, or string (default = ‘sqrt’) Ratio of number of features (columns) to consider per node split. ■ If type int then max_features is the absolute count of features to be used ■ If type float then max_features is used as a fraction. ■ If 'sqrt' then max_features=1/sqrt(n_features). ■ If 'log2' then max_features=log2(n_features)/n_features. Changed in version 24.06: The default of max_features changed from "auto" to "sqrt". n_binsint (default = 128) Maximum number of bins used by the split algorithm per feature. For large problems, particularly those with highly-skewed input data, increasing the number of bins may improve accuracy. n_streamsint (default = 4) Number of parallel streams used for forest building. min_samples_leafint or float (default = 1) The minimum number of samples (rows) in each leaf node. ■ If type int, then min_samples_leaf represents the minimum number. ■ If float, then min_samples_leaf represents a fraction and ceil(min_samples_leaf n_rows) is the minimum number of samples for each leaf node. min_samples_splitint or float (default = 2) The minimum number of samples required to split an internal node. ■ If type int, then min_samples_split represents the minimum number. ■ If type float, then min_samples_split represents a fraction and max(2, ceil(min_samples_split * n_rows)) is the minimum number of samples for each split. min_impurity_decreasefloat (default = 0.0) Minimum decrease in impurity required for node to be split. max_batch_sizeint (default = 4096) Maximum number of nodes that can be processed in a given batch. random_stateint (default = None) Seed for the random number generator. Unseeded by default. Does not currently fully guarantee the exact same results. Specifies the cuml.handle that holds internal CUDA state for computations in this model. Most importantly, this specifies the CUDA stream that will be used for the model’s computations, so users can run different models concurrently in different streams by creating handles in several streams. If it is None, a new one is created. verboseint or boolean, default=False Sets logging level. It must be one of cuml.common.logger.level_. See Verbosity Levels for more info. output_type{‘input’, ‘array’, ‘dataframe’, ‘series’, ‘df_obj’, ‘numba’, ‘cupy’, ‘numpy’, ‘cudf’, ‘pandas’}, default=None Return results and set estimator attributes to the indicated output type. If None, the output type set at the module level (cuml.global_settings.output_type) will be used. See Output Data Type Configuration for more info. Known Limitations This is an early release of the cuML Random Forest code. It contains a few known limitations: ☆ GPU-based inference is only supported with 32-bit (float32) datatypes. Alternatives are to use CPU-based inference for 64-bit (float64) datatypes, or let the default automatic datatype conversion occur during GPU inference. ☆ While training the model for multi class classification problems, using deep trees or max_features=1.0 provides better performance. For additional docs, see scikitlearn’s RandomForestClassifier. >>> import cupy as cp >>> from cuml.ensemble import RandomForestClassifier as cuRFC >>> X = cp.random.normal(size=(10,4)).astype(cp.float32) >>> y = cp.asarray([0,1]5, dtype=cp.int32) >>> cuml_model = cuRFC(max_features=1.0, ... n_bins=8, ... n_estimators=40) >>> cuml_model.fit(X,y) >>> cuml_predict = cuml_model.predict(X) >>> print("Predicted labels : ", cuml_predict) Predicted labels : [0. 1. 0. 1. 0. 1. 0. 1. 0. 1.] convert_to_fil_model([output_class, ...]) Create a Forest Inference (FIL) model from the trained cuML Random Forest model. convert_to_treelite_model() Converts the cuML RF model to a Treelite model fit(X, y[, convert_dtype]) Perform Random Forest Classification on the input data get_detailed_text() Obtain the detailed information for the random forest model, as text get_json() Export the Random Forest model as a JSON string get_summary_text() Obtain the text summary of the random forest model predict(X[, predict_model, threshold, algo, ...]) Predicts the labels for X. predict_proba(X[, algo, convert_dtype, ...]) Predicts class probabilities for X. score(X, y[, threshold, algo, ...]) Calculates the accuracy metric score of the model for X. convert_to_fil_model(output_class=True, threshold=0.5, algo='auto', fil_sparse_format='auto')[source]# Create a Forest Inference (FIL) model from the trained cuML Random Forest model. output_classboolean (default = True) This is optional and required only while performing the predict operation on the GPU. If true, return a 1 or 0 depending on whether the raw prediction exceeds the threshold. If False, just return the raw prediction. algostring (default = ‘auto’) This is optional and required only while performing the predict operation on the GPU. ★ 'naive' - simple inference using shared memory ★ 'tree_reorg' - similar to naive but trees rearranged to be more coalescing-friendly ★ 'batch_tree_reorg' - similar to tree_reorg but predicting multiple rows per thread block ★ 'auto' - choose the algorithm automatically. Currently ★ 'batch_tree_reorg' is used for dense storage and ‘naive’ for sparse storage thresholdfloat (default = 0.5) Threshold used for classification. Optional and required only while performing the predict operation on the GPU. It is applied if output_class == True, else it is ignored fil_sparse_formatboolean or string (default = auto) This variable is used to choose the type of forest that will be created in the Forest Inference Library. It is not required while using predict_model=’CPU’. ★ 'auto' - choose the storage type automatically (currently True is chosen by auto) ★ False - create a dense forest ★ True - create a sparse forest, requires algo=’naive’ or algo=’auto’ A Forest Inference model which can be used to perform inferencing on the random forest model. Converts the cuML RF model to a Treelite model tl_to_fil_modelTreelite version of this model fit(X, y, convert_dtype=True)[source]# Perform Random Forest Classification on the input data Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas yarray-like (device or host) shape = (n_samples, 1) Dense matrix of type np.int32. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas DataFrame/Series. convert_dtypebool, optional (default = True) When set to True, the method will automatically convert the inputs to np.float32. convert_dtypebool, optional (default = True) When set to True, the fit method will, when necessary, convert y to be of dtype int32. This will increase memory used for the method. Obtain the detailed information for the random forest model, as text Export the Random Forest model as a JSON string Obtain the text summary of the random forest model predict(X, predict_model='GPU', threshold=0.5, algo='auto', convert_dtype=True, fil_sparse_format='auto') CumlArray[source]# Predicts the labels for X. Xarray-like (device or host) shape = (n_samples, n_features) Dense matrix. If datatype is other than floats or doubles, then the data will be converted to float which increases memory utilization. Set the parameter convert_dtype to False to avoid this, then the method will throw an error instead. Acceptable formats: CUDA array interface compliant objects like CuPy, cuDF DataFrame/Series, NumPy ndarray and Pandas predict_modelString (default = ‘GPU’) ‘GPU’ to predict using the GPU, ‘CPU’ otherwise. algostring (default = 'auto') This is optional and required only while performing the predict	{"url":"https://docs.rapids.ai/api/cuml/latest/api/","timestamp":"2024-11-05T07:43:21Z","content_type":"text/html","content_length":"1049025","record_id":"<urn:uuid:13ea4048-b503-4a40-8c3e-cbbd35aad2eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00147.warc.gz"}
How to Convert a Monthly IRR Into an Annual IRR \| Sapling When embarking on a new endeavor or investing in development to increase revenue or cut down costs, a company will determine the annualized IRR before committing. Both monthly and annual IRR analyze various investments, especially under the circumstances with multiple cash investments. IRR stands for internalized rate of return. According to the U.S. Securities and Exchange Commission, it is the annualized effective compounded rate of return of an investment, accounting for money's time value and expressed as a percentage. Essentially, an annualized IRR is the expected annual rate of return on an investment or project. More often than not, taking on a new project requires investing in development. When a company decides to either cut down on costs or increase their revenue by developing a new project, they will research the return on the investments to gauge whether it's worth taking on and how to prioritize projects. Internal rate of return is most commonly used when analyzing investments in private equity and venture capital, Corporate Finance Institute writes. It's particularly advantageous in situations with multiple cash investments over a business's life as well as cash flow through an IPO or business sale. After determining the IRR, it is compared to the company's minimum acceptable rate of return (MARR), also known as the hurdle rate, to determine whether the project is a viable investment. However, the MARR is not the only metric a company considers before investing. Calculating Internal Rate of Return To calculate annualized IRR, the cash investment for the beginning period will equal the present value of future cash flow, making the net present value (NPV) equal zero. According to Corporate Finance Institute, the NPV is the value of all future cash flows over the investment's life, discounted to the present and expressed as either a positive or negative value. To perform the calculation for annualized IRR, one would need to know the initial investment and the cash flows for each period. Calculating the internal rate of return can get complicated, so it's typically recommended to calculate it using the IRR function in Excel. However, it's essential to note that the IRR function in Excel assumes equal time periods, meaning it ends the following year on the same month it started. If you require flexibility in the time period, use the XIRR function in Excel instead, which allows you to manually input the time periods you wish to use in the equation. When using either IRR function in Excel, the resulting value is an annual rate of return. According to Ablebits, you can calculate the monthly internal rate of return by using the XIRR function in Excel. First, calculate the annual internal rate of return by inputting the cash flow beside the corresponding date. Once you have the annual XIRR value, input it into the following equation: (1 + Annual XIRR)^(1/12) - 1. Converting Monthly to Annualized IRR Essentially, the annualized IRR is the amount of money made when the monthly internal rate of return continues each month for a year. To convert a monthly IRR to annual IRR, use the formula above, except instead of raising to the power of 1/12, you will raise to the power of 12: Annual IRR = (1 + Monthly IRR)^12 - 1. For example, suppose the monthly IRR is 5 percent. The calculation will go as follows: (1 + .05)^12 -1 = 79.59 percent.	{"url":"https://www.sapling.com/12202281/convert-monthly-irr-annual-irr","timestamp":"2024-11-13T12:00:17Z","content_type":"text/html","content_length":"241741","record_id":"<urn:uuid:050eee46-5fae-41c9-8f35-2d2a9cb345d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00517.warc.gz"}
Please find below a range of free books on the subject of Vedic Mathematics. N.B. these books are mostly in English, except for the "Vedic Mathematics Teacher's Manual - Elementary Level", which has been translated into six languages (See Book tab). A good free introductory ebook in Spanish can be found here. [We would like to point that when searching for free material on Vedic Mathematics, you may need to be cautious, as we found that often the material would not be provided unless you carried out one of the following:- • Provide your email address • Have to sign up to one of several offers • Provide your credit card details to verify your identity • Required to install a program to make the book available N.B. for the last option we often found Adobe Acrobat books with file names like "<Book Title>.pdf.exe". The ".exe" implying installation of a program on your computer which is not necessary for a document to be readable by the Adobe Acrobat reader i.e. file names should just end with ".pdf". From our point of view, the best situation you can end up with is receiving spam email etc, with the worst scenario's being someone using your credit card or having virus's or spyware installed onto your computer. It should be noted that the above requirements are completely unnecessary to make material available for free and imply some non monetary cost to yourself.] The free books available below are all simple Adobe Acrobat PDF documents. We only ask that you do not upload them to any other place on the Internet without consulting us first (and where the document is hosted on another website, please consult with the original author Vedic Mathematics Teacher's Manual - Elementary Level MULTIPLYING BY 4, 8 DIVIDING BY 4, 8 MULTIPLYING BY 5, 50, 25 DIVIDING BY 5, 50, 25 DIVIDING BY 5 DIVIDING BY 50, 25 LESSON 3 DIGIT SUMS LESSON 4 LEFT TO RIGHT LESSON 5 ALL FROM 9 AND THE LAST FROM 10 ALL FROM 9 AND THE LAST FROM 10 NUMBERS CLOSE TO 100 NUMBERS OVER 100 THE FIRST BY THE FIRST AND THE LAST BY THE LAST DIVISIBILITY BY 11 ALL FROM 9 AND THE LAST FROM 10 THE FIRST BY THE FIRST AND THE LAST BY THE LAST LESSON 12 SQUARING SQUARING NUMBERS NEAR 50 3 AND 4 FIGURE NUMBERS LESSON 13 EQUATIONS LESSON 14 FRACTIONS DIVISION BY 9 DIVISION BY 8 ETC. DIVISION BY 99, 98 ETC. LESSON 16 THE CROWNING GEM This book is designed for teachers of children in grades 3 to 7. It shows how Vedic Mathematics can be used in a school course but does not cover all school topics (see contents). The book can be used for teachers who wish to learn the Vedic system or to teach courses on Vedic mathematics for this level. The Manual contains many topics that are not in the other Manuals that are suitable for this age range and many topics that are also in Manual 2 are covered in greater detail here. ENGLISH version Homework exercises for Manual 1 (extracted from the book) - (click below) Full solutions to the exercises for Manual 1 (click below) HINDI version (click below) GERMAN version (click below) FRENCH Translation (click below) ROMANIAN Translation (click below) TAMIL Translation (click below) JAPANESE Translation (click below) Japanese Workbook (click below) NORWEGIAN Translation (click below) Vertically and Crosswise Preface to 'Vedic Mathematics' 9 SINE, COSINE AND INVERSE TANGENT 10 INVERSE SINE AND COSINE AND TANGENT This is an advanced book of sixteen chapters on one Sutra ranging from elementary multiplication etc. to the solution of non-linear partial differential equations. It deals with (i) calculation of common functions and their series expansions, and (ii) the solution of equations, starting with simultaneous equations and moving on to algebraic, transcendental and differential equations. The text contains exercises and answers. Part A: Congruence, magnitudes and Lines Part B: Angles, parallels, triangles and quadrilaterals Part C: Concerning area equalities and similar triangles Part D: Elementary properties of a circle Part I: Some basics Part II: Language and reason Part III: Comparisons with Euclid's Elements Part IV: Movement in geometry Part V: The valid use of figures Summary and Conclusions Appendix 1: Application of the sixteen sutras to the present system of geometry Appendix 2: Alternative proofs and sequences in Part D Appendix 3: Further definitions This book demonstrates the kind of system that could have existed before literacy was widespread and takes us from first principles to theorems on elementary properties of circles. It presents direct, immediate and easily understood proofs. These are based on only one assumption (that magnitudes are unchanged by motion) and three additional provisions (a means of drawing figures, the language used and the ability to recognise valid reasoning). It includes discussion on the relevant philosophy of mathematics and is written both for mathematicians and for a wider audience. Authors: A. P. Nicholas, J. Pickles, K. Williams, 1982. Paperback, 166 pages, A4 size. Currently out of print. Chapter 8 is on Addition. Chapter 9 is on Subtraction and combined addition and subtraction. Following various lecture courses in London an interest arose for printed material containing the course material. This book of 12 chapters was the result covering a range topics from elementary arithmetic to cubic equations. FREE PRACTICE SHEETS (& Sutras list etc.) These worksheets are designed for use with the DVD Basic Course, anyone is welcome to download and use them. Answers are given at the end of each sheet. Mental Math Workouts Vedic Math Genius Free books Hosted on other websites Fundamentals & Applications of Vedic Mathematics Published by : State Council of Educational Research & Training, New Delhi and printed at Educational Stores, S-5, Bsr. Road Ind. Area, Ghaziabad (U.P.) Varun Marg, Defence Colony, New Delhi-110024 Chief Advisor Anita Satia Director, SCERT Dr. Pratibha Sharma, Joint Director, SCERT Dr. Anil Kumar Teotia (Sr. Lecturer, DIET Dilshad Garden) Neelam Kapoor (Retired PGT, Directorate of Education) Chander Kanta Chabria (PGT, RPVV Tyagraj Nagar, Lodhi Road) Rekha Jolly (TGT, RPVV Vasant Kunj) Dr. Satyavir Singh (Principal SNI College Pilana) Dr. Anil Kumar Teotia Sr. Lecturer, DIET Dilshad Garden Publication Officer Ms. Sapna Yadav Publication Team Navin Kumar, Ms. Radha, Jai Baghwan Preface 03-04 Introduction 07-11 Chapter-1 Addition and Subtraction 12-24 1. Addition - Completing the whole 2. Addition from left to right 3. Addition of list of numbers - Shudh method 4. Subtraction - Base method 5. Subtraction - Completing the whole 6. Subtraction from left to right Chapter-2 Digit Sums, Casting out 9s, 9-Check Method 25-28 Chapter-3 11-Check method 29-31 Chapter-4 Special Multiplication methods 32-52 1. Base Method 2. Sub Base Method 3. Vinculum 4. Multiplication of complimentary numbers 5. Multiplication by numbers consisting of all 9s 6. Multiplication by 11 7. Multiplication by two-digit numbers from right to left 8. Multiplication by three and four-digit numbers from right to left. Chapter-5 Squaring and square Roots 53-57 1. Squaring numbers ending in 5 2. Squaring Decimals and Fraction 3. Squaring Numbers Near 50 4. Squaring numbers near a Base and Sub Base 5. General method of Squaring - from left to right 6. Number splitting to simplify Squaring Calculation 7. Algebraic Squaring Square Roots 1. Reverse squaring to find Square Root of Numbers ending in 25 2. Square root of perfect squares 3. General method of Square Roots Chapter-6 Division 58-64 1. Special methods of Division 2. Straight Division Vedic Math Presentation Vedic Mathematics Vedic Mathematics Methods Vedic Mathematics - Methods Preface 1 I. Why Vedic Mathematics? 3 II. Vedic Mathematical Formulae 5 1. Ekadhikena Purvena 7 2. Nikhilam navatascaramam Dasatah 18 3. Urdhva - tiryagbhyam 31 4. Paravartya Yojayet 41 5. Sunyam Samya Samuccaye 53 6. Anurupye - Sunyamanyat 64 7. Sankalana - Vyavakalanabhyam 65 8. Puranapuranabhyam 67 9. Calana - Kalanabhyam 68 10. Ekanyunena Purvena 69 11. Anurupyena 75 12. Adyamadyenantya - mantyena 82 13. Yavadunam Tavadunikrtya Varganca Yojayet 86 14. Antyayor Dasakepi 93 15. Antyayoreva 96 16. Lopana Sthapanabhyam 101 17. Vilokanam 106 18. Gunita Samuccayah : Samuccaya Gunitah 113 III Vedic Mathematics - A briefing 115 1. Terms and Operations 116 2. Addition and Subtraction 130 3. Multiplication 139 4. Division 144 5. Miscellaneous Items 151 IV Conclusion 158 Vedic Mathematical Concepts and Their Application to Unsolved Mathematical Problems: Three Proofs of Fermat's Last Theorem Author John M Muehlman, 1993. Publisher: Dissertation Information Service Pages 72 (although pages are numbered from 36 for some reason) ASIN: B0006PAVSA Introduction 39 Maharishi's Description of the Veda 41 Self-Interacting Dynamics of Consciousness 41 Laws of Nature 42 Pure Knowledge and Infinite Organizing Power 43 Development of Consciousness through MaharishiÕs Vedic Technologies 43 Experience of Pure Consciousness 45 Higher States of Consciousness 47 Nature of All Knowingness in MaharishiÕs Teaching 49 Infinite Correlation 49 Principle of Least Action 50 Mathematics Without Steps 50 Introduction to Vedic Sªtra Based Computation 55 Literature Review on Characteristics of Vedic Sªtra Based Computation Relevant to the Experimental Study 61 Efficient Algorithms and Ease of Learning 61 Flexible Format 62 Conducive to Mental Computation 63 Upsurges of Joy 63 Pilot Research on Vedic Sªtra Based Computation 63 Pilot Study Number One 63 Pilot Study Number Two 64 Pilot Study Number Three 65 An Experimental Study on Vedic Sªtra Based Multiplication and Checking 69 Subjects and Setting 69 Overview of the Curricula 70 Comparison of the Algorithms 71 Instruments Used to Evaluate Hypotheses 75 Design 76 Discussion of Results from the Study 76 Hypothesis One: Multiplication Skill 77 Hypothesis Two: Checking Skill 80 Hypothesis Three: Multiplication and Checking Affect 81 Hypothesis Four: Mental Mathematics 86 Extension of Results to Explain Possible Growth in the Direction of All Knowingness 87 Appendix: Review of Additional Characteristics of a Full Vedic Sªtra Based Computation Program 90 Improving achievement, affect, and mental mathematics through Vedic sutra based computation Vedic Mathematics: Vedic or Mathematics: A Fuzzy and Neutrosophic Analysis W. B. Vasantha Kandasamy (Author) Florentin Smarandache (Author) Meena Kandasamy (Illustrator) Perfect Paperback: 220 pages Publisher: Automaton; 1 edition (December 5, 2006) Language: English ISBN-10: 1599730049 ISBN-13: 978-1599730042 Product Dimensions: 8.7 x 6 x 0.5 inches Preface 5 Chapter One Chapter Two 2.1 Views of Prof. S.G.Dani about Vedic Mathematics from Frontline 33 2.2 Neither Vedic Nor Mathematics 50 2.3 Views about the Book in Favour and Against 55 2.4 Vedas: Repositories of Ancient Indian Lore 58 2.5 A Rational Approach to Study Ancient Literature 59 2.6 Shanghai Rankings and Indian Universities 60 2.7 Conclusions derived on Vedic Mathematics and the Calculations of Guru Tirthaji - Secrets of Ancient Maths 61 Chapter Three 3.1 Introduction to FCM and the Working of this Model 65 3.2 Definition and Illustration of Fuzzy Relational Maps (FRMS) 72 3.3 Definition of the New Fuzzy Dynamical System 77 4 3.4 Neutrosophic Cognitive Maps with Examples 78 3.5 Description of Neutrosophic Relational Maps 87 3.6 Description of the new Fuzzy Neutrosophic model 92 Chapter Four 4.1 Views of students about the use of Vedic Mathematics in their curriculum 97 4.2 Teachers views on Vedic Mathematics and its overall influence on the Students Community 101 4.3 Views of Parents about Vedic Mathematics 109 4.4 Views of Educationalists about Vedic Mathematics 114 4.5 Views of the Public about Vedic Mathematics 122 Chapter Five OBSERVATIONS 165 5.1 Students’ Views 165 5.2 Views of Teachers 169 5.3 Views of Parents 180 5.4 Views of the Educated 182 5.5 Observations from the Views of the Public 193 REFERENCE 197 INDEX 215 ABOUT THE AUTHORS 220 The Vedas are considered divine in origin and are assumed to be revelations from God. In traditional Hinduism, the Vedas were to be learnt only by the upper caste Hindus. The lower castes (Sudras) and so-called untouchables (who were outside the Hindu social order) were forbidden from even hearing to its recitation. In recent years, there have been claims that the Vedas contain the cure to AIDS and the production of electricity. Here the authors probe into Vedic Mathematics (that gained renown during the revivalist Hindutva rule in India and was introduced into school syllabus in several states); and explore if it is really Vedic in origin or Mathematics in content. To gain a better understanding of its imposition, we interviewed students, teachers, parents, educationists and activists. We analyze this problem using models like Fuzzy Cognitive Maps (FCM), Fuzzy Relational Maps (FRM) and newly constructed Fuzzy Dynamical System (and their Neutrosophic Analogues). The issue of imposition of Vedic Mathematics into the school curriculum involves religious politics, caste supremacy, apart from elementary arithmetic so we use fuzzy and neutrosophic techniques to gain acute insight into how students have been affected because of this politically motivated syllabus revision. Our Comments We believe this paper was a reaction by some in the academic community to certain elements of the Hindu community trying to use Vedic Mathematics to promote Hinduism by teaching Vedic Mathematics in schools. It is our understanding that much of this initial teaching of Vedic Mathematics may have been implemented poorly by people who did not have a good grounding in their subject matter as they may have been more interested in promoting Hinduism than Vedic Mathematics, hence the academic reaction. This paper was produced in 2006 and often appears in searches for Vedic Mathematics. The understanding and teaching of Vedic Mathematics has moved on a lot since this point in time.	{"url":"https://www.vedicmaths.org/free","timestamp":"2024-11-02T03:07:08Z","content_type":"application/xhtml+xml","content_length":"164580","record_id":"<urn:uuid:286bf6ce-1a6a-4cbc-b9fa-86ebfd89a2c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00266.warc.gz"}
Price Estimation - Assignment Help Central Price Estimation you are trying to estimate the proper price to charge a market for the firm that sells beer in Lancaster Pennsylvania. They estimated that the demand curve for the market is Quantity demanded=20-2P. The firm currently prices the good at 8 dollars. They want to move the price to 6 dollars in a attempt to increase profits. This question is worth 5 points (please show work for full credit) 1) What is the elasticity for this move in price (Use the midpoint method for elasticity) make sure you show your work? (1.5) 2) Now you are contacted by a different branch of this company that is in a lower priced (Scranton P.A.) market with the same demand curve. They want to move the price from 2 to 5 dollars. What is the elasticity of this change (Use the midpoint method)? (1.5) 3) Now suppose you find out that there is a similar product that impacts the Quantity demanded for a product. The relationship is Qi=15+Pb. Now suppose the price (Pb) of the other good is 5 and goes to 6. What is the Cross-price elasticity of the good? what type of relationship is this (complement or substitute) (2) Price Estimation you are trying to estimate the proper price to charge a market for the firm that sells beer in Lancaster Pennsylvania. They estimated that the demand curve for the market is Quantity demanded=20-2P. The firm currently prices the good at 8 dollars. They want to move the price to 6 dollars in a attempt to increase profits.This question is worth 5 points (please show work for full credit) 1) What is the elasticity for this move in price (Use the midpoint method for elasticity) make sure you show your work? (1.5) 2) Now you are contacted by a different branch of this company that is in a lower priced (Scranton P.A.) market with the same demand curve. They want to move the price from 2 to 5 dollars. What is the elasticity of this change (Use the midpoint method)? (1.5) 3) Now suppose you find out that there is a similar product that impacts the Quantity demanded for a product. The relationship is Qi=15+Pb. Now suppose the price (Pb) of the other good is 5 and goes to 6. What is the Cross-price elasticity of the good? what type of relationship is this (complement or substitute) (2) you are trying to estimate the proper price to charge a market for the firm that sells beer in Lancaster Pennsylvania. They estimated that the demand curve for the market is Quantity demanded=20-2P. The firm currently prices the good at 8 dollars. They want to move the price to 6 dollars in a attempt to increase profits. This question is worth 5 points (please show work for full credit) Price Estimation 1) What is the elasticity for this move in price (Use the midpoint method for elasticity) make sure you show your work? (1.5) 2) Now you are contacted by a different branch of this company that is in a lower priced (Scranton P.A.) market with the same demand curve. They want to move the price from 2 to 5 dollars. What is the elasticity of this change (Use the midpoint method)? (1.5) 3) Now suppose you find out that there is a similar product that impacts the Quantity demanded for a product. The relationship is Qi=15+Pb. Now suppose the price (Pb) of the other good is 5 and goes to 6. What is the Cross-price elasticity of the good? what type of relationship is this (complement or substitute) (2) Use APA referencing style.	{"url":"https://assignmenthelpcentral.com/price-estimation/","timestamp":"2024-11-06T22:02:28Z","content_type":"text/html","content_length":"86216","record_id":"<urn:uuid:5d353190-63bb-49fc-aed0-c2a82dd0b653>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00515.warc.gz"}
[Pl-seminar] [CHANGE OF DATE] van Horn at NU PL-seminar Mitchell Wand wand at ccs.neu.edu Sun May 20 09:49:11 EDT 2007 CHANGE OF DATE NU Programming Languages Seminar *TUESDAY 5/22/07* (* NOTE CHANGE OF DATE *) Room 366 WVH (http://www.ccs.neu.edu/home/wand/directions.html) Relating Complexity and Precision in Control Flow Analysis David Van Horn We analyze the computational complexity of kCFA, a hierarchy of control flow analyses that determine which functions may be applied at a given call-site. This hierarchy specifies related decision problems, quite apart from any algorithms that may implement their solutions. We identify a simple decision problem answered by this analysis and prove that in the 0CFA case, the problem is complete for polynomial time. The proof is based on a nonstandard, symmetric implementation of Boolean logic within multiplicative linear logic (MLL). We also identify a simpler version of 0CFA related to eta-expansion, and prove that it is complete for LOGSPACE, using arguments based on computing paths and For any fixed k>0, it is known that kCFA (and the analogous decision problem) can be computed in time exponential in the program size. For k=1, we show that the decision problem is NP-hard, and sketch why this remains true for larger fixed values of k. The proof technique depends on using the approximation of CFA as an essentially nondeterministic computing mechanism, as distinct from the exactness of normalization. When k=n, so that the ``depth'' of the control flow analysis grows linearly in the program length, we show that the decision problem is complete for EXPTIME. In addition, we sketch how the analysis presented here may be extended naturally to languages with control operators. All of the insights presented give clear examples of how straightforward observations about linearity, and linear logic, may in turn be used to give a greater understanding of functional programming and program analysis. (joint work with Harry Mairson) Upcoming Events: # 5/30 Torben Amtoft # Don't you want to speak at pl-seminar? More information about the pl-seminar mailing list	{"url":"http://lists.ccs.neu.edu/pipermail/pl-seminar/2007/000386.html","timestamp":"2024-11-14T10:38:19Z","content_type":"text/html","content_length":"4906","record_id":"<urn:uuid:c800998f-e601-40ba-bcd4-326e48b699e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00423.warc.gz"}
Navier-Stokes equations \| Description, Example & Application Navier-Stokes equations Introduction to Navier-Stokes Equations Navier-Stokes equations are a set of partial differential equations that describe the motion of fluid in both the laminar and turbulent regimes. They were first developed by Claude-Louis Navier and George Gabriel Stokes in the 19th century and have since been used to model a wide range of fluid flow problems in various fields, including engineering, physics, and biology. The equations describe how the pressure, velocity, and density of a fluid interact with each other and with external forces, such as gravity and friction. Derivation and Properties of the Equations The Navier-Stokes equations are derived from the principles of conservation of mass, momentum, and energy. They consist of four equations, one for the conservation of mass (continuity equation) and three for the conservation of momentum (Navier-Stokes equations). The equations are nonlinear and coupled, which makes them difficult to solve analytically, except for a few simple cases. However, numerical methods, such as finite element and finite volume methods, can be used to solve them numerically. The properties of the solutions to the Navier-Stokes equations, such as turbulence and vortices, have been studied extensively and are still an active area of research. Applications and Limitations of Navier-Stokes Equations The Navier-Stokes equations have numerous applications in engineering and science, such as in the design of aircraft, ships, and pipelines, as well as in the simulation of weather patterns and ocean currents. However, the equations have limitations, such as the assumption of continuum fluid, which breaks down at the molecular scale. In addition, the equations are computationally expensive to solve, especially for problems involving turbulence or multi-phase flows. Therefore, simplified models are often used, such as the Reynolds-averaged Navier-Stokes equations, which average over the turbulent fluctuations. Example: Solving Fluid Dynamics Problems with Navier-Stokes Equations One example of using the Navier-Stokes equations to solve a fluid dynamics problem is the simulation of blood flow in arteries. This problem involves complex fluid dynamics, such as flow separation, recirculation, and wall shear stresses, which can lead to the development of atherosclerosis. The Navier-Stokes equations can be used to model the blood flow and predict the flow patterns and wall shear stresses. This information can aid in the diagnosis and treatment of cardiovascular diseases. However, the accuracy of the simulations depends on the assumptions and boundary conditions used, as well as the computational resources available.	{"url":"https://your-physicist.com/navier-stokes-equations/","timestamp":"2024-11-15T03:15:44Z","content_type":"text/html","content_length":"54005","record_id":"<urn:uuid:7b1820ba-299d-48ca-8f00-4eb3d0ba0b67>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00312.warc.gz"}
Financial Ratios - Balance Sheet and Income Statement \| AccountingCoach Financial Ratios Using Amounts from the Balance Sheet and Income Statement In this section, we will discuss five financial ratios which use an amount from the balance sheet and an amount from the income statement. Specifically, we will discuss the following: • Ratio #10 Receivables turnover ratio • Ratio #11 Days’ sales in receivables (average collection period) • Ratio #12 Inventory turnover ratio • Ratio #13 Days’ sales in inventory (days to sell) • Ratio #14 Return on stockholders’ equity The first four of the above ratios inform us about a company’s speed in: • Collecting (turning over) its accounts receivable • Selling (turning over) its goods in inventory The speed at which a company is able to convert its accounts receivable and inventory into cash is crucial for the company to meet its payroll, pay its suppliers, and pay other current liabilities when the amounts are due. In other words, a company could have a huge amount of working capital and an impressive current ratio, but it requires that the current assets be converted to cash to pay the bills. Therefore, a higher receivables turnover ratio (Ratio #10) and a higher inventory turnover ratio (Ratio #12) are better than lower ratios. These higher turnover ratios mean there will be less days’ sales in receivables (Ratio #11) and less days’ sales in inventory (Ratio #13). Having less days in receivables and inventory are better than a higher number of days. Recall that the amounts reported on the balance sheet are as of an instant or point in time, such as the final moment of an accounting year. Therefore, a balance sheet dated December 31 provides a “snapshot” of the pertinent general ledger account balances (assets, liabilities, equity) as of the final moment of December 31. Also recall that the income statement reports the cumulative amounts of revenues, expenses, gains, and losses that occurred during the entire 12 months that ended on December 31. To overcome this mismatch of comparing an income statement amount (such as the cumulative sales for the entire year) to a balance sheet amount (such as the accounts receivable balance at the final moment of the year), we need the balance sheet amount to be an average amount that is representative of all the days during the year. Graphing the daily (or perhaps weekly) balances during the year and then computing an average of those many data points will provide a representative average. Unfortunately, people outside of the company do not have access to those details. As an alternative, outsiders often compute an average based on the end-of-the-year moment for the current year and the previous year. (They do this without regard to whether these end-of-the-year balances are much lower than the balances during the year.) Ratio #10 Receivables Turnover Ratio The receivables turnover ratio (or accounts receivable turnover ratio) relates a company’s net credit sales from the most recent year to the amount the customers owed the company during that year. Net credit sales = gross credit sales (cash sales are excluded) minus any related sales discounts, sales returns, and sales allowances. If a new startup company makes its first sale with credit terms of net 30 days, the company records the sale by increasing Accounts Receivable and increasing Sales on Credit. If the customer pays in 30 days, the company will increase Cash and will decrease Accounts Receivable. This means that the company will be turning over its receivables in 30 days. If that occurs with every sale, the receivables turnover ratio will be approximately 12.2 times per year (365 days / 30 days). However, if all customers take 40 days to pay the amount owed, the receivables turnover ratio will be approximately 9.1 times per year (365 days / 40 days). The higher the receivables turnover ratio, the faster the receivables are turning into cash (which is necessary for the company to pay its current liabilities on time). Therefore, a higher receivables turnover ratio is better than a lower ratio. Some people categorize the receivables turnover ratio as an efficiency ratio since it indicates the speed in which the company had collected its accounts receivables and turned them into cash. The receivables turnover ratio is calculated as follows: Receivables turnover ratio = net credit sales for the year / average amount in accounts receivable Example 10 Assume that a company competes in an industry where customers are given credit terms of net 30 days. Also assume that the company had $570,000 of net credit sales during the most recent year and on average it had accounts receivable during the year of $60,000. Given this information, the company’s receivables turnover ratio is: Receivables turnover ratio = net credit sales for the year / avg. amount of accounts receivable Receivables turnover ratio = $570,000 / $60,000 Receivables turnover ratio = 9.5 times To determine if this company’s receivables turnover ratio of 9.5 is acceptable or not acceptable, you could do the following: • Look at the average receivables turnover ratio for the company’s industry • Calculate a competitor’s receivables turnover ratio • Compare it to the company’s past receivables turnover ratios • Compare it to the expected ratio for the credit terms given to its customers The larger the number of times that the receivables turn over during the year, the more often the company collects the cash it needs to pay its current liabilities. If you are computing the receivables turnover ratio by using a corporation’s published (external) financial statements, you should be aware of the following: • Typically, the income statement does not report the amount of net credit sales as a separate amount. Instead, only the amount of net sales (credit sales + cash sales) will be available. In our examples, we will provide the amount of a corporation’s net credit sales. • The balance sheet reports the corporation’s accounts receivable only at the final moment of the accounting year (and usually the balance at the final moment of the previous year). The average balance in accounts receivable throughout the year is not reported. As a result, an average balance in accounts receivable must be calculated. Since people outside of the corporation do not have access to the daily, weekly, or monthly balances, they often calculate a simple average based on the two balances as of the final moment of each accounting year. (This average could be much lower than the balances throughout the year since U.S. corporations often end their accounting years when their business activity is at the lowest levels.) In our examples, we will provide the average amount of a corporation’s accounts receivable throughout the accounting year. Ratio #11 Days’ Sales in Receivables (Average Collection Period) The days’ sales in receivables (also known as the average collection period) indicates the average amount of time it took in the past year for a company to collect its accounts receivable. An easy way to determine the number of days’ sales in receivables is to divide 365 (the days in a year) by the receivables turnover ratio, which was explained in Ratio #10. In other words, the formula for the days’ sales in receivables is: Days’ sales in receivables = 365 days / receivables turnover ratio Example 11 Assume that a company had $570,000 of net credit sales during the most recent year. During the year it had an average of $60,000 of accounts receivable. As a result, its receivables turnover ratio was 9.5 times per year ($570,000 / $60,000). Since the company’s receivables turnover ratio was determined to be 9.5, the days’ sales in receivables is calculated as follows: Days’ sales in receivables = 365 days / receivables turnover ratio Days’ sales in receivables = 365 days / 9.5 Days’ sales in receivables = 38.4 days To determine whether this company’s days’ sales in receivables of 38.4 days is acceptable (or not acceptable), you could do the following: • Look at the average receivables turnover ratio for the company’s industry • Calculate a competitor’s receivables turnover ratio • Compare it to the company’s past receivables turnover ratios • Compare it to the expected ratio for the credit terms given to its customers Having a smaller number of days’ sales in receivables means that on average, the company is converting its receivables into the cash needed to pay its current liabilities. The days’ sales in receivables (such as the 38.4 days we just calculated) was based on all customers’ transactions and unpaid balances. It includes the credit sales made a few days ago, 25 days ago, 50 days ago, 75 days ago, etc. Therefore, the average of 38.4 could be concealing some slow-paying customers’ accounts. Instead of using one of the receivables ratios, it would be better to have an aging of accounts receivable (which is readily available with accounting software). The aging of accounts receivable sorts each customer’s unpaid balance into columns which have headings such as: Current, 1-30 days past due, 31-60 days past due, 61-90 days past due, 91+ days past due. This aging report allows a company’s personnel to see the exact amount(s) owed by each customer. As a result, the company can take action to collect the past due amounts. Ratio #12 Inventory Turnover Ratio The inventory turnover ratio indicates the speed at which a company’s inventory of goods was sold during the past year. Since inventory is reported on a company’s balance sheet at its cost (not selling prices), it is necessary to relate the inventory cost to the cost of goods sold (not sales) reported on the company’s income statement. Since the cost of goods sold is the cumulative cost for all 365 days during the year, it is important to relate it to the average inventory cost throughout the year. Because a company’s published balance sheet reports only the inventory cost at the final moment of the accounting year and the final moment of the prior accounting year, the average of these two data points may not be representative of the inventory levels throughout the 365 days of the year. (The reason is that many U.S. corporations end their accounting year at the lowest levels of activity.) In our examples, we will provide you with the company’s average cost of inventory that is representative of the entire year. Here is the formula for the inventory turnover ratio: Inventory turnover ratio = cost of goods sold for the year / average cost of inventory during the year Since there are risks and costs associated with holding inventory, companies strive for a high inventory turnover ratio, so long as its inventory items are never out of stock. Example 12 Assume that during the most recent accounting year, a company had sales of $420,000 and its cost of goods sold was $280,000. Also assume that the company’s balance sheet at the end of the year reported the cost of its inventory as $75,000 and was $65,000 at the end of the previous year. An analysis of the company’s inventory records indicates that inventory cost increased steadily throughout the year. Based on the analysis, the average inventory cost during the accounting year was determined to be $70,000. Given this information, the company’s inventory turnover ratio for the recent accounting year was: Inventory turnover ratio = cost of goods sold for the year / avg. cost of inventory during the year Inventory turnover ratio = $280,000 / $70,000 Inventory turnover ratio = 4 times in the year To determine whether this company’s inventory turnover ratio of 4 is acceptable or not acceptable, you could do the following: • Look at the average inventory turnover ratio for the company’s industry • Calculate a competitor’s inventory turnover ratio • Compare it to the company’s past inventory turnover ratios • Compare it to the expected inventory turnover ratio The inventory turnover ratio is an average of perhaps hundreds of different products and component parts carried in inventory. Some items in inventory may not have had any sales in more than a year, some may not have had sales in six months, some may sell within weeks of arriving from the suppliers, etc. Here’s a Tip Rather than relying on the average turnover ratio for the entire inventory, a company’s managers could calculate a turnover ratio for each product it has in inventory. For example, the average quantity/units of its Item #123 in inventory would be compared to the quantity/units of Item #123 that were sold during the year. A simple worksheet would list every item in inventory and then calculate each item’s approximate inventory turnover ratio. The formula is: the number of units sold during the past year / the number of units in inventory. The slow-moving items (those with low inventory turnover ratios) would then be reviewed to determine whether it is profitable to continue carrying these items. An additional column could be added to the worksheet to show the days’ sales in inventory (Ratio #13 which follows). Ratio #13 Days’ Sales in Inventory (Days to Sell) The days’ sales in inventory (also known as days to sell) indicates the average number of days that it took for a company to sell its inventory. The goal is to have the fewest number of days of inventory on hand because of the high cost of carrying items in inventory (including the risk of items spoiling or becoming obsolete). Of course, there is also a cost for being out of stock. Therefore, managing inventory levels is important. An easy way to calculate the number of days’ sales in inventory is to divide 365 (the days in a year) by the inventory turnover ratio (Ratio #12). Here is the formula for calculating the days’ sales in inventory: Days’ sales in inventory = 365 days / inventory turnover ratio Example 13 Assume that a company’s cost of goods sold for the year was $280,000 and its average inventory cost for the year was $70,000. Therefore, its inventory turnover ratio was 4 times during the year ($280,000 / $70,000). Given that the company’s inventory turnover ratio was 4, the days’ sales in inventory is calculated as follows: Days’ sales in inventory = 365 days / inventory turnover ratio Days’ sales in inventory = 365 days / 4 Days’ sales in inventory = 91.25 days A smaller number of days’ sales in inventory is preferred, since it indicates the company will be converting its inventory to cash sooner. (It may get cash immediately for cash sales, or it will get cash when the resulting receivables are collected.) The days’ sales in inventory is an average of the many products that a company had in inventory. Some of the products may not have been sold in more than a year, some may not have been sold in 10 months, some were sold shortly after arriving from the suppliers, etc. Since we used the inventory turnover ratio to calculate the days’ sales in inventory, a mistake in calculating the inventory turnover ratio will result in an incorrect number of days’ sales in inventory. (For instance, if someone uses sales instead of the cost of goods sold to calculate the inventory turnover ratio, the days’ sales in inventory will not be accurate.) Ratio #14 Return on Stockholders’ Equity For a corporation that has only common stock (no preferred stock) outstanding, the return on stockholders’ equity is calculated by dividing its earnings (net income after tax) for a year by the average amount of stockholders’ equity during the same year. The amount of stockholders’ equity reported on a corporation’s balance sheet is the amount as of the final moment of the accounting year. On the other hand, the net income after tax is the cumulative amount earned throughout the entire year. Therefore, the calculation of the return on stockholders’ equity ratio should use the average amount of stockholders’ equity throughout the year. The formula for the annual return on stockholders’ equity for a corporation with only common stock is: Return on stockholders’ equity = net income after tax / average stockholders’ equity Example 14 Assume that during the past year a corporation had net income after tax (earnings) of $560,000. It was determined that a representative average amount of stockholders’ equity during the year was $2,800,000. Given this information, the corporation’s return on stockholders’ equity for the past year was: Return on stockholders’ equity = net income after tax / average stockholders’ equity Return on stockholders’ equity = $560,000 / $2,800,000 Return on stockholders’ equity = 20% To determine whether a corporation’s return on stockholders’ equity is reasonable, you could do the following: • Look at the average return on stockholders’ equity for the corporation’s industry • Calculate a competitor’s return on stockholders’ equity • Compare it to the corporation’s return on stockholders’ equity in recent years • Compare it to the planned return on stockholders’ equity Send Feedback Please let us know how we can improve this explanation No Thanks	{"url":"https://www.accountingcoach.com/financial-ratios/explanation/4","timestamp":"2024-11-02T15:32:24Z","content_type":"text/html","content_length":"126942","record_id":"<urn:uuid:ff3e3985-ef3d-42ef-9593-694da2045e37>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00297.warc.gz"}
What is the Dot Product of Perpendicular Vectors? Problems related to the multiplication of vectors are common in Physics and Mathematics. You may be asked to find the dot product of two vectors in your textbooks or assignments. The questions related to the dot product of perpendicular vectors are the simplest ones when compared to the cross product of the same vector’s type or others. Are you struggling to understand this multiplication? Read this blog till the end as it will discuss this topic in detail. By the end, you will learn about perpendicular vectors and their multiplication as well as check the solved example related to it. What are perpendicular vectors? Two vectors are said to be perpendicular if they are making an angle of 90 degrees with each other. In simple words, if vector A is pointing up along with y-axis and the other vector is pointing horizontally along with x-axis, those vectors are said to be perpendicular. What is the dot product of two perpendicular vectors? To learn about the dot product of 2 vectors, you must understand the formula of the dot product of two vectors first. For two vectors, A and B, the dot product formula will be written as: A . B = AB Cos𝜽 In the above formula, A and B on the left-hand side represent the vectors while “A” and “B” on the right side show the magnitude of the vectors. “𝜽” is the angle between two vectors being multiplied to get the resultant one. The dot product of two perpendicular vectors is “zero” because the angle between them is “90” degrees. It means there will be no resultant vector when two perpendicular vectors are multiplied. Why the dot product of perpendicular vectors is zero? The above formula for the dot product of perpendicular vectors involves “Cos 𝜽” and the value of “Cos 90” is “0”. Resultantly, the resultant of the above formula will become “0” when two perpendicular vectors are multiplied. If you have other types of vectors and facing problems while finding their multiplication, you can use the dot product calculator. This calculator can assist you in solving your questions quickly to complete your assignment. It shows you a step-by-step solution by following which you can learn the way to find this product easily. Example of the dot product of 2 perpendicular vectors Find the dot product of two vectors having coordinates (2, 3) and (4, 8) while the angle between them is 90 degrees. Suppose the vectors are “A” and “B” with vertices given in the question respectively. By putting the values in the above formula of the dot product, A . B = AB Cos𝜽 = [(2 x 4) + (3 x 8)] Cos (90) Cos 90 = 0 A . B = 0 By reading this guide and checking the example, we hope you have learned about the dot product of two perpendicular vectors. If you still have doubts, you can use the dot product calculator to check the solution for your required question and understand its steps. What is the dot product of perpendicular vectors? The dot product of perpendicular vectors is always “Zero”. What is the dot product of two perpendicular vectors? If two given vectors are perpendicular, the dot product of those vectors will be “0” because of the “Cosine” involvement in the formula for angle calculation. Is the dot product of perpendicular vectors 0? Yes, the vectors are said to be perpendicular only when their dot product is “0”. What does a parametric equation show you? The parametric equation describes the position of the point on the circumference of the circle for which the equation has been written. What is the dot product of two perpendicular vectors A and B equal to? The dot product of two perpendicular vectors A and B can be calculated as, A . B = AB Cos𝜽	{"url":"https://calculatorsbag.com/blogs/what-is-dot-product-of-perpendicular-vectors","timestamp":"2024-11-12T18:20:43Z","content_type":"text/html","content_length":"41611","record_id":"<urn:uuid:77b69b96-770a-434f-93ad-11c3caef6de5>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00306.warc.gz"}
Pat Quillen - MATLAB Central of 295,295 0 Questions 1 Answer of 20,205 0 Files of 153,527 0 Problems 66 Solutions Generate a random matrix A of (1,-1) Input n: is an positive integer which serves as the dimension of the matrix A; Output: A=(Aij),where each entry Aij is either... 12 months ago Sum all integers from 1 to 2^n Given the number x, y must be the summation of all integers from 1 to 2^x. For instance if x=2 then y must be 1+2+3+4=10. 1 year ago Magic is simple (for beginners) Determine for a magic square of order n, the magic sum m. For example m=15 for a magic square of order 3. 1 year ago Make a random, non-repeating vector. This is a basic MATLAB operation. It is for instructional purposes. --- If you want to get a random permutation of integer... 1 year ago Roll the Dice! Description Return two random integers between 1 and 6, inclusive, to simulate rolling 2 dice. Example [x1,x2] =... 1 year ago Number of 1s in a binary string Find the number of 1s in the given binary string. Example. If the input string is '1100101', the output is 4. If the input stri... 1 year ago Return the first and last characters of a character array Return the first and last character of a string, concatenated together. If there is only one character in the string, the functi... 1 year ago Getting the indices from a vector This is a basic MATLAB operation. It is for instructional purposes. --- You may already know how to <http://www.mathworks.... 1 year ago Check if number exists in vector Return 1 if number _a_ exists in vector _b_ otherwise return 0. a = 3; b = [1,2,4]; Returns 0. a = 3; b = [1,... 1 year ago Swap the input arguments Write a two-input, two-output function that swaps its two input arguments. For example: [q,r] = swap(5,10) returns q = ... 1 year ago Reverse the vector Reverse the vector elements. Example: Input x = [1,2,3,4,5,6,7,8,9] Output y = [9,8,7,6,5,4,3,2,1] 1 year ago Length of the hypotenuse Given short sides of lengths a and b, calculate the length c of the hypotenuse of the right-angled triangle. <<https://i.imgu... 1 year ago Generate a vector like 1,2,2,3,3,3,4,4,4,4 Generate a vector like 1,2,2,3,3,3,4,4,4,4 So if n = 3, then return [1 2 2 3 3 3] And if n = 5, then return [1 2 2 3 3 3 4... 1 year ago Maximum value in a matrix Find the maximum value in the given matrix. For example, if A = [1 2 3; 4 7 8; 0 9 1]; then the answer is 9. 1 year ago Vector creation Create a vector using square brackets going from 1 to the given value x in steps on 1. Hint: use increment. 1 year ago Doubling elements in a vector Given the vector A, return B in which all numbers in A are doubling. So for: A = [ 1 5 8 ] then B = [ 1 1 5 ... 1 year ago Create a vector Create a vector from 0 to n by intervals of 2. 1 year ago Flip the vector from right to left Flip the vector from right to left. Examples x=[1:5], then y=[5 4 3 2 1] x=[1 4 6], then y=[6 4 1]; Request not ... 1 year ago Find max Find the maximum value of a given vector or matrix. 1 year ago Answered Why is dst and idst 'not recommended'? These functions are not recommended mostly because of where they live---namely the PDE toolbox. This discouragment is a step in... 5 years ago \| 0 \| accepted Fibonacci sequence Calculate the nth Fibonacci number. Given n, return f where f = fib(n) and f(1) = 1, f(2) = 1, f(3) = 2, ... Examples: Inpu... 12 years ago Target sorting Sort the given list of numbers \|a\| according to how far away each element is from the target value \|t\|. The result should return... 12 years ago Make one big string out of two smaller strings If you have two small strings, like 'a' and 'b', return them put together like 'ab'. 'a' and 'b' => 'ab' For extra ... 12 years ago QWERTY coordinates Given a lowercase letter or a digit as input, return the row where that letter appears on a <http://en.wikipedia.org/wiki/Keyboa... 12 years ago Love triangles Given a vector of lengths [a b c], determines whether a triangle with non-zero area (in two-dimensional Euclidean space, smarty!... 12 years ago	{"url":"https://ch.mathworks.com/matlabcentral/profile/authors/536326","timestamp":"2024-11-15T03:26:31Z","content_type":"text/html","content_length":"104562","record_id":"<urn:uuid:d5a36c2a-61f2-4894-b85a-caa2ab6b72b8>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00542.warc.gz"}
Information hiding, enforced Code should be reusable. An expression traversing a data structure shouldn’t be written multiple times, it should be pulled out into a generic traversal function. At a larger scale, a random number generator shouldn’t be written multiple times, but rather pulled out into a module that can be used by others. It is important that such abstractions must be done carefully. Often times a type is visible to the caller, and if the type is not handled carefully the abstraction can leak. For example, a set with fast random indexing (useful for random walks on a graph) can be implemented with a sorted Vector. However, if the Vector type is leaked, the user can use this knowledge to violate the invariant. import scala.annotation.tailrec /** (i in repr, position of i in repr) / def binarySearch(i: Int, repr: Vector[Int]): (Boolean, Int) = / elided / object IntSet { type Repr = Vector[Int] def empty: Repr = Vector.empty def add(i: Int, repr: Repr): Repr = { val (isMember, indexOf) = binarySearch(i, repr) if (isMember) repr else { val (prefix, suffix) = repr.splitAt(indexOf) prefix ++ Vector(i) ++ suffix def contains(i: Int, repr: Repr): Boolean = binarySearch(i, repr)._1 import IntSet._ // import IntSet._ val good = add(1, add(10, add(5, empty))) // good: IntSet.Repr = Vector(1, 5, 10) val goodResult = contains(10, good) // goodResult: Boolean = true val bad = good.reverse // We know it's a Vector! // bad: scala.collection.immutable.Vector[Int] = Vector(10, 5, 1) val badResult = contains(10, bad) // badResult: Boolean = false val bad2 = Vector(10, 5, 1) // Alternatively.. // bad2: scala.collection.immutable.Vector[Int] = Vector(10, 5, 1) val badResult2 = contains(10, bad2) // badResult2: Boolean = false The issue here is the user knows more about the representation than they should. The function add enforces the sorted invariant on each insert, and the function contains leverages this to do an efficient look-up. Because the Vector definition of Repr is exposed, the user is free to create any Vector they wish which may violate the invariant, thus breaking contains. In general, the name of the representation type is needed but the definition is not. If the definition is hidden, the user is only able to work with the type to the extent the module allows. This is precisely the notion of information hiding. If this can be enforced by the type system, modules can be swapped in and out without worrying about breaking client code. It turns out there is a well understood principle behind this idea called existential quantification. Contrast with universal quantification which says “for all”, existential quantification says “there is a.” Below is an encoding of universal quantification via parametric polymorphism. trait Universal { def apply[A]: A => A Here Universal#apply says for all choices of A, a function A => A can be written. In the Curry-Howard Isomorphism, a profound relationship between logic and computation, this translates to “for all propositions A, A implies A.” It is therefore acceptable to write the following, which picks A to be Int. def intInstantiatedU(u: Universal): Int => Int = (i: Int) => u.apply(i) // intInstantiatedU: (u: Universal)Int => Int Existential quantification can also be written in Scala. trait Existential { type A def apply: A => A Note that this is just one way of encoding existentials - for a deeper discussion, refer to the excellent Type Parameters and Type Members blog series. The type parameter on apply has been moved up to a type member of the trait. Practically, this means every instance of Existential must pick one choice of A, whereas in Universal the A was parameterized and therefore free. In the language of logic, Existential#apply says “there is a” or “there exists some A such that A implies A.” This “there is a” is the crux of the error when trying to write a corresponding intExistential function. def intInstantiatedE(e: Existential): Int => Int = (i: Int) => e.apply(i) // <console>:19: error: type mismatch; // found : i.type (with underlying type Int) // required: e.A // (i: Int) => e.apply(i) // ^ In code, the type in Existential is chosen per-instance, so there is no way of knowing what the actual type chosen is. In logical terms, the only guarantee is that there exists some proposition that satisfies the implication, but it is not necessarily the case (and often is not) it holds for all propositions. Abstract types In the ML family of languages (e.g. Standard ML, OCaml), existential quantification and thus information hiding, is achieved through type members. Programs are organized into modules which are what contain these types. In Scala, this translates to organizing code with the object system, using the same type member feature to hide representation. The earlier example of IntSet can then be written: /* Abstract signature / trait IntSet { type Repr def empty: Repr def add(i: Int, repr: Repr): Repr def contains(i: Int, repr: Repr): Boolean /* Concrete implementation */ object VectorIntSet extends IntSet { type Repr = Vector[Int] def empty: Repr = Vector.empty def add(i: Int, repr: Repr): Repr = { val (isMember, indexOf) = binarySearch(i, repr) if (isMember) repr else { val (prefix, suffix) = repr.splitAt(indexOf) prefix ++ Vector(i) ++ suffix def contains(i: Int, repr: Repr): Boolean = binarySearch(i, repr)._1 As long as client code is written against the signature, the representation cannot be leaked. def goodUsage(set: IntSet) = { import set._ val s = add(1, add(10, add(5, empty))) contains(5, s) // goodUsage: (set: IntSet)Boolean If the user tries to assert the representation type, the type checker prevents it at compile time. def badUsage(set: IntSet) = { import set._ val s = add(10, add(1, empty)) // Maybe it's a Vector contains(10, Vector(10, 5, 1)) // <console>:23: error: value reverse is not a member of set.Repr // s.reverse // ^ // <console>:24: error: type mismatch; // found : scala.collection.immutable.Vector[Int] // required: set.Repr // contains(10, Vector(10, 5, 1)) // ^ Abstract types enforce information hiding at the definition site (the definition of IntSet is what hides Repr). There is another mechanism that enforces information hiding, which pushes the constraint to the use site. Consider implementing the following function. def foo[A](a: A): A = ??? Given nothing is known about a, the only possible thing foo can do is return a. If instead of a type parameter the function was given more information.. def bar(a: String): String = "not even going to use `a`" ..that information can be leveraged to do unexpected things. This is similar to the first IntSet example when knowledge of the underlying Vector allowed unintended behavior to occur. From the outside looking in, foo is universally quantified - the caller gets to pick any A they want. From the inside looking out, it is existentially quantified - the implementation knows only as much about A as there are constraints on A (in this case, nothing). Consider another function listReplace. def listReplace[A, B](as: List[A], b: B): List[B] = ??? Given the type parameters, listReplace looks fairly constrained. The name and signature suggests it takes each element of as and replaces it with b, returning a new list. However, even knowledge of List can lead to type checking implementations with strange behavior. // Completely ignores the input parameters def listReplace[A, B](as: List[A], b: B): List[B] = List.empty[B] Here, knowledge of List allows the implementation to create a list out of thin air and use that in the implementation. If instead listReplace only knew about some F[_] where F is a Functor, the implementation becomes much more constrained. trait Functor[F[_]] { def map[A, B](fa: F[A])(f: A => B): F[B] implicit val listFunctor: Functor[List] = new Functor[List] { def map[A, B](fa: List[A])(f: A => B): List[B] = def replace[F[_]: Functor, A, B](fa: F[A], b: B): F[B] = implicitly[Functor[F]].map(fa)(_ => b) replace(List(1, 2, 3), "typelevel") // res8: List[String] = List(typelevel, typelevel, typelevel) Absent any knowledge of F other than the ability to map over it, replace is forced to do the correct thing. Put differently, irrelevant information about F is hidden. The fundamental idea behind this is known as parametricity, made popular by Philip Wadler’s seminal Theorems for free! paper. The technique is best summarized by the following excerpt from the paper: Write down the definition of a polymorphic function on a piece of paper. Tell me its type, but be careful not to let me see the function’s definition. I will tell you a theorem that the function Why types matter Information hiding is a core tenet of good program design, and it is important to make sure it is enforced. Underlying information hiding is existential quantification, which can manifest itself in computation through abstract types and parametricity. Few languages support defining abstract type members, and fewer yet support higher-kinded types used in the replace example. It is therefore to the extent that a language’s type system is expressive that abstraction can be enforced. This blog post was tested with Scala 2.11.7 using tut.	{"url":"https://typelevel.org/blog/2016/03/13/information-hiding.html","timestamp":"2024-11-02T08:36:03Z","content_type":"text/html","content_length":"19012","record_id":"<urn:uuid:57dc99dd-f7b0-477a-8f6c-7139e66aec3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00246.warc.gz"}
Class VectorVectorMult_DDRM public class VectorVectorMult_DDRM extends Object Operations that involve multiplication of two vectors. • Method Summary Modifier and Type static void Adds to A ∈ ℜ ^m × n the results of an outer product multiplication of the two vectors. static void Multiplies a householder reflection against a vector: y = (I + γ u u^T)x static double Computes the inner product of the two vectors. static double static double static void Sets A ∈ ℜ ^m × n equal to an outer product multiplication of the two vectors. static void Performs a rank one update on matrix A using vectors u and w. static void Performs a rank one update on matrix A using vectors u and w. Methods inherited from class java.lang.Object clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait • Constructor Details □ VectorVectorMult_DDRM public VectorVectorMult_DDRM() • Method Details □ innerProd Computes the inner product of the two vectors. In geometry this is known as the dot product. ∑[k=1:n] x[k] * y[k] where x and y are vectors with n elements. These functions are often used inside of highly optimized code and therefor sanity checks are kept to a minimum. It is not recommended that any of these functions be used directly. x - A vector with n elements. Not modified. y - A vector with n elements. Not modified. The inner product of the two vectors. □ innerProdA x - A vector with n elements. Not modified. A - A matrix with n by m elements. Not modified. y - A vector with m elements. Not modified. The results. □ innerProdTranA x - A vector with n elements. Not modified. A - A matrix with n by n elements. Not modified. y - A vector with n elements. Not modified. The results. □ outerProd Sets A ∈ ℜ ^m × n equal to an outer product multiplication of the two vectors. This is also known as a rank-1 operation. A = x * y' where x ∈ ℜ ^m and y ∈ ℜ ^n are vectors. Which is equivalent to: A[ij] = x[i]y[j] These functions are often used inside of highly optimized code and therefor sanity checks are kept to a minimum. It is not recommended that any of these functions be used directly. x - A vector with m elements. Not modified. y - A vector with n elements. Not modified. A - A Matrix with m by n elements. Modified. □ addOuterProd Adds to A ∈ ℜ ^m × n the results of an outer product multiplication of the two vectors. This is also known as a rank 1 update. A = A + γ x y^T where x ∈ ℜ ^m and y ∈ ℜ ^n are vectors. Which is equivalent to: A[ij] = A[ij] + γ x[i]*y[j] These functions are often used inside of highly optimized code and therefor sanity checks are kept to a minimum. It is not recommended that any of these functions be used directly. gamma - A multiplication factor for the outer product. x - A vector with m elements. Not modified. y - A vector with n elements. Not modified. A - A Matrix with m by n elements. Modified. □ householder Multiplies a householder reflection against a vector: y = (I + γ u u^T)x The Householder reflection is used in some implementations of QR decomposition. u - A vector. Not modified. x - a vector. Not modified. y - Vector where the result are written to. □ rank1Update Performs a rank one update on matrix A using vectors u and w. The results are stored in B. B = A + γ u w^T This is called a rank1 update because the matrix u w^T has a rank of 1. Both A and B can be the same matrix instance, but there is a special rank1Update for that. gamma - A scalar. A - A m by m matrix. Not modified. u - A vector with m elements. Not modified. w - A vector with m elements. Not modified. B - A m by m matrix where the results are stored. Modified. □ rank1Update Performs a rank one update on matrix A using vectors u and w. The results are stored in A. A = A + γ u w^T This is called a rank1 update because the matrix u w^T has a rank of 1. gamma - A scalar. A - A m by m matrix. Modified. u - A vector with m elements. Not modified.	{"url":"https://ejml.org/javadoc/org/ejml/dense/row/mult/VectorVectorMult_DDRM.html","timestamp":"2024-11-09T06:49:47Z","content_type":"text/html","content_length":"27661","record_id":"<urn:uuid:d7b2996c-74f3-415b-9c2f-0b58e1e7af66>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00313.warc.gz"}
967 research outputs found We examine a unitarity of a particular higher-derivative extension of general relativity in three space-time dimensions, which has been recently shown to be equivalent to the Pauli-Fierz massive gravity at the linearized approximation level, and explore a possibility of generalizing the model to higher space-time dimensions. We find that the model in three dimensions is indeed unitary in the tree-level, but the corresponding model in higher dimensions is not so due to the appearance of non-unitary massless spin-2 modes.Comment: 10 pages, references adde We consider a locally supersymmetric theory where the Planck mass is replaced by a dynamical superfield. This model can be thought of as the Minimal Supersymmetric extension of the Brans-Dicke theory (MSBD). The motivation that underlies this analysis is the research of possible connections between Dark Energy models based on Brans-Dicke-like theories and supersymmetric Dark Matter scenarios. We find that the phenomenology associated with the MSBD model is very different compared to the one of the original Brans-Dicke theory: the gravitational sector does not couple to the matter sector in a universal metric way. This feature could make the minimal supersymmetric extension of the BD idea phenomenologically inconsistent.Comment: 6 pages, one section is adde We consider the electromagnetic and gravitational interactions of a massive Rarita-Schwinger field. Stueckelberg analysis of the system, when coupled to electromagnetism in flat space or to gravity, reveals in either case that the effective field theory has a model-independent upper bound on its UV cutoff, which is finite but parametrically larger than the particle's mass. It is the helicity-1/2 mode that becomes strongly coupled at the cutoff scale. If the interactions are inconsistent, the same mode becomes a telltale sign of pathologies. Alternatively, consistent interactions are those that propagate this mode within the light cone. Studying its dynamics not only sheds light on the Velo-Zwanziger acausality, but also elucidates why supergravity and other known consistent models are pathology-free.Comment: 18 pages, cutoff analysis improved, to appear in PR We show that the graviton acquires a mass in a de Sitter background given by $m_{g}^{2}=-{2/3}\Lambda.$ This is precisely the fine-tuning value required for the perturbed gravitational field to mantain its two degrees of freedom.Comment: Title changed and few details added, without any changes in the conclusio We present a Lagrangian for a massive, charged spin 3/2 field in a constant external electromagnetic background, which correctly propagates only physical degrees of freedom inside the light cone. The Velo-Zwanziger acausality and other pathologies such as loss of hyperbolicity or the appearance of unphysical degrees of freedom are avoided by a judicious choice of non-minimal couplings. No additional fields or equations besides the spin 3/2 ones are needed to solve the problem.Comment: 10 pages, references added. To appear in PR It is a general belief that the only possible way to consistently deform the Pauli-Fierz action, changing also the gauge algebra, is general relativity. Here we show that a different type of deformation exists in three dimensions if one allows for PT non-invariant terms. The new gauge algebra is different from that of diffeomorphisms. Furthermore, this deformation can be generalized to the case of a collection of massless spin-two fields. In this case it describes a consistent interaction among them.Comment: 21+1 pages. Minor corrections and reference adde The order parameter of a finite system with a spontaneously broken continuous global symmetry acts as a quantum mechanical rotor. Both antiferromagnets with a spontaneously broken $SU(2)_s$ spin symmetry and massless QCD with a broken $SU(2)_L \times SU(2)_R$ chiral symmetry have rotor spectra when considered in a finite volume. When an electron or hole is doped into an antiferromagnet or when a nucleon is propagating through the QCD vacuum, a Berry phase arises from a monopole field and the angular momentum of the rotor is quantized in half-integer units.Comment: 4 page The equivalence of inertial and gravitational masses is a defining feature of general relativity. Here, we clarify the status of the equivalence principle for interactions mediated by a universally coupled scalar, motivated partly by recent attempts to modify gravity at cosmological distances. Although a universal scalar-matter coupling is not mandatory, once postulated, it is stable against classical and quantum renormalizations in the matter sector. The coupling strength itself is subject to renormalization of course. The scalar equivalence principle is violated only for objects for which either the graviton self-interaction or the scalar self-interaction is important---the first applies to black holes, while the second type of violation is avoided if the scalar is Galilean-symmetric.Comment: 4 pages, 1 figur We have analysed here the equivalence of RVB states with $u=1/2$ FQH states in terms of the Berry Phase which is associated with the chiral anomaly in 3+1 dimensions. It is observed that the 3-dimensional spinons and holons are characterised by the non-Abelian Berry phase and these reduce to 1/2 fractional statistics when the motion is confined to the equatorial planes. The topological mechanism of superconductivity is analogous to the topological aspects of fractional quantum Hall effect with $u=1/2$.Comment: 12 pages latex fil	{"url":"https://core.ac.uk/search/?q=author%3A(M.%20Fierz)","timestamp":"2024-11-05T00:16:36Z","content_type":"text/html","content_length":"142472","record_id":"<urn:uuid:affcddb8-baa4-473a-9063-a25a583e06ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00865.warc.gz"}
Archery Draw Length Calculator - Calculator City Archery Draw Length Calculator Enter your wingspan or other measurements into the calculator to determine your draw length. Archery Draw Length Calculation Formula The following formula is used to calculate your draw length based on your wingspan or other measurements. Draw Length = Wingspan / 2.5 • Draw Length is the length of the draw from your bow (inches) • Wingspan is the length from fingertip to fingertip with arms extended (inches) To calculate the draw length, divide your wingspan by 2.5. What is Archery Draw Length? Archery draw length refers to the distance from the nocking point of the bowstring to the back of the bow when the bow is fully drawn. This measurement is crucial for ensuring proper form, accuracy, and comfort while shooting. A draw length that is too short or too long can affect your performance and may cause physical strain. Knowing your correct draw length helps you choose the right bow and customize your archery setup to suit your needs. How to Calculate Draw Length? The following steps outline how to calculate your draw length using the given formula. 1. First, measure your wingspan by extending your arms and measuring from fingertip to fingertip. 2. Next, divide your wingspan by 2.5 to get your draw length. 3. If you prefer using your height and arm length, add these two measurements and divide by 2.5. 4. Finally, use the calculated draw length to set up your bow or confirm with an archery professional. 5. After inserting the variables and calculating the result, check your answer with the calculator above. Example Problem: Use the following variables as an example problem to test your knowledge. Wingspan = 70 inches Height = 68 inches Arm Length = 32 inches 1. What is draw length? Draw length is the distance measured from the nocking point to the back of the bow when fully drawn. It determines how far you pull the bowstring back before releasing an arrow. 2. Why is draw length important? Proper draw length ensures that you can shoot accurately and comfortably. An incorrect draw	{"url":"https://calculator.city/archery-draw-length-calculator/","timestamp":"2024-11-05T22:15:09Z","content_type":"text/html","content_length":"74093","record_id":"<urn:uuid:53b1e756-31ff-455f-9d0c-c5c09b6064b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00696.warc.gz"}
Today, Maxime Augier gave a great talk about the state of security of the internet PKI infrastructure. The corresponding paper written by Lenstra, Hughes, Augier, Bos, Kleinjung, and Wachter was uploaded to eprint.iacr.org archive a few weeks ago. In a nutshell, they found out that some RSA keys, that is often used in the SSL/TLS protocol to secure internet traffic, are generated by bad pseudo random number generators and can be easily recovered, thous providing no security at all. The RSA cryptosystem RSA is one of the oldest and most famous asymmetric encrytion schmes. The key generation for RSA can be summarized as follows: For a given bitlength l (for example $latex l = 1024$ or $latex l = 2048$ bits), choose randomly two prime numbers p and q of of bitlength $latex l/2$. Choose a number $latex 1 < e < (p-1)(q-1)$, that has no divisor in common with $latex (p-1)(q-1)$. Many people choose $latex e = 2^{16+1}$ here for performance reasons, but other choices are valid as well. Now, the number $latex n = pq$ and e form the public key, while $latex d = e^{-1} \bmod (p-1)(q-1)$ is the private key. Sometimes, the numbers p and q are stored with the private key, because they can be used to accelerated To encrypt a message m, one just computes $latex c = m^e \bmod n$, and to decrypt a message, one computes $latex m = c^d \bmod n$. However, we don’t need that for the rest of this text can safely ignore that. How random do these numbers need to be? When generatingÂ cartographicÂ keys, we need toÂ distinguishÂ between just random numbers and cryptographically secure random numbers. Many computers cannot generate real random numbers, so they generate random numbers in software. For many applications like computer games and for example simmulations of experiments, we only need number that seem to be random. Functions like “rand()” from the standard c library provide such numbers and the generation of these numbers is often initialized from the current system time only. For cryptographic applications, we need cryptographically secure random numbers. These are numbers that a generated in a way, that there is no efficient algorithm, that distinguish them from real random numbers. Generating such random numbers on a computer can be very hard. In fact, there have been a lot of breaches of devices and programs, that used a bad random number generator for cryptographic applications. What has been found out? From my point of view, the paper contains two noteably results: Many keys are shared by several certificates 6 185 228 X.509 certificates have been collected by the researchers. About 4.3% of them contained an RSA public key, that was also used in another certificate. There could be several reasons for • After a certificate has expired, another certificate is issued, that contains the same public key. From my point of view, there is nothing wrong with doing that. • A company changes their name or is taken over by another company. To reflect that change, a new certificate is issued, that contains another company name, but still uses the same key. I don’t see any problems here either. • A product comes with a pre-installed key, and the consumer has to request an certificate for that key. The same key is shipped to several customers. From my point of view, this is really a bad • Or there might be really a bad random number generator in some key generation routines, that two entities that are not related come up with the same RSA public (and private) key. This is a security nightmare. Some keys share a common divisor This is definitely not supposed to happen. If two RSA keys are generated, that share a common divisor by the same or by different key generation routines, the private key for both public keys can be easily determined, and the key generation routine is deeply flawed. What are the consequences? For those, who use an RSA public key, that shares a modulus with another different RSA public key, their key provides no protection at all. All implementations, that generated these keys definitely need to be updated and the certificates using the weak keys need to be revoked. Which devices and vendors are affected? Because disclosing the list of affected devices and vendors would compromise the security of these systemsÂ immediately, and allow everyone to recover their secret RSA keys, this has not been	{"url":"https://cryptanalysis.eu/blog/2012/04/","timestamp":"2024-11-03T02:44:19Z","content_type":"text/html","content_length":"38062","record_id":"<urn:uuid:573b5ee4-b28f-4c0a-93a8-666a42f28b26>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00045.warc.gz"}
Documents For An Access Point Click the serial number on the left to view the details of the item. # Author Title Accn# Year Item Type Claims 11 Mantica, Giorgio Emergent Complexity from Nonlinearity, in Physics, Engineering and the Life Sciences I09877 2017 eBook 12 Lehnert, Judith Controlling Synchronization Patterns in Complex Networks I09860 2016 eBook 13 Helias, Moritz Statistical Field Theory for Neural Networks I09583 2020 eBook 14 Hutt, Axel Synergetics I09580 2020 eBook 15 Czischek, Stefanie Neural-Network Simulation of Strongly Correlated Quantum Systems I09120 2020 eBook 16 Lubashevsky, Ihor Physics of the Human Mind I08732 2017 eBook (page:2 / 2) [#16] First Page Previous Page Title Emergent Complexity from Nonlinearity, in Physics, Engineering and the Life Sciences : Proceedings of the XXIII International Conference on Nonlinear Dynamics of Electronic Systems, Como, Italy, 7-11 September 2015 Author(s) Mantica, Giorgio;Stoop, Ruedi;Stramaglia, Sebastiano Publication Cham, Springer International Publishing, 2017. Description XXV, 222 p. 113 illus., 86 illus. in color : online resource Abstract This book collects contributions to the XXIII international conference ???Nonlinear dynamics of electronic systems???. Topics range from non-linearity in electronic circuits to Note synchronisation effects in complex networks to biological systems, neural dynamics and the complex organisation of the brain. Resting on a solid mathematical basis, these investigations address highly interdisciplinary problems in physics, engineering, biology and biochemistry ISBN,Price 9783319478104 Keyword(s) 1. BIOCHEMISTRY 2. Biochemistry, general 3. COMPLEX SYSTEMS 4. DYNAMICAL SYSTEMS 5. EBOOK 6. EBOOK - SPRINGER 7. ELECTRONICS 8. Electronics and Microelectronics, Instrumentation 9. Mathematical Models of Cognitive Processes and Neural Networks 10. MICROELECTRONICS 11. Neural networks (Computer science)?? 12. STATISTICAL PHYSICS 13. Statistical Physics and Dynamical Systems 14. Systems biology Item Type eBook Multi-Media Links Please Click here for eBook Circulation Data Accession# Call# Status Issued To Return Due On Physical Location I09877 On Shelf Title Controlling Synchronization Patterns in Complex Networks Author(s) Lehnert, Judith Publication Cham, Springer International Publishing, 2016. Description XV, 203 p : online resource Abstract This research aims to achieve a fundamental understanding of synchronization and its interplay with the topology of complex networks. Synchronization is a ubiquitous phenomenon observed Note in different contexts in physics, chemistry, biology, medicine and engineering. Most prominently, synchronization takes place in the brain, where it is associated with several cognitive capacities but is - in abundance - a characteristic of neurological diseases. Besides zero-lag synchrony, group and cluster states are considered, enabling a description and study of complex synchronization patterns within the presented theory. Adaptive control methods are developed, which allow the control of synchronization in scenarios where parameters drift or are unknown. These methods are, therefore, of particular interest for experimental setups or technological applications. The theoretical framework is demonstrated on generic models, coupled chemical oscillators and several detailed examples of neural networks ISBN,Price 9783319251158 Keyword(s) 1. Applications of Graph Theory and Complex Networks 2. DYNAMICAL SYSTEMS 3. DYNAMICS 4. EBOOK 5. EBOOK - SPRINGER 6. Mathematical Models of Cognitive Processes and Neural Networks 7. Neural networks (Computer science)?? 8. PHYSICAL CHEMISTRY 9. PHYSICS 10. SYSTEM THEORY 11. Systems Theory, Control 12. VIBRATION 13. Vibration, Dynamical Systems, Control Item Type eBook Multi-Media Links Please Click here for eBook Circulation Data Accession# Call# Status Issued To Return Due On Physical Location I09860 On Shelf Title Statistical Field Theory for Neural Networks Author(s) Helias, Moritz;Dahmen, David Publication Cham, Springer International Publishing, 2020. Description XVII, 203 p. 127 illus., 5 illus. in color : online resource Abstract This book presents a self-contained introduction to techniques from field theory applied to stochastic and collective dynamics in neuronal networks. These powerful analytical techniques, Note which are well established in other fields of physics, are the basis of current developments and offer solutions to pressing open problems in theoretical neuroscience and also machine learning. They enable a systematic and quantitative understanding of the dynamics in recurrent and stochastic neuronal networks. This book is intended for physicists, mathematicians, and computer scientists and it is designed for self-study by researchers who want to enter the field or as the main text for a one semester course at advanced undergraduate or graduate level. The theoretical concepts presented in this book are systematically developed from the very beginning, which only requires basic knowledge of analysis and linear algebra ISBN,Price 9783030464448 Keyword(s) 1. EBOOK 2. EBOOK - SPRINGER 3. MACHINE LEARNING 4. Mathematical Models of Cognitive Processes and Neural Networks 5. MATHEMATICAL STATISTICS 6. Neural networks (Computer science)?? 7. Neurosciences 8. Probability and Statistics in Computer Science 9. STATISTICAL PHYSICS 10. Statistical Physics and Dynamical Systems Item Type eBook Multi-Media Links Please Click here for eBook Circulation Data Accession# Call# Status Issued To Return Due On Physical Location I09583 On Shelf Title Synergetics Author(s) Hutt, Axel;Haken, Hermann Publication New York, NY, Springer US, 2020. Description 223 illus., 95 illus. in color. eReference : online resource Abstract This volume of the ???Encyclopedia of Complexity and Systems Science, Second Edition??? (ECSS), introduces the fundamental physical and mathematical concepts underlying the theory of Note complex physical, chemical, and biological systems. Numerous applications illustrate how these concepts explain observed phenomena in our daily lives, which range from spatio-temporal patterns in fluids from atmospheric turbulence in hurricanes and tornadoes to feedback dynamics of laser intensity to structures in cities and rhythms in the brain. The spontaneous formation of well-organized structures out of microscopic system components and their interactions is one of the most fascinating and challenging phenomena for scientists to understand. Biological systems may also exhibit organized structures emanating from interactions of cells and their networks. For instance, underlying structures in the brain emerge as certain mental states, the ability to coordinate movement, or pathologies such as tremor or epileptic seizures. When we try to explain or understand these extremely complex biological phenomena, it is natural to ask whether analogous processes of self-organization may be found in much simpler systems of the inanimate world. In recent decades, it has become increasingly evident that there exist numerous examples in physical and chemical systems in which well-organized spatio-temporal structures arise out of disordered states. As in living organisms, the functioning of these systems can be maintained only by a flux of energy (and matter) through them. Synergetics combines elements from physics and mathematics to explain how a diversity of systems obey the same basic principles. All chapters in this volume have been thoroughly revised and updated from the first edition of ECSS. The second edition also includes new or expanded coverage of such topics as chaotic dynamics in laser systems and neurons, novel insights into the relation of classical chaos and quantum dynamics, and how noise in the brain tunes observed neural activity and controls animal and human behavior. ISBN,Price 9781071604212 Keyword(s) 1. Applications of Nonlinear Dynamics and Chaos Theory 2. COMPLEX SYSTEMS 3. COMPLEXITY 4. COMPUTATIONAL COMPLEXITY 5. EBOOK 6. EBOOK - SPRINGER 7. Mathematical Models of Cognitive Processes and Neural Networks 8. Neural networks (Computer science)?? 9. STATISTICAL PHYSICS 10. Statistical Physics and Dynamical Systems 11. SYSTEM THEORY 12. Systems biology Item Type eBook Multi-Media Links Please Click here for eBook Circulation Data Accession# Call# Status Issued To Return Due On Physical Location I09580 On Shelf Title Neural-Network Simulation of Strongly Correlated Quantum Systems Author(s) Czischek, Stefanie Publication Cham, Springer International Publishing, 2020. Description XV, 205 p. 51 illus., 48 illus. in color : online resource Abstract Quantum systems with many degrees of freedom are inherently difficult to describe and simulate quantitatively. The space of possible states is, in general, exponentially large in the Note number of degrees of freedom such as the number of particles it contains. Standard digital high-performance computing is generally too weak to capture all the necessary details, such that alternative quantum simulation devices have been proposed as a solution. Artificial neural networks, with their high non-local connectivity between the neuron degrees of freedom, may soon gain importance in simulating static and dynamical behavior of quantum systems. Particularly promising candidates are neuromorphic realizations based on analog electronic circuits which are being developed to capture, e.g., the functioning of biologically relevant networks. In turn, such neuromorphic systems may be used to measure and control real quantum many-body systems online. This thesis lays an important foundation for the realization of quantum simulations by means of neuromorphic hardware, for using quantum physics as an input to classical neural nets and, in turn, for using network results to be fed back to quantum systems. The necessary foundations on both sides, quantum physics and artificial neural networks, are described, providing a valuable reference for researchers from these different communities who need to understand the foundations of both ISBN,Price 9783030527150 Keyword(s) 1. CONDENSED MATTER 2. CONDENSED MATTER PHYSICS 3. EBOOK 4. EBOOK - SPRINGER 5. MACHINE LEARNING 6. Mathematical Models of Cognitive Processes and Neural Networks 7. Neural networks (Computer science)?? 8. QUANTUM PHYSICS Item Type eBook Multi-Media Links Please Click here for eBook Circulation Data Accession# Call# Status Issued To Return Due On Physical Location I09120 On Shelf Title Physics of the Human Mind Author(s) Lubashevsky, Ihor Publication Cham, Springer International Publishing, 2017. Description XIV, 380 p. 83 illus., 41 illus. in color : online resource Abstract This book tackles the challenging question which mathematical formalisms and possibly new physical notions should be developed for quantitatively describing human cognition and behavior, Note in addition to the ones already developed in the physical and cognitive sciences. Indeed, physics is widely used in modeling social systems, where, in particular, new branches of science such as sociophysics and econophysics have arisen. However, many if not most characteristic features of humans like willingness, emotions, memory, future prediction, and moral norms, to name but a few, are not yet properly reflected in the paradigms of physical thought and theory. The choice of a relevant formalism for modeling mental phenomena requires the comprehension of the general philosophical questions related to the mind-body problem. Plausible answers to these questions are investigated and reviewed, notions and concepts to be used or to be taken into account are developed and some challenging questions are posed as open problems. This text addresses theoretical physicists and neuroscientists modeling any systems and processes where human factors play a crucial role, philosophers interested in applying philosophical concepts to the construction of mathematical models, and the mathematically oriented psychologists and sociologists, whose research is fundamentally related to modeling mental processes ISBN,Price 9783319517063 Keyword(s) 1. Cognitive psychology 2. Data-driven Science, Modeling and Theory Building 3. EBOOK 4. EBOOK - SPRINGER 5. ECONOPHYSICS 6. Mathematical Methods in Physics 7. Mathematical Models of Cognitive Processes and Neural Networks 8. Neural networks (Computer science)?? 9. PHILOSOPHY OF MIND 10. PHYSICS 11. 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Literal Equation Worksheet - Equations Worksheets Literal Equation Worksheet If you are looking for Literal Equation Worksheet you’ve come to the right place. We have 30 worksheets about Literal Equation Worksheet including images, pictures, photos, wallpapers, and more. In these page, we also have variety of worksheets available. Such as png, jpg, animated gifs, pic art, logo, black and white, transparent, etc. 1275 x 1664 · jpeg literal equations worksheet answers chessmuseum template library from chessmuseum.org 271 x 350 · jpeg algebra solving literal equations worksheet eric pattersons store from www.teacherspayteachers.com 270 x 350 · jpeg solving literal equations practice worksheet magical math teacher from www.teacherspayteachers.com 494 x 640 · jpeg literal equations worksheet infinite algebra tessshebaylo from www.tessshebaylo.com 1352 x 1750 · jpeg solving literal equations connect activity student approved from www.pinterest.com 270 x 350 · jpeg literal equations worksheet epic math teachers pay teachers from www.teacherspayteachers.com 270 x 350 · jpeg literal equations worksheet answer notutahituq worksheet information from notutahituq.blogspot.com 271 x 350 · jpeg worksheet solving literal equations frills math practice tpt from www.teacherspayteachers.com Don’t forget to bookmark Literal Equation Worksheet using Ctrl + D (PC) or Command + D (macos). If you are using mobile phone, you could also use menu drawer from browser. Whether it’s Windows, Mac, iOs or Android, you will be able to download the worksheets using download button. Leave a Comment	{"url":"https://www.equationsworksheets.com/literal-equation-worksheet/","timestamp":"2024-11-13T22:31:23Z","content_type":"text/html","content_length":"75457","record_id":"<urn:uuid:17d92ee3-4426-43d9-bee6-54df8917d839>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00185.warc.gz"}
Relative commutativity degree of some dihedral groups Abdul Hamid, M. and Mohd. Ali, N. M. and Sarmin, N. H. and Abd. Manaf, F. N. (2013) Relative commutativity degree of some dihedral groups. In: Proceedings Of The 20th National Symposium On Mathematical Sciences (SKSM20): Research In Mathematical Sciences: A Catalyst For Creativity And Innovation, PTS A And B. Full text not available from this repository. Official URL: http://dx.doi.org/10.1063/1.4801214 The commutativity degree of a finite group G was introduced by Erdos and Turan for symmetric groups, finite groups and finite rings in 1968. The commutativity degree, P(G), is defined as the probability that a random pair of elements in a group commute. The relative commutativity degree of a group G is defined as the probability for an element of subgroup, H and an element of G to commute with one another and denoted by P(H,G). In this research the relative commutativity degree of some dihedral groups are determined. Item Type: Conference or Workshop Item (Paper) Uncontrolled Keywords: dihedral groups Subjects: Q Science Divisions: Science ID Code: 51277 Deposited By: Haliza Zainal Deposited On: 27 Jan 2016 01:53 Last Modified: 18 Sep 2017 00:45 Repository Staff Only: item control page	{"url":"http://eprints.utm.my/51277/","timestamp":"2024-11-10T21:02:31Z","content_type":"application/xhtml+xml","content_length":"16936","record_id":"<urn:uuid:76cc3664-47a2-4fa9-803f-b5f256ededbe>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00784.warc.gz"}
Exponential Functions - Formula, Properties, Graph, Rules What is an Exponential Function? An exponential function measures an exponential decrease or increase in a particular base. Take this, for example, let us assume a country's population doubles yearly. This population growth can be portrayed as an exponential function. Exponential functions have many real-world applications. In mathematical terms, an exponential function is written as f(x) = b^x. Today we will review the fundamentals of an exponential function in conjunction with important examples. What’s the equation for an Exponential Function? The common equation for an exponential function is f(x) = b^x, where: 1. b is the base, and x is the exponent or power. 2. b is fixed, and x is a variable As an illustration, if b = 2, we then get the square function f(x) = 2^x. And if b = 1/2, then we get the square function f(x) = (1/2)^x. In cases where b is greater than 0 and not equal to 1, x will be a real number. How do you plot Exponential Functions? To plot an exponential function, we need to discover the points where the function crosses the axes. These are known as the x and y-intercepts. Considering the fact that the exponential function has a constant, one must set the value for it. Let's take the value of b = 2. To find the y-coordinates, one must to set the worth for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4. In following this approach, we determine the range values and the domain for the function. After having the values, we need to graph them on the x-axis and the y-axis. What are the properties of Exponential Functions? All exponential functions share identical characteristics. When the base of an exponential function is more than 1, the graph is going to have the following properties: • The line crosses the point (0,1) • The domain is all positive real numbers • The range is more than 0 • The graph is a curved line • The graph is rising • The graph is smooth and continuous • As x approaches negative infinity, the graph is asymptomatic towards the x-axis • As x advances toward positive infinity, the graph increases without bound. In instances where the bases are fractions or decimals within 0 and 1, an exponential function exhibits the following attributes: • The graph passes the point (0,1) • The range is greater than 0 • The domain is all real numbers • The graph is descending • The graph is a curved line • As x nears positive infinity, the line within graph is asymptotic to the x-axis. • As x gets closer to negative infinity, the line approaches without bound • The graph is flat • The graph is continuous There are some basic rules to remember when working with exponential functions. Rule 1: Multiply exponential functions with an equivalent base, add the exponents. For instance, if we have to multiply two exponential functions that posses a base of 2, then we can write it as 2^x * 2^y = 2^(x+y). Rule 2: To divide exponential functions with an identical base, deduct the exponents. For example, if we have to divide two exponential functions that posses a base of 3, we can note it as 3^x / 3^y = 3^(x-y). Rule 3: To grow an exponential function to a power, multiply the exponents. For instance, if we have to increase an exponential function with a base of 4 to the third power, we are able to write it as (4^x)^3 = 4^(3x). Rule 4: An exponential function with a base of 1 is consistently equivalent to 1. For example, 1^x = 1 regardless of what the value of x is. Rule 5: An exponential function with a base of 0 is always equivalent to 0. For instance, 0^x = 0 regardless of what the value of x is. Exponential functions are usually utilized to denote exponential growth. As the variable grows, the value of the function increases quicker and quicker. Example 1 Let's look at the example of the growing of bacteria. Let us suppose that we have a cluster of bacteria that doubles each hour, then at the close of the first hour, we will have 2 times as many At the end of hour two, we will have 4x as many bacteria (2 x 2). At the end of hour three, we will have 8 times as many bacteria (2 x 2 x 2). This rate of growth can be portrayed an exponential function as follows: f(t) = 2^t where f(t) is the number of bacteria at time t and t is measured hourly. Example 2 Similarly, exponential functions can represent exponential decay. If we have a dangerous substance that decomposes at a rate of half its volume every hour, then at the end of the first hour, we will have half as much material. After two hours, we will have a quarter as much substance (1/2 x 1/2). At the end of hour three, we will have 1/8 as much substance (1/2 x 1/2 x 1/2). This can be shown using an exponential equation as follows: f(t) = 1/2^t where f(t) is the quantity of material at time t and t is measured in hours. As you can see, both of these illustrations pursue a comparable pattern, which is why they can be represented using exponential functions. As a matter of fact, any rate of change can be denoted using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base remains constant. This indicates that any exponential growth or decay where the base is different is not an exponential function. For example, in the matter of compound interest, the interest rate continues to be the same while the base varies in regular time periods. An exponential function is able to be graphed employing a table of values. To get the graph of an exponential function, we have to input different values for x and calculate the matching values for Let us review the example below. Example 1 Graph the this exponential function formula: y = 3^x First, let's make a table of values. As shown, the worth of y increase very rapidly as x rises. Imagine we were to draw this exponential function graph on a coordinate plane, it would look like this: As seen above, the graph is a curved line that goes up from left to right ,getting steeper as it continues. Example 2 Plot the following exponential function: y = 1/2^x To begin, let's draw up a table of values. As you can see, the values of y decrease very rapidly as x rises. The reason is because 1/2 is less than 1. If we were to plot the x-values and y-values on a coordinate plane, it is going to look like what you see below: This is a decay function. As you can see, the graph is a curved line that gets lower from right to left and gets smoother as it goes. The Derivative of Exponential Functions The derivative of an exponential function f(x) = a^x can be displayed as f(ax)/dx = ax. All derivatives of exponential functions present unique properties by which the derivative of the function is the function itself. This can be written as following: f'x = a^x = f(x). Exponential Series The exponential series is a power series whose expressions are the powers of an independent variable number. The general form of an exponential series is: Grade Potential Can Help You Learn Exponential Functions If you're struggling to comprehend exponential functions, or merely require a little extra assistance with math as a whole, consider working with a tutor. At Grade Potential, our Centennial math tutors are experts in their subjects and can offer you with the individualized attention you need to thrive. Call us at (303) 578-1372 or contact us today to learn more about your options for us assist you in reaching your academic potential.	{"url":"https://www.centennialinhometutors.com/blog/exponential-functions-formula-properties-graph-rules","timestamp":"2024-11-10T01:37:20Z","content_type":"text/html","content_length":"82283","record_id":"<urn:uuid:f2612697-25ba-4e23-8f14-f7f36d3bec51>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00227.warc.gz"}
Net Loss Ratio Formula Excel The Net Loss Ratio is a financial metric used in the insurance industry to evaluate the ratio of an insurance company’s net losses to its earned premiums. The metric helps to determine the firm’s profitability and overall financial health. We calculate the Net Loss Ratio by dividing an insurance company’s net losses (claims paid plus reserves for outstanding claims, less any recoveries or salvages) by its earned premiums (the total amount of premiums earned by an insurance company over a particular period). The resulting ratio is expressed as a percentage and represents the amount of each premium dollar that is used to pay for losses. A high Net Loss Ratio shows that the insurance company is paying out a large percentage of its premiums in claims, which can indicate a higher level of exposure or risk. In contrast, a low Net Loss Ratio suggests that the insurance company retains more of its premiums as profit. Insurance companies strive to balance the premiums they collect and the claims they pay to remain financially stable and competitive. How to Compute Net Loss Ratio in Excel The following is the formula for computing the Net Loss Ratio in Excel: 1 = (Net Losses / Earned Premiums) * 100 Let’s consider the following dataset showing the total claims a particular insurance company paid out in 2022 and the total premiums the firm earned in the same year. We want to compute the company’s Net Loss Ratio for 2022 and display it in cell E3. We use the steps below: 1. Select cell E3 and type in the following formula: 2. Press Enter. The Net Loss Ratio expressed as a percentage is shown in cell E3. 3. Select cell E3 and click Home >> Number >> Decrease Decimal several times to decrease the decimal numbers of the Net Loss Ratio to two. The Net Loss Ratio with two decimal places is displayed in cell E3. The ratio indicates that the insurance company spent 55.88% of each premium dollar in 2022 to settle claims and retained 44.12% as profit. Use Named Ranges in Excel’s Net Loss Ratio Formula Instead of using cell references, we can use named ranges in Excel’s Net Loss Ratio formula. Named ranges will make the formula more readable. We can name cell B6 containing the total paid claims, “Net Losses,” and cell C6, total earned premiums, “Earned Premiums.” Name Cells B6 and C6 We use the following steps to name cells B6 and C6: 1. Select cell B6. 2. Click Formulas >> Define Names >> Define Name. 3. On the New Name dialog box, type Net_Losses on the Name box and click OK. Notice that there should be no spaces in the range name. 4. Select cell C6 and click Formulas >> Define Names >> Define Name. 5. On the New Name dialog box, type Earned_Premiums on the Name box and click OK. Use the Named Ranges in the Net Loss Ratio Formula Now we can use the named ranges in the formula using the below steps: 1. Select cell E3 and type in the below formula: 1 =(Net_Losses/Earned_Premiums)*100 Notice that the full name appears in the drop-down as you begin to type the name Net_Losses. Instead of continuing to type the name, press the tab key to enter the name. 2. Press Enter. The Net Loss Ratio expressed as a percentage is displayed in cell E3. This tutorial showed how to use the Net Loss Ratio formula in Excel. In addition, it explained how to use named ranges instead of cell references to make the formula more readable. We hope you found the tutorial helpful.	{"url":"https://officetuts.net/excel/formulas/net-loss-ratio-formula/","timestamp":"2024-11-07T01:02:44Z","content_type":"text/html","content_length":"154237","record_id":"<urn:uuid:0662764d-2994-4855-a441-6e47bcb640da>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00247.warc.gz"}
Collection of Solved Problems Task number: 2217 A capillary tube has an inner radius 0.10 mm. Evaluate: a) When you immerse one end in water, how high will the water in the tube rise? b) How great the hydrostatic pressure created by the water column is? c) How does the result change when we use a capillary tube with a double radius? d) How would the result change if we performed the experiment on the Moon? e) How would the experiment go in a satellite in a state of weightlessness? • Hint Height of water column in the tube can be evaluated from the equilibrium of forces acting on the water in the tube. Gravity is acting down, and it is the surface force acting up that is “holding” the water in the tube. For simplicity, let us assume that the surface force is pointing straight up and its magnitude is proportional to the inner circumference of the tube (which is the circumference of the water • Notation r = 0.1 mm = 1.0·10^−4 m inner circumference of the capillary h = ? height of the water in the capillary Tabulated values: σ = 73 mN m^−1 = 73·10^−3 N m^−1 surface tension of water ρ = 1 000 kg m^−3 density of water g = 9.81 m s^−2 gravitational acceleration • Analysis of part a) When solving this task we shall assume that water perfectly adheres to the tube. This means that the water surface takes on the shape of a half of a sphere and that the force given by surface tension points down. According to the third Newton’s law, glass is acting on glass with the same force of opposite direction – up. Then there is also gravitational force acting down. When water surface is in a state of equilibrium, magnitudes of both of these forces must equal. We determine the elevation difference of water from this equality. • Solution of part a) There are two forces acting upon the water: gravitational force and the force, which is glass’ reaction to surface forces. Firstly, let’s determine the magnitude of gravitational force: \[F_G\,=\,mg\,=\,V \varrho g\,=\,\pi r^2h \varrho g\,.\] In previous equation, we used water density to express mass, and a formula for cylinder volume. Now we determine force given by surface tension. This force acts on inner circumference o of the capillary, so for its magnitude following applies: \[F\,=\,\sigma o \,=\,\sigma 2\pi r\,.\] The magnitudes of both forces must equal: \[F_G \,=\, F\] \[\pi r^2h \varrho g \,=\, \sigma 2\pi r\] Now we express h from the last equation, and input given values: \[h \,=\,\frac{ 2\sigma}{r \varrho g} \,=\,\frac{ 2\,\cdot\,0.073}{1.0\,\cdot\,10^{-4}\,\cdot\,1000\,\cdot\,9.81}\,\mathrm{m}\] \[h \,=\,0.149 \,\mathrm{m}\,\dot=\,15\,\mathrm{cm}\] • Solution of parts b) – e) b) Hydrostatic pressure of water column in the tube can be expressed and evaluated \[p\,=\,h\varrho g \,=\,\frac{ 2\sigma}{r \varrho g}\,\varrho g \,=\,\frac{ 2\sigma}{r }\] \[p\,\dot=\,1.5\,\mathrm{kPa}\] This pressure corresponds to the so-called capillary pressure generated under the curved surface of the liquid as a result of surface tension. The previous task could be solved by equilibrium of hydrostatic and capillary pressure. c) The height of the water in the capillary is inversely proportional to the radius of the capillary, so in a capillary with a double radius it will only reach half the height. ... \[h_b \,=\,\frac{ 2\sigma}{r_b \varrho g}\,=\,\frac{ 2\sigma}{2r \varrho g} \,=\,\frac{ 1}{2 }\,h\] d) Gravitational acceleration is six times smaller on the Moon. Height of the water in the capillary is inversely proportional to this acceleration, so the water height would be six times \[h_d \,=\,\frac{ 2\sigma}{r_b \varrho g_{Mesic}}\,=\,\frac{ 2\sigma}{r \varrho \frac{ g}{6 }} \,=\,6h\] e) In a state of weightlessness, there would be no force acting upon water, so water would, due to its wettability, reach the top of arbitrarily long capillary. • Answer In the given capillary the water will rise to the height of approximately 15 cm, and water column will cause hydrostatic pressure 1.5 kPa. In a capillary with the double radius, water would rise to the half of the height. However, on the Moon the height would be six times greater. In a state of weightlessness, water would fill arbitrarily long capillary.	{"url":"https://physicstasks.eu/2217/capillary","timestamp":"2024-11-11T16:59:47Z","content_type":"text/html","content_length":"29789","record_id":"<urn:uuid:9b9795b8-d9ec-47dc-b291-66c752fa4a66>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00804.warc.gz"}
One to one function - Explanation & Examples One to One Function – Explanation & Examples one to one functions special? This article will help you learn about their properties and appreciate these functions. Let’s start with this quick definition of one to one functions: One to one functions are functions that return a unique range for each element in their domain. Since one to one functions are special types of functions, it’s best to review our knowledge of functions, their domain, and their range. This article will help us understand the properties of one to one functions. We’ll also learn how to identify one to one functions based on their expressions and graphs. Let’s go ahead and start with the definition and properties of one to one functions. What is a one to one function? To easily remember what one to one functions are, try to recall this statement: “for every y, there is a unique x.” The next two sections will show you why this phrase helps us remember the core concept behind one to one functions. One to one function definition The function, f(x), is a one to one function when one unique element from its domain will return each element of its range. This means that for every value of x, there will be a unique value of y or Why don’t we visualize this by mapping two pairs of values to compare functions that are not in one to one correspondence? Let’s take a look at g(x) first, g(4) and g(-4) share a common y value of 16. This is also true for g(-2) and g(2). You guessed it right; g(x) is a function that does not have a one to one Now, observe f(x). Notice how for each f(x) value, there is only one unique value of x? When you observe functions having that correspondence, we call those functions one to one functions. One to one function graph To better understand the concept of one to one functions, let’s study a one to one function’s graph. Remember that for one to one functions, each x is expected to have a unique value of y. Since each x will have a unique value for y, one to one functions will never have ordered pairs that share the same y-coordinate. Now that we’ve studied the definition of one to one functions, do you now understand why “for every y, there is a unique x” is a helpful statement to remember? One to one function properties What are other important properties of one-to-one functions we should keep in mind? Here are some properties that can help you understand different types of functions with a one to one • If two functions, f(x) and g(x), are one to one, f ◦ g is a one to one function as well. • If a function is one to one, its graph will either be always increasing or always decreasing. • If g ◦ f is a one to one function, f(x) is guaranteed to be a one to one function as well. Try to study two pairs of graphs on your own and see if you can confirm these properties. Of course, before we can apply these properties, it will be important for us to learn how we can confirm whether a given function is a one to one function or not. How to determine if a function is one to one? The next two sections will show you how we can test functions’ one to one correspondence. We are sometimes given a function’s expression or graph, so we must learn how to identify one-to-one functions algebraically and geometrically. Let’s go ahead and start with the latter! Testing one to one functions geometrically Remember that for functions to be one to one functions. Each x-coordinate must have a unique y-coordinate? We can check for one to one functions using the horizontal line test. • When given a function, draw horizontal lines along with the coordinate system. • Check if the horizontal lines can pass through two points. • If the horizontal lines pass through only one point throughout the graph, the function is a one to one function. What if it passes two or more points of a function? Then, as you may have guessed, they are not considered one to one functions. To better understand the process, let’s go ahead and study these two graphs shown below. The reciprocal function, f(x) = 1/x, is known to be a one to one function. We can also verify this by drawing horizontal lines across its graph. See how each horizontal line passes through a unique ordered pair each time? When this happens, we can confirm that the given function is a one to one function. What happens then when a function is not one to one? For example, the quadratic function, f(x) = x^2, is not a one to one function. Let’s look at its graph shown below to see how the horizontal line test applies to such functions. As you can see, each horizontal line drawn through the graph of f(x) = x^2 passes through two ordered pairs. This further confirms that the quadratic function is not a one to one function. Testing one to one functions algebraically Let’s refresh our memory on how we define one to one functions. Recall that functions are one to one functions when: • f(x[1]) = f(x[2]) if and only if x[1] = x[2] • f(x[1]) ≠ f(x[2]) if and only if x[1] ≠ x[2] We’ll use this algebraic definition to test whether a function is one to one. How do we do that, then? • Use the given function and find the expression for f(x[1]). • Apply the same process and find the expression for f(x[2]). • Equate both expressions and show that x[1] = x[2]. Why don’t we try proving that f(x) = 1/x is a one to one function using this method? Let’s first substitute x[1] and x[2] into the expression. We’ll have f(x[1]) = 1/x[1] and f(x[2]) = 1/x[2]. To confirm the function’s one to one correspondence, let’s equate f(x[1]) and f(x[2]). 1/x[1 ]= 1/x[2] Cross-multiply both sides of the equation to simplify the equation. x[2 ]= x[1] x[1] = x[2] We’ve just shown that x[1] = x[2] when f(x[1]) = f(x[2]), hence, the reciprocal function is a one to one function. Example 1 Fill in the blanks with sometimes, always, or never to make the following statements true. • Relations can _______________ be one to one functions. • One to one functions are ______________ functions. • When a horizontal line passes through a function that is not a one to one function, it will ____________ pass through two ordered pairs. When answering questions like this, always go back to the definitions and properties we just learned. • Relations can sometimes be functions and, consequently, can sometimes represent a one to one function. • Since one to one functions are a special type of function, they will always be, first and foremost, functions. • Our example may have shown the horizontal lines passing through the graph of f(x) = x^2 twice, but the horizontal lines can pass through more points. Hence, it sometimes passes through two ordered pairs. Example 2 Let A = {2, 4, 8, 10} and B = {w, x, y, z}. Which of the following sets of ordered pairs represent a one to one function? • {(2, w), (2, x), (2, y), (2,z)} • {(4,w), (2,x), (10,z), (8, y)} • {(4,w), (2,x), (8,x), (10, y)} For a function to be a one to one function, each element from A must pair up with a unique element from B. • The first option has the same value for x for each value of y, so it’s not a function and, consequently, not a one-to-one function. • The third option has different values of x for each ordered pair, but 2 and 8 share the same range of x. Hence, it does not represent a one to one function. • The second option uses a unique element from A for every unique element from B, representing a one-to-one function. This means that {(4,w), (2,x), (10,z), (8, y)} represent a one to one function. Example 3 Which of the following sets of values represent a one to one function? Always go back to the statement, “for every y, there is a unique x.” For each set, let’s inspect whether each element from the right is paired with a unique value from the left. • For the first set, f(x), we can see that each element from the right side is paired up with a unique element from the left. Hence, f(x) is a one to one function. • The set, g(x), shows a different number of elements on each side. This alone will tell us that the function is not a one to one function. • Some values from the left side correspond to the same element found on the right, so m(x) is not a one to one function as well. • Each of the elements on the first set corresponds to a unique element on the next, so n(x) represents a one to one function. Example 4 Graph f(x) = \|x\| + 1 and determine whether f(x) is a one to one function. Construct a table of values for f(x) and plot the generated ordered pairs. Connected these points to graph f(x). The table alone can already give you a clue on whether f(x) is a one to one function [Hint: f(1) = 2 and f(-1) =2]. But let’s go ahead and plot these points on the xy-plane and graph f(x). Once we’ve set up the graph of f(x) = \|x\| + 1, draw horizontal lines across the graph and see if it passes through one or more points. From the graph, we can see that the horizontal lines we’ve constructed pass through two points each, so the function is not a one to one function. Example 5 Determine if f(x) = -2x^3 – 1 is a one to one function using the algebraic approach. Recall that for a function to be a one to one function, f(x[1]) = f(x[2]) if and only if x[1] = x[2]. For us to check if f(x) is a one to one function, let’s find the respective expressions for x[1] and x[2] first. f(x[1]) = -2 x[1]^3 – 1 f(x[2]) = -2 x[2]^3 – 1 Equate both expressions and see if it reduces to x[1] = x[2]. -2 x[1]^3 – 1 = -2 x[2]^3 – 1 -2 x[1]^3 = -2 x[2]^3 (x[1])^3 = (x[2])^3 Taking the cube root of both sides of the equation will lead us to x[1] = x[2]. Hence, f(x) = -2x^3 – 1 is a one to one function. Example 6 Show that f(x) = -5x^2 + 1 is not a one to one function. Another important property of one to one functions is that when x[1] ≠ x[2], f(x[1]) must not be equal to f(x[2]). A quick way to prove that f(x) is not a one to one function is by thinking of a counterexample showing two values of x where they return the same value for f(x). Let’s see what happens when x[1] = -4 and x[2] = 4. f(x[1]) = -5(-4)^2 + 1 f(x[2]) = -5(4)^2 + 1 = -80 + 1 = -80 + 1 = -79 = -79 We can see that even when x[1] is not equal to x[2], it still returned the same value for f(x). This shows that the function f(x) = -5x^2 + 1 is not a one to one function. Example 7 Given that a and b are not equal to 0 show that all linear functions are one-to-one functions. Remember that the general form of linear functions can be expressed as ax + b, where a and b are nonzero constant. We apply the same process by substituting x[1] and x[2 ]into the general expression for linear functions. f(x[1]) = a x[1] + b f(x[2]) = a x[2] + b Equate both equations and see if they can be reduced to x[1] = x[2]. Since b represents a constant, we can subtract b from both sides of the equation. a x[1] + b = a x[2] + b a x[1] = a x[2] Divide both sides of the equation by a, and we’ll have x[1] = x[2]. From this, we can conclude that all linear functions are one-to-one functions. Practice Questions 1. Fill in the blank with sometimes, always, or never to make the statement true. Cosine functions can _______________ be one to one functions. 2. Fill in the blank with sometimes, always, or never to make the statement true. If $f(x)$ is a one to one function, its domain will ______________ have the same number of elements as its range. 3. Fill in the blank with sometimes, always, or never to make the statement true. When a horizontal line passes through a function that is a one to one function, it will ____________ pass through two ordered pairs. 4. Let $M = \{3, 6, 9, 12\}$ and $N = \{a, b, c, d\}$. Which of the following sets of ordered pairs represent a one to one function? 5. Which of the following sets of values represent a one-to-one function? 6. The graph of the function, $f(x) = x^2 – 4$, is as shown below. Is the function is a one-to-one function or not? 7. The graph of the function, $g(x) = -4x + 1$, is as shown below. Is the function is a one-to-one function or not? 8. The graph of the function, $h(x) = e^x$, is as shown below. Is the function is a one-to-one function or not? 9. Check whether the function, $f(x) = 2x – 1$, is a one to one function using the algebraic approach. 10. Check whether the function, $g(x) = \dfrac{1}{x^2}$, is a one to one function using the algebraic approach. 11. Check whether the function, $h(x) = \|x\| + 4$, is a one to one function using the algebraic approach. Open Problems 1. Show that $g(x) = \|x\| – 4$ is not a one to one function. 2. Show that all quadratic expressions are not one-to-one functions. Open Problem Solutions 1. The absolute value function, $g(x) = \|x\| – 4$, has the curve as shown below. Using the horizontal line test, we can see that each horizontal line passes through more than one point. We can also use the algebraic approach to show that $g(x)$ is not an absolute value function. Simply think of a pair of points sharing the same $y$-coordinate and satisfying the function such as $ (-6, 2)$ and $(6, 2)$. 2. The graph of a quadratic function will always be a U-shaped curve, so the curve will always fail the horizontal line test. This means that quadratic functions will never be a one-to-one function. Images/mathematical drawings are created with GeoGebra.	{"url":"https://www.storyofmathematics.com/one-to-one-function/","timestamp":"2024-11-08T09:07:31Z","content_type":"text/html","content_length":"212340","record_id":"<urn:uuid:10526d1f-c902-41c2-a520-b85377930920>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00200.warc.gz"}
EV of the following BJ Jackpot promotion Hi everyone, in a local casino here in Germany they offer the following Jackpot Promotion for Blackjack from time to time: At the beginning of a certain month the Jackpot is set to 100€. In order to receive it you have to get dealt a certain 3-card-combination that equals 21, for example, King of diamonds/7 of spades/4 of clubs (Kd7s4c). If the Jackpot is not hit during this first day then another 100€ are added to it up to 3000€ for a 30-day-month. The Jackpot will remain 3000€ until it is hit also past this month. If it is hit while holding for example 2000€ it is reset to 100€. Now I have two questions: 1. Is it possible to play with an advantage during this promotion with the Jackpot having a certain size? If yes, what size does it need to have if you are betting the minimum flat (5€)? 2. Should you deviate from basic strategy in order to maximize your chances of hitting the jackpot? Examples with the Kd7s4c mentioned above: --> split any two face cards with at least one Kd in them? --> hit hard 17 consisting of Kd7s --> hit hard 14 against dealer 6 consisting of Kd4c --> list goes on... For me it was always fun playing it especially because I did deviate from basic strategy thus splitting 20 which i normally never do thinking that the jackpot justifies it. Thank you for reading and possibly helping me with this problem! 6 Decks Infinite Re-Split Double on 9-11 only No Surrender Split Tens can become Blackjack Let's assume a single deck. You have a (3/52)(2/51) chance of getting dealt a jackpot-eligible combination. This simplifies to 6/2652, or one in 442. When you get one such combination, you now have a 1/49 chance of hitting it. So assuming you completely ignore basic strategy, you will hit the jackpot once every (442)(49) hands, or once every 21,658 hands. So let's say the jackpot is 1000 euro, and you are betting 5 euro. The jackpot is worth an extra 200 bets every 21,658 hands. This is an approximately 0.9% boost to your bottom line. That would be enough to overcome the house edge at just about any blackjack game, even with bad rules. An accurate answer to your questiion can't be given without knowing what the rules and the number of decks are. To answer your second question, given that you have a 1/49 chance of hitting the jackpot, you should deviate from basic strategy any time the cost of doing so is less than the gain from going for the jackpot. Obviously, if the jackpot was 50 times your bet or more (in this case, 250 euro), you should ALWAYS hit. If the jackpot was less than 50 times your bet, you should scale back (for instance, if the jackpot was only 100 euro, you probably shouldn't do anything radical like hit hard 17). The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw Quote: dr3dd 1. Is it possible to play with an advantage during this promotion with the Jackpot having a certain size? If yes, what size does it need to have if you are betting the minimum flat (5€)? 2. Should you deviate from basic strategy in order to maximize your chances of hitting the jackpot? Examples with the Kd7s4c mentioned above: --> split any two face cards with at least one Kd in them? --> hit hard 17 consisting of Kd7s --> hit hard 14 against dealer 6 consisting of Kd4c --> list goes on... ... Using the example of Kd7s4c, assuming multiple decks, no-peek rule, S17, double on 9-11 only, and double after split, I find you have an advantage on the second day (betting 5€ with the jackpot at 200€). I find that you should always hit Kd4c. And except on the first day, you should always hit Kd7s. On the first day, I find that you should stand on Kd7s only vs the dealer's two through seven. Here are my splits (only one card with the special suit) with corresponding minimum jackpot sizes: K vs 4 2800 K vs 5 2200 K vs 6 1800 7 vs 8 300 7 vs 9 1700 4 vs 2 2000 4 vs 3 1400 4 vs 4 700 4 vs 7 2900 And here are my splits of specially suited pairs: K vs 2 1900 K vs 3 1700 K vs 4 1400 K vs 5 1100 K vs 6 900 7 vs A 3000 7 vs 8 200 7 vs 9 900 7 vs 10 1600 4 vs 2 1000 4 vs 3 700 4 vs 4 400 4 vs 7 1500 4 vs 8 1600 4 vs 9 1800 4 vs 10 2600 Thank you for this analyses! I appreciate it a lot. Could you tell me how you came up with the numbers? Maybe you can pick out one example and put me thorugh? I have added the rules of the game in the first post. Quote: dr3dd Thank you for this analyses! I appreciate it a lot. Could you tell me how you came up with the numbers? Maybe you can pick out one example and put me thorugh? I have added the rules of the game in the first post. Thanks! Also, can split aces become blackjack? Quote: ChesterDog Thanks. Also, can split aces become blackjack? No, they would as usual count as 21. Quote: dr3dd ...Split Tens can become Blackjack ... Applying this rule, I would split pairs with the specially suited king at lower jackpots. Here are my latest minimum jackpots for splits with one specially suited king: K vs 2 2500 K vs 3 2000 K vs 4 1500 K vs 5 900 K vs 6 500 K vs 7 2400 And here are my splits of a pair of specially suited kings: K vs 2 1300 K vs 3 1000 K vs 4 800 K vs 5 500 K vs 6 300 K vs 7 1200 K vs 8 2100 I used an infinite deck model to do an analysis of the game. An example of the strategy would be standing or hitting hard 17 vs 2. Standing would have an EV of -15%, and hitting with a jackpot of 100 would have an EV of -17%, so I would stand. However, when the prize got to 200, I get an EV of +22%, so I would hit then. Using the infinite deck model and allowing only one split instead of infinite resplitting and ignoring jackpots on split hands, I find that the game has a positive EV betting 5 euros with a jackpot of at least 200 euros. To do the splitting strategies, I compared the EV's of standing/hitting vs splitting. For example, standing on ten-ten vs 2 has an EV of 64%, and splitting has an EV of 46%. The probability of getting a jackpot with one king is the probability of getting a special 7 and a special 4, which is about 2(1/52)(1/52), so the EV of splitting a Kd-ten would be about 46% + 2(1/52)(1/52)* (Jackpot/bet). A jackpot of 1300 would raise that sum to above 64% which would make splitting better than standing then. Quote: ChesterDog AThe probability of getting a jackpot with one king is the probability of getting a special 7 and a special 4, which is about (1/52)(1/52) I think, it's actually twice that, no? "When two people always agree one of them is unnecessary" Quote: weaselman I think, it's actually twice that, no? Yes, thanks! I'll edit my post. Quote: weaselman I think, it's actually twice that, no? Thanks to weaselman! Here is my revised splitting strategy for pairs with one special card: K vs 2 1300 K vs 3 1000 K vs 4 800 K vs 5 500 K vs 6 300 K vs 7 1200 K vs 8 2100 7 vs A 3000 7 vs 8 200 7 vs 9 900 7 vs 10 1600 4 vs 2 1000 4 vs 3 700 4 vs 4 400 4 vs 7 1500 4 vs 8 1600 4 vs 9 1800 4 vs 10 2600 And here are my splits of specially suited pairs: K vs 2 700 K vs 3 500 K vs 4 400 K vs 5 300 K vs 6 200 K vs 7 600 K vs 8 1100 7 vs A 1500 7 vs 8 100 7 vs 9 500 7 vs 10 800 4 vs 2 500 4 vs 3 400 4 vs 4 200 4 vs 7 800 4 vs 8 800 4 vs 9 900 4 vs 10 1300 prob of hitting jackpot =p House Edge on the game =h minimum bet = b Game is +EV when JackpotValue > hb/p p= 3/52 2/511/50(for SD), rest you should know.. Based on a 10 euro table, the HA only is reduced by a miniscule .0427 (infinite deck) to .0452 (single deck) percent for every 100 euros in the jackpot. Assuming that the house advantage is usually about .50 percent, it becomes a positive EV game at 1,200 Euros. The combination hits every 22,100 - 23,434 hands on average. If the combination always has to add up to 21 and you have two cards in the combination to make the jackpot, I would consider that the odds of hitting that jackpot are 52 to 1. The EV on the jackpot at that point is 1.92 Euros per 100 in the jackpot. You would make a decision I guess adding the EV of the jackpot (which you could get by hitting or doubling down only) to the EV of the decision to calculate the total EV and then compare that to the EV you would get on the correct decision. Walking through the real life decision, if you are betting 10, and the jackpot is 200, and the hand dealt is K-7, and you need a 4 of spades, and you don't know how many 4 of spades are left in the deck, and the dealer has an 8 up. The expected value of staying on 10-7 (8 decks, S17) is -.383 x 10 = -3.83 The expected value of hitting a 10-7 is -.502 x 10 = -5.02, but the jackpot EV is 1.92 x 2 = 3.84, making the total EV, -1.18, better than the -3.83 you get on staying. ----- You want the truth! You can't handle the truth!	{"url":"https://wizardofvegas.com/forum/gambling/blackjack/3085-ev-of-the-following-bj-jackpot-promotion/","timestamp":"2024-11-04T09:21:40Z","content_type":"text/html","content_length":"72166","record_id":"<urn:uuid:b1a178dc-b3b2-4609-b18b-9fba96072796>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00260.warc.gz"}
Area Solar Radiation Available with Spatial Analyst license. Derives incoming solar radiation from a raster surface. • Calculating insolation can be very time consuming, where the calculations for a large digital elevation model (DEM) can take several hours, and for a very large DEM, even days. You may wish to do some test runs with a coarser resolution or subset of your data to ensure the settings are correct before committing a run with the full-resolution data. • The output radiation rasters will always be floating-point type and have units of watt hours per square meter (WH/m^2). The direct duration raster output will be integer with unit hours. • The latitude for the site area (units: decimal degree, positive for the north hemisphere and negative for the south hemisphere) is used in calculations such as solar declination and solar The analysis is designed only for local landscape scales, so it is generally acceptable to use one latitude value for the whole DEM. With larger datasets, such as for states, countries, or continents, the insolation results will differ significantly at different latitudes (greater than 1 degree). To analyze broader geographic regions, it is necessary to divide the study area into zones with different latitudes. • For multiday time configurations, the maximum range of days is a total of one year (365 days, or 366 days for leap years). If the start day is greater than the end day, the time calculations will proceed into the following year. For example, [start day, end day] = [365, 31], represents December 31 to January 31 of the following year. For an example of [1, 2], the time is inclusive for the first day from 0:00 hours (January 1) to 0:00 (January 2). The start day and end day cannot be equal. • The year value for time configuration is used to determine a leap year. It does not have any other influence on the solar radiation analysis as the calculations are a function of the time period determined by Julian days. • For within-day time configurations, the maximum range of time is one day (24 hours). Calculations will not be performed across days (for instance, from 12:00 p.m. to 12:00 p.m. the next day). The start time must be less than the end time. • For within-day time configurations, the start and end times are displayed as solar time (units: decimal hours). Use the time conversion dialog box window to convert the local standard time and local solar time (HMS). When converting local standard time to solar time, the program accounts for equation of time. • The use of a z-factor is essential for correcting calculations when the surface z units are expressed in units different from the ground x,y units. To get accurate results, the z units should be the same as the x,y ground units. If the units are not the same, use a z-factor to convert z units to x,y units. For example, if your x,y units are meters and your z units are feet, you could specify a z-factor of 0.3048 to convert feet to meters. • It is recommended to have your data in a projected coordinate system with units of meters. If you choose to run the analysis with a spherical coordinate system, you will need to specify an appropriate z-factor for that latitude. Following is a list of some appropriate z-factors to use if the x,y units are decimal degrees and the z units are meters: Latitude Z-factor 0 0.00000898 10 0.00000912 20 0.00000956 30 0.00001036 40 0.00001171 50 0.00001395 60 0.00001792 70 0.00002619 80 0.00005156 • The latitude for the site area (units: decimal degree, positive for the north hemisphere and negative for the south hemisphere) is used in calculations such as solar declination and solar position. Because the solar analysis is designed for landscape scales and local scales, it is acceptable to use one latitude value for the whole DEM. For broader geographic regions, it is necessary to divide the study area into zones with different latitudes. • For input surface rasters containing a spatial reference, the mean latitude is automatically calculated; otherwise, latitude will default to 45 degrees. When using an input layer, the spatial reference of the data frame is used. • Sky size is the resolution of the viewshed, sky map, and sun map rasters that are used in the radiation calculations (units: cells per side). These are upward-looking, hemispherical raster representations of the sky and do not have a geographic coordinate system. These rasters are square (equal number of rows and columns). Increasing the sky size increases calculation accuracy but also increases calculation time considerably. • When the day interval setting is small (for example, < 14 days), a larger sky size should be used. During analysis the sun map (determined by the sky size) is used to represent sun positions (tracks) for particular time periods to calculate direct radiation. With smaller day intervals, if the sky size resolution is not large enough, sun tracks may overlap, resulting in zero or lower radiation values for that track. Increasing the resolution provides a more accurate result. • The maximum sky size value is 10,000. A value of 200 is default and is sufficient for whole DEMs with large day intervals (for example, > 14 days). A sky size value of 512 is sufficient for calculations at point locations where calculation time is less of an issue. At smaller day intervals (for example, < 14 days), it is recommended to use higher values. For example, to calculate insolation for a location at the equator with day interval = 1, it is recommended to use a sky size of 2,800 or more. • Day intervals greater than 3 are recommended as sun tracks within three days typically overlap, depending on sky size and time of year. For calculations of the whole year with monthly interval, day interval is disabled and the program internally uses calendar month intervals. The default value is 14. • Because the viewshed calculation can be highly intensive, horizon angles are only traced for the number of calculation directions specified. Valid values must be multiples of 8 (8, 16, 24, 32, and so on). Typically, a value of 8 or 16 is adequate for areas with gentle topography, whereas a value of 32 is adequate for complex topography. The default value is 32. • The number of calculation directions needed is related to the resolution of the input DEM. Natural terrain at 30 meters resolution is usually quite smooth, so fewer directions are sufficient for most situations (16 or 32). With finer DEMs, and particularly with man-made structures incorporated in the DEMs, the number of directions needs to increase. Increasing the number of directions will increase accuracy but will also increase calculation time. • The Create outputs for each interval check box provides the flexibility to calculate insolation integrated over a specified time period or insolation for each interval in a time series. For example, for the within-day time period with an hour interval of one, checking this box will create hourly insolation values; otherwise, insolation integrated for the entire day is calculated. • The Create outputs for each interval parameter affects the format and number of output radiation files. When checked, the output raster will contain multiple bands that correspond to the radiation or duration values for each time interval (hour interval when time configuration is less than one day, or day interval when multiple days). • The diffuse proportion is the fraction of global normal radiation flux that is diffuse. Values range from 0 to 1. This value should be set according to atmospheric conditions. Typical values are 0.2 for very clear sky conditions and 0.3 for generally clear sky conditions. • The amount of solar radiation received by the surface is only a portion of what would be received outside the atmosphere. Transmittivity is a property of the atmosphere that is expressed as the ratio of the energy (averaged over all wavelengths) reaching the earth's surface to that which is received at the upper limit of the atmosphere (extraterrestrial). Values range from 0 (no transmission) to 1 (complete transmission). Typically observed values are 0.6 or 0.7 for very clear sky conditions and 0.5 for only a generally clear sky. The value for the energy received at the earth's surface is at the shortest path through the atmosphere (that is, the sun is at the zenith, or directly overhead) and for sea level. For areas beyond Tropic of Capricorn and Tropic of Cancer, the sun can never be at the exact zenith, not even at noon; however, this value still refers to the moment when the sun is at the zenith. Because the algorithm corrects for elevation effects, transmittivity should always be given for sea level. Transmittivity has an inverse relation with the diffuse proportion parameter. • See Analysis environments and Spatial Analyst for additional details on the geoprocessing environments that apply to this tool. AreaSolarRadiation (in_surface_raster, {latitude}, {sky_size}, {time_configuration}, {day_interval}, {hour_interval}, {each_interval}, {z_factor}, {slope_aspect_input_type}, {calculation_directions}, {zenith_divisions}, {azimuth_divisions}, {diffuse_model_type}, {diffuse_proportion}, {transmittivity}, {out_direct_radiation_raster}, {out_diffuse_radiation_raster}, {out_direct_duration_raster}) Parameter Explanation Data Type in_surface_raster Input elevation surface raster. Raster Layer latitude The latitude for the site area. The units are decimal degrees, with positive values for the northern hemisphere and negative for the southern. (Optional) For input surface rasters containing a spatial reference, the mean latitude is automatically calculated; otherwise, latitude will default to 45 degrees. sky_size The resolution or sky size for the viewshed, sky map, and sun map rasters. The units are cells. (Optional) The default creates a raster of 200 by 200 cells. Specifies the time configuration (period) used for calculating solar radiation. The Time class objects are used to specify the time configuration. The different types of time configurations available are TimeWithinDay, TimeMultiDays, TimeSpecialDays, and TimeWholeYear. The following are the forms: Time (Optional) configuration • TimeWithinDay({day},{startTime},{endTime}) • TimeMultiDays({year},{startDay},{endDay}) • TimeSpecialDays() • TimeWholeYear({year}) The default time configuration is TimeMultiDays with the startDay of 5 and endDay of 160 for the current Julian year. day_interval The time interval through the year (units: days) used for calculation of sky sectors for the sun map. (Optional) The default value is 14 (biweekly). hour_interval Time interval through the day (units: hours) used for calculation of sky sectors for sun maps. (Optional) The default value is 0.5. Specifies whether to calculate a single total insolation value for all locations or multiple values for the specified hour and day interval. • NOINTERVAL —A single total radiation value will be calculated for the entire time configuration. This is default. Boolean (Optional) • INTERVAL —Multiple radiation values will be calculated for each time interval over the entire time configuration. The number of outputs will depend on the hour or day interval. For example, for a whole year with monthly intervals, the result will contain 12 output radiation values for each location. The number of ground x,y units in one surface z unit. The z-factor adjusts the units of measure for the z units when they are different from the x,y units of the input surface. The z-values of the input surface are multiplied by the z-factor when calculating the final output surface. If the x,y units and z units are in the same units of measure, the z-factor is 1. This is the default. Double If the x,y units and z units are in different units of measure, the z-factor must be set to the appropriate factor, or the results will be incorrect. For example, if your z units are feet and your x,y units are meters, you would use a z-factor of 0.3048 to convert your z units from feet to meters (1 foot = 0.3048 meter). slope_aspect_input_type How slope and aspect information are derived for analysis. (Optional) • FROM_DEM — The slope and aspect rasters are calculated from the input surface raster. This is the default. String • FLAT_SURFACE — Constant values of zero are used for slope and aspect. calculation_directions The number of azimuth directions used when calculating the viewshed. (Optional) Valid values must be multiples of 8 (8, 16, 24, 32, and so on). The default value is 32 directions, which is adequate for complex topography. zenith_divisions The number of divisions used to create sky sectors in the sky map. (Optional) The default is eight divisions (relative to zenith). Values must be greater than zero and less than half the sky size value. azimuth_divisions The number of divisions used to create sky sectors in the sky map. (Optional) The default is eight divisions (relative to north). Valid values must be multiples of 8. Values must be greater than zero and less than 160. diffuse_model_type Type of diffuse radiation model. (Optional) • UNIFORM_SKY — Uniform diffuse model. The incoming diffuse radiation is the same from all sky directions. This is the default. String • STANDARD_OVERCAST_SKY — Standard overcast diffuse model. The incoming diffuse radiation flux varies with zenith angle. diffuse_proportion The proportion of global normal radiation flux that is diffuse. Values range from 0 to 1. (Optional) This value should be set according to atmospheric conditions. The default value is 0.3 for generally clear sky conditions. transmittivity The fraction of radiation that passes through the atmosphere (averaged over all wavelengths). Values range from 0 (no transmission) to 1 (all transmission). (Optional) The default is 0.5 for a generally clear sky. out_direct_radiation_raster The output raster representing the direct incoming solar radiation for each location. (Optional) The output has units of watt hours per square meter (WH/m^2). Dataset out_diffuse_radiation_raster The output raster representing the diffuse incoming solar radiation for each location. (Optional) The output has units of watt hours per square meter (WH/m^2). Dataset out_direct_duration_raster The output raster representing the duration of direct incoming solar radiation. Raster (Optional) The output has units of hours. Return Value Name Explanation Data Type The output raster representing the global radiation or total amount of incoming solar insolation (direct + diffuse) calculated for each location of the input out_global_radiation_raster surface. Raster The output has units of watt hours per square meter (WH/m^2). Code Sample AreaSolarRadiation example 1 (Python window) The following Python window script demonstrates how to use this tool. import arcpy from arcpy.sa import * from arcpy import env env.workspace = "C:/sapyexamples/data" outGlobalRadiation = AreaSolarRadiation("dem30", "", "400", TimeMultipleDays(2008,91,152)) AreaSolarRadiation example 2 (stand-alone script) Calculate the amount of incoming solar radiation over a geographic area. # Name: AreaSolarRadiation_example02.py # Description: Derives incoming solar radiation from a raster surface. # Outputs a global radiation raster and optional direct, diffuse and direct duration rasters # for a specified time period. (April to July). # Requirements: Spatial Analyst Extension # Import system modules import arcpy from arcpy import env from arcpy.sa import * # Set environment settings env.workspace = "C:/sapyexamples/output" # Check out the ArcGIS Spatial Analyst extension license # Set local variables inRaster = "C:/sapyexamples/data/solar_dem" latitude = 35.75 skySize = 400 timeConfig = TimeMultipleDays(2008, 91, 212) dayInterval = 14 hourInterval = 0.5 zFactor = 0.3048 calcDirections = 32 zenithDivisions = 16 azimuthDivisions = 16 diffuseProp = 0.7 transmittivity = 0.4 outDirectRad = "" outDiffuseRad = "" outDirectDur = Raster("C:/sapyexamples/output/dir_dur") # Execute AreaSolarRadiation outGlobalRad = AreaSolarRadiation(inRaster, latitude, skySize, timeConfig, dayInterval, hourInterval, "NOINTERVAL", zFactor, "FLAT_SURFACE", calcDirections, zenithDivisions, azimuthDivisions, "UNIFORM_SKY", diffuseProp, transmittivity, outDirectRad, outDiffuseRad, outDirectDur) # Save the output Licensing Information • ArcGIS for Desktop Basic: Requires Spatial Analyst • ArcGIS for Desktop Standard: Requires Spatial Analyst • ArcGIS for Desktop Advanced: Requires Spatial Analyst Feedback on this topic?	{"url":"https://desktop.arcgis.com/en/arcmap/10.3/tools/spatial-analyst-toolbox/area-solar-radiation.htm","timestamp":"2024-11-14T07:58:42Z","content_type":"text/html","content_length":"58032","record_id":"<urn:uuid:64c7016e-5700-4f8f-8d4f-280a1b6c2a8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00301.warc.gz"}
seminars - Gromov-Witten invariants and mirror symmetry QSMS Summer School 조직위원: 조철현, 조윤형 연사; 오정석(Imperial College London) 일시: 8/21월 ~ 8/23수 ( 매일 14:00 ~ 15:15 Lecture, 15:15 ~ 15:45, Break, 15:45~ 17:00 Lecture) 장소: 서울대학교 (상산수리과학관 129 동 104호) 강연제목: Gromov-Witten invariants and mirror symmetry 초록: Mirror symmetry or its understanding seems to get better even at this moment. But on the other hand it makes it looks too diverse to follow other's progress. In this talk I would like to introduce one old fashioned understanding in an enumerative geometer's point of view, following the work of Bumsig Kim. The simplest version of mirror symmetry could be a symmetry of Hodge numbers of a pair of Calabi-Yau 3-folds. It says dimensions of tangent spaces of one's K"ahler moduli and the other's complex moduli are the same. This predicts these two moduli spaces are isomorphic in local neighbourhoods. Over these two neighbourhoods, two different D-modules are naturally defined on each. Then an advanced version of mirror symmetry could be stated with an isomorphism between the two D-modules. These two define differential equations on the spaces of sections. Then mirror symmetry gives a relationship between the solutions, which are known as J and I-functions, respectively. The coefficients are generating functions of genus 0 Gromov-Witten invariants and period integrals, In the above story, the former is completely understood in terms of genus 0 Gromov-Witten theory. Hence it can be generalised beyond Calabi-Yau 3-folds and genus >0. The latter is hard for enumerative geometers to understand because it is not developed with moduli spaces. But interestingly I-function can be written as a generating function of genus 0 quasimap invariants (Givental and Ciocan-Fontanine--Kim) though it is not fully understood why. The relationship between J and I-functions can be understood as a wall-crossing phenomenon of moduli spaces (Givental, Ciocan-Fontanine--Kim and others). So it seems quasimap theory plays some role in mirror symmetry. Now quasimap theory defines a cohomological field theory for gauged linear sigma model (Favero--Kim). In other words, there is a curve counting theory for certain LG models, which can hopefully be connected to other progress in mirror symmetry.	{"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&l=en&sort_index=Time&order_type=desc&page=56&document_srl=1105275","timestamp":"2024-11-03T00:19:33Z","content_type":"text/html","content_length":"52754","record_id":"<urn:uuid:66eee04c-fbc8-47c5-8ca0-e6a1de44c25b>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00278.warc.gz"}
How to Calculate the Cubic Volume of a Log ••• catalby/iStock/GettyImages A straight log might not be a perfect cylinder, but it's very close. That means that if you're being asked to find the volume of a log, you can use the formula for finding volume of a cylinder to make a very close approximation. But before you can use the formula, you also need to know the log's length and either its radius or its diameter. TL;DR (Too Long; Didn't Read) Apply the formula for volume of a cylinder, V = π × r^2 × h, where V is the log's volume, r is the radius of the log and h is its height (or if you prefer, its length; the straight-line distance from one end of the log to the other). If you already know the log's radius, skip straight to Step 2. But if you've measured or been given the log's diameter, you must first divide it by 2 to get the log's radius. For example, if you've been told that the log has a diameter of 1 foot, its radius would be: \frac{1\text{ foot}}{2}=0.5\text{ ft} □ Note that in this case, the radius could be expressed in either inches or feet. Leaving it in feet is a judgment call because the log's length is likely to be expressed in feet as well. Both measurements must use the same unit, or the formula won't work. In order to work the formula for volume of a cylinder, you'll also need to know the cylinder's height, which for a log is really its length straight from one end to the other. For this example, let the log's length be 20 feet. The formula for volume of a cylinder is: V=\pi r^2 h where V is the volume, r is the radius of the log and h is its height (or in this case, the length of the log). After substituting the radius and length of your example log into the formula, you have: V=\pi (0.5)^2\times 20 Simplify the equation to find the volume, V. In most cases, you can substitute 3.14 for π, which gives you: V=3.14\times (0.5)^2\times 20=3.14\times 0.25\times 20=15.7\text{ ft}^3 The volume of the example log is 15.7 ft^3. About the Author Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! -- math subjects like algebra and	{"url":"https://sciencing.com/calculate-cubic-volume-log-7711573.html","timestamp":"2024-11-05T09:53:50Z","content_type":"text/html","content_length":"405770","record_id":"<urn:uuid:86f4a363-ef1d-4507-a1ad-34109a4e0b09>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00787.warc.gz"}
Finding the Equation of a Straight Line given the Points It Passes Through Question Video: Finding the Equation of a Straight Line given the Points It Passes Through Mathematics • First Year of Secondary School Given π ΄(β 3, β 2), π ΅(0, 5), and π Ά(2, β 6), find the equation of the straight line that passes through the vertex π ΄ and bisects the line segment π ΅π Ά. Video Transcript Given π ΄: negative three, negative two; π ΅: zero, five; and π Ά: two, negative six, find the equation of the straight line that passes through the vertex π ΄ and bisects the line segment π ΅π So, firstly, weβ ve been given the coordinates of one point that this line passes through: the point negative three, negative two. We also know that it bisects the line segment π ΅π Ά, which means we can work out the coordinates of a second point on the line. If the line bisects the line segment π ΅π Ά, then it passes through its midpoint. And we know that the midpoint of the line segment of the points π ₯ one, π ¦ one and π ₯ two, π ¦ two is π ₯ one plus π ₯ two over two, π ¦ one plus π ¦ two over two. Essentially, the π ₯-coordinate of the midpoint is the mean of the two π ₯-coordinates, and the π ¦-coordinate of the midpoint is the mean of the two π ¦-coordinates. The midpoint of π ΅π Ά then is zero plus two over two for the π ₯-coordinate and five plus negative six over two for the π ¦-coordinate, which simplifies to one, negative one-half. We now know two points that lie on the line weβ re looking for, and so we can use the coordinates of these two points to calculate the slope of our line. The slope of the line connecting the points π ₯ one, π ¦ one and π ₯ two, π ¦ two can be calculated as π ¦ two minus π ¦ one over π ₯ two minus π ₯ one. Itβ s change in π ¦ over change in π ₯. Substituting the values for our two points, we have negative a half minus negative two for the change in π ¦ in the numerator and then one minus negative three for the change in π ₯ in the denominator. In the numerator, negative a half minus negative two is negative a half plus two, which is one and a half. And in the denominator, one minus negative three. Thatβ s one plus three, which is four. Now this looks very untidy because we have a mixed number in the numerator of our fraction. So we can convert that mixed number, one and a half, to a top-heavy fraction of three over two and think of dividing by four as multiplying by one-quarter. So this simplifies to three over two multiplied by one over four. And then multiplying the numerators and multiplying the dominators gives three-eighths, so the slope of the line is three-eighths. We now know the slope of our line and the coordinates of two points it passes through, so we can use the pointβ slope form of the equation of a straight line to calculate its equation: π ¦ minus π ¦ one equals π π ₯ minus π ₯ one. It doesnβ t matter which of our two points we choose to be π ₯ one, π ¦ one. Iβ m going to use the point π ΄ negative three, negative two because both of its coordinates are integer values. So we substitute negative two for π ¦ one, three-eighths for π , and negative three for π ₯ one to give π ¦ minus negative two is equal to three-eighths π ₯ minus negative three. That is, of course, π ¦ plus two is equal to three-eighths of π ₯ plus three. Next, we want to deal with this fraction, and as we have an eight in the denominator, if we multiply the entire equation by eight, we can eliminate this. Doing so gives eight π ¦ plus 16 on the left-hand side. And on the right-hand side, weβ re left with three multiplied by π ₯ plus three. The next step is going to be to distribute the parentheses on the right-hand side, so we now have eight π ¦ plus 16 is equal to three π ₯ plus nine. Finally, weβ re going to group all of the terms on the same side of the equation. By subtracting both three π ₯ and nine from each side of the equation, we can group all the terms on the left-hand side. And this gives our answer to the problem. For the three given points, the equation of the straight line that passes through point π ΄ and bisects the line segment π ΅π Ά is eight π ¦ minus three π ₯ plus seven is equal to zero.	{"url":"https://www.nagwa.com/en/videos/954187679541/","timestamp":"2024-11-12T07:36:00Z","content_type":"text/html","content_length":"254622","record_id":"<urn:uuid:312fefbf-ce8b-47ba-9b74-5c356e64fc5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00899.warc.gz"}
Lorenz Attractor 3D Model was written in NetLogo 3D 6.0.1 • Viewed 981 times • Downloaded 49 times • Run 0 times Do you have questions or comments about this model? Ask them here! (You'll first need to log in.) NB: Since NetLogo Web does not support ‘.nlogo3d’ files you can run this model only on your PC after downloading it. ## WHAT IS IT? This is a model of the phase-space of a system consisting of three ordinary differential equations, known as Lorenz system. It has chaotic solutions for certain parameter values. A particular set of chaotic solutions of the Lorenz system when plotted, resembles a butterfly or figure eight and is presented as Lorenz The model is intended to visualize Lorenz attractor in 3D view with the possibility of changing one of the key parameters (i.e. rho-parameter) ## HOW IT WORKS It is a system of three ordinary differential equations: dX/dt = σ (Y-X) dY/dt = X (ρ - Z) - Y dZ/dt = XY - βZ where σ, ρ and β are positive system parameters. Lorenz used the values ρ=28, σ=10, β=8/3 In the actual model σ=10, β=8/3 and ρ can take a value between 10 and 50. Governed by these equations and calculations a phase-space is created: with each calculation a turtle is generated and plotted in space with respective coordinates (X, Y, Z). ## HOW TO USE IT (1) Setup: creates basic conditions for the model to run (i.e. erases data from previous runs, generates X, Y and Z axes, etc.). (2) Go: starts running the model with generation of new points (turtles) in accordance with numeric values as a result of calculations, performed every time-step. (3) 'Rho-slider' is used to change ρ-value (before a new run) Buttons (3) and (4) Zoom, (5) and (6) Orbit L&R, (7) and (8) Orbit Up&Down can be used for adjusting the 3D view (9) The plot shows X, Y, Z values on each time-step/tick. ## THINGS TO NOTICE The system was developed by Edward Lorenz as a simplified mathematical model for atmospheric convection. It is a toy-model, but a very interesting one in a mathematical sense and can help to understand some very complex behavior. After the model starts it looks like points are generated in a chaotic manner but after some time it becomes evident that their trajectories are placed around two attractors. ## THINGS TO TRY You can change rho-parameter value and perform new runs. How this influences the 'butterfly' shape? ## EXTENDING THE MODEL An optimization of the code for a faster model run would be an option. ## NETLOGO FEATURES Initially the model was built with the use of NetLogo System Dynamics Modeler, then it was converted from ‘System Dynamics’ version to a ‘regular’ one by recompiling the code and adding new pieces of the code with respective changes/additions in the model interface (buttons, sliders, etc.). Finally it was wrapped using NetLogo 3D-application. ## RELATED MODELS * Turtle and Observer Motion Example 3D * Turtle Perspective Example 3D * Rossler Attractor Model First two models can be seen in NetLogo library. The last one is part of a suit of models created to visualize some key concepts of Chaos Theory and Dynamical Systems. Most of the models are available on http://modelingcommons.org/account/models/2495 This simple abstract model was developed by Victor Iapascurta, MD. At time of development he was in the Department of Anesthesia and Intensive Care at University of Medicine and Pharmacy in Chisinau, Moldova / ICU at City Emergency Hospital in Chisinau. Please email any questions or comments to viapascurta@yahoo.com The model was created in NetLogo 6.0.1, Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. This model was inspired by Introduction to Dynamical Systems and Chaos (Fall, 2017) MOOC by David Feldman @ Complexity Explorer (https://www.complexityexplorer.org/courses). Comments and Questions globals [ to setup set mylist-x list 0 (X) set mylist-y list 0 (Y) set mylist-z list 0 (Z) zoom 20 orbit-up 5 orbit-right 10 to draw-axes create-turtles 1 [ set shape "line" set heading 90 set color red set size world-width die ] create-turtles 1 [ set shape "line" set color yellow set heading 0 set size world-height die ] create-turtles 1 [ set shape "line" set pitch 90 set color blue set size world-depth die ] ask patch max-pxcor 0 0 [ set plabel "x-axis" ] ask patch 0 max-pycor 0 [ set plabel "y-axis" ] ask patch 0 0 max-pzcor [ set plabel "z-axis" ] to system-dynamics-setup set dt 0.01 set g 10 set X 1 set Y 0 set z 0 to go set mylist-x lput result-x mylist-x set mylist-y lput result-y mylist-y set mylist-z lput result-z mylist-z crt 1 [ set color green set xcor (last mylist-x ) * 0.2 set ycor (last mylist-y ) * 0.2 set zcor (last mylist-z ) * 0.2 set size 0.2 set shape "circle" to system-dynamics-go let local-b b let local-r r let local-inflow inflow let local-inflow1 inflow1 let local-inflow2 inflow2 let new-X ( X + local-inflow1 ) let new-Y ( Y + local-inflow ) let new-Z ( Z + local-inflow2 ) set X new-X set Y new-Y set Z new-Z tick-advance dt to-report result-x report X to-report result-y report Y to-report result-z report Z to-report inflow report ( X * ( r - Z ) - Y ) * dt to-report inflow1 report ( g * ( Y - X ) ) * dt to-report inflow2 report ( X * Y - b * Z ) * dt to-report b report 8 / 3 to-report r report rho-slider to system-dynamics-do-plot if plot-pen-exists? "X" [ set-current-plot-pen "X" plotxy ticks X if plot-pen-exists? "Y" [ set-current-plot-pen "Y" plotxy ticks Y if plot-pen-exists? "z" [ set-current-plot-pen "z" plotxy ticks z There is only one version of this model, created almost 7 years ago by Victor Iapascurta. Attached files File Type Description Last updated Lorenz Attractor 3D.png preview Preview for 'Lorenz Attractor 3D' almost 7 years ago, by Victor Iapascurta Download This model does not have any ancestors. This model does not have any descendants.	{"url":"http://modelingcommons.org/browse/one_model/5260","timestamp":"2024-11-03T23:05:29Z","content_type":"application/xhtml+xml","content_length":"21500","record_id":"<urn:uuid:fba380cc-cf4f-4deb-8e24-aa49c6cedb14>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00560.warc.gz"}
Detail proof of Universal Approximation Theorem—Part 1 The magic of Artificial Neural Network— Part 1 Photo by Jonathan Kemper on Unsplash This is an article for those who are interested in Artificial Neural Network and want to know why it works. In 1989, George Cybenko has published a paper to explain why it works and proved the following theorem, Universal Approximation Theorem: Universal approximation theorem from wikipedia This theorem states that for any given continuous function over an interval of [0, 1], it is guaranteed that there exists a neural network that can approximate it within the given accuracy. This theorem does not tell you how to find the neural network, but it tells you that you can find it anyway. It is very cool, isn’t it? I will describe the proof from George Cybenko using abstract mathematics (Topology and Functional Analysis) instead of a visual proof. After reading this article, you will get a feeling of how the abstract mathematics is used in solving real world problem. C(Im) is a Vector Space First, I want to talk about the properties of C(Im) that will be used in the proof. C(Im) denotes the space of real-valued continuous function on Im. Actually, it is a Vector Space (also called Function Space) over field R in which every function f is represented as a point. The operations are defined point-wise, that is, for any f,g : Im → R, and any c in R : (f+g)(x) = f(x) + g(x), (c·f)(x) = c·f(x) The above operation definition is valid, f+g and c·f are still continuous function. Proof- When f and g are continuous at a in Im, we have: ∀ ε > 0, ∃ δ1 > 0 s.t. ∀ x [\|x-a\|<δ1 ⇒ \|f(x)-f(a)\| < ε/2] and ∃ δ2 > 0 s.t. ∀ x [\|x-a\|<δ2 ⇒ \|g(x)-g(a)\| < ε/2] ∃ δ = min(δ1, δ2) > 0 s.t. ∀ x[\|x-a\|<δ ⇒ \|(f+g)(x)-(f+g)(a)\|= \|f(x)-f(a)+g(x)-g(a)\| < \|f(x)-f(a)\| + \|g(x)-g(a)\| < ε/2 + ε/2 = ε] f+g is continuous at a ∀ ε > 0, ∃ δ > 0 s.t. ∀ x [\|x-a\|<δ ⇒ \|f(x)-f(a)\| < ε/\|c\|] ∃ δ > 0 s.t. ∀ x[\|x-a\|<δ ⇒ \|(c·f)(x)-(c·f)(a)\|= \|c·f(x)-c·f(a)\|= \|c\|·\|f(x)-f(a)\| < \|c\|·ε/\|c\| = ε] c·f is continuous at a Now, to prove that C(Im) is a Vector Space, we have to prove that the following properties are satisfied. Proof- Let f, g and h be arbitrary functions in C(Im), a and b scalars in R associativity: f + (g + h) = (f + g) + h commutativity: f + g = g + f identity element: 0(x) = 0, (f + 0)(x) = f(x) + 0(x) = f(x) + 0 = f(x) inverse elements: (−f)(x) = −f(x) such that f+ (−f) = f−f = 0 compatibility: a(b·f) = (ab)f , 1·f = f distributivity: a·(f + g) = a·f + a·g, (a + b)f = a·f + b·f C(Im) is a Normed Space C(Im) is a Vector Space. Furthermore, it is a Normed Space. A Normed Space is an ordered pair (V, ∥·∥) where V is a vector space over the real or complex numbers, on which a norm ∥·∥ is defined. A norm is the intuitive notion of “length” in the real world. A norm is a real-valued function defined on the vector space ∥·∥: V → ℝ, for any x∈V and has the following properties: N1: ∥x∥ ≥ 0 N2: ∥x∥ = 0 ⇔ x = 0 N3: ∥αx∥ = \|α\|∥x∥ N4: ∥x+y∥ ≤ ∥x∥ + ∥y∥ Define ∥·∥: C(Im) → ℝ, for any f ∈ C(Im) ∥f∥ = sup{ \|f(x)\| : x ∈ Im} Let’s prove that ∥·∥ is a norm. Proof- Let f, g ∈ C(Im) N1: ∥f∥ = sup{ \|f(x)\| : x ∈ Im} ≥ 0 N2: ∥f∥ = 0 ⇔ sup{ \|f(x)\| : x ∈ Im}= 0 ⇔ f(x) = 0 ∀x ∈ Im ⇔ f = 0 N3: ∥αf∥ = sup{ \|αf(x)\| : x ∈ Im} = \|α\| sup{ \|f(x)\| : x ∈ Im} = \|α\|∥f∥ N4:∥f∥ + ∥g∥ = sup{\|f(x)\| : x ∈ Im}+ sup{\|g(x)\| : x ∈ Im}≥ sup{\|f(x)\|+\|g(x)\| : x ∈ Im} ≥ sup{\|f(x)+g(x)\| : x ∈ Im} = ∥f+g∥ ∥·∥ is a norm C(Im) is a Metric Space and Topological Space When a vector space is a Normed Space, it will be a Metric Space and Topological Space. Normed Vector Space, Metric Space and Topological Space (from Wikipedia) A Metric Space is an ordered pair (M, d) where M is a set on which a metric d is defined. A metric is the intuitive notion of “distance” in the real world. A metric is a non-negative function defined on the set d: M x M → [0, ∞), for any f, g ∈ M and has the following properties: M1: d(f, g) = 0 ⇔ f = g M2: d(g, f) = d(f, g) M3: d(f, g)+d(g, h)≥d(f, h) Define d(f, g): M x M → [0, ∞), for any f, g ∈ M and M is a Normed Space d(f, g) = ∥f - g∥ Let’s prove that d(·,·) is a metric. Proof- Let f, g, h ∈ M M1: d(f, g) = 0 ⇔ \|\|f-g\|\| = 0 ⇔ f-g = 0 ⇔ f = g M2: d(g, f) = ∥g-f∥= ∥(-1)(f-g)∥ = \|-1\|∥f-g∥ = d(f, g) M3: d(f, g)+d(g, h) = ∥f-g∥ + ∥g-h∥ ≥ ∥f-h∥ = d(f, h) d(·,·) is a metric Since C(Im) is a Normed Space, it is also a Metric Space. A Topological Space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets ui of X, called Open Set, satisfying the following axioms: T1: ø, X ∈ τ T2: any (finite or infinite) union of sets in τ is itself in τ, i.e. ∪ui ∈ τ ∀ ui ∈ τ T3: any finite intersection of sets in τ is itself in τ, i.e. ∩ui ∈ τ ∀ ui ∈ τ To understand why a Metric Space (M, d) is also a Topological Space, we need a tool called Open Ball: Define Open Ball B(x, r), for any x ∈ M, r > 0 B(x, r) = {p\|p ∈ M, d(x, p) < r} Define Open Set o and collection τ τ ={o\|o ⊆ M, ∀x∈o ∃r>0 (B(x,r) ⊆ o)} Let’s prove that (M, d) is a Topological Space ø contains no point ⇒ ø ∈ τ ∀x∈M ∃r>0 B(x,r)⊆M⇒ M ∈ τ o[i] ∈ τ ⇒ ∪o[i] ⊆ M and ∪o[i] = {x\|∃o∈τ ∃r>0 (x∈o and B(x,r) ⊆o ⊆ ∪o[i])} ⇒ ∪o[i] ⊆ M and ∪o[i] = {x\|∃r>0 (B(x,r) ⊆ ∪o[i])} ⇒ ∪o[i] ∈ τ o1, o2 ∈ τ ⇒ o1 ∩ o2 ⊆ M and o1 ∩ o2 = {x\|∃r1>0 ∃r2>0 (x∈o1 and B(x,r1) ⊆o1 and x∈o2 and B(x,r2) ⊆o2)} ⇒ o1 ∩ o2 ⊆ M and o1 ∩ o2 = {x\|∃min(r1,r2)>0 (x∈o1 ∩ 02 and B(x,min(r1,r2)) ⊆ B(x,r1) ⊆ o1 and B(x,min(r1,r2)) ⊆ B(x,r2) ⊆ o2)} ⇒ o1 ∩ o2 ⊆ M and o1 ∩ o2 = {x\|∃min(ru)>0 (B(x,min(r1,r2)) ⊆ o1 ∩ o2)} ⇒ o1 ∩ o2 ∈ τ ∴(M, d) is a Topological Space Functions of the form F(x) is dense in C(Im) ? Return to the Universal Approximation Theorem, it says that “functions of the form F(x) is Dense in C(Im)”. Let S be the set of functions of the form F(x). Clearly, set S is a subset of C(Im). Dense is a topological feature. It means that for any point x in C(Im), then every Neighbourhood N of x contains a point from S. A Neighbourhood N of x is a subset of C(Im) that includes an Open Set u containing x. x ∈ u ⊆ N Dense set S, Neighbourhood N and Open Set u containing point x In Metric Space, it can be proved that any Open Ball can be an Open Set and neighbourhood at the same time [See Appendix]: x∈B(x, r)=u=N We have proved that C(Im) is a Topological Space, so we can now use the Topology Theory to help us to prove the theorem. Now, we can express the Universal Approximation Theorem in the language of Topology Theory: S is Dense in C(Im) ⇔ For any f ∈ C(Im), N ∩ S ≠ ∅ ∀ N where N is a neighbourhood of f ⇒ For any f ∈ C(Im), B(f, ε) ∩ S ≠ ∅ ∀ε > 0 ⇒ For any f ∈ C(Im), ∀ε > 0, ∃ F ∈ S s.t. d(f, F) < ε ⇒ For any f ∈ C(Im), ∀ε > 0, ∃ F ∈ S s.t. ∥f-F∥ < ε ⇒ For any f ∈ C(Im), ∀ε > 0, ∃ F ∈ S s.t. sup{\|F(x)-f(x)\|\|∀ x ∈ Im} < ε ⇒ For any function f ∈ C(Im), ∀ε > 0, ∃ F ∈ S s.t. \|F(x)-f(x)\| < ε ∀ x ∈ Im So, once we have proved that S is dense in C(Im), the Universal Approximation Theorem is finally proved. To prove this, we use the following Topology theorem and then prove by contradiction: S is Dense in C(Im) ⇔ Closure(S) = C(Im) Suppose Closure(S) ≠ C(Im) ⇒ Closure(S) ⊂ C(Im) ⇒ contradiction I will explain this in details in Part 2. In this article, the Universal Approximation Theorem is introduced. It is a theorem that can explain why the Artificial Neural Network works. The set of all real-valued continuous function on Im, C (Im), is a normed vector space which is also a topological space. We find that the Universal Approximation Theorem will be proven when we can show that the Closure of S is dense in C(Im). The detail of the proof will be explained in Part 2. If you want to know more about topology, please visit my series of articles on this topic: Your feedback is highly appreciated and will help me to continue my articles. Please give this post a clap if you like this post. Thanks!! Open Ball B(x, r) is an Open Set and neighbourhood for any point x in Metric Space. ∀ p ∈ B(x, r) ⇒ d(p, x)= l < r Consider Open Ball B(p, t) with t = r-l > 0 ∀ y ∈ B(p, t) ⇒ d(y, p) < t ⇒ d(y, x) < d(y, p) + d(p, x) < t + l = r ⇒ y ∈ B(x, r) ⇒ ∃t >0 s.t. B(p, t) ⊂ B(x, r) ⇒ B(x, r) ∈ τ Clearly, B(x, r)⊆B(x, r), so Open Ball B(x, r) contains an Open Set and is thus a neighbourhood of point x.	{"url":"https://simonkwan-35335.medium.com/demystifying-the-universal-approximation-theorem-part-1-6605d3d1dd73","timestamp":"2024-11-06T09:35:48Z","content_type":"text/html","content_length":"201771","record_id":"<urn:uuid:a126d1d6-0dfa-4423-8e86-4fc1d42132ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00283.warc.gz"}
Duration & volatility \| Debt securities \| Achievable Series 6 When interest rates change, bond market values fluctuate in the market. Bonds withlonger maturities and lower coupons tend to see the most price volatility. When a bond has a long maturity, it tends to be more sensitive to interest rate changes. Time has a compounding effect on market values. Assume you own a 1 year bond and a 20 year bond in your portfolio. When interest rates rise, market values for both bonds willfall. The 20 year bond’s price will fall more in price. Let’s talk about why. When interest rates rise, itmakes current bonds less valuable. Existing bond values are dependent on interest rates of new bonds. When interest rates rise, new bonds become more valuable (they’re being issued with higher rates than before), leading to existing bonds falling in value. While both the 1 year and the 20 year bond will fall in value, the1 year bond won’t fall as much. Within one year, the investor will receive their par value back. At that point in time, the investor can reinvest their money back into a new bond with a higher rate of interest. The other bond has a 20 yearwait until that can happen. It’s locked in at the lower rate of interest until it matures or is sold. This is why bonds with longer maturities fall further in price when interest rates rise. When interest rates fall, long term bonds rise further in price for the same reasons. Going back to our comparison, the 1 year bond will rise in price, but not by much. It matures within one year; if the investor decides to reinvest their money back in the market, they will be buyinga bond with a lower rate of return. The other bond has a higher interestrate that’s locked in for the next 20 years. Because of this, the market value of the 20 year bond will rise much further than the 1 year bond. Bonds with lower coupons have more price volatility than bonds with higher coupons. To understand this, assume you own two 10 year bonds. One has a 2% coupon andthe other has a 10% coupon. When interest rates rise, the value of both bonds will fall. The 2% coupon bond will fall further in price because it has less interest to reinvest back into the market at the new, higher rate of interest. The 10% coupon bond pays much more interest and gives more money to the bondholder toreinvest back into the market at the new, higher rate of interest. The lower the coupon of a bond, the more likely it was sold at a discount. If a bond’s value is mostly from its discount, the investor must wait until maturity to makemoney from the bond’s discount. The 10% bond is more valuable in this situation because the 10% bond pays more interest that can be reinvested at higher rates right now. When interest rates fall, the value of both bonds will rise. The 2% coupon bond will rise further in price because its value is likely tied to a discount. Remember, the lower the coupon, the more likely the bond was sold at a discount. When much of the bond’s value is achieved at maturity when the investor receives the par value of the bond, it is not required to reinvest largesums of money at lower rates of return. The 10% bond pays much more interest to its bondholder. If the bondholder decides to reinvest their interest back into the market, they are forced to now buy bonds with lower rates of return as interest rates fell. The 10% bond is less valuable in this situation because the 10% bond pays more interest that would be reinvested at lowerrates right now. Here’s avideo breakdown of a practice question regarding price volatility: The concept of duration is part of the same “family of ideas” as price volatility. In fact, the debt security with the longest maturity andthe lowest coupon will maintain the highest duration. So, how is duration unique? In addition to measuring how quickly asecurity responds to interest rate changes, it also measures the amount of time necessary for an investor to recoup their original investment. For example, let’s assume we’re analyzing the following bond: 20 year, $1,000 par, 10% debenture tradingfor 120 This bond will pay $100 in annual interest (10% of $1,000) over a 20 year period. It currently costs $1,200 (tradingat 120% of $1,000). Bonds are typically quoted on a percentage of par basis. Meaning, a quote of 120 means a bond is trading for 120% of par ($1,000). For test purposes, you can simply add a zero to the end of the bond quote tofind its price. You should not expect to see detailed questions related to bond quotes on the Series 6 exam. If this bond pays $100in annual interest and currently costs $1,200, how long will it take an investor to recoup their original investment? If we assume the interest is not reinvested, it will take 12 years (12 years x $100 annual interest). Therefore, the duration of this debenture is roughly 12 years. Duration calculations often assume future cash flows are discounted to present value andreinvested. The details are not important for test purposes, but we’re calling this out because the duration calculation above is very oversimplified. However, test questions tend to focus on the fundamental concepts of duration. Know the basics and you’ll be fine! Now, let’s assumewe’re analyzing this bond: 20 year, $1,000 par, zerocoupon bond trading for 45 This bond doesnot pay interest until maturity (same with all zero coupon bonds), which is in 20 years. It currently costs $450 (trading at 45% of $1,000). Zero coupon bonds do not pay interest until the very end of the bond, therefore it will take the entire length of the bond for the investor to recouptheir original investment. Or, another way of saying a zero coupon bond’s duration is equal to its maturity. Therefore, this bond’s duration is 20 years. Let’snow compare the two bonds: 20 year, $1,000 par, 10% debenture trading for120 20 year, $1,000par, zero coupon bond trading for 45 As we discussed at the beginning of this section, duration and price volatility both measure price volatility on a bond. The longer the maturity and the lower the coupon of the bond, thehigher the price volatility and longer the duration. These two bonds align with this concept. Both are 20 year bonds, but the zero coupon bond’s market price is more volatile and reflects a longer duration.	{"url":"https://app.achievable.me/study/finra-series-6/learn/debt-securities-duration-and-volatility","timestamp":"2024-11-11T00:04:25Z","content_type":"text/html","content_length":"185294","record_id":"<urn:uuid:e69301b7-6c50-408c-822f-b3f33ff799bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00430.warc.gz"}
Zero entanglement entropy for Kitaev Chain at Delta=-1 I tested some calculations with Kitaev Chain. The energy levels should not depend on the sign of @@\Delta@@ (the coefficient of @@c^\dagger c^\dagger + cc@@ term). ITensor gives correct energy, but there is some problems for the entanglement entropy. The following code works with ITensor3. But ITensor2 gives different entanglement entropy (close to zero). #include "itensor/all.h" using namespace itensor; auto N = 100; auto sites = Spinless(N,{"ConserveNf",false}); auto ampo = AutoMPO(sites); for(int b = 1; b <= N-1; ++b) ampo += -1,"N",b; ampo += -1,"Cdag",b,"C",b+1; ampo += -1,"Cdag",b+1,"C",b; ampo += -1,"Cdag",b,"Cdag",b+1; ampo += -1,"C",b+1,"C",b; ampo += -1,"N",N; auto H = toMPO(ampo); auto state = InitState(sites); for(int is = 1; is <= N; ++is) state.set(is,is%2==1 ? "1" : "0"); auto psi0 = randomMPS(state); auto sweeps = Sweeps(20); sweeps.maxdim() = 20,20,40,40,100,200,300,400; sweeps.cutoff() = 1E-10; auto [energy0, psi] = dmrg(H, psi0, sweeps, {"Quiet=",true}); return 0; When Jordan-Wigner transformed to spin-1/2. Neither ITensor3 nor ITensor2 gives correct entanglement entropy. #include "itensor/all.h" using namespace itensor; auto N = 100; auto sites = SpinHalf(N, {"ConserveSz=",false}); auto ampo = AutoMPO(sites); for(int b = 1; b <= N-1; ++b) ampo += -1,"Sz",b; ampo += -1,"S+",b,"S-",b+1; ampo += -1,"S+",b+1,"S-",b; ampo += -1,"S+",b,"S+",b+1; ampo += -1,"S-",b+1,"S-",b; ampo += -1,"Sz",N; auto H = toMPO(ampo); auto state = InitState(sites); for(int is = 1; is <= N; ++is) state.set(is,is%2==1 ? "Up" : "Dn"); auto psi0 = randomMPS(state); auto sweeps = Sweeps(200); sweeps.maxdim() = 20,20,40,40,100,200,300,400; sweeps.cutoff() = 1E-10; auto [energy,psi] = dmrg(H,psi0,sweeps,{"Quiet",true,"EnergyErrgoal=",1E-8,"EntropyErrgoal=",1E-7}); return 0; But everything works well when the coefficient of @@c^\dagger c^\dagger + cc@@ term reverses sign. Is there something wrong? Thanks. Hi Jin, I am not an expert on the Kitaev model, but if I understand correctly you are saying that in ITensor v3, both the original Hamiltonian and the Jordan-Wignered version give the correct ground state energy, but for Delta = -1 the Jordan-Wignered version gives an entanglement entropy that you are not expecting. Is it possible that in that case, DMRG is giving you one of two degenerate ground states (i.e. maybe a Majorana mode sitting on either one edge or the other), and mixing the two would give a state with the entanglement entropy you expect? Hi Matt, thanks for the comment. For ITensor v3, yes, that is what I mean. I'm not sure. Maybe it is possible that dmrg gives one zero entanglement ground state. But why this only happens for spin model? For fermion model, ITensor v2 also gives an entanglement entropy close to zero. What changes made for ITensor v3 gives correct one for fermion? It seems like the discrepancy you are seeing is based on the symmetries that are being imposed. For example, in ITensor v3, you can impose spin parity symmetry like this: auto sites = SpinHalf(N, {"ConserveSz=",false,"ConserveParity=",true}); In that case, I see an entanglement of approximately log(2) (which I guess is what you were hoping to see) for both Delta = 1 or Delta = -1. It seems like what is happening is the parity conservation both in the case of fermions and spins enforces that the ground state you find is the more entangled state (probably a superposition of the states with different Majorana modes on each end, if I were to guess). Not imposing this symmetry may mean that DMRG is able to find the less entangled state with only one Majorana mode on either end of the system (again, this is some conjecture since I have not studied the Kitaev chain very much and don't remember what the ground state is supposed to be). Perhaps the issue in your ITensor v2 code is that you are not imposing parity symmetry. In general, DMRG will be biased towards finding less entangled states. Does that make sense to you? A lot of this is conjecture on my part based on what I am seeing from DMRG. Hi Matt, Yes, it makes sense. The two ground states have different parity. I'm also new to Kitaev Chain, But I think that is the answer. Thanks for your help. See the comments for the answer (because there is a degenerate groundspace, different states are found by DMRG depending on whether or not parity symmetry is imposed in both the fermionic and spin Hi Matt, using {"ConserveParity=",true} is 2.5 times slower than using {"ConserveParity=",false}, is this normal? For what bond dimension? For small bond dimensions that is possible since there is a bit of overhead when using quantum numbers (since they are implemented with block sparse tensors, so there is some overhead to analyzing which blocks contract with which other block), but for larger bond dimensions that should not be the case. Just a note also about the physics of the Kitaev chain: the ground states of the fermionic model versus the spin model are different (independent of DMRG or how you compute them). This is because for fermion systems, ground states should have a well-defined fermion parity. For this parity to be well-defined, the ground states have to be macroscopic superpositions of the ground states of the spin model. The spin model is just a simple ferromagnet, so has two product-state ground states. So the behavior you are seeing is actually correct.	{"url":"https://www.itensor.org/support/1874/zero-entanglement-entropy-for-kitaev-chain-at-delta-1?show=1877","timestamp":"2024-11-08T15:55:31Z","content_type":"text/html","content_length":"42485","record_id":"<urn:uuid:d2d474f9-7f7d-475a-940d-20e3b3e6c83d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00640.warc.gz"}
Changing the way we teach math - Technique I like math, I really do. I do not, however, like the way we are forced to learn math. This week, I had to call my mother, a middle school math teacher, to ask her what the quadratic formula was. Now, I know what the quadratic formula is. I know when to use it. I know how to use it. I just could not for the life of me remember where the 4ac went. Do you want to know why I could not remember? Because other than the once-every -five-year problem when I need to factor some complicated equation, no one ever uses the quadratic formula. Do bankers use it? Do engineers use it? Do calculus professors us it? Heck, I don’t even think professional factor-ers would use it. I mean, we have Ti-89’s for a reason, guys. So why exactly did we spend months, maybe even years, of our middle school lives memorizing and rememorizing a nearly useless formula? Or a better question would be, why were we not using that time and effort to learn parts of math that would later be useful or relevant or even just used more than once in our entire adulthoods. I do not mean to come across as whiny; I just believe that my issue with the quadratic formula is a good example of a larger problem within the American teaching system. And no, I’m not talking about the battle with the Common Core or the differences between public and private school. What I believe is that we need to change the way math is viewed as a subject in Often, in school, and even at Tech, math seems like mindless work that is meant just to get through. But this is a failing. Math is important and useful and should not be a bunch of near meaningless numbers that highschoolers cram into their brains before an AP test. There are several easy (well, seemingly easy) solutions to this predicament. Schools could focus on math that will be useful in the students’ futures— such as how does one calculate their expected taxes and how much is too high of an interest rate and how much should an apartment rent be, based on one’s income? Math is a huge part of adulthood and honestly that is the math I do not know. I would like to see changes beyond that. I would like to see, and I hope to one day see, a change in the way Americans view math. There is no reason for students to be afraid of math. There is no reason for thousands of smart students to become baristas in part because they think calculus or statistics is too hard. The problem is not the difficulty of the subject, it is the difficulty we create in teaching it. For now, I will rest easy knowing I will most likely not have to use the quadratic formula for years … well, probably ever.	{"url":"https://nique.net/opinions/2015/01/23/changing-the-way-we-teach-math/","timestamp":"2024-11-06T08:10:22Z","content_type":"text/html","content_length":"134245","record_id":"<urn:uuid:6a1a5bac-115c-4459-8c15-9d02fda08c33>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00585.warc.gz"}
Electrons take a phonon bath Aug. 28, 2015 Research Highlight Physics / Astronomy Electrons take a phonon bath A theoretical model enables the first exact and universal description of electrons moving in a ‘bath’ of atomic vibrations In fundamental physics, it is relatively easy to describe the motion of a single moving particle, but it is much more challenging to develop a reliable theoretical description of a particle such as an electron moving in an environment where it interacts with many other particles. Now, Naoto Nagaosa and Andrey Mishchenko of the RIKEN Center for Emergent Matter Science with colleagues in Italy have succeeded in constructing a comprehensive and mathematically exact description of the movement of particles within such an interacting environment as a function of parameters such as temperature The movement of electrons in crystals is one of the defining characteristics of materials and determines their behavior in many practical applications. As electrons move through a crystal, they interact with surrounding atoms via atomic vibrations known as phonons. A particle moving in such a ‘phonon bath’ is known as a polaron (Fig. 1). “Polarons occur in almost every transport phenomenon in solids,” explains Nagaosa. Despite their ubiquity, however, deriving a mathematical description of polarons has proved a challenge that has confounded even some of our most famous physicists. The problem is the difficulty of reducing the complexity of interactions that make up a polaron to a few basic simplifications. Although this strategy works well for many problems in physics, phonon systems are so complex they defy simplification. Consequently, previous approaches have been limited to approximations only. In contrast, Nagaosa and his colleagues used mathematically exact computational methods without approximations to calculate results for specific scenarios. They then mapped the information gained from these calculations onto a two-dimensional ‘phase diagram’ of temperature versus strength of interaction between the electron and the surrounding phonon bath. The phase diagram revealed a strong dependence of polaron transport on temperature and strength of interaction, with several distinct transport regimes that explain many of the observed fundamental properties of materials, such as the electrical conductivity of metals and semiconductors. “The study of polarons and particularly polaron mobility is important for technology because polarons are carriers in many modern electronic devices,” says Mishchenko. Although derived in general terms, the researchers’ calculations and the resultant phase diagram are so far limited to describing polarons in one dimension. Extending the theory to higher dimensions will allow a more realistic description of polaron behavior and could lead to a fundamental model that describes a broad range of important effects in materials, such as magnetism and • 1. Mishchenko, A. S., Nagaosa, N., De Filippis, G., de Candia, A. & Cataudella, V. Mobility of Holstein polaron at finite temperature: An unbiased approach. Physical Review Letters 114, 146401 (2015). doi: 10.1103/PhysRevLett.114.146401	{"url":"https://www.riken.jp/en/news_pubs/research_news/rr/8052/index.html","timestamp":"2024-11-02T23:41:28Z","content_type":"text/html","content_length":"19536","record_id":"<urn:uuid:259ed027-441f-4dce-a5c2-f59e9df76b94>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00224.warc.gz"}
Course angles and distance between the two points on the orthodrome(great circle) Calculates the distance between two points of the Earth specified geodesic (geographical) coordinates along the shortest path - the great circle (orthodrome). Calculates the initial and final course angles and azimuth at intermediate points between the two given. As we mentioned before, in Course angle and the distance between the two points on loxodrome (rhumb line)., if you are traveling the Earth surface from point A to point B, maintaining the same course angle, your current path won't be the shortest distance between these points. To achieve your target with the shortest path, you have to correct your course angle so your movement's trajectory will be close to the great circle (orthodromy), which will be the shortest distance between these two points. The following calculator calculates the distance between two coordinates, the initial course angle, the final course angle, and the course angles for the intermediate points. The difference between this calculator from the earlier version Distance calculator is that this one uses a quite precise algorithm developed by Polish scientist Thaddeus Vincenty. The calculation error is less than 0.5mm. Digits after the decimal point: 2 Distance in nautical miles Distance between the waypoints (km) Distance between the waypoints (nm) The file is very large. Browser slowdown may occur during loading and creation. Firstly, the inverse position computation was solved - the distance between the two points was calculated, and the initial and final grid azimuths were found. Then the acquired distance was divided into an equal number of segments following a predetermined number of waypoints. For every segment, the common survey computation was solved - the given directional angle found the coordinates of the next point and the previous point's coordinates. For this solution, Vincenty's algorithm was used (It's described here Direct and Inverse Solutions of Geodesics on the Ellipsoid with the application of nested equations, Survey Review, April 1975) URL copiado para a área de transferência Calculadoras similares PLANETCALC, Course angles and distance between the two points on the orthodrome(great circle)	{"url":"https://pt.planetcalc.com/722/","timestamp":"2024-11-11T11:03:38Z","content_type":"text/html","content_length":"47229","record_id":"<urn:uuid:fb7b5775-c0a9-49f3-a37e-5eb381efe2a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00625.warc.gz"}
Logistic Regression in Python using scikit-learn Package - Finance Train Logistic Regression in Python using scikit-learn Package Using the scikit-learn package from python, we can fit and evaluate a logistic regression algorithm with a few lines of code. Also, for binary classification problems the library provides interesting metrics to evaluate model performance such as the confusion matrix, Receiving Operating Curve (ROC) and the Area Under the Curve (AUC). Hyperparameter Tuning in Logistic Regression in Python In the Logistic Regression model (as well as in the rest of the models), we can change the default parameters from scikit-learn implementation, with the aim of avoiding model overfitting or to change any other default behavior of the algorithm. For the Logistic Regression some of the parameters that could be changed are the following: • Penalty: (string) specify the norm used for penalizing the model when its complexity increases, in order to avoid overfitting. The possible parameters are “l1”, “l2” and “none”. “l1” is the Lasso Regression and “l2” is the Ridge Regression that represents two different ways to increase the magnitude of the loss function. Default value is “l2”. “none” means no regularization parameter. l1, lasso regression, adds “absolute value of magnitude” of coefficient as penalty term to the loss function. l2, ridge regression, adds “squared magnitude” of coefficient as penalty term to the loss function. • C: (float): the default value is 1. With this parameter we manage the λ value of regularization as C = 1/λ. Smaller values of C mean strong regularization as we penalize the model hard. • multi_class: (string) the default value is “ovr” that will fit a binary problem. To fit a multiple classification should pass “multinomial”. • solver: (string) algorithm to use in the optimization problem: “newton-cg”, “lbfgs”,”liblinear”,”sag”, and ”saga”. Default is “liblinear”. These algorithms are related to how the optimization problem achieves the global minimum in the loss function. □ “liblinear”: is a good choice for small datasets □ “Sag” or “saga”: useful for large datasets □ “lbfgs”,”sag”,”saga”, or “newton-cg”: handle multinomial loss, so they are suitable for multinomial problems. □ “liblinear” and “saga”: handle “l1” and “l2” penalty The penalty has the objective to introduce regularization which is a method to penalize complexity in models with high amount of features by adding new terms to the loss function. It is possible to tune the model with the regularization parameter lambda (λ) and handle collinearity (high correlation among features), filter out noise and prevent overfitting. By increasing the value of lambda (λ) in the loss function we can control how well the model is fit to the training data. As stated above, the value of λ in the logistic regression algorithm of scikit learn is given by the value of the parameter C, which is 1/λ. To show these concepts mathematically, we write the loss function without regularization and with the two ways of regularization: “l1” and “l2” where the term are the predictions of the model. Loss Function without regularization Loss Function with l1 regularization Loss Function with l2 regularization Data Science in Finance: 9-Book Bundle Master R and Python for financial data science with our comprehensive bundle of 9 ebooks. What's Included: • Getting Started with R • R Programming for Data Science • Data Visualization with R • Financial Time Series Analysis with R • Quantitative Trading Strategies with R • Derivatives with R • Credit Risk Modelling With R • Python for Data Science • Machine Learning in Finance using Python Each book includes PDFs, explanations, instructions, data files, and R code for all examples. Get the Bundle for $39 (Regular $57) JOIN 30,000 DATA PROFESSIONALS Free Guides - Getting Started with R and Python Enter your name and email address below and we will email you the guides for R programming and Python.	{"url":"https://financetrain.com/logistic-regression-in-python-using-scikit-learn-package","timestamp":"2024-11-12T20:49:22Z","content_type":"text/html","content_length":"123940","record_id":"<urn:uuid:f5a6c005-aeda-4ee6-80bc-17eed1006375>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00648.warc.gz"}
Cosecant: Introduction to the Cosecant Function in Mathematica (subsection CscInMathematica/05) Operations under special Mathematica functions Series expansions Calculating the series expansion of a cosecant function to hundreds of terms can be done in seconds. Mathematica comes with the add‐on package DiscreteMath`RSolve` that allows finding the general terms of the series for many functions. After loading this package, and using the package function SeriesTerm, the following term of can be evaluated. This result can be verified by the following process. Mathematica can evaluate derivatives of the cosecant function of an arbitrary positive integer order. Finite products Mathematica can calculate some finite symbolic and nonsymbolic products that contain the cosecant function. Here are two examples. Indefinite integration Mathematica can calculate a huge number of doable indefinite integrals that contain the cosecant function. The results can contain special functions. Here are some examples. Definite integration Mathematica can calculate wide classes of definite integrals that contain the cosecant function. Here are some examples. Limit operation Mathematica can calculate limits that contain the cosecant function. Here are some examples. Solving equations The next inputs solve two equations that contain the cosecant function. Because of the multivalued nature of the inverse cosecant function, a printed message indicates that only some of the possible solutions are returned. A complete solution of the previous equation can be obtained using the function Reduce. Solving differential equations Here is a nonlinear first-order differential equation that is obeyed by the cosecant function. Mathematica can find the general solution of this differential equation. In doing so, the generically multivariate inverse of a function is encountered, and a message is issued that a solution branch is potentially missed. Mathematica has built‐in functions for 2D and 3D graphics. Here are some examples.	{"url":"https://functions.wolfram.com/ElementaryFunctions/Csc/introductions/CscInMathematica/05/ShowAll.html","timestamp":"2024-11-10T21:11:00Z","content_type":"text/html","content_length":"52112","record_id":"<urn:uuid:7430fb0a-382b-4f26-8b08-bda8257ac66b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00213.warc.gz"}
next → ← prev Keras Convolutional Layers It refers to a one-dimensional convolutional layer. For example, temporal convolution which generates a convolution kernel for creating a tensor of outputs. The convolution kernel is convolved with input layer over a single temporal (spatial) dimension. A bias vector will be developed and included to the outputs, if use_bias is True. It will be applied to the output, if activation is set None. When we utilize the conv1D layer as the initial layer in our model, it provides us with an input_shape argument, which is either a tuple of integers or None. It does not incorporate the batch axis. • filters: Filters refers to an integer in which the output space dimensionality or the number of output filters present in the convolution. • kernel_size: It is an integer or tuple/list of an individual integer that specifies the length of the 1D convolution window. • strides: It is an integer or tuple/list of an individual integer that specifies the stride length of the convolution. Determining any stride value != 1 is incompatible with specifying any dilation_rate value != 1. • padding: It is one of "valid", "casual" or "same", where valid implies to no padding, same means padding the input in such a way that it generates the output having the same length as that of the original input and casual results in dilated output, i.e., output[t] is independent of output[t + 1:]. To get the output length similar to the input, a zero-padding can be used. The concept of padding is useful while modeling a temporal data in order to make sure that the model does not violate temporal • data_format: It is a string of "channels_last" or "channels_first", which is the order of input dimensions. Here the "channels_last" links to the input shape (batch, steps, channels), which is the default format for temporal data in Keras. However, the "channels_first" is used to relate the input shape (batch, channels, steps). • dilation_rate: It is an integer or tuple/ list of an individual integer that relates to the dilation rate of a dilated convolution. It currently relates any dilation_rate value != 1 is incompatible by specifying any strides value != 1. • activation: It refers to an activation function to be used. When nothing is specified, then by defaults, it is a linear activation a(x) = x, or we can say no activation function is applied. • use_bias: It represents a Boolean that shows whether the layer utilizes a bias vector. • kernel_initializer: It can be defined as an initializer for the kernel weights matrix. • bias_initializer: It refers to an initializer for bias vector. • kernel_regularizer: It refers to a regularizer function, which is applied to the kernel weights matrix. • bias_regularizer: It can be defined as a regularizer function, which is applied to the bias vector. • activity_regularizer: It refers to a regularizer function that is applied to the activation (output of the layer). • kernel_constraint: It is a constraint function applied to the kernel matrix. • bias_constraint: It can be defined as a constraint function applied to the bias vector. Input shape It refers to a 3D tensor of shape (batch, steps, channels). Output shape The output shape is a 3D tensor of shape (batch, new_steps, filters) steps. The values might differ due to strides and padding. It refers to a two-dimensional convolution layer, like a spatial convolution on images. It develops a convolution kernel, which can be convolved with the input layer for the generation of the tensors output. If we set use_bias to True, it will create a bias vector, which will be added to the output. Similarly, if we set activation to None, then also it will be added to the output. The layer can be used as an initial layer in the model by using the input_shape keyword argument, which is a tuple of integers, and it does not include the batch axis. • filter: It is an integer that signifies the output space dimensionality or a total number of output filters present in a convolution. • kernel_size: It can either be an integer or tuple/list of 2 integers to represent the height and width of a 2D convolution window. It can also exist as a single integer that signifies the same value for rest all of the spatial domain. • strides: It is either an integer or a tuple/list of 2 integers that represents the convolution strides along with height and width. It might exist as a single integer that indicates the same value for the spatial dimension. If we signify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • padding: One of "valid" or "same," where the same reflects some inconsistency across the backend with strides !=1. • data_format: It is a string of "channels_last" or "channels_first", which is the order of input dimensions. Here the "channels_last" describes the input shape (batch, height, width, channels), and the "channels_first" describes the input shape (batch, channels, height, width). It defaults to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder, then it is residing at "channels_last". • dilation_rate: It can be an integer or tuple/ list of 2 integers that relates to the dilation rate to be used for dilated convolution. It might have an individual integer that indicates the same value for a spatial dimension. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • activation: It refers to an activation function to be used. When nothing is specified, then by defaults, it is a linear activation a(x) = x, or we can say no activation function is applied. • use_bias: It represents a Boolean that shows whether the layer utilizes a bias vector. • kernel_initializer: It can be defined as an initializer for the kernel weights matrix. • bias_initializer: It refers to an initializer for bias vector. • kernel_regularizer: It refers to a regularizer function, which is applied to the kernel weights matrix. • bias_regularizer: It can be defined as a regularizer function, which is applied to the bias vector. • activity_regularizer: It refers to a regularizer function that is applied to the activation (output of the layer). • kernel_constraint: It is a constraint function applied to the kernel matrix. • bias_constraint: It can be defined as a constraint function applied to the bias vector. Input shape If the data_format is "channels_first", then the input shape of a 4D tensor is (batch, channels, rows, cols), else if data_format is "channels_last" the input shape of a 4D tensor is (batch, rows, cols, channels). Output shape If the data_format is "channels_first", the output shape of a 4D tensor will be (batch, filters, new_rows, new_cols), else if the data_format is "channels_last" the output will be (batch, new_rows, new_cols, filters). The values of rows and cols might variate due to the effect of padding. It is a Depthwise separable 1D convolution. Firstly, it accomplishes a depthwise spatial convolution on an single channel and then pointwise convolution to mix the resultant channels output. The argument depth_multiplier manages the generation of number of outputs channels per input channel in a depthwise manner. The Separable Convolutions can be easily understood by means of factorizing a convolution kernel into two smaller kernels. • filter: It is an integer that signifies the output space dimensionality or a total number of output filters present in a convolution. • kernel_size: It can either be an integer or tuple/list of single integer to represent the length of a 1D convolution window. • strides: It is either an integer or a tuple/list of a single integer that represents the convolution strides length. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • padding: One of "valid" or "same," where the same shows some inconsistency across the backend with strides !=1. • data_format: It is in either mode, i.e., 'channels_first' that corresponds to input shape: (batch, channels, steps) or 'channels_last' corresponds to (batch, steps, channels). • dilation_rate: It can be an integer or tuple/ list of a single integer that relates to the dilation rate to be used for dilated convolution. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • depth_multiplier: It represents the total number of depthwise convolution channels each of the respective input channels, which is equivalent to filters_in * depth_multiplier. • activation: It refers to an activation function to be used. When nothing is specified, then by defaults, it is a linear activation a(x) = x, or we can say no activation function is applied. • use_bias: It represents a Boolean that shows whether the layer utilizes a bias vector. • depthwise_initializer: It refers to an initializer for the depthwise kernel matrix. • pointwise_initializer: It refers to an initializer for the pointwise kernel matrix. • bias_initializer: It refers to an initializer for bias vector. • depthwise_regularizer: It refers to a regularizer function that is applied to the depthwise kernel matrix. • pointwise_regularizer: It refers to a regularizer function that is applied to the pointwise kernel matrix. • bias_regularizer: It can be defined as a regularizer function, which is applied to the bias vector. • activity_regularizer: It refers to a regularizer function that is applied to the activation (output of the layer). • depthwise_constraint: It can be defined as a constraint function applied to the depthwise kernel matrix. • pointwise_constraint: It can be defined as a constraint function applied to the pointwise kernel matrix. • bias_constraint: It can be defined as a constraint function applied to the bias vector. Input shape If the data_format is "channels_first", then the input shape of a 3D tensor is (batch, channels, steps), else if the data_format is "channels_last," the input shape of a 3D tensor is (batch, steps, Output shape If the data_format is "channels_first", then the output shape of a 3D tensor will be (batch, filters, new_steps), else if the data_format is "channels_last" the output shape of a 3D tensor will be (batch, new_steps, filters). The value of new_steps may vary due to the padding or strides. It is a depthwise separable 2D convolution. Firstly, it performs a depthwise spatial convolution on an individual channel and then pointwise convolution to mix the output of the resultant channel. The argument depth_multiplier manages the generation of the number of outputs channels per input channel in a depthwise manner. The Separable Convolutions can be easily understood by means of factorizing a convolution kernel into two smaller kernels or as an extension of an Inception block. • filter: It is an integer that signifies the output space dimensionality or the total number of output filters present in a convolution. • kernel_size: It can either be an integer or tuple/list of 2 integers to represent the height and width of a 2D convolution window. It can also exist as a single integer that signifies the same value for rest all of the spatial domain. • strides: It is either an integer or a tuple/list of 2 integers that represents the convolution strides along with the height and width. It can also exist as a single integer that signifies the same value for rest all of the spatial domain. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • padding: One of "valid" or "same," where the same shows some inconsistency across the backend with strides !=1. • data_format: It is in either mode, i.e., 'channels_first' that corresponds to input shape: (batch, channels, height, width) or 'channels_last' corresponds to (batch, height, width, channels). It defaults to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder, then it is residing at "channels_last". • dilation_rate: It can be an integer or tuple/ list of 2 integers that relates to the dilation rate to be used for dilated convolution. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • depth_multiplier: It represents the total number of depthwise convolution channels for each of the respective input channels, which is equivalent to filters_in * depth_multiplier. • activation: It refers to an activation function to be used. When nothing is specified, then by defaults, it is a linear activation a(x) = x, or we can say no activation function is applied. • use_bias: It represents a Boolean that shows whether the layer utilizes a bias vector. • depthwise_initializer: It refers to an initializer for the depthwise kernel matrix. • pointwise_initializer: It refers to an initializer for the pointwise kernel matrix. • bias_initializer: It refers to an initializer for bias vector. • depthwise_regularizer: It refers to a regularizer function that is applied to the depthwise kernel matrix. • pointwise_regularizer: It refers to a regularizer function that is applied to the pointwise kernel matrix. • bias_regularizer: It can be defined as a regularizer function, which is applied to the bias vector. • activity_regularizer: It refers to a regularizer function that is applied to the activation (output of the layer). • depthwise_constraint: It can be defined as a constraint function applied to the depthwise kernel matrix. • pointwise_constraint: It can be defined as a constraint function applied to the pointwise kernel matrix. • bias_constraint: It can be defined as a constraint function applied to the bias vector. Input shape If the data_format is "channels_first", then the input shape of a 4D tensor is (batch, channels, rows, cols), else if the data_format is "channels_last" the input shape of a 4D tensor is (batch, rows, cols, channels). Output shape If the data_format is "channels_first", then the output shape of a 4D tensor will be (batch, filters, new_rows, new_cols), else if the data_format is "channels_last" the output shape of a 4D tensor will be (batch, new_rows, new_cols, filters). The value of rows and cols may vary due to the padding or strides. It is a depthwise 2D convolution layer that firstly performs a similar action as that of the depthwise spatial convolution in which it separately performs on each input channel. The argument depth_multiplier manages the generation of the number of outputs channels per input channel in a depthwise manner. • kernel_size: It can either be an integer or tuple/list of 2 integers to represent the height and width of a 2D convolution window. It can also exist as a single integer that signifies the same value for all of the spatial domain. • strides: It is either an integer or a tuple/list of 2 integers that represents the convolution strides along with the height and width. It can exist as a single integer that signifies the same value for rest all of the spatial domain. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • padding: One of "valid" or "same," where the same shows some inconsistency across the backend with strides !=1. • data_format: It is in either mode, i.e., 'channels_first' that corresponds to input shape: (batch, channels, height, width) or 'channels_last' corresponds to (batch, height, width, channels). It defaults to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder, then it is residing at "channels_last". • dilation_rate: It can be an integer or tuple/ list of 2 integers that relates to the dilation rate to be used for dilated convolution. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • depth_multiplier: It represents the total number of depthwise convolution channels for each of the respective input channels, which is equivalent to filters_in * depth_multiplier. • activation: It refers to an activation function to be used. When nothing is specified, then by defaults, it is a linear activation a(x) = x, or we can say no activation function is applied. • use_bias: It represents a Boolean that shows whether the layer utilizes a bias vector. • depthwise_initializer: It refers to an initializer for the depthwise kernel matrix. • bias_initializer: It refers to an initializer for bias vector. • depthwise_regularizer: It refers to a regularizer function that is applied to the depthwise kernel matrix. • bias_regularizer: It can be defined as a regularizer function, which is applied to the bias vector. • activity_regularizer: It refers to a regularizer function that is applied to the activation (output of the layer). • depthwise_constraint: It can be defined as a constraint function applied to the depthwise kernel matrix. • bias_constraint: It can be defined as a constraint function applied to the bias vector. Input shape If the data_format is "channels_first", then the input shape of a 4D tensor is (batch, channels, rows, cols), else if the data_format is "channels_last" the input shape of a 4D tensor is (batch, rows, cols, channels). Output shape If the data_format is "channels_first", then the output shape of a 4D tensor will be (batch, channels * depth_multiplier, new_rows, new_cols), else if the data_format is "channels_last" the output shape of a 4D tensor will be (batch, new_rows, new_cols, channels * depth_multiplier). The value of rows and cols may vary due to the padding or strides. It is a Transpose convolution layer, which is sometimes incorrectly known as Deconvolution. But in reality, it does not perform Deconvolution. The Conv2DTranspose layer is mainly required when the transformation moves in the opposite direction to that of a normal convolution, or simply we can say when the transformation goes from something that has an output shape of some convolution to the one that has input shape of convolution. The layer can be used as an initial layer by using an argument input_shape, which is nothing but a tuple of integers and does not encompass the batch axis. • filter: It is an integer that signifies the output space dimensionality or a total number of output filters present in a convolution. • kernel_size: It can either be an integer or tuple/list of 2 integers to represent the height and width of a 2D convolution window. It can also exist as a single integer that signifies the same value for all of the spatial domain. • strides: It is either an integer or a tuple/list of 2 integers that represents the convolution strides along with the height and width. It can exist as a single integer that signifies the same value for rest all of the spatial domain. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • padding: One of "valid" or "same," where the same shows some inconsistency across the backend with strides !=1. • output_padding: It can either be an integer or tuple/list of 2 integers to represent the height and width of a 2D convolution window. It can also exist as a single integer that signifies the same value for all of the spatial domain. The amount of output data padding along any specified dimension should be given less than the stride along the same dimension. By default, it is set to None, which states that the output shape is inferred. • data_format: It is in either mode, i.e., 'channels_first' that corresponds to input shape: (batch, channels, height, width) or 'channels_last' corresponds to (batch, height, width, channels). It defaults to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder, then it is residing at "channels_last". • dilation_rate: It can be an integer or tuple/ list of 2 integers that relates to the dilation rate to be used for dilated convolution. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • activation: It refers to an activation function to be used. When nothing is specified, then by defaults, it is a linear activation a(x) = x, or we can say no activation function is applied. • use_bias: It represents a Boolean that shows whether the layer utilizes a bias vector. • kernel_initializer: It can be defined as an initializer for the kernel weights matrix. • bias_initializer: It refers to an initializer for bias vector. • kernel_regularizer: It refers to a regularizer function, which is applied to the kernel weights matrix. • bias_regularizer: It can be defined as a regularizer function, which is applied to the bias vector. • activity_regularizer: It refers to a regularizer function that is applied to the activation (output of the layer). • kernel_constraint: It is a constraint function applied to the kernel matrix. • bias_constraint: It can be defined as a constraint function applied to the bias vector. Input shape If the data_format is "channels_first" then input shape of a 4D tensor is (batch, channels, rows, cols), else if the data_format is "channels_last" the input shape of a 4D tensor is (batch, rows, cols, channels). Output shape If the data_format is "channels_first" then the output shape of a 4D tensor will be (batch, filters, new_rows, new_cols), else if the data_format is "channels_last" the output shape of a 4D tensor will be (batch, new_rows, new_cols, filters). The value of rows and cols may vary due to the padding. If output_padding is specified: It is a 3D convolution layer; for example, spatial convolution over volumes helps in the creation of the convolution kernel, which is convolved with the input layer in order to generate outputs of a tensor. It creates a bias vector if the use_bias is set to True, and then the bias vector is added to the output. It is applied outputs only if the activation is set to None. The layer can be used as the first layer in the model by using the input_shape keyword argument, which is nothing but a tuple of integers and does not embrace the batch axis. • filter: It is an integer that signifies the output space dimensionality or a total number of output filters present in a convolution. • kernel_size: It can either be an integer or tuple/list of 3 integers to represent the depth, height, and width of a 3D convolution window. It can also exist as a single integer that signifies the same value for all of the spatial domain. • strides: It is either an integer or a tuple/list of 3 integers that represents the convolution strides along with the depth, height, and width. It can exist as a single integer that signifies the same value for rest all of the spatial domain. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • padding: One of "valid" or "same," where the same shows some inconsistency across the backend with strides !=1. • data_format: It is in either mode, i.e. 'channels_first' that corresponds to input shape: (batch, channels, spatial_dim1, spatial_dim2, spatial_dim3) or 'channels_last' corresponds to (batch, spatial_dim1, spatial_dim2, spatial_dim3, channels). It defaults to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder, then it is residing at "channels_last". • dilation_rate: It can be an integer or tuple/ list of 3 integers that relates to the dilation rate to be used for dilated convolution. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • activation: It refers to an activation function to be used. When nothing is specified, then by defaults, it is a linear activation a(x) = x, or we can say no activation function is applied. • use_bias: It represents a Boolean that shows whether the layer utilizes a bias vector. • kernel_initializer: It can be defined as an initializer for the kernel weights matrix. • bias_initializer: It refers to an initializer for bias vector. • kernel_regularizer: It refers to a regularizer function, which is applied to the kernel weights matrix. • bias_regularizer: It can be defined as a regularizer function, which is applied to the bias vector. • activity_regularizer: It refers to a regularizer function that is applied to the activation (output of the layer). • kernel_constraint: It is a constraint function applied to the kernel matrix. • bias_constraint: It can be defined as a constraint function applied to the bias vector. Input shape If the data_format is "channels_first" then input shape of a 5D tensor is (batch, channels, conv_dim1, conv_dim2, conv_dim3), else if the data_format is "channels_last" the input shape of a 5D tensor is (batch, conv_dim1, conv_dim2, conv_dim3, channels). Output shape If the data_format is "channels_first" then the output shape of a 5D tensor will be (batch, filters, new_conv_dim1, new_conv_dim2, new_conv_dim3), else if the data_format is "channels_last" the output shape of a 5D tensor will be (batch, new_conv_dim1, new_conv_dim2, new_conv_dim3, filters). The value of new_conv_dim1, new_conv_dim2 and new_conv_dim3 may vary due to the padding. Conv3D Transpose It is a transposed convolution layer, which is sometimes also called as Deconvolution. This layer is mainly required when the transformation moves in the opposite direction to that of a normal convolution, or simply we can say when the transformation goes from something that has an output shape of some convolution to the one that has input shape of convolution. The layer can be used as an initial layer by using an argument input_shape, which is nothing but a tuple of integers and does not encompass the batch axis. • filter: It is an integer that signifies the output space dimensionality or a total number of output filters present in a convolution. • kernel_size: It can either be an integer or tuple/list of 3 integers to represent the depth, height, and width of a 3D convolution window. It can also exist as a single integer that signifies the same value for all of the spatial domain. • strides: It is either an integer or a tuple/list of 3 integers that represents the convolution strides along with the depth, height, and width. It can exist as a single integer that signifies the same value for rest all of the spatial domain. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • padding: One of "valid" or "same," where the same shows some inconsistency across the backend with strides !=1. • output_padding: It can either be an integer or tuple/list of 3 integers to represent the depth, height, and width of a 3D convolution window. It can also exist as a single integer that signifies the same value for all of the spatial domain. The amount of output data padding along any specified dimension should be given less than the stride along the same dimension. By default, it is set to None, which states that the output shape is inferred. • data_format: It is in either mode, i.e., 'channels_first' that corresponds to input shape: (batch, channels, depth, height, width) or 'channels_last' corresponds to (batch, depth, height, width, channels). It defaults to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder, then it is residing at "channels_last". • dilation_rate: It can be an integer or tuple/ list of 3 integers that relates to the dilation rate to be used for dilated convolution. If we specify any stride value!=1, it relates to its incompatibility with specifying the dilation_rate value!=1. • activation: It refers to an activation function to be used. When nothing is specified, then by defaults, it is a linear activation a(x) = x, or we can say no activation function is applied. • use_bias: It represents a Boolean that shows whether the layer utilizes a bias vector. • kernel_initializer: It can be defined as an initializer for the kernel weights matrix. • bias_initializer: It refers to an initializer for bias vector. • kernel_regularizer: It refers to a regularizer function, which is applied to the kernel weights matrix. • bias_regularizer: It can be defined as a regularizer function, which is applied to the bias vector. • activity_regularizer: It refers to a regularizer function that is applied to the activation (output of the layer). • kernel_constraint: It is a constraint function applied to the kernel matrix. • bias_constraint: It can be defined as a constraint function applied to the bias vector. Input shape If the data_format is "channels_first" then input shape of a 5D tensor is (batch, channels, depth, rows, cols), else if the data_format is "channels_last" the input shape of a 5D tensor is (batch, depth, rows, cols, channels). Output shape If the data_format is "channels_first" then the output shape of a 5D tensor will be (batch, filters, new_depth, new_rows, new_cols), else if the data_format is "channels_last" the output shape of a 5D tensor will be (batch, new_depth, new_rows, new_cols, filters). The value of depth, rows and cols may vary due to the padding. If output_padding is specified:: It is a cropping layer for 1Dimension input, for example, a temporal sequence that crops alongside axis 1, i.e., time dimension. • cropping: It is a tuple, which is of int length 2 ensures a total number of units to be trimmed at the beginning and end of axis 1(cropping dimension). In case if you provide a single int, then the same value will be utilized at the beginning and end. Input shape It is a 3D tensor of shape (batch, axis_to_crop, features). Output shape It is a 3D tensor of shape (batch, cropped_axis, features). It is a 2Dimension cropping layer for example picture that yields along the spatial dimensions such as height and width. • cropping: It is a int, or tuple of 2 ints, or a tuple of 2 tuples of 2 int, such that if int, which is the same cropping symmetric is applied to height and width and if tuple of 2 int is interpreted as two different symmetric cropping value for height and width: (symmetric_height_crop, symmetric_width_crop) • data_format: It is in either mode, i.e. 'channels_first' that corresponds to input shape: (batch, channels, height, width) or 'channels_last' corresponding to (batch, height, width, channels). It is default to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder then it is residing at "channels_last". Input shape If the data_format is "channels_first" then input shape of a 4D tensor is (batch, channels, rows, cols), else if the data_format is "channels_last" the input shape of a 4D tensor is (batch, rows, cols, channels). Output shape If the data_format is "channels_first" then the output shape of a 4D tensor will be (batch, channels, cropped_rows, cropped_cols), else if the data_format is "channels_last" the output shape of a 4D tensor will be (batch, cropped_rows, cropped_cols, channels). It is a 3D cropping layer just like spatio-temporal or spatial. • cropping: It is an int, or a tuple of 3 ints, or a tuple of 3 tuples of 2ints, such that; If int is the same symmetric cropping that is applied to depth, height and width, If tuple of 3 ints is interpreted as three distinct values of symmetric cropping for depth, height and width: (symmetric_dim1_crop, symmetric_dim2_crop, symmetric_dim3_crop) and If tuple of 3 tuples of 2 ints is interpreted as ((left_dim1_crop, right_dim1_crop), (left_dim2_crop, right_dim2_crop), (left_dim3_crop, right_dim3_crop)). • data_format: It is a string of either mode, i.e. 'channels_first' that corresponds to input shape: (batch, channels, spatial_dim1, spatial_dim2, spatial_dim3) or 'channels_last' corresponding to (batch, spatial_dim1, spatial_dim2, spatial_dim3, channels). It is default to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder then it is residing at "channels_last". Input shape If the data_format is "channels_first" then input shape of a 5D tensor is (batch, depth, first_axis_to_crop, second_axis_to_crop, third_axis_to_crop), else if the data_format is "channels_last" the input shape of a 5D tensor is (batch, first_axis_to_crop, second_axis_to_crop, third_axis_to_crop, depth). Output shape If the data_format is "channels_first" then the output shape of a 5D tensor will be (batch, depth, first_cropped_axis, second_cropped_axis, third_cropped_axis), else if the data_format is "channels_last" the output shape of a 5D tensor will be (batch, first_cropped_axis, second_cropped_axis, third_cropped_axis, depth). It is an Upsampling layer for 1 Dimensional inputs that repeat each individual temporal steps in terms of size times alongside time axis. • size: It is an integer, which is an Upsampling factor. Input shape It is a 3D tensor of shape: (batch, steps, features). Output shape It is a 3D tensor with shape: (batch, upsampled_steps, features). It is an Upsampling layer for 2D input that repeats the rows of the data by size [0] and columns of the data by size [1]. • size: It is an int or tuple of 2 integers, which is an upsampling factor for rows and columns. • data_format: It is a string of either mode, i.e., 'channels_first' that corresponds to input shape: (batch, channels, height, width) or 'channels_last' corresponding to (batch, height, width, channels). It defaults to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder, then it is residing at "channels_last". • interpolation: It is a string one of nearest or bilinear. It should be illustrated that CNTK does not support yet the bilinear upscaling and that with Theano, only size=(2, 2) is possible. Input shape If data_format is "channels_last", then the input shape of a 4D tensor is (batch, rows, cols, channels), else if data_format is "channels_first", then the input shape of a 4D tensor is (batch, channels, rows, cols). Output shape If data_format is "channels_last", then the output shape of a 4D tensor is (batch, upsampled_rows, upsampled_cols, channels), else if the data_format is "channels_first", then the output shape of a 4D tensor is (batch, channels, upsampled_rows, upsampled_cols). It refers to an Upsampling layer for 3 dimensional input that repeats 1^st dimension of the data by size [0], 2^nd dimension of the data by size [1], and 3^rd dimension of the data by size [2]. • size: It is an int or tuple of 3 integers, which is an upsampling factor for dim1, dim2, and dim3. • data_format: It is a string of either mode, i.e. 'channels_first' that corresponds to input shape: (batch, channels, spatial_dim1, spatial_dim2, spatial_dim3) or 'channels_last' corresponding to (batch, spatial_dim1, spatial_dim2, spatial_dim3, channels). It defaults to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder, then it is residing at "channels_last". Input shape If data_format is "channels_last", then the input shape of a 5D tensor is (batch, dim1, dim2, dim3, channels), else if data_format is "channels_first", then the input shape of a 5D tensor is (batch, channels, dim1, dim2, dim3). Output shape If data_format is "channels_last", then the output shape of a 4D tensor is (batch, upsampled_dim1, upsampled_dim2, upsampled_dim3, channels), else if the data_format is "channels_first", then the output shape of a 4D tensor is (batch, channels, upsampled_dim1, upsampled_dim2, upsampled_dim3). It refers to a zero-padding layer for one-dimensional input. For example, a temporal sequence. • padding: It is an int, or tuple of int (length 2) or dictionary, such that If int demonstrates a total number of zeros to be added at the beginning as well as at the end of the padding dimension (axis 1), whereas in case of If a tuple of int (length 2) the zeros are added at the beginning and end of the padding dimension ((left_pad, right_pad)). Input shape It is a 3D tensor of shape (batch, axis_to_pad, features). Output shape It refers to a 3 dimensional tensor of shape (batch, padded_axis, features). It refers to a two-dimensional input zero-padding layer (for example, picture) that supports the addition of zero rows and columns containing zeros at the top, bottom, left, and right of a tensor • padding: It is an int, or tuple of 2 ints, or tuple of 2 tuples of 2 ints; where If int is the similar symmetric padding being applied to height and width, If a tuple of 2 ints is taken as two distinct values symmetric padding values for height and width of: (symmetric_height_pad, symmetric_width_pad), whereas If a tuple of 2 tuples of 2 ints is understood as ((top_pad, bottom_pad), (left_pad, right_pad)). • data_format: It is a string of either mode, i.e., 'channels_first' that corresponds to input shape: (batch, channels, height, width) or 'channels_last' corresponding to (batch, height, width, channels). It defaults to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder, then it is residing at "channels_last". Input shape If data_format is "channels_last", then the input shape of a 4D tensor is (batch, rows, cols, channels), else if data_format is "channels_first", then the input shape of a 4D tensor is (batch, channels, rows, cols). Output shape If data_format is "channels_last", then the output shape of a 4D tensor is (batch, padded_rows, padded_cols, channels), else if the data_format is "channels_first", then the output shape of a 4D tensor is (batch, channels, padded_rows, padded_cols). It is a three-dimensional zero-padding layer. For example, spatial or Spatio-temporal. • padding: It is an int, or a tuple of 3 ints, or a tuple of 3 tuples of 2ints, such that; If int is the same symmetric padding that is applied to depth, height and width, If tuple of 3 ints is interpreted as three distinct values of symmetric padding values for depth, height and width: (symmetric_dim1_pad, symmetric_dim2_pad, symmetric_dim3_pad) and If tuple of 3 tuples of 2 ints is interpreted as ((left_dim1_pad, right_dim1_pad), (left_dim2_pad, right_dim2_pad), (left_dim3_pad, right_dim3_pad)). • data_format: It is a string of either mode, i.e. 'channels_first' that corresponds to input shape: (batch, channels, spatial_dim1, spatial_dim2, spatial_dim3) or 'channels_last' corresponding to (batch, spatial_dim1, spatial_dim2, spatial_dim3, channels). It is default to the image_data_format value that is found in Keras config at ~/.keras/keras.json. If you cannot find it in that folder then it is residing at "channels_last". Input shape If the data_format is "channels_first" then input shape of a 5D tensor is (batch, depth, first_axis_to_pad, second_axis_to_pad, third_axis_to_pad), else if the data_format is "channels_last" the input shape of a 5D tensor is (batch, first_axis_to_pad, second_axis_to_pad, third_axis_to_pad, depth). Output shape If the data_format is "channels_first" then the output shape of a 5D tensor will be (batch, depth, first_padded_axis, second_padded_axis, third_axis_to_pad), else if the data_format is "channels_last" the output shape of a 5D tensor will be (batch, first_padded_axis, second_padded_axis, third_axis_to_pad, depth). 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igraph Reference Manual The following short example program demonstrates the basic usage of the igraph library. #include <igraph.h> int main() { igraph_integer_t num_vertices = 1000; igraph_integer_t num_edges = 1000; igraph_real_t diameter; igraph_t graph; igraph_rng_seed(igraph_rng_default(), 42); igraph_erdos_renyi_game(&graph, IGRAPH_ERDOS_RENYI_GNM, num_vertices, num_edges, IGRAPH_UNDIRECTED, IGRAPH_NO_LOOPS); igraph_diameter(&graph, &diameter, NULL, NULL, NULL, NULL, IGRAPH_UNDIRECTED, /* unconn= / true); printf("Diameter of a random graph with average degree %g: %g\n", 2.0 igraph_ecount(&graph) / igraph_vcount(&graph), (double) diameter); return 0; This example illustrates a couple of points: • First, programs using the igraph library should include the igraph.h header file. • Second, igraph uses the igraph_integer_t type for integers instead of int or long int, and it also uses the igraph_real_t type for real numbers instead of double. Depending on how igraph was compiled, and whether you are using a 32-bit or 64-bit system, igraph_integer_t may be a 32-bit or 64-bit integer. • Third, igraph graph objects are represented by the igraph_t data type. • Fourth, the igraph_erdos_renyi_game() creates a graph and igraph_destroy() destroys it, i.e. deallocates the memory associated to it. For compiling this program you need a C compiler. Optionally, CMake can be used to automate the compilation. It is convenient to use CMake because it can automatically discover the necessary compilation flags on all operating systems. Many IDEs support CMake, and can work with CMake projects directly. To create a CMake project for this example program, create a file name CMakeLists.txt with the following contents: cmake_minimum_required(VERSION 3.18) find_package(igraph REQUIRED) add_executable(igraph_test igraph_test.c) target_link_libraries(igraph_test PUBLIC igraph::igraph) To compile the project, create a new directory called build in the root of the igraph source tree, and switch to it: mkdir build cd build Run CMake to configure the project: cmake .. If igraph was installed at a non-standard location, specify its prefix using the -DCMAKE_PREFIX_PATH=... option. The prefix must be the same directory that was specified as the CMAKE_INSTALL_PREFIX when compiling igraph. If configuration has succeeded, build the program using cmake --build . C++ must be enabled in igraph projects Parts of igraph are implemented in C++; therefore, any CMake target that depends on igraph should use the C++ linker. Furthermore, OpenMP support in igraph works correctly only if C++ is enabled in the CMake project. The script that finds igraph on the host machine will throw an error if C++ support is not enabled in the CMake project. C++ support is enabled by default when no languages are explicitly specified in CMake's project command. If you do specify some languages explicitly, make sure to also include CXX. On most Unix-like systems, the default C compiler is called cc. To compile the test program, you will need a command similar to the following: cc igraph_test.c -I/usr/local/include/igraph -L/usr/local/lib -ligraph -o igraph_test The exact form depends on where igraph was installed on your system, whether it was compiled as a shared or static library, and the external libraries it was linked to. The directory after the -I switch is the one containing the igraph.h file, while the one following -L should contain the library file itself, usually a file called libigraph.a (static library on macOS and Linux), libigraph.so (shared library on Linux), libigraph.dylib (shared library on macOS), igraph.lib (static library on Windows) or igraph.dll (shared library on Windows). If igraph was compiled as a static library, it is also necessary to manually link to all of its dependencies. If your system has the pkg-config utility you are likely to get the necessary compile options by issuing the command pkg-config --libs --cflags igraph (if igraph was built as a shared library) or pkg-config --static --libs --cflags igraph (if igraph was built as a static library). On most systems, the executable can be run by simply typing its name like this: If you use dynamic linking and the igraph library is not in a standard place, you may need to add its location to the LD_LIBRARY_PATH (Linux), DYLD_LIBRARY_PATH (macOS) or PATH (Windows) environment The functions generating graph objects are called graph generators. Stochastic (i.e. randomized) graph generators are called “games”. igraph can handle directed and undirected graphs. Most graph generators are able to create both types of graphs and most other functions are usually also capable of handling both. E.g., igraph_get_shortest_paths(), which calculates shortest paths from a vertex to other vertices, can calculate directed or undirected paths. igraph has sophisticated ways for creating graphs. The simplest graphs are deterministic regular structures like star graphs (igraph_star()), ring graphs (igraph_ring()), lattices ( igraph_square_lattice()) or trees (igraph_kary_tree()). The following example creates an undirected regular circular lattice, adds some random edges to it and calculates the average length of shortest paths between all pairs of vertices in the graph before and after adding the random edges. (The message is that some random edges can reduce path lengths a lot.) #include <igraph.h> int main() { igraph_t graph; igraph_vector_int_t dimvector; igraph_vector_int_t edges; igraph_vector_bool_t periodic; igraph_real_t avg_path_len; igraph_vector_int_init(&dimvector, 2); igraph_vector_bool_init(&periodic, 2); igraph_vector_bool_fill(&periodic, true); igraph_square_lattice(&graph, &dimvector, 0, IGRAPH_UNDIRECTED, /* mutual= / false, &periodic); igraph_average_path_length(&graph, &avg_path_len, NULL, IGRAPH_UNDIRECTED, / unconn= / true); printf("Average path length (lattice): %g\n", (double) avg_path_len); igraph_rng_seed(igraph_rng_default(), 42); / seed RNG before first use / igraph_vector_int_init(&edges, 20); for (igraph_integer_t i=0; i < igraph_vector_int_size(&edges); i++) { VECTOR(edges)[i] = RNG_INTEGER(0, igraph_vcount(&graph) - 1); igraph_add_edges(&graph, &edges, NULL); igraph_average_path_length(&graph, &avg_path_len, NULL, IGRAPH_UNDIRECTED, / unconn= / true); printf("Average path length (randomized lattice): %g\n", (double) avg_path_len); return 0; This example illustrates some new points. igraph uses igraph_vector_t and its related types (igraph_vector_int_t, igraph_vector_bool_t and so on) instead of plain C arrays. igraph_vector_t is superior to regular arrays in almost every sense. Vectors are created by the igraph_vector_init() function and, like graphs, they should be destroyed if not needed any more by calling igraph_vector_destroy() on them. A vector can be indexed by the VECTOR() function (right now it is a macro). The elements of a vector are of type igraph_real_t for igraph_vector_t, and of type igraph_integer_t for igraph_vector_int_t. As you might expect, igraph_vector_bool_t holds igraph_bool_t values. Vectors can be resized and most igraph functions returning the result in a vector automatically resize it to the size they need. igraph_square_lattice() takes an integer vector argument specifying the dimensions of the lattice. In this example we generate a 30x30 two dimensional periodic lattice. See the documentation of igraph_square_lattice() in the reference manual for the other arguments. The vertices in a graph are identified by a vertex ID, an integer between 0 and N-1, where N is the number of vertices in the graph. The vertex count can be retrieved using igraph_vcount(), as in the The igraph_add_edges() function simply takes a graph and a vector of vertex IDs defining the new edges. The first edge is between the first two vertex IDs in the vector, the second edge is between the second two, etc. This way we add ten random edges to the lattice. Note that this example program may add loop edges, edges pointing a vertex to itself, or multiple edges, more than one edge between the same pair of vertices. igraph_t can of course represent loops and multiple edges, although some routines expect simple graphs, i.e. graphs which contain neither of these. This is because some structural properties are ill-defined for non-simple graphs. Loop and multi-edges can be removed by calling igraph_simplify(). In our next example we will calculate various centrality measures in a friendship graph. The friendship graph is from the famous Zachary karate club study. (Do a web search on "Zachary karate" if you want to know more about this.) Centrality measures quantify how central is the position of individual vertices in the graph. #include <igraph.h> int main() { igraph_t graph; igraph_vector_int_t v; igraph_vector_int_t result; igraph_vector_t result_real; igraph_integer_t edges[] = { 0,1, 0,2, 0,3, 0,4, 0,5, 0,6, 0,7, 0,8, 0,10, 0,11, 0,12, 0,13, 0,17, 0,19, 0,21, 0,31, 1, 2, 1, 3, 1, 7, 1,13, 1,17, 1,19, 1,21, 1,30, 2, 3, 2, 7, 2,27, 2,28, 2,32, 2, 9, 2, 8, 2,13, 3, 7, 3,12, 3,13, 4, 6, 4,10, 5, 6, 5,10, 5,16, 6,16, 8,30, 8,32, 8,33, 9,33, 13,33, 14,32, 14,33, 15,32, 15,33, 18,32, 18,33, 19,33, 20,32, 20,33, 22,32, 22,33, 23,25, 23,27, 23,32, 23,33, 23,29, 24,25, 24,27, 24,31, 25,31, 26,29, 26,33, 27,33, 28,31, 28,33, 29,32, 29,33, 30,32, 30,33, 31,32, 31,33, 32,33 }; igraph_vector_int_view(&v, edges, sizeof(edges) / sizeof(edges[0])); igraph_create(&graph, &v, 0, IGRAPH_UNDIRECTED); igraph_vector_int_init(&result, 0); igraph_vector_init(&result_real, 0); igraph_degree(&graph, &result, igraph_vss_all(), IGRAPH_ALL, IGRAPH_LOOPS); printf("Maximum degree is %10" IGRAPH_PRId ", vertex %2" IGRAPH_PRId ".\n", igraph_closeness(&graph, &result_real, NULL, NULL, igraph_vss_all(), IGRAPH_ALL, / weights= / NULL, / normalized= / false); printf("Maximum closeness is %10g, vertex %2" IGRAPH_PRId ".\n", (double) igraph_vector_max(&result_real), igraph_betweenness(&graph, &result_real, igraph_vss_all(), IGRAPH_UNDIRECTED, / weights= */ NULL); printf("Maximum betweenness is %10g, vertex %2" IGRAPH_PRId ".\n", (double) igraph_vector_max(&result_real), return 0; This example demonstrates some new operations. First of all, it shows a way to create a graph a list of edges stored in a plain C array. Function igraph_vector_view() creates a view of a C array. It does not copy any data, which means that you must not call igraph_vector_destroy() on a vector created this way. This vector is then used to create the undirected graph. Then the degree, closeness and betweenness centrality of the vertices is calculated and the highest values are printed. Note that the vector result, into which these functions will write their result, must be initialized first, and also that the functions resize it to be able to hold the result. Notice that in order to print values of type igraph_integer_t, we used the IGRAPH_PRId format macro constant. This macro is similar to the standard PRI constants defined in stdint.h, and expands to the correct printf format specifier on each platform that igraph supports. The igraph_vss_all() argument tells the functions to calculate the property for every vertex in the graph. It is shorthand for a vertex selector, represented by type igraph_vs_t. Vertex selectors help perform operations on a subset of vertices. You can read more about them in one of the following chapters.	{"url":"https://igraph.org/c/html/0.10.0/igraph-Tutorial.html","timestamp":"2024-11-14T15:14:19Z","content_type":"text/html","content_length":"38849","record_id":"<urn:uuid:b3f4ba83-8c10-45ef-abf3-a45274b9f275>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00403.warc.gz"}