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Items where Subject is "Q Science > QA Mathematics" Number of items at this level: 155. 1Alabi, Nurudeen Olawale and 2Are, Stephen Olusegun (2017) Smoothing non-stationary noise of the Nigerian Stock Exchange All-Share Index data using variable coefficient functions. Mathematical Theory and Modeling, 7 (7). pp. 34-45. ISSN 2224-5804 ADEBOYE, N.O and AJIBODE, I. A (2016) CLASSIFICATION OF NIGERIA BANKS BASED ON FINANCIAL STABILITY USING LINEAR DISCRIMINANT ANALYSIS. Ilaro Journal of Science and Technology (ILJST)., 2 (1). pp. AJIBODE, I. A and Adeboye, Nurain Olawale ARIMA AND ARIMAX IN THE MODELLING OF COVID-19 MORTALITY IN NIGERIA. Journal of Business and Educational Policies. pp. 1-25. ISSN 0794-3210 12 AJIBODE, I. A and BUSARI, I. A (2020) Volatility and Nigeria Stock Market Performance: Evidence from Naira-Dollar Exchange Rate and Market Capitalization. Journal of Mathematics and Statistics Studies (JMSS), 1 (2). pp. 38-45. ISSN 2709-4200 AJIBODE, I. A and Sikiru, O. A. (2021) MODELING AND FORECASTING ANTE-NATAL CARE ATTENDANCE USING BOX AND JENKINS METHOD. In: Presented at the 5th National Conference of the School of Pure & Applied Sciences Federal Polytechnic Ilaro held between 29 and 30th September, 2021. Theme: Food Security and Safety: A Foothold for Development of Sustainable Economy in Nigeria, 29th – 30th September, 2021, The Federal Polytechnic, Ilaro. ALAWODE, A.J and Adegboye, A. J. (2019) AUTOMATIC SEATING ARRANGEMENT SYSTEM USING TABU SEARCH ALGORITHM. FEPI-JOPAS, 1 (1). pp. 127-133. ISSN 2714-2531 Aako, O. L. and AJIBODE, I. A (2020) Regression Control Chart for Monitoring Road Accident Fatalities. In: 1ST International Conference, Federal Polytechnic Ekowe, Bayelsa State, 9th – 10th Dec., 2020., 9th – 10th Dec., 2020., Federal Polytechnic Ekowe, Bayelsa State.. Aako, O. L. and AJIBODE, I. A (2019) Shewhart Individual Control Charts for Monitoring Monthly Rainfall in South-West Nigeria. In: Proceedings of the 16th iSTEAMS Multidisciplinary Research Nexus Conference, 9th – 11th June, 2019., The Federal Polytechnic, Ilaro. Aako, O. L. and Adewara, J. A. and Adekeye, K. S. and Nkemnole, E. B. (2020) Robust Scale Estimator-Based Control Charts for Marshall-Olkin Inverse Log-Logistic Distribution. BENIN JOURNAL OF STATISTICS, 3. pp. 33-65. ISSN 2682-5767 Aako, Olubisi and Are, Stephen Olusegun Modeling Mode Of Childbirth Delivery Using Dummy Dependent Variable Models. Journal of Pure and Applied Sciences, 3 (1). pp. 32-37. ISSN 2714-2531 Aako, Olubisi Lawrence and Adewara, J. A. and Are, Stephen Olusegun (2022) RISK FACTOR ANALYSIS OF BREAST CANCER PATIENTS IN A NIGERIAN TERTIARY HOSPITAL. FUDMA Journal of Sciences, 6 (3). pp. 95-99. ISSN 2645 - 2944 Aako, Olubisi Lawrence and Adewara, J. A. and Nkemnole, E. B. (2022) Marshall-Olkin Generalized Inverse Log-Logistic Distribution: Its properties and applications. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 8 (2). pp. 79-93. ISSN 2150-6103 Aako, Olubisi Lawrence and Agbolade, O. A. (2022) Use of a generalized gamma additive model to determine the effect of monetary policy on the Nigerian stock market. International Journal of Mathematical Analysis and Modelling, 5 (4). pp. 50-57. ISSN 2682 - 5694 Adaramola Ojo Jayeola, Adaramola Ojo Jayeola and Olasina, J.R (2019) Design and Construction of Microcontroller Based, Solar- Powered Automated Energy for Office Use. In: 16th iSTEAMS Multidisciplinary Research Nexus Conference, 9th – 11th June, 2019., THE FEDERAL POLYTECHNIC, ILARO. Adebesin, A. A and Olaiya, O.O (2020) E- Learning and the Covid-19: Issues, Challenges and Observations. In: NSE Ilaro Branch, 1st National Conference, Ilaro, 2-3 November, 2020, 2-3 November, 2020, NSE Ilaro Branch, 1st National Conference, Ilaro. Adeboye, N. O and Ajibode, I. A (2019) SPATIAL AUTOCORRELATION ESTIMATION OF MALARIA MORBIDITY AMONG POLYTECHNIC STUDENTS IN NIGERIA. Journal of Pure and Applied Science, FPI, 1 (1). pp. 110-115. ISSN 2714-2531 Adeboye, N.O and Akinbo, R.Y and Ajibode, I.A (2013) Modeling and forecasting of rare events in Nigeria. Caribbean Journal of Science and Technology, 1. pp. 98-104. ISSN 07993757 Adeboye, Nurain Olawale (2018) IMPACT OF ICT IN ACHIEVING A COMPETENCY-BASED TVET EDUCATION AMONG POLYTECHNIC STUDENTS: A PRINCIPAL COMPONENT ANALYSIS APPROACH. In: 40th International Conference of all Polytechnics and Technical Universities in Africa, 2018, Nicon Luxury, Abuja Nigeria. Adeboye, Nurain Olawale (2017) MODELLING AND FORECASTING MORTALITY RATE DUE TO MALARIA INFECTION, USING AUTOREGRESSIVE MOVING AVERAGE (ARMA) MODELS A PANACEA TO NIGERIAN SOCIO-ECONOMIC CHALLENGES. In: National Conference on Science, Technology and Communication, 2017, Federal Polytechnic, Ilaro. Adeboye, Nurain Olawale and Agunbiade, Dawud Adebayo (2017) ESTIMATING THE HETEROGENEITY EFFECTS IN A PANEL DATA REGRESSION MODEL. Anale. Seria Informatică, 15 (1). pp. 149-158. Adeboye, Nurain Olawale and Agunbiade, Dawud Adebayo (2019) Monte Carlo Estimation of Heterogeneity Effects in a Panel Data Regression Model. International Journal of Mathematics and Computation, 30 (1). pp. 62-77. ISSN 0974-5718 Adeboye, Nurain Olawale and Agunbiade, Dawud Adebayo (2019) Panel Data Regression Modeling with Heteroscedasticity and Periodicity Effects. In: 3nd PSSN (formerly NSS) International Conference, 2019, Ahmadu Bello University of Zaria, Nigeria. Adeboye, Nurain Olawale and Agunbiade, Dawud Adebayo (2017) STATISTICAL EFFECTS OF SOCIAL MEDIA AND ICT ON THE ACADEMIC PERFORMANCE: AN APPLICATION OF PRINCIPAL COMPONENT ANALYSIS. In: 1st International Conference of Faculty of Science, 2017, Olabisi Onabanjo University Ago-Iwoye. Adeboye, Nurain Olawale and Agunbiade, Dawud Adebayo (2017) Testing for Periodicity Effects in a Panel Data Regression Model. Caribbean Journal of Science and Technology, 5. 040-050. ISSN 0799-3757 Adeboye, Nurain Olawale and Alabi, Nurudeen Olawale (2022) Deep-LearningModellingofDynamicPanelDataforAfricanEconomicGrowth. Journal of Econometrics and Statistics, 2 (1). pp. 47-60. ISSN 2583-0473 Adeboye, Nurain Olawale and Badru, J. O. (2021) OPTIMIZATION CONCEPT AS A PANACEA TO PROFITABLE FOOD SUFFICIENCY IN A DEVELOPING ECONOMY. In: 5th National Conference of the School of Pure & Applied Sciences, 29 and 30th September, 2021, The Federal Polytechnic Ilaro, Ogun State, Nigeria. Adeboye, Nurain Olawale and Dawud, A. Agunbiade (2019) Monte Carlo Estimation of Heteroscedasticity and Periodicity Effects in a Panel Data Regression Model. International Journal of Mathematical and Computational Sciences, 13 (6). pp. 157-166. Adeboye, Nurain Olawale and Fagoyinbo, Idowu (2017) Fitting of Seasonal Autoregressive Integrated Moving Average to the Nigerian Stock Exchange Trading Activities. In: 1st NSS International Conference, 2017, University of Ibadan, Nigeria. Adeboye, Nurain Olawale and Ilesanmi, A. Ajibode and Olubisi, L. Aako (2020) On the Survival Assessment of Asthmatic Patients Using Parametric and Semi-Parametric Survival Models. Occupational Diseases and Environmental Medicine, 8 (2). pp. 50-63. ISSN 2333-3561 Adeboye, Nurain Olawale and Ogunnusi, O.N (2019) On Fractional Time Domain Modelingof Nigerian Monthly Rainfall Statistics. In: 4th National Development Conference of The School of Pure and Applied Science, 2nd – 5th December, 2019, The Federal Polytechnic Ilaro, Ogun State. Adeboye, Nurain Olawale and Ogunnusi, O.N (2020) On the Predictive Ability of Time-Domain Modeling of Long Memory Data. In: 4th International Conference of Professional Statisticians Society of Nigeria, 2020, Nigeria. Adeboye, Nurain Olawale and Olayiwola, M. Oyedunsi (2020) Big Data Affluence in Statistics Application: A Comparison of Real Life and Simulated Open Data. In: The Roundtable conference of the International Association for Statistical Education (IASE),, July 2020, The Netherlands.. Adeniyan, A. and Fatunmbi, E. O. (2018) Effects of Viscous Dissipation and Micropolar Heat Conduction on MHD Thermally Radiating Flow Along A Vertical Plate. In: 37th Annual Conference Nigerian Mathematical Society, 8-11, May, 2018, Bayero University, Kano. Adewara, Johnson A. and Adekeye, Kayode S. and Aako, Olubisi L. (2020) On Performance of Two-Parameter Gompertz-Based X Control Charts. Hindawi Journal of Probability and Statistics, 2020. ISSN Adewole, Adekanmi and Babailo, Yetunde (2019) KNOWLEDGE AND TREATMENT SEEKING BEHAVIOUR OF PEOPLE IN NIGERIA-BENIN BOARDER COMMUNITY OF OJA- ODAN,YEWA NORTH LG, NIGERIA. In: 1st National Conference of WITED, Ilaro Chapter, August 13-16, 2019, The Federal Polytechnic, Ilaro. Afolabi, L.O. and Olaiya, O.O (2019) Effect of Meteorological Paramrters on Tropospheric Refractivity in Jos, North Central of Nigeria. International Journal of Latest Technology in Engineering, Management & Applied Science (IJLTEMAS), VIII (XI). pp. 77-82. ISSN 2278-2540 Agbaje, Wale Henry and Busari, Ganiyu Adeniran and Adeboye, Nureni Olawale (2014) Effects of Accounting Information Management on Profitability of Nigerian Banking Industry. International Journal of Humanities Social Sciences and Education (IJHSSE), 1 (9). pp. 100-104. ISSN 2349-0373 Agbolade, O. A. (2020) Solution to Nonlinear First and Second Order Differential Equations Using Mahgoub Transform Decomposition Method. FEPI-JOPAS, 2 (1). pp. 66-80. Agbolade, O. A. and Anake, T.A (2017) Solutions of First-Order Volterra Type Linear Integrodifferential Equations by Collocation Method. Journal of Applied Mathematics. pp. 1-6. Agbolade, O. A. and Fatumbi, E.O (2021) Stagnation-Point Flow of a Radiative Tangent Hyperbolic Nanofluid over a Nonlinear Surface with Variable Thermal Conductivity. International Journal of Latest Research in Engineering and Management" (IJLREM), 5 (3). pp. 19-28. ISSN 2456-0766 Agunbiade, D.A and Adeboye, N.O (2012) Estimation of Heteroscedasticity Effects in A Classical Linear Regression Model Of a Cross-Sectional Data. Journal of Progress in Applied Mathematics, CSCANADA, 4 (2). pp. 18-28. ISSN 1925-251X Agunbiade, Dawud Adebayo and Adeboye, Nurain Olawale (2012) Estimation under Heteroscedasticity: A Comparative Approach Using Cross-Sectional Data. Mathematical Theory and Modeling, 2 (11). pp. 1-8. ISSN 2224-5804 Aiyelabegan, Adijat Bukola and Adeoye, Akeem Olarenwaju and Suleiman, Sikiru (2020) Estimating the Effect of Money Supply on Economy Growth. Journal of Women in Technical Education and Employment, 1 (2). pp. 8-12. ISSN 2734-3227 Akanbi, O. O. and Edeki, S. O. and Agbolade, O. A. (2017) Continuous-time model and physical simulation of population dynamics of sickle cell anaemia. International Journal of Advanced and Applied Sciences, 4 (6). pp. 14-18. Akanbi, Olumuyiwa O. APPLICATION OF RENEWAL EQUATION AND MARKOV CHAIN TO THE INHERITANCE PATTERN OF SICKLE CELL ANAEMIA (SCA). In: The Federal Polytechnic, offa conference, The Federal Polytechnic, Akanbi, Olumuyiwa O. A COMPARISON OF TWO CLASSES OF CONSTRAINED OPTIMIZATION MODELS IN OPTIMIZING THE COST OF HUMAN CAPACITY BUILDING: A FOCUS ON TERTIARY INSTITUTION. In: Kano State polytechnic Conference, Kano State polytechnic, Kano.. Akanbi, Olumuyiwa O. (2018) PROFIT MAXIMIZATION USING LINEAR PROGRAMMING AND INTEGER LINEAR PROGRAMMING MODELS: A FOCUS ON NIGERIAN BOTTLING COMPANY OTA PLANT, OGUN STATE. In: ASUP National Conference, ADO-EWI, Monday, 27th – Thursday, 30th August, 2018, Federal Polytechnic, Ado-Ekiti, Ekiti State. Akanbi, Olumuyiwa O. and Agbolade, Olumuyiwa A and Shomoye, Idowu A. (2017) PHYSICAL AND MONTE CARLOS SIMULATION OF CONTINUOUS TIME MODEL OF SICKLE CELL ANAEMIA. In: SPASCIT, 4th – 7th December, 2017., Federal Polytechnic Ilaro, Ogun State. Akanbi, Olumuyiwa O. and Are, Stephen Olusegun (2019) A Mathematical Model for Optimal Allocation of Security Personnel on Campus Streets. THE INTERNATIONAL JOURNAL OF SCIENCE & TECHNOLEDGE, 7 (11). pp. 7-11. ISSN 2321 – 919X Akanbi, Olumuyiwa O. and Edeki, Sunday O. and Agbolade, Olumuyiwa A. (2017) Monte Carlos Simulation Approach to Population Dynamics of Sickle Cell Anaemia. American Journal of Applied Sciences, 14 (3). pp. 358-364. ISSN 358.364 Akanbi, Olumuyiwa O. and Fatunmbi, E. O. (2019) Hydromagnetic Micropolar Fluid Flow Over a Nonlinear Stretching Surface In uenced by Non-uniform Heat Source/Sink and Slip Effects. In: 30th annual colloquium and congress of the Association of Mathematical Physics, 25th-8th Nov., 2019, Igbinedion University Okada, Nigeria. Akanbi, Olumuyiwa O. and Shomoye, I.A (2019) OPTIMIZATION OF LAYERS MASH FROM SOME LOCALLY SELECTED POULTRY FEED INGREDIENTS. FEPI-JOPAS, 1 (1). pp. 109-218. ISSN 2714-2531 Akanbi, Olumuyiwa O. and Shomoye, Idowu. A (2018) APPLICATION OF LINEAR PROGRAMMING AS AN INNOVATION IN FEED FORMULATION FOR GLOBAL COMPETITIVENESS IN POULTRY FARMING. In: International Conference Federal Polytechnic Ilaro, Ogun State, 5th – 8th November, 2018, The Federal Polytechnic, Ilaro. Akanbi, Olumuyiwa O. and Shomoye, Idowu. A (2019) OPTIMIZATION OF FINISHED FEED MIX FOR COMMERCIAL LAYERS AS A PANACEA FOR SCIENTIFIC ADVANCEMENT TOWARDS NATIONAL DEVELOPMENT. In: 4th National Development Conference of The School of Pure and Applied Science, 2nd – 5th December, 2019, The Federal Polytechnic, Ilaro. Akinbo, R.Y and Adeboye, Nurain Olawale and Akinde, M. A (2013) STATISTICAL ANALYSIS OF INFLATION RATES IN NIGERIAN ECONOMY. Journal of Business and Educational Policies, 9 (1). pp. 196-203. ISSN Alaba, K.E and Alabi, N. O. (2021) Effects of Socio-Economic Characteristics on Nutritional Status of the Elderly in Ilaro, Ogun State, Nigeria. Further Nutrients, 1 (1). pp. 3-13. Alabi, N. O. and Akanbi, Olumuyiwa O. (2022) Estimatingregressionparametersinthepresenceofextremeinfluentialobservations:AcaseofNigeriaExchangeRate. Journal of Econometrics and Statistics, 2 (2). pp. Alabi, Nurudeen Olawale and Bada, Olatunbosun (2021) Breakpoint Unit Root Tests on Select Macroeconomic Variables in Nigeria. Global Journal of Science Frontier Research, 21 (2). pp. 33-37. ISSN Alabi, Nurudeen Olawale and Bada, Olatunbosun (2018) Can A Decision Tree Forecast Real Economic Growth from Relative Depth of Financial Sector in Nigeria? Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, 18 (4). pp. 55-67. ISSN 2249-4626 Alabi, Nurudeen Olawale and Bada, Olatunbosun (2019) Investigating the Causality between Unemployment Rate, Major Monetary Policy Indicators and Domestic Output using an Augmented Var Approach: A Case of Nigeria. Global Journal of HUMAN-SOCIAL SCIENCE: E Economics, 19 (6). pp. 11-21. ISSN 2249-460x Alabi, Nurudeen Olawale and Bada, Olatunbosun (2022) An application of Hamilton Model of Switching with Autoregressive Dynamics on Exchange Rate Movement in Nigeria. International Journal of Women in Technical Education and Employment, 3 (1). pp. 66-76. ISSN 2811-1567 Are, S.O and Alabi, N. O. (2016) Modeling Domestic Output from Selected Macroeconomic Variables in Nigeria Using Cross-Validated Scatterplot Smoothers. In: Academic Staff Union of Polytechnics (ASUP) Ilaro Chapter 5th National Conference, 2016, The Federal Polytechnic, Ilaro. Aweda, Nurudeen Olawale and Akinsanya, Taofik and Akingbade, Adekunle and Are, Stephen Olusegun (2014) Empirical analysis of the elasticity of real money demand to macroeconomic variables in the United Kingdom with 2008 financial crisis effects. Journal of Economics and International Finance, 6 (8). pp. 190-202. ISSN 2141-6672 Aweda, Nurudeen Olawale and Are, Stephen Olusegun and Akinsanya, Taofik (2014) Statistically Significant Relationships between Returns on FTSE 100, S&P 500 Market Indexes and Macroeconomic Variables with Emphasis on Unconventional Monetary Policy. International Journal of Statistics and Applications, 4 (6). pp. 249-268. Ayoola, Femi Joshua and Adeboye, Nurain Olawale and Balogun, Kayode (2017) On the Investigation of Awareness Level of Family Planning among Rural Dwellers in Nigeria (Principal Component Analysis Approach). Open Access Library Journal, 4. pp. 1-11. ISSN ISSN Online: 2333-9721 ISSN Print: 2333-9705 Bada, Olatunbosun and Alabi, Nurudeen Olawale (2018) Smoothing Parameter and the Performance of Exponentially Weighted Moving Average and Variance Control Scheme. In: School of Information and Communication Technology (SICTCON), Auchi Polytechnic, July 2018, Auchi Polytechnic. Busari, Ganiyu Adeniran and Akinbo, Rasaq Olayinka and Adeboye, Nureni Olawale (2013) Impact of Internally Generated Revenue (IGR) on the Growth of Local Governments in Nigeria. Indian Journal of Research, 2 (11). pp. 267-268. ISSN 2250-1991 Ezekiel, I.D. and Alabi, N. O (2018) Boosted Regression Tree for Modeling Evaporation Piche Using Other Climatic Factors Over Ilorin. Academic Journal of Applied Mathematical Sciences, 4 (9). pp. 98-106. ISSN 2415-2188 Fatumbi, E.O and Agbolade, O. A. (2020) Soret And Dufour Effects In Hydromagnetic Micropolar Fluid Flow Passing A Permeable Stretching Shhet In A Porous Medium. International Journal of All Research Education and Scientific Methods (IJARESM),, 8 (12). pp. 2008-2027. ISSN 2455-6211 Fatunmbi, E. O. (2016) COMPARISON OF DIFFERENTIAL TRANSFORM METHOD AND MATCHED ASYMPTOTIC EXPANSION FOR SOME BOUNDARY VALUE PROBLEMS. In: 5th national Conference of Academic Staff Union Of Polytechnic, Ilaro Chapter, 16-19, Oct. 2016, The Federal Polytechnic, Ilaro. Fatunmbi, E. O. and Adeniyan, A. (2018) Heat and Mass Transfer in MHD Micropolar Fluid Flow over a Stretching Sheet with Velocity and Thermal Slip Conditions. Open Journal of Fluid Dynamics, 8. pp. 195-215. ISSN 2165-3852 Fatunmbi, E. O. and Adeniyan, A. (2018) Heat and Mass Transfer in MHD Micropolar Fluid Flow over a Stretching Sheet with Velocity and Thermal Slip Conditions. In: 1st International Conference and Exhibition on Technological Innovation and Global Competitiveness, 5-8, Nov. 2018, The Federal Polytechnic, Ilaro. Fatunmbi, E. O. and Adeniyan, A. (2019) Heat transfer Analysis of Magneto-Micropolar Fluid Flow over an Inclined Non-linearly Permeable Stretching Sheet with Variable Fluid Properties. In: 38th Annual Conference of Nigerian Mathematical Society, 18th-21, June 2019, University of Nigeria, Nsukka. Fatunmbi, E. O. and Adeniyan, A. (2018) Hydromagnetic flow and heat transfer of a micropolar fluid over an exponentially stretching sheet through a porous medium with slip effects. In: 11 iSteam mart mind Conference, 26-28, June, 2018, YABATECH, Lagos, Nigeria. Fatunmbi, E. O. and Adeniyan, A. (2018) MHD Stagnation Point-flow of Micropolar Fluids Past a Permeable Stretching Plate in Porous Media with Thermal Radiation, Chemical Reaction and Viscous Dissipation. Journal of Advances in Mathematics and Computer Science, 26 (1). pp. 1-19. ISSN 2456-9968 Fatunmbi, E. O. and Adeniyan, A. (2020) Nonlinear thermal radiation and entropy generation on steady flow of magneto-micropolar fluid passing a stretchable sheet with variable properties. Results in Engineering, 6 (2020). pp. 1-10. ISSN 100142 Fatunmbi, E. O. and Adeniyan., A. (2017) Magnetohydromagnetic Stagnation Point-Flow of Micropolar fluids Past a Permeable Stretching Plate in Porous Media with Thermal Radiation, Chemical Reaction and Viscous Dissipation. In: Annual National Conference on Science, Technology and communication, 4-7, Dec. 2017, The Federal Polytechnic Ilaro, Ogun State. Fatunmbi, E. O. and Adeosun, Adeshina Taofeeq (2020) Nonlinear radiative Eyring-Powell nanofluid flow along a vertical Riga plate with exponential varying viscosity and chemical reaction. International Communications in Heat and Mass Transfer, 19 (2020). pp. 1-10. Fatunmbi, E. O. and Agbolade, A. O. (2019) Heat and Mass Transfer of Thermophoretic Magneto-Micropolar Fluid Passing an Inclined Plate with Chemical Reaction in Porous Medium. School of Pure and Applied Sciences Journal (SPAS), 1 (1). pp. 15-22. ISSN 2714-2531 Fatunmbi, E. O. and Agbolade, O. A. (2019) Numerical Investigation of MHD Micropolar Fluid Flow Along an Inclined Permeable Surface with Variable Electric Conductivity and Variable Heat Flux. In: 16 iSteam mart Conference in conjunction with ASUP Ilaro, 9-11th JUNE, 2029, The Federal Polytechnic, Ilaro. Fatunmbi, E. O. and Akanbi, Olumuyiwa O. (2019) HEAT AND MASS TRANSFER OF MAGNETO-MICROPOLAR FLUID PAST A NONLINEAR STRETCHING SHEET IN A POROUS MEDIUM WITH CHEMICAL REACTION. In: MOUNT TOP CONFERENCE, 25th, July, 2019, MOUNTAIN TOP UNIVERSITY, NIGERIA.. Fatunmbi, E. O. and Akanbi, Olumuyiwa O. (2019) Heat and Mass Transfer of Magneto-Micropolar Fluid Flow Past a Nonlinear Stretching Sheet in a Porous Medium with Chemical Reaction. In: 4th Annual Conference of Institute of Operation Research and Management, 23rd-26th, July, 2019, Mountain Top University, Mowe, Ogun State. Fatunmbi, E. O. and Are, Stephen Olusegun (2020) Irreversibility Analysis of Magneto-Micropolar Fluid Flow Past an Inclined Stretchable Sheet with Viscous Dissipation. In: Proceedings of the 25th SMART-iSTEAMS Trans-Atlantic Multidisciplinary Conference, June – July, 2020, Universite Grenoble, Alpes, France. Fatunmbi, E. O. and Are, Stephen Olusegun (2019) NUMERICAL INVESTIGATION OF ENTROPY GENERATION IN HYDROMAGNETIC DISSIPATIVE MICROPOLAR FLUID FLOW ALONG A NONLINEAR STRETCHING SHEET. In: 4th National Development Conference of The School of Pure and Applied Science, 2nd – 5th December, 2019, The Federal Polytechnic, Ilaro. Fatunmbi, E. O. and Are, Stephen Olusegun (2019) Reactive Flow of Magneto-Micropolar Fluid Along a Nonlinear Permeable Stretching Sheet in a Porous Medium. In: 30th annual colloquium and congress of the Association of Mathematical Physics, 5th-8th Nov., 2019, Igbinedion University Okada, Nigeria. Fatunmbi, E. O. and Badru, J. O. and Oke, A.S (2020) Magneto-Reactive Jeffrey Nanofluid Flow over a Stretching Sheet with Activation Energy, Nonlinear Thermal Radiation and Entropy Analysis. In: NSE Ilaro Branch, 1st National Conference, Ilaro, 2-3 November, 2020., 2-3 November, 2020., NSE Ilaro Branch, 1st National Conference, Ilaro.. Fatunmbi, E. O. and Bello, S. O (2019) Thermophoretic Mixed Convection Flow of MHD Micropolar Fluid Along an Inclined Surface with Soret-Dufour Effects and Variable Properties. In: 1st International Conference of College of Physical Science, 27th-30th, Aug. 2019, University of Agriculture, Abeokuta. Fatunmbi, E. O. and Bello, S. O. (2019) Entropy Generation in Hydromagnetic Thermal Boundary Layer Flow of Micropolar Fluid Over a Convectively Heated Nonlinear Stretching Sheet. In: 4th National Conference Of the School of Engineering, NOVEMBER, 2019, International Conference Centre, The Federal Polytechnic, Ilaro, Ogun State, Nigeria.. Fatunmbi, E. O. and Fenuga, O. J. (2018) Heat and Mass Transfer of a Chemically Reacting MHD Micropolar Fluid Flow over an Exponentially Stretching Sheet with Slip Effects. Physical Science International Journal, 18 (3). pp. 1-15. ISSN 2348-0130 Fatunmbi, E. O. and Fenuga, O. J. (2017) MHD Micropolar Fluid Flow Over a Permeable Stretching Sheet in the presence of Variable Viscosity and Thermal Conductivity with Soret and Dufour Effects. INTERNATIONAL JOURNAL OF MATHEMATICAL ANALYSIS AND OPTIMIZATION: THEORY AND APPLICATIONS, 2017. pp. 211-232. Fatunmbi, E. O. and Mabood, Fazle and Elmonser, Hedi and Tlili, Iskander (2020) Magnetohydrodynamic nonlinear mixed convection flow of reactive tangent hyperbolic nano fluid passing a nonlinear stretchable surface. physica scripta, 96. pp. 1-13. ISSN 14024896 Fatunmbi, E. O. and Odesola, A. S. (2018) MHD Free Convective Heat and Mass Transfer of a Micropolar Fluid Flow over a Stretching Permeable Sheet with Constant Heat and Mass Flux. Asian Research Journal of Mathematics, 9 (3). pp. 1-15. ISSN 2456-477X Fatunmbi, E. O. and Odesola, A.S (2016) MHD STAGNATION POINT FLOW AND HEAT TRANSFER OF MICROPOLAR FLUID IN POROUS MEDIUM OVER A STRETCHING SURFACE WITH THERMAL RADIATION, HEAT GENERATION AND DISSIPATION. In: 27th Annual Colloquium and Congress of Nigerian Association of Mathematical Physics (NAMP), 1st-4th November, 2016, Michael Okpara University, Umudike,Abia State. Fatunmbi, E. O. and Odesola, A.S. (2017) MHD Free Convective Heat and Mass Transfer of a Micropolar Fluid Flow over a Stretching Permeable Sheet with Constant Heat and Mass Flux. In: 28th Annual Colloquium and Congress of Nigerian Association of Mathematical Physics (NAMP), 31 Oct.-3rd Nov. 2017, Bayero University, Kano, Nigeria. Fatunmbi, E. O. and Ogunseye, Hammed Abiodun and Sibanda, Precious (2020) Magnetohydrodynamic micropolar fluid flow in a porous medium with multiple slip conditions. International Communication in Heat and Mass Transfer, 115 (2020). pp. 1-10. Fatunmbi, E. O. and Okoya, S. S. and Adeniyan, A. (2017) FLOW OF HEAT AND MASS TRANSFER IN MHD MICROPOLAR FLUID PAST STRETCHING PERMEABLE SURFACE IN A POROUS MEDIUM WITH SORET AND DUFOUR EFFECTS. In: 6th International Conference on Mathematical Analysis and Optimization: Theory and Application, 8-11, March,2017, University of Lagos. Fatunmbi, E. O. and Okoya, Samuel S. (2020) Heat Transfer in Boundary Layer Magneto-Micropolar Fluids with Temperature-Dependent Material Properties over a Stretching Sheet. Hindawi Advances in Materials Science and Engineering, 2020 (1). pp. 1-11. Fatunmbi, E. O. and Okoya, Samuel S. (2021) Quadratic Mixed Convection Stagnation-Point Flow in Hydromagnetic Casson Nanofluid over a Nonlinear Stretching Sheet with Variable Thermal Conductivity. Defect and Diffusion Forum, 409. pp. 95-109. ISSN 1662-9507 Fatunmbi, E. O. and Okoya, Samuel Segun and Makinde, Oluwole Daniel (2020) Convective Heat Transfer Analysis of Hydromagnetic Micropolar Fluid Flow Past an Inclined Nonlinear Stretching Sheet with Variable ThermoPhysical Properties. Diffusion Foundations, 26. pp. 63-77. ISSN 2296-3642 Fatunmbi, E. O. and Salawu, S. O. (2020) Thermodynamic second law analysis of magnetomicropolar fluid flow past nonlinear porous media with non-uniform heat source. Propulsion and Power Research, 9 (3). pp. 281-288. Fatunmbi, E. O. and Salawu, Sulyman Olakunle (2020) Analysis of Entropy Generation in Hydromagnetic Micropolar Fluid Flow over an Inclined Nonlinear Permeable Stretching Sheet with Variable Viscosity. Journal of Applied Computational Mechanics, 7 (1). pp. 21-35. ISSN 2383-4536 Fatunmbi, E. O. and Sikiru, O. A. (2019) Technological Impact of Hydromagnetic Micropolar Fluid Flow Over a Stretching Permeable Sheet with Thermal Radiation and Joule Heating Effects. In: 1st National Conference of Women in Technical Education & Employment (WITED) Ilaro chapter, 12-15 Aug. 2019, The Federal Polytechnic, Ilaro. Fenuga, O.J. and Fatunmbi, E. O. and Adeniyan, A. (2013) Effects of thermogenesis parameters and Biot numbers on a nonlinear heat conduction model of the human head. International Journal of Advanced Scientific and Technical Research, 3 (5). pp. 290-299. ISSN 2249-9954 Hammed, Mudasiru and Soyemi, Jumoke (2019) Information Leakage Prevention Using Public Key Encryption System and Fingerprint Augmented with Apriori Algorithm. International Journal of Computer Science and Security (IJCSS), 13 (3). pp. 90-100. Ilori, B.Y. and Alabi, N. O. and Ogun, C. A and Awofodu, J.O (2016) Empirical Models for Forecasting Global Solar Radiation on Horizontal Surface using Sunshine Hour and Temperature data over Ikeja, Lagos State. In: 39th Nigerian Statistical Association (NSA) Annual Conference. Nigerian Statistical Association, 2016, Nigeria. Isinkaye, F.O. and Soyemi, Jumoke and Arowosegbe, O.I (2020) An Android-based Face Recognition System for Class Attendance and Malpractice Control. International Journal of Computer Science and Information Security (IJCSIS), 180 (1). pp. 78-83. ISSN 1947-5500 Isinkaye, F.O. and Soyemi, Jumoke and Awosupin, S.O. (2017) A Mobile Based Expert System for Disease Diagnosis and Medical Advice Provisioning. International Journal of Computer Science and Information Security (IJCSIS), 15 (1). pp. 568-572. ISSN 1947-5500 Isinkaye, F.O. and Soyemi, Jumoke and Oluwafemi, O.P (2017) A Mobile-based Neuro-fuzzy System for Diagnosing and Treating Cardiovascular Diseases. International Journal of Information Engineering and Electronic Business(IJIEEB), 9 (6). pp. 19-26. ISSN 2074-9031 Joseph, E.A and Olaiya, O.O (2018) Artificial Intelligence Application in Transportation System Control: An Effective Way To Minimize Global Warming Effect. In: 11th International Science, Technology, Arts, Education, Management & the Social Sciences Conference, June, 2018, Lagos, Nigeria. Joseph, E.A and Olasina, J.R (2017) Mathematical Model for Heat Control in Rotary Kiln System. Multidisciplinary Research and Development, 3 (7). pp. 155-161. ISSN 2454-6615 Lawal, Ganiyu Omoniyi and Aweda, Nurudeen Olawale and Oyeyemi, Gafar Matanmi (2015) A Conditional Restricted Equilibrium Correction Model on Nigerian Stock Exchange All-Share Index and Macroeconomic Indicators with 2008 Global Financial Crisis Effects: A Univariate Framework Approach. American Journal of Mathematics and Statistics, 5 (3). pp. 150-162. Lawal, Ganiyu Omoniyi, and Aweda, Nurudeen Olawale (2015) An Application of ARDL Bounds Testing Procedure to the Estimation of Level Relationship between Exchange Rate, Crude Oil Price and Inflation Rate in Nigeria. International Journal of Statistics and Applications, 5 (2). pp. 81-90. Lawal, Omoniyi Ganiyu and Alabi, Nurudeen Olawale and Ige, Sikiru Ajibade and Ibraheem, Rahmat Abisola (2016) The Nexus between Nigerian Government Spending and Domestic Output in the Presence of Long-Term Crude Oil Price Shock: A Conditional Unrestricted Equilibrium Correction Model Approach. Open Journal of Statistics, 6. pp. 412-425. Lawanson, A. A. and Oduntan, O.E. (2020) EMERGING TRENDS IN ANIMAL REPRODUCTIVE TECHNOLOGY – A REVIEW. In: Proceedings of the 2nd International Conference, The Federal Polytechnic, Ilaro, 10th – 11th Nov., 2020, 10th – 11th Nov., 2020, The Federal Polytechnic, Ilaro. Nureni, Olawale Adeboye and Agunbiade, Dawud Adebayo (2020) A Simultaneous Technique Estimation of Keynesian Economic Growth Model. In: International Webinar on Recent Advancements in Mathematical Sciences and its Applications, organized by Department of Mathematics, Chakdaha College, Nadia, West Bengal, India, 3rd & 4th September 2020., West Bengal, India. Nureni, Olawale Adeboye and Olawale, Victor Abimbola (2020) An overview of cardiovascular disease infection: A comparative analysis of boosting algorithms and some single based classifiers. Statistical Journal of the IAOS 36. pp. 1189-1198. Nureni, Olawale Adeboye and Olawale, Victor Abimbola and Sakinat, Oluwabukola Folorunso (2020) Malaria patients in Nigeria: Data exploration approach. Data in Brief, Elsevier, 28. pp. 1-9. ISSN Nureni, Olawale Adeboye and Osuolale, Peter Popoola and Iyabode, Favour Oyenuga (2021) Building Robust Collaboration between the Producers and the Users of the Official Statistics in Africa. In: 63rd International Statistical Institute World Statistics Congress, 11-16 July 2021, Nigeria. Nureni, Olawale Adeboye and Peter, Osuolale Popoola and Ogunnusi, O.N (2020) Data science skills: Building partnership for efficient school curriculum delivery in Africa. Statistical Journal of the IAOS. pp. 49-62. ISSN 200693 OLANIRAN-AKINYELE, O.F and YUSUFF, M.A (2019) Women in Science, Technology, Engineering and Mathematics (STEM); Implications for Growth in Nigeria. In: 1st National Conference of WITED, Ilaro Chapter, August 13-16, 2019, THE FEDERAL POLYTECHNIC, ILARO, OGUN STATE. Obafemi, O. S and Alabi, N. O. (2018) An Alternative Method of Detecting Outlier in Multivariate Data using Covariance Matrix. Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, 19 (4). pp. 37-48. ISSN 2249-4626 Odeyemi, Joseph Bamidele (2021) Critical Evaluation of Learning Duration and Students’ Performance in Secondary School Mathematics. International Journal of Women in Technical Education and Employment (IJOWITED), The Federal Polytechnic, 2 (1). pp. 29-34. ISSN 2734-3227 Ogunnusi, O.N and Ojo, G and Sikiru, O. A. (2021) On the Partial Least Square Regression Modeling to Collinear Regressors: Contribution of Transportation Sector to Nigeria Economic Growth. IJISET - International Journal of Innovative Science, Engineering & Technology, 8 (4). pp. 231-241. ISSN 2348 – 7968 Oke, A.S and Fatunmbi, E. O. (2015) Pattern Formation in Competition-Diffusion equation. International Journal of Advanced Scientific and Technical Research, 5 (7). pp. 52-59. ISSN 2249-9954 Okosodo, E.F and Orimaye, J. O and Ogunyemi, O. O and Kolawole, O.O. (2019) Diversity and Abundance of Bird Species in Akure Forest Reserve South Western Nigeria. kukula, 4 (6). Olaiya, O.O and Oduntan, O.E. and Frederick, O.E. (2020) A SHORT OVERVIEW OF PIPELINE MONITORING TECHNOLOGIES FOR VANDALISM PREVENTION WITH A PROPOSED FRAMEWORK FOR A GSM BASED SYSTEM. JOURNAL OF ENGINEERING & RESEARCH TECH., 13 (5). pp. 175-188. ISSN 0428-3123 Olatayo, T.O. and Adeboye, Nurain Olawale (2013) Predicting Population Growth through Births and Deaths Rate in Nigeria. Journal Of Mathematical Theory And Modelling, 3 (1). pp. 96-101. ISSN Onawola, H.J. and Longe, O.B. and Adebayo, S. and Olasina, J.R (2019) Acceptability of Smart Grid Technologies for Sustainable Energy Distribution in Developing Nations. In: 19th iSTEAMS Multidisciplinary Conference, 7th – 9th August 2019, The Federal Polytechnic, Offa, Kwara State, Nigeria. Osuolale, Peter Popoola and Nureni, Olawale Adeboye (2020) Opportunities, Challenges and Building Partnerships for Official Statistics in the Era of Big Data in Nigeria. International Journal of Research, 7 (8). pp. 152-163. ISSN 2348-795X Oyelade, Jelili and Isewon, Itunuoluwa and Ogunbona, Olanrewaju and Aromolaran, Olufemi and Soyemi, Jumoke (2017) Modeling of Metabolic Pathways Using Petri Net. In: 2017 International Conference on Computational Science and Computational Intelligence (CSCI), December,, USA. Oyelade, Jelili and Isewon, Itunuoluwa and Ogunbona, Olanrewaju and Aromolaran, Olufemi and Soyemi, Jumoke (2017) Modeling of Metabolic Pathways using Petri Net. In: 2017 International Conference on Computational Science and Computational Intelligence (CSCI), 14-16 Dec. 2017, Las Vegas, NV, USA. Saka, K and Badru, J. O. (2020) A STRUCTURAL MODELING OF ECONOMIC IMPLICATIONS OF BANKRUPTCY IN POST COVID-19 ERA: EVIDENCE FROM NIGERIA. In: 2nd International Conference, The Federal Polytechnic Ilaro, November, 2020, The Federal Polytechnic Ilaro, Ogun State, Nigeria. Salawu, S. O. and Fatunmbi, E. O. (2017) Dissipative Heat Transfer of Micropolar Hydromagnetic Variable Electric Conductivity Fluid Past Inclined Plate with Joule Heating and Non-uniform Heat Generation. Asian Journal of Physical & Chemical Sciences, 2 (1). pp. 1-10. Salawu, S. O. and Fatunmbi, E. O. (2017) Inherent Irreversibility of Hydromagnetic Third-Grade Reactive Poiseuille Flow of a Variable Viscosity in Porous Media with Convective Cooling. Journal of the Serbian Society for Computational Mechanics, 11 (1). pp. 46-58. Salawu, S. O. and Fatunmbi, E. O. and Ayanshola, A.M. (2020) On the diffusion reaction of fourth-grade hydromagnetic fluid flow and thermal criticality in a plane Couette medium. Results in Engineering, 8 (2020). pp. 1-8. Salawu, S. O. and Fatunmbi, E. O. and Okoya, S. S. (2021) MHD heat and mass transport of Maxwell Arrhenius kinetic nanofluid flow over stretching surface with nonlinear variable properties. Results in Chemistry. pp. 1-15. ISSN 2211-7156 Salawu, S.O. and Fatunmbi, E. O. (2020) Current density and criticality branch-chain for a reactive Poiseuille second-grade hydromagnetic flow with variable electrical conductivity. International Journal of Thermofluids, 3 (4). pp. 1-7. Soyemi, Jumoke (2018) Student Project Quality Assurance In Academic Institutions Using Plagiarism Software Checker. In: 11th International Science, Technology, Arts, Education, Management & the Social Sciences Conference, June, 2018, Lagos, Nigeria. Soyemi, Jumoke and Adegboye, James (2018) Database Record Duplicate Detection System using Simil Algorithm. International Journal on Computer Science and Engineering (IJCSE), 10 (2). pp. 55-61. ISSN Soyemi, Jumoke and Adesi, Adesola Bolaji (2019) Software Piracy Detection and Prevention Using Sift Algorithm with Two-Way Authentication Mechanism. In: 16th iSTEAMS TRIP Conference, 9th - 11th June, 2019, The Federal Polytechnic, Ilaro, Ogun State, Nigeria. Soyemi, Jumoke and Adesi, Adesola Bolaji (2018) A Web-based Decision Support System with SMS-based Technology for Agricultural Information and Weather Forecasting. International Journal of Computer Applications, 180 (16). pp. 1-6. Soyemi, Jumoke and Akinode, J.L. and Oloruntoba, S.A. (2017) Automated Lecture Time-tabling System for Tertiary Institutions. nternational Journal of Applied Information Systems (IJAI, 12 (5). pp. 20-27. ISSN 2249-0868 Soyemi, Jumoke and Akinode, J.L. and Oloruntoba, S.A. (2017) Electronic Lecture Time-Table Scheduler Using Genetic Algorithm. In: 15th IEEE International Conference on Dependable, Autonomic and Secure Computing (IEEE DASC, 2017), 6-10 November, 2017, Orlando, Florida, USA. Soyemi, Jumoke and Hammed, Mudasiru (2019) Handling Mobile Network Congestion with Assembly line Control Algorithm. International Journal of Computer Science and Information Security, 17 (2). pp. 167-174. ISSN 947-5500 Soyemi, Jumoke and Isewon, Itunuoluwa and Ogunlana, Olubanke and Rotimi, Solomion and Oyelade, Jelili and Adebiyi, Ezekiel (2018) Computational analysis of Plasmodium falciparum RNA-Seq data reveals PPIs that might be implicated in the invasion of the RBCs. In: Computational Intelligence in Bioinformatics and computational Biology (CIBCB)., April 30 to May 2, 2018, St. Louis, MO, USA. Soyemi, Jumoke and Isewon, Itunuoluwa and Oyelade, Jelili and Adebiyi, Ezekiel (2017) Functional enrichment of human protein complexes in malaria parasites. In: Conference: 2017 International Conference on Computing Networking and Informatics (ICCNI), 29-31 Oct. 2017, Lagos, Nigeria. Soyemi, Jumoke and Isewon, Itunuoluwa and Oyelade, Jelili and Adebiyi, Ezekiel (2018) Identification of Important Interacting Proteins responsible for Merozoites Invasion in Human RBCs. In: 11TH CONFERENCE OF THE AFRICAN SOCIETY OF HUMAN GENETICS (AFSHG) AND H3AFRICA CONSORTIUM, 19 SEPT -21 SEPT 2018, KIGALI, RWANDA. (Submitted) Soyemi, Jumoke and Isinkaye, F.O. (2017) A Web-Based Final Year Student Project Duplication Detection System. International Journal of Computer Application, 7 (1). pp. 1-7. ISSN 2250-1797 Soyemi, Jumoke and Itunuoluwa, Isewon and Jelili, Oyelade and Ezekiel, Adebiyi (2018) Inter-Species/Host-Parasite Protein Interaction Predictions Reviewed. PubMed, 13 (4). pp. 396-406. Soyemi, Jumoke and Olasina, J.R (2016) Assessment of the Impact of Social Networking Media on Students’ Academic Performance in Higher Institutions: A case Study of Federal Polytechnic, Ilaro. In: International Conference on Computing Research and Innovations (CoRI), Sept 7–9, 2016, Ibadan, Nigeria. Soyemi, Jumoke and Oloruntoba, S.A. and Okafor, Blessing (2015) Analysis of Mobile Phone Impact on Student Academic Performance in Tertiary Institution. International Journal of Emerging Technology and Advanced Engineering, 5 (1). pp. 361-365. Soyemi, Jumoke and Sanjay, Misra and Omoregbe, Nicholas (2015) Towards e-Healthcare Deployment in Nigeria: The Open Issues. Communications in Computer and Information Science. pp. 588-599. Soyemi, Jumoke and Soyemi, Olugbenga Babajide (2019) Fostering Female Participation in Science and Technology through Enhanced Curriculum Delivery Using Information Technology. In: First National WITED Conference, Ilaro Chapter, 13th -16th August, 2019, The Federal Polytechnic, Ilaro, Ogun State, Nigeria. Soyemi, Jumoke and Soyemi, Olugbenga Babajide (2019) Revamping TVET Programmes through Industrial Collaborations for Youth Empowerment and Employment Opportunities. In: 4thNational Development Conference of The Schoolof Pure and Applied Science, 2nd –5th December, 2019, The Federal Polytechnic Ilaro, Ogun State. Soyemi, Jumoke and Soyemi, Olugbenga Babajide and Hammed, Mudasiru (2015) Nigeria Cashless Culture: The Open Issues. Nigeria Cashless Culture: The Open Issues, 4 (4). pp. 51-56. 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Mathematics & Statistics Presentations Mathematics & Statistics Presentations Presentation Schedules Room 301 Presentations: Join us on Zoom. Glucose Regulation Using an Intelligent PID Controller 1:00 p.m. Team Members: Parker Willmon Advisors: Dr. Katie Evans The SIR Models, Their Applications, and Approximations of Their Rates p.m. Team Members: Christopher Cano Advisor: Dr. Stacey McAdams A Novel Method for Computations of Ratios of Jet Cross Sections in Perturbative Quantum Chromodynamic 1:30 p.m. Team Members: Connor Waits Advisors: Dr. Markus Wobisch Forecasting Daily Stock Market Return with Multiple Linear Regression 1:45 p.m. Team Members: Shengxuan Chen Advisor: Dr. Xiao Zhong The Shallow Water Equations 2:00 p.m. Team Members: Chase Jones Advisor: Dr. Weizhong Dai The Theory of Cryptography In BTcoin 2:15 p.m. Team Members: Can Hong Advisor: Dr. John Doyle Impact of Eating and Sleeping Prior to Test Taking 2:45 p.m. Team Members: Cassidy Meadows Advisor: Dr. Brian Barron Periodic Points and Sharkovsky’s Theorem 3:00 p.m. Team Members: Luke J. Seaton Advisor: Dr. John Doyle The Prediction of Fantasy Football 3:15 p.m. Team Members: Chelsea Robinson Advisor: Mr. Stanley McCaa The Axiom of Choice and Related Topics 3:30 p.m. Team Members: Bryan McCormick Advisor: Dr. John Doyle Strategies and Algorithms of Sudoku 3:45 p.m. Team Members: Callie Weaver Advisor: Dr. Stacey McAdams Bridge to Bulldogs: A Student and Financial Analysis 4:00 p.m. Team Members: Rebekah Moss Advisor: Cassidi Jacobs Predicting and Comparing the Stock Value of Chick-fil-A 4:15 p.m. Team Members: Mark Yates Advisor: Mr. Stan McCaa Glucose Regulation Using an Intelligent PID Controller Type 1 diabetes is a condition characterized by a lack of insulin production. This lack of insulin causes glucose concentration in the blood to increase after meals. In order to maintain blood glucose levels, diabetics must inject insulin using needles or an insulin pump. Additionally, the lack of insulin can cause glucose levels to decrease overnight. This project uses a proportional-integralderivative (PID) controller to modify the rate of insulin and glucagon infusion when glucose levels are increasing or decreasing, respectively. A system of 12 differential equations was used to anticipate changes in glucose concentration as insulin and glucagon were injected. The system was simulated for virtual patients over a 24-hour time span to test its feasibility in human patients. The PID controller uses the current, past, and anticipated future glucose levels, respectively, to determine the best course of treatment for the virtual patient. One of the many difficulties in medical technology, however, is everyone is different. These differences are a result of metabolism and other factors. To account for these differences, the controller is designed to change the gain of the different controller components to better tailor the treatment to each patient. The SIR Models, Their Applications, and Approximations of Their Rates The SIR (susceptible-infected-recovered) models are used to help predict the spread of diseases. The goals of this paper are: elaborating on the methods of approximating the recovery rate, infection rate, and loss of immunity rate; comparing the SIR models with these approximation methods to real-world data, and determining the most accurate combination of the approximation methods for each SIR model. There are several SIR models such as the Kermack-McKendrick, SIRS, and SI models that are designed for specific diseases. Understanding the parameters of these models will assist us in maximizing their accuracy. For example, there is no explicit formula for any of the rates within the models. Therefore, those rates must be approximated. Using these models to represent real-world situations will explain why each disease needs to be represented by a specific model. Understanding the content and the rate approximations of each model can help determine the level of accuracy the model will have in predicting the spread of the disease. A Novel Method for Computations of Ratios of Jet Cross Sections in Perturbative Quantum Chromodynamic The strong interaction is the force responsible for binding quarks to form hadrons, such as protons and neutrons, and also for binding protons and neutrons to form the nuclei of atoms. The properties of the strong interaction can be studied in particle collisions from measurements of the production rates of collimated sprays of particles, called jets. In particular, the ratio of the number of collisions that produce three jets over the number of collisions that produce two jets is a direct measure of the strength of the strong interaction, which is quantified by the strong coupling constant. Determinations of the strong coupling constant from particle collider data require theoretical calculations. In this paper, a new approach for the theoretical calculations that differs from the commonly used approach is investigated. Computations of the results are presented for different ratio measurements performed at the CERN Large Hadron Collider and the Fermilab Tevatron Collider. The results of the two different approaches are compared to each other and to the results of the experimental measurements. It is discussed in which kinematical regions the two approaches agree and where they differ. Forecasting Daily Stock Market Return with Multiple Linear Regression The purpose of this project is to use data mining and big data analytic techniques to forecast daily stock market returns with multiple linear regression. Using mathematical and statistical models to analyze the stock market is important and challenging. The accuracy of the final results relies on the quality of the input data and the validity of the methodology. In the report, within a 5-year period, the data regarding eleven financial and economical features are observed and recorded on each trading day. After preprocessing the raw data with the statistical method, we use the multiple linear regression to predict the daily return of the S&P 500 Index ETF (SPY). A model selection procedure is also completed to find the most parsimonious forecasting model. The Shallow Water Equations For this project, we are doing research on the shallow water equations: a set of hyperbolic partial differential equations. These equations exist as a set of three primary equations. However, there is another version of the shallow water equations called the Saint Venant’s equations. These equations are similar to the standard shallow water equations but are reduced to one-dimension. The primary goal of our research is to investigate the behavior and mathematical construction of the Saint Venant’s equations and model these equations using COMSOL. Regardless of the equation type, standard or Saint Venant’s, it is useful to note that these equations are only applicable under some restrictions such as hydrostatic balance and the distance from one crest to another, on any two waves, must be greater than the distance from the free surface to the sea floor (bottom topography). These restrictions, along with initial conditions, are also a target in this research, and these conditions and equations can help with flood predictions and regulations not only now, but also in the future. The Theory of Cryptography in BTcoin Bitcoin is a well known virtual currency, or cryptocurrency. It was created by a group of people using the name Satoshi Nakamoto in 2008. Currently, many people are utilizing Bitcoin for personal gains and transactions. To keep transactions secure requires techniques from modern cryptography. In this paper, we explain certain aspects of the cryptography of Bitcoin. We are going to discuss two components of the cryptography of Bitcoin— hash functions and signatures. We will describe what the hash function and signature are, give some examples of hash functions, and discuss certain criteria that good hash functions should satisfy. Impact of Eating and Sleeping Prior to Test Taking This paper addresses an ongoing issue that many high schools nationwide are having with low test scores in mathematics. There are many different factors that could be contributing to this problem. Questions that we must ask in solving this problem are whether what a student eats and how much sleep they receive are factors?” If the answers to these questions are yes, how beneficial would it be to be able to assist students in their academics by teaching them about the best times to eat and how to improve sleep habits to improve their test scores? Students long for an easy yet efficient way to improve their mathematics test scores, and knowing the best times to eat and sleep could lead to a simple plan that could help without adding additional classroom work or study time. A survey is given to students prior to testing to identify whether they ate and how much sleep they received prior to their exam. The goal of this project is to research and extract data on whether or not eating prior to taking a test is associated with higher mathematics test scores among high school students while also taking sleep into account. Periodic Points and Sharkovsky’s Theorem The number of periodic points of a function depends on the context. The number of complex periodic points and rational periodic points have been shown to be infinite and finite, respectively, if f is a polynomial of degree at least 2. However, the number of real periodic points can be either finite or infinite. Sharkovsky’s Theorem states that if p is left of q in the “Sharkovsky ordering” and the continuous function f has a point of period p, then f also has a point of period q. This statement becomes very powerful when considering a function that has points of period 3, all the way to the left side of the Sharkovsky ordering, since having a point of period 3 implies the existence of points of all periods. We explore a continuous function with points of period 3 where the function can be restricted to an interval containing points of period all other natural numbers. The Prediction of Fantasy Football In this paper, we consider the game fantasy football, which allows people to simulate being a National Football League team owner. Imaginary owners select from the best players in the NFL and compete on a weekly basis based upon player performances on the field. Fantasy football has become popular over the years. In 2011, according to the Fantasy Sports Trade Association, there were 35 million people that played fantasy sports online in the United States and Canada. Some of the major companies that use fantasy football are Yahoo, ESPN, and the NFL, although there are more platforms. Many people use these platforms to view NFL reporting, preseason rankings, player statistics, fantasy points projections, and expert opinions on drafts. Even though fantasy sports have increased over time and there are various platforms to view stats and predictions, there is no method that provides a strategy to predict the entire fantasy football league. During this project, we will predict NFL players’ performances on the field and calculate their fantasy points for the next season using the autoregression integrated moving average (ARIMA) models using players’ historical data. We will use the data from these predictions and an algebraic equation to rank players by overall fantasy prediction points for the 2020 fantasy draft. The Axiom of Choice and Related Topics This project covers the axiom of choice and two mathematical statements which are equivalent to it. The axiom of choice is an axiom of Zermelo-Fraenkel set theory that states that given a collection of non-empty sets, there exists a choice function which selects one element from each set to form a new set. The equivalents of the axiom of choice that are discussed in this project include Zorn’s Lemma, which states that a partially ordered set with every chain being bounded above contains a maximal element, and the Well-Ordering Theorem, which states that every set has a well ordering. In addition to proving the equivalence of these statements, this project explains the mathematics required to prove them individually, as well as various mathematical consequences of the statements. Strategies and Algorithms of Sudoku This paper discusses different strategies for the game of Sudoku and how those strategies relate to other problem-solving techniques while also attempting to use those other techniques in a way that improves the strategies for Sudoku. This includes a thorough analysis of the general algorithm and an algorithm that is formed by the Occupancy Theorem and Preemptive Sets. This paper also compares these algorithms that directly relate to Sudoku with algorithms to similar combinatorial problems such as the Traveling Salesman problem and more. With the study of game theory becoming more popular, these strategies have also been shown to help students in various ways in the classroom. To understand Sudoku on a deeper level, this paper demonstrates ways to model a puzzle by using permutation matrices and different symmetries. Bridge to Bulldogs: A Student and Financial Analysis In this paper, we discuss the statistical analysis of the Bridge to Bulldogs program. The program provides prospective students, who do not meet all of the admission requirements, an alternate route of admission to Louisiana Tech University. It is offered over two consecutive quarters, either summer/fall or fall/winter. During the program, students focus on building their math skills through tutoring and special advising. We compare the Bridge students to other first-time freshmen in relation to scores in freshman-level math classes. We also compare composite and Math ACT scores. Finally, we perform a financial analysis, including retention rates, to determine if the Bridge to Bulldogs program is financially beneficial to the university. Predicting and Comparing the Stock Value of Chick-fil-A This project focuses on estimating the stock value of Chick-fil-A as if it were a publicly-traded company using a comparable analysis method or CAM. We begin by obtaining financial information from Chick-fil-A as well as the number of locations there are chain-wide. Next, we find two publicly traded fast food companies, one that is larger than, and another that is smaller than Chick-fil-A and obtain the same information from them. The idea is that Chick-fil-A will lie between these two companies, and we can use the CAM to estimate their stock value. The CAM gives us a multiple of the valuation of Chick-fil-A in comparison to the companies we use and that information is used to estimate the stock value. Lastly, we can compare Chick-fil-A with the larger company and then with the smaller company and average the two values which will give us a more accurate estimate.	{"url":"https://coes.latech.edu/senior-projects-conference/mathematics-statistics-presentations/","timestamp":"2024-11-08T11:31:59Z","content_type":"text/html","content_length":"90054","record_id":"<urn:uuid:d461cb93-6211-4ca1-8896-c7439ecc1ef7>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00120.warc.gz"}
pposite of 2 8/10 Calculate the opposite of 2 8/10 Use our calculator to get the opposite of 2 8/10. Also, learn what is an opposite of a number as use our step-by-step calculator to find the opposite of any real number or fraction. Opposite of a Number or of a Fraction Calculator What Does the Opposite of a Number Mean? Definition 1: Opposite number or additive inverse of any number (n) is a number which, if added to, results in 0, the identity element of addition. The opposite number for n is written as −n. Definition 2: The opposite of a number is its additive inverse, which means that its sign is reversed. A number and its additive inverse equal zero when added. Definition 3: The opposite of a number is the number on the other side of the 0 number line and the same distance from 0. Examples: The opposite of 3 is -3 and vice-versa. Video on How To Find The Opposite of Whole Numbers, Fractions and, Decimal Numbers This excellent video explains how to find the reciprocal of a whole number, a fraction, as well as, the reciprocal of a mixed number. While every effort is made to ensure the accuracy of the information provided on this website, neither this website nor its authors are responsible for any errors or omissions. Therefore, the contents of this site are not suitable for any use involving risk to health, finances or property.	{"url":"https://coolconversion.com/math/opposite-of/What-is-the-Opposite-of_2-8/10_%3F","timestamp":"2024-11-04T22:09:22Z","content_type":"text/html","content_length":"76552","record_id":"<urn:uuid:c8ab4a23-ed5b-4811-a89c-ac1fdfec2ebe>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00134.warc.gz"}
RD Sharma Class 8 Maths Solutions Chapter 9 - Linear Equations in One Variable RD Sharma Solutions for Class 8 Maths - Linear Equations in One Variable - Free PDF Download Free PDF download of RD Sharma Solutions for Class 8 Maths Chapter 9 - Linear Equations in One Variable solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 9 - Linear Equations in One Variable Exercise Questions with Solutions to help you to revise complete Syllabus and Score More marks. Vedantu is a platform that provides free NCERT Solutions, RD Sharma Solutions and other study materials for students. Download RD Sharma Solution for Class 8 Math to help you to revise the complete syllabus and score more marks in your examinations. At Vedantu, All chapter-wise solutions of RD Sharma for Class 8 Math textbook are free to download. These are created by the best Teachers at Vedantu so the Math Students who are looking for the best solutions, can download Class 8 Math RD Solutions to help you to revise the complete syllabus and score more marks in your examinations. FAQs on RD Sharma Class 8 Maths Solutions Chapter 9 - Linear Equations in One Variable 1. Why should I select Vedantu's solutions for my child's growth? Vedantu is a fantastic online educational platform that makes learning simple. It also includes a number of exercises that students may use to assess their abilities. Master Teachers also provide live classes to clear your doubts. On Vedantu, you can get all of RD Sharma's Maths solutions. These answers cover all of the topics in each Class Mathematics according to the updated CBSE syllabus. These solutions are designed by verified teachers at Vedantu in such a way that students can understand the solutions in easy and multiple ways which helps for better scores and laying foundation for future classes. 2. How do you find RD Sharma Maths solutions for Class 8 useful? Every learner desires to be at the top in today's competitive environment. This, however, can only be accomplished if he or she has a thorough comprehension of the subject. Regardless, students can only improve their marks in high-scoring topics. RD Sharma Solutions for Class 8 Maths has been selected as the subject in which complete marks can only be attained with the help of RD Sharma This book explains everything step by step, making it easy for students to comprehend. For instance, if a student is learning percentages, this book might be quite useful because it breaks out the topic step by step. All that is required of the student is to start with step 1, thoroughly grasp it, then continue on to step 2, and lastly to the finish. 3. Is it adequate to study RD Sharma Class 8 Maths Solutions for the exam? To do well in Class 8th mathematics, you must study the subject thoroughly and gain a thorough understanding of the various ideas. So, if you're reading Class 8 Maths answers to help you prepare for the subject, you'll be able to give your studies a big boost. These answers are self-contained and allow you to keep on top of your test preparation. Students may use the thorough RD Sharma Solutions for Class 8 to assist them to build a firm foundation for complicated subjects. The Features RD Sharma Maths Solutions for Class 8 are: • Created by our subject matter specialists. • It's completely free. • Based on the most recent CBSE curriculum pattern • Chapter-by-chapter solutions • A simple way of preparing 4. What are the steps to attempting a Class 8 Maths Sample Paper? To get the most out of this R D. Sharma Mathematics sample test for Class 8, everyone must understand how to practice it. • First, you must be familiar with all concepts present in the book. • Go over each section in your book and make a mental note of all of the key points. • Note and Revise all of the formulae. • Set a timer on your phone to remind you to complete the exam before the practice paper. • This is a test to evaluate how good you are at analytical and reasonable thinking. • The exam should be timed to coincide with your school's actual evaluations. • Adhere to the guidelines on the paper to address the problem.	{"url":"https://www.vedantu.com/rd-sharma-solutions/class-8-maths-chapter-9-solutions","timestamp":"2024-11-05T21:41:46Z","content_type":"text/html","content_length":"221021","record_id":"<urn:uuid:45644bd3-858d-4782-a775-ef31d96b938a>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00629.warc.gz"}
Many people tell you to trust your intuition. Is this a good idea? Listen to this: Suppose you have a sack of potatoes that weighs 100 pounds (not including the sack, which you can forget about). Further suppose I tell you that those potatoes are 99% water. Several days later, after some of the water has evaporated, I now tell you that the potatoes are 98% water. How much do the potatoes weigh now? Go ahead and try to figure it out before you turn the page. And I promise you, it’s not a trick question. Having trouble? Then just use your intuition and take a wild guess. Now click here for the answer. How well did your intuition serve you? The math is pretty simple once it’s explained, and you might even have an "Aha!" moment. Fun, no? Here’re a couple more. Suppose you have a stack of nickels so high, it reaches to the very top of Mt. Everest. How big a box would it take to hold all these nickels? [Click here for the answer] Suppose you live in a 3,500 square foot house. And all your ceilings are 8 feet high, which is about standard. Now let’s say you filled up your house with cardboard boxes shaped like cubes, a foot on each side. A lot of boxes. How high do you figure they’d reach if you took them outside and stacked them on top of one another. [Click here for the answer] These are simple. But sometime it’s not so simple. Try to remain calm as you read the following: You’re playing Let’s Make a Deal, and there are three big doors onstage. One of the doors has a Rolls Royce behind it. The other two are empty. Your job is to pick one of the doors. If the Rolls is behind it, it’s yours. Simple game. But there’s one small variation. You tell Monty Hall which door you plan to pick. He then opens up one of the doors you didn’t pick, and shows it to be empty. Then Monty does an interesting thing: He offers you the option of changing your mind, and switching to the other unopened door instead. What should you do? Even though the studio audience is mindlessly screaming at you to do one thing or the other, if you’re reasonably intelligent, you realize that it doesn’t make any difference what you do. Your odds of picking the correct door were one out of three, now they’re one out of two, and that’s all there is to it. Changing your mind from one of the unopened doors to the other one couldn’t possibly make any difference. That’s your intuition talking, see? And you should trust it, right? Nope. The fact is — and I know you’re going to start yelling and screaming about this, so hang on a little — your odds of winning the Rolls double if you change your mind and open the door you didn’t pick originally. Now you think I’m completely nuts. Any idiot can see that changing your mind doesn’t make any difference. Your intuition simply won’t let it go. My guess is, at this point, you’d even be willing to bet a substantial portion of your bank account against me. Maybe even your whole bank account. You’d lose. That’s not an opinion, it’s a fact, and I’ll show you why soon. But what’s important right now is that your intuition would cost you an awful lot of money. The answer to the question concerning whether you should trust your intuition is this: only trust your intuition if you have a solid enough background in the topic at hand to have developed an intuition worth trusting. If you’ve been fixing cars successfully for twenty years and you hear a weird sound coming from the bowels of your ’56 Chevy BelAir, a sound you’ve never heard before but you are strongly convinced is a fluttering carburetor valve, you probably won’t go too far wrong ordering the part even before you’ve taken off the air filter to look at the thing. But if you’ve never been to the track in your life and don’t know a fetlock from a sulky, then that really strong hunch you have about the prospects of Pookie’s Potato in the ninth at Aqueduct is probably worth about as much as your intuition was on the Let’s Make a Deal problem. And that’s all I have to say on the subject. Wait a minute. I have a hunch that you don’t still don’t believe me on the game show thing. I know I promised to explain it to you, but you know what? I have a hunch that you still won’t believe me! So try this: Get a friend to help you simulate the Let’s Make a Deal scenario. You can use opaque envelopes, or cups turned upside down, instead of doors. Your friend puts a prize slip in one of the three envelopes without your looking. Then you get to choose one of the envelopes. Then your friend opens one of the remaining two envelopes he knows is empty. At that point, you can stick with your original choice or change your mind to the one remaining envelope. Finally, write down whether you won or lost. Play the game two hundred times. The first hundred times, never change your mind. The second hundred times, always change your mind. You will discover that you won the prize about twice as often when you changed your mind than when you stuck to the first envelope you picked. If you thought about it real hard while you were playing, you might even discover that it is beginning to dawn on you why it works that way. (This is what Martin Gardner calls the "Aha!" moment.) If you don’t feel like going through all those games, I’ve got a little program that will play them for you. It’s on my Website at http://LeeGruenfeld.com/Software/monty.htm [or click here to get there automatically if you're reading this on-line], and I promise it doesn’t contain a computer virus. So now that you really do believe me, because you proved it to yourself, I’ll give you the technical explanation in the back of the book. [or click here] But do yourself a favor and think real hard the next time you’re tempted to trust your intuition.	{"url":"https://leegruenfeld.com/good-stuffs/intuition/","timestamp":"2024-11-11T11:06:28Z","content_type":"text/html","content_length":"183854","record_id":"<urn:uuid:be3d4552-f6bd-4231-b00c-5e7b116fa6ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00298.warc.gz"}
A characterization of the Nash bargaining solution - Springer Link - P.PDFKUL.COM Soc Choice Welfare (2002) 19: 811–823 A characterization of the Nash bargaining solution Nir Dagan1, Oscar Volij2, Eyal Winter3 1 Academic Priority Ltd., Rashi 31, 52015 Ramat-Gan, Israel (e-mail: [email protected] ; http://www.nirdagan.com) 2 Department of Economics, 260 Heady Hall, Iowa State University, Ames, Iowa 50011, USA (e-mail address: [email protected] ; http://volij.co.il) 3 Department of Economics, Hebrew University, Jerusalem 91905, Israel (e-mail: [email protected] . Web site: http://www.ma.huji.ac.il/~eyalw/) Received: 4 September 2000/Accepted: 6 September 2001 Abstract. We characterize the Nash bargaining solution replacing the axiom of Independence of Irrelevant Alternatives with three independent axioms: Independence of Non-Individually Rational Alternatives, Twisting, and Disagreement Point Convexity. We give a non-cooperative bargaining interpretation to this last axiom. 1 Introduction Since Nash (1950), a bargaining problem is usually deﬁned as a pair ðS; d Þ where S is a compact, convex subset of R2 containing both d and a point that strictly dominates d. Points in S are interpreted as feasible utility agreements and d represents the status-quo outcome. A bargaining solution is a rule that assigns a feasible agreement to each bargaining problem. Nash (1950) proposed four independent properties and showed that they are simultaneously satisﬁed only by the Nash bargaining solution. While three of Nash’s axioms are quite uncontroversial, the fourth one (known as independence of irrelevant alternatives (IIA)) raised some criticisms, which lead to two di¤erent lines of research. Some authors looked for characterizations of alternative solutions which do not use the controversial axiom (see for instance, Kalai and Smorodinsky (1975), and Perles and Maschler (1981)) while other papers provided alternative characterizations of the Nash solution without appealing to the IIA axiom. Examples of this second line of research are Peters (1986b), Chun and Thomson (1990), Peters and van Damme (1991), Mariotti (1999), Mariotti (2000), and Lensberg (1988). The ﬁrst three We thank Marco Mariotti, two anonymous referees and an associate editor for helpful comments. N. Dagan et al. papers replace IIA by several axioms in conjunction with some type of continuity. The next two papers replace IIA and other axioms by one axiom. Lastly, Lensberg (1988) replaces IIA with consistency, and consequently a domain with a variable number of agents is needed. In this paper, we provide an alternative characterization of the Nash bargaining solution in which the axiom of independence of irrelevant alternatives is replaced by three di¤erent axioms. While all three of these axioms are known in the literature, they have never been used in combination. One of the axioms is independence of non-individually rational alternatives, which requires a solution to be insensitive to changes in the feasible set that involve only non-individually rational outcomes. This axiom neither implies nor is implied by IIA, but is weaker than IIA and Individual Rationality together.1 The second axiom is twisting, which is a weak monotonicity requirement that is implied by IIA. The third axiom is disagreement point convexity which requires that the solution be insensitive to movements of the disagreement point towards the proposed compromise. This last axiom does not imply nor is implied by IIA. Further, the three axioms together do not imply IIA. All of the axioms used in this paper have a straightforward interpretation except, perhaps, for disagreement point convexity. This axiom, however, has an interpretation that is closely related to non-cooperative models of bargaining. Assume that the solution recommends f ðS; d Þ when the bargaining problem is ðS; d Þ. The players may postpone the resolution of the bargaining for t periods getting f ðS; d Þ only after t periods of disagreement. From today’s point of view, knowing that one has the alternative of reaching agreement t periods later is as if the new disagreement point was f ðS; d Þ paid t periods later. Disagreement point convexity requires that the solution be insensitive to this kind of manipulation. Our result, though not its proof, is closely related to Peters and van Damme (1991). The main di¤erence is that we replace their disagreement point continuity axiom by the twisting axiom. In this way, we get rid of a mainly technical axiom and replace it by a more intuitive and reasonable one. Needless to say, disagreement point continuity and twisting, are not equivalent. Further, neither of them implies the other. The paper is organized as follows: In Sect. 2, we present the preliminary deﬁnitions and the axioms used in the characterization. Section 3 gives the main result. Section 4 shows that the axioms are independent. Finally, Sect. 5 discusses the related literature. 2 Basic deﬁnitions In this section, we present some basic deﬁnitions. Since most of them are standard, we do not provide their interpretation. 1 A solution is individually rational if it assigns each player a utility level that is not lower than its disagreement level. See next section. The Nash bargaining solution A bargaining problem is a pair ðS; d Þ where S J R2 is a compact, convex set, d A S and there is s A S with s g d.2 We denote by B the set of all bargain2 ing problems. A bargaining solution is a set-valued function f : B ! 2R nq such that for every bargaining problem B ¼ ðS; d Þ, f ðBÞ J S. We allow for set-valued solutions to highlight the role of some of the axioms in the present characterization. Let ðS; d Þ be a bargaining problem. We say that s A S is individually rational if s b d. We say that s A S is weakly e‰cient if there is no s 0 A S such that s 0 g s and that s is e‰cient if there is no s 0 A S, s 0 0 s, such that s 0 b s. We denote by IRðS; d Þ the set of individually rational points in ðS; d Þ. 2 The Nash bargaining solution is the solution n : B ! 2R nq that for each bargaining problem ðS; d Þ selects the singleton fðs1 ; s2 ÞgJS that contains the only point in IRðS; d Þ which satisﬁes ðs1 d1 Þðs2 d 2 Þ b ðs1 d1 Þðs2 d 2 Þ, for all ðs1 ; s2 Þ A IRðS; d Þ. We now turn to properties of bargaining solutions. A bargaining problem ðS; d Þ is symmetric if . d ¼ d and . ðs ; s Þ A S implies ðs ; s Þ A S. 1 We say that ðS 0 ; d 0 Þ is obtained from the bargaining problem ðS; d Þ by the transformations si ! ai si þ b i , for i ¼ 1; 2, if di0 ¼ ai di þ bi , for i ¼ 1; 2 and S 0 ¼ fða1 s1 þ b 1 ; a2 s2 þ b2 Þ A R2 : ðs1 ; s2 Þ A Sg: The following properties are standard: Symmetry. A bargaining solution f satisﬁes symmetry if for all symmetric bargaining problems ðS; d Þ, ðs1 ; s2 Þ A f ðS; d Þ , ðs2 ; s1 Þ A f ðS; d Þ: Weak Pareto optimality. A bargaining solution f satisﬁes weak Pareto optimality if for all bargaining problems ðS; d Þ, f ðS; d Þ is a subset of the weakly e‰cient points in S. It satisﬁes Pareto optimality if for all bargaining problems ðS; d Þ, f ðS; d Þ is a subset of the e‰cient points in S. Invariance. A bargaining solution satisﬁes invariance if whenever ðS 0 ; d 0 Þ is obtained from the bargaining problem ðS; d Þ by means of the transformations si ! ai si þ bi , for i ¼ 1; 2, where ai > 0 and b i A R, we have that fi ðS 0 ; d 0 Þ ¼ ai fi ðS; d Þ þ b i , for i ¼ 1; 2. IIA. A bargaining solution f satisﬁes independence of irrelevant alternatives if f ðS 0 ; d Þ ¼ f ðS; d Þ X S 0 whenever S 0 J S and f ðS; d Þ X S 0 0 q. Since we do not require solutions to be single-valued, the above properties are not enough to characterize the Nash bargaining solution. In order to establish what is essentially Nash’s characterization we need the following property. 2 We adopt the following conventions for vector inequalities: x g y $ xi > yi for all i, and x b y $ xi b yi for all i. N. Dagan et al. Single-valuedness in symmetric problems. A bargaining solution f satisﬁes single-valuedness in symmetric problems if for every symmetric problem B A B, f ðBÞ is a singleton. As stated in the introduction, we shall replace the axiom of IIA by the following three independent properties: Independence of non-individually rational alternatives. A bargaining solution satisﬁes independence with respect to non-individually rational alternatives if for every two problems ðS; d Þ and ðS 0 ; d Þ such that IRðS; d Þ ¼ IRðS 0 ; d Þ we have f ðS; d Þ ¼ f ðS 0 ; d Þ. Independence of non-individually rational alternatives requires that the solution be insensitive to changes in the feasible set that do not involve individually rational outcomes. It clearly implies that the solution always chooses a subset of the individually rational agreements. It can be checked that if a solution always chooses a subset of the individually rational agreements and also satisﬁes IIA then the solution satisﬁes independence of non-individually rational alternatives. This axiom was ﬁrst discussed in Peters (1986a). The following axiom says the following. Assume that the point s^ ¼ ð^ s1 ; s^2 Þ is chosen by the solution when the problem is ðS; d Þ. Assume further that the feasible set is modiﬁed so that all the subtracted points are preferred by one player to s^ while s^ is preferred by the same player to each of the added points. Then the axiom requires that s^ be weakly preferred by that same player to at least one point selected by the solution in the new problem ðS 0 ; d Þ. Twisting. A bargaining solution f satisﬁes twisting if the following holds: Let ðS; d Þ be a bargaining problem and let ð^ s1 ; s^2 Þ A f ðS; d Þ. Let ðS 0 ; d Þ be another bargaining problem such that for some agent i ¼ 1; 2 S nS 0 J fðs1 ; s2 Þ : si > s^i g S 0 nS J fðs1 ; s2 Þ : si < s^i g: Then, there is ðs10 ; s20 Þ A f ðS 0 ; d Þ such that si0 a s^i . Twisting is a mild monotonicity condition, which was introduced (in its singlevalued version) by Thomson and Myerson (1980) who also showed that it is implied by IIA. Twisting is satisﬁed by most solutions discussed in the literature. The next axiom was used in Peters and van Damme (1991). Thomson (1994), who calls it star-shaped inverse, succinctly summarizes this axiom as saying ‘‘that the move of the disagreement point in the direction of the desired compromise does not call for a revision of this compromise’’. Disagreement point convexity. A bargaining solution f satisﬁes disagreement point convexity if for every bargaining problem B ¼ ðS; d Þ, for all s A f ðS; d Þ and for every l A ð0; 1Þ we have s A f ðS; ð1 lÞd þ lsÞ. This axiom has a non-cooperative ﬂavor and it is related to one of the properties of the Nash equilibrium concept for extensive form games, namely The Nash bargaining solution the property that one can ‘‘fold back the tree’’. Consider an extensive form game and ﬁx a Nash equilibrium s in it. For every node n in the tree, s determines an outcome, zðn; sÞ, which is the outcome that would result if s was played in the subgame that starts at node n. In particular, s determines a Nash equilibrium outcome zðn 0 ; sÞ, where n 0 denotes the root of the tree. Now, zðn 0 ; sÞ remains a Nash equilibrium outcome if we replace any given node n by the outcome zðn; sÞ. This ‘‘tree folding property’’ is also satisﬁed by the Subgame Perfect equilibrium concept. However, we want to stress that this property is so basic that it is even satisﬁed by the Nash equilibrium concept. The axiom of disagreement point convexity tries to capture the tree folding property when applied to the subgame perfect equilibrium of a speciﬁc class of bargaining games, which we turn to describe. Many non-cooperative models of bargaining are represented by an inﬁnite-horizon stationary extensive form game with common discount factor d, Rubinstein’s (1982) alternating o¤ers model being the most prominent example. Further, the solution concept used is subgame perfect equilibrium. All these games have the following properties: 1. The disagreement outcome corresponds to the inﬁnite history in which the current proposal is rejected at every period. 2. There is an agreement a such that the unique subgame perfect equilibrium of the game dictates that a is immediately agreed upon. Further, a is immediately agreed upon at every subgame that is equivalent to the original game. To see an application of the tree folding property to one such game, consider a stationary extensive form bargaining game G with the properties 1 and 2 above3 and ﬁx a period t. Assume that at period t the proposer is the same one as in the ﬁrst period so that all subgames that start at the beginning of period t are identical to G. Build a new game by replacing each subgame of G that starts at the beginning of period t by the subgame perfect equilibrium outcome of that subgame. (Note that an outcome will typically have the format of ‘‘disagreement until period t 0 and agreement a at t 0 ’’.4). By property 2 above, this outcome is ‘‘disagreement until period t, and agreement a at t’’. The resulting game, GðtÞ, is a ﬁnite horizon extensive form game in which a history of constant rejections leads to a at period t. That is, in this new game disagreement leads to the subgame perfect equilibrium outcome a , but delayed by t periods during which there is disagreement. Still, the subgame perfect equilibrium outcome of this modiﬁed game GðtÞ is an immediate agreement on a , which is what the tree folding property says. Going back to the cooperative bargaining problem, let d be the present value of the utility stream of disagreement forever, and let s be the vector of 3 The reader may ﬁnd it convenient to consider Rubinstein’s (1982) game. 4 We have in mind bargaining over a per-period payo¤ rather than over a stock. Both approaches are equivalent since every constant ﬂow is equivalent to a stock and vice versa. N. Dagan et al. utilities that correspond to the equilibrium outcome a . Then, the shifted disagreement point ð1 lÞd þ ls in the disagreement point convexity axiom corresponds precisely to the disagreement outcome of the amended game GðtÞ, l being d t . To see this, note that the present value of a stream of t periods of disagreement and then agreement on a at t is ð1 d t Þdi þ d t si for player i, for i ¼ 1; 2.5 Using this interpretation, disagreement point convexity simply says that if we amend the bargaining problem so that the consequence of no agreement is that players disagree for t periods, and receive f ðS; d Þ afterwards (yielding a payo¤ of ð1 d t Þd þ d t f ðS; d Þ), then they should agree on f ðS; d Þ to be paid from the outset. Note that for the disagreement point to move along the segment that connects d and s when we replace the subgame with its equilibrium outcome, it is essential to assume a common discount factor. Disagreement point convexity seems to be an appropriate requirement, especially if one has in mind a stationary bargaining game. Dagan et al. (1999) exploit this axiom to give a characterization of the time-preference Nash solution in a setting with physical outcomes.6 3 The main result We can now present the main result. Theorem 1. A bargaining solution satisﬁes weak Pareto optimality, symmetry, invariance, single-valuedness in symmetric problems, independence with respect to non-individually rational allocations, twisting, and disagreement point convexity if and only if it is the Nash bargaining solution. Proof. It is known that the Nash solution satisﬁes weak Pareto optimality, symmetry, invariance and single-valuedness in symmetric problems (see Nash 1950). By its deﬁnition, the Nash solution also satisﬁes independence of nonindividually rational alternatives. Also, the Nash solution satisﬁes twisting, since twisting is weaker than IIA (see Thomson and Myerson 1980 or the Appendix for the set valued version used here), which is in turn satisﬁed by the Nash solution. Finally, Peters and van Damme (1991) showed that it also satisﬁes disagreement point convexity. This shows that the Nash solution satisﬁes all the axioms in the theorem. We now show that no other solution satisﬁes all of them together. Suppose that a solution f satisﬁes all the axioms. First step. Consider ﬁrst a triangular problem ðS; d Þ where S ¼ cofðd1 ; d 2 Þ; ðb1 ; d 2 Þ; ðd1 ; b2 Þg with bi > di for i ¼ 1; 2, and for any set A J R2 , co A is the convex hull of A. Since there are a‰ne transformations by means of which ðS; d Þ is obtained from ðcofð0; 0Þ; ð1; 0Þ; ð0; 1Þg; ð0; 0ÞÞ 1 ðI ; ð0; 0ÞÞ and since 5 If one considers a model without impatience but where after each rejected o¤er there is a probability 1 d of negotiations breakdown, resulting in d, then ð1 d t Þd þ d t s is the expected utility pair associated with a history of agreement on a after t rejections. 6 See Binmore et al. (1986) for the di¤erence between what they call the standard and the time-preference Nash solutions. The Nash bargaining solution both f and n satisfy invariance, we have that f ðS; d Þ ¼ nðS; d Þ if and only if f ðI ; ð0; 0ÞÞ ¼ nðI ; ð0; 0ÞÞ. But by single-valuedness in symmetric problems, weak Pareto optimality and symmetry of f we know that f ðI ; ð0; 0ÞÞ ¼ fð1=2; 1=2Þg ¼ nðI ; ð0; 0ÞÞ. Second step. Consider a general bargaining problem ðS; d Þ and let s^ A f ðS; d Þ. Since both n and f satisfy independence of non-individually rational alternatives, we can assume without loss of generality that IRðS; d Þ ¼ S. Case 1. s^g d: In this case, by invariance we can assume without loss of generality that d ¼ ð0; 0Þ and s^ ¼ ð1=2; 1=2Þ. It is enough to show that s^ A nðS; d Þ. Assume by contradiction that s^ B nðS; d Þ and consider the triangular problem ðcofð0; 0Þ; ð1; 0Þ; ð0; 1Þg; ð0; 0ÞÞ ¼ ðI ; ð0; 0ÞÞ. We know that nðI ; ð0; 0ÞÞ ¼ f^ sg. Since n satisﬁes IIA, we have that SUI . That is there exists s ¼ ðs1 ; s2 Þ A S nI . By weak Pareto optimality of f , s^ is a weakly e‰cient point of S. Therefore it cannot be the case that s g s^. Also, we cannot have s a s^ because otherwise s would be in I. Therefore, either s1 > s^1 or s2 > s^2 . Assume without loss of generality that s1 > s^1 and s2 < s^2 (if s1 > s^1 and s2 ¼ s^2 , then there must be another point s ¼ ðs1 ; s2 Þ A S nI , close enough to s with s1 > s^1 and s2 < s^2 ). Also, since any convex combination of s and s^ is in S nI , we can choose s g d. We now build two bargaining problems, both of which have ðs2 ; s2 Þ as disagreement point. The ﬁrst problem is ðS 0 ; ðs2 ; s2 ÞÞ, where S 0 ¼ IRðS; ðs2 ; s2 ÞÞ. The second problem is the individually rational region of the triangle whose hypothenuse is the line connectingn s and s^ (see Fig. 1). Formally, theoproblem is ðD; ðs2 ; s2 ÞÞ where D ¼ co ðs2 ; s2 Þ; ðs1 ; s2 Þ; s^2 s ðs2 ; s2 þ s ^s21 ðs1 s2 Þ . 1 By disagreement point convexity and independence of non-individually rational alternatives of f , we have s^ ¼ ð1=2; 1=2Þ A f ðS 0 ; ðs2 ; s2 ÞÞ: Further, we claim that S 0 nD J fðs1 ; s2 Þ A R2 : s1 > s^1 g and DnS 0 J fðs1 ; s2 Þ A R2 : s1 < s^1 g: Indeed, if there was a point ðs1 ; s2 Þ A S 0 nD with s1 a s^1 ¼ 1=2, then we would have that ðs1 ; s2 Þ is above the straight line that connects s^ and s . Therefore, the line segment that connects ðs1 ; s2 Þ with s is also above this line. But then, there would be a point in this segment which belongs to S and which dominates s^, which is impossible given that s^ is a weakly e‰cient point of S. Similarly, if there was a point ðs1 ; s2 Þ A DnS 0 with s1 b s^1 , then ðs1 ; s2 Þ would be on or below the straight line that connects s^ and s . Therefore, it would be a convex combination of s ^, s and ðs2 ; s2 Þ. Since the three points are in S 0 , so would ðs1 ; s2 Þ, which contradicts the fact that ðs1 ; s2 Þ B S 0 . Therefore, by twisting of f we have bðs1 ; s2 Þ A f ðD; ðs2 ; s2 ÞÞ such that s1 a s^1 ¼ 1=2: N. Dagan et al. Fig. 1. The two auxiliary problems On the other hand, since ðD; ðs2 ; s2 ÞÞ is a triangular problem, by the ﬁrst step in the proof f ðD; ðs2 ; s2 ÞÞ ¼ nðD; ðs2 ; s2 ÞÞ which implies that f ðD; ðs2 ; s2 ÞÞ ¼ fðs1 ; s2 Þg ¼ nðD; ðs2 ; s2 ÞÞ: By construction of D, the Nash solution awards player 1 in ðD; ðs2 ; s2 ÞÞ more than 1/2, that is s1 > 1=2 which contradicts (2). Case 2. s^g 6 d: Again, without loss of generality assume d ¼ ð0; 0Þ. In this case either s^ ¼ ðb1 ; 0Þ or s^ ¼ ð0; b2 Þ. Assume without loss of generality that s^ ¼ ð0; b2 Þ with b2 > 0. Pick any l A ð0; 1Þ and let SðlÞ ¼ IRðS; l^ sÞ. Since l^ s is an interior point of S in the space R2þ , we can ﬁnd a triangular set D ¼ cofl^ s; s^; ðc1 ; l^ s2 Þg that is contained in SðlÞ. Consider now the following two bargaining problems: ðSðlÞ; l^ sÞ and ðD; l^ sÞ (see Fig. 2). By disagreement point convexity and independence of non-individually rational alternatives f ðSðlÞ; l^ sÞ ¼ s^ ¼ ð0; b2 Þ. Since ðD; l^ sÞ is a triangular problem, by the ﬁrst step in the proof we have f ðD; l^ sÞ ¼ nðD; l^ sÞ ¼ ðs10 ; s20 Þ g ð0; 0Þ: By construction, we have SðlÞnD J fðs1 ; s2 Þ : s1 > s^1 g DnSðlÞ J fðs1 ; s2 Þ : s1 < s^1 g: Therefore, by twisting we must have s10 a s^1 ¼ 0 which contradicts Eq. 3. r The Nash bargaining solution Fig. 2. Case 2 Remark. It should be clear that the statement of the theorem still holds if we restrict attention to the family of bargaining problems ðS; d Þ that are comprehensive with respect to d. Namely, those bargaining problems ðS; d Þ such that if s b s 0 b d and s A S, then s 0 A S. 4 Independence of the axioms The following examples show that the seven axioms used in the characterization are independent. Beside each axiom there is a solution that fails to satisfy that axiom but which satisﬁes the other six. Weak Pareto optimality. The disagreement point solution: f : ðS; d Þ ! fd g. Symmetry. Any asymmetric Nash solution. Invariance. The Lexicographic Egalitarian solution (see, Chun and Peters 1988). Single-valuedness in symmetric problems. The set of weakly e‰cient and individually rational points. Independence of non-individually rational alternatives. The Kalai-Rosenthal solution: it selects the maximal point of S in the segment connecting d and bðS; d Þ, where bi ðS; d Þ 1 maxfxi : x A Sg (see Kalai and Rosenthal 1978). Twisting. If B can be obtained by means of a pair of a‰ne transformations from a bargaining problem B 0 ¼ ðS 0 ; d 0 Þ, where d 0 ¼ ð0; 0Þ and IRðB 0 Þ ¼ cofð0; 0Þ; ð1; 0Þ; ð1=3; 2=3Þg, then f ðBÞ is the point that is obtained by means of these transformations from ð1=3; 2=3Þ. Otherwise, f coincides with the Nash bargaining solution. N. Dagan et al. Disagreement point convexity. The Kalai-Smorodinsky bargaining solution: it selects the maximal point of S in the segment connecting d and aðS; d Þ, where ai ðS; d Þ 1 maxfxi : x A IRðS; d Þg (see, Kalai and Smorodinsky 1975). The reader may have noticed that we could have restricted solutions to be single valued instead of imposing single-valuedness in symmetric problems as an axiom. We chose this presentation to highlight the role of singlevaluedness. There are many bargaining solutions that satisfy all the axioms except for single-valuedness. As mentioned above, the set of e‰cient and individually rational outcomes is one example but there are many more. For instance, if f a is the asymmetric Nash solution that maximizes the asymmetric Nash product s1a s21 a , for a A ð0; 1Þ, then the solution that selects for every ðS; d Þ, the set f a ðS; d Þ W f 1 a ðS; d Þ also satisﬁes all the axioms except for single-valuedness. Further, it can be easily checked that if f fg gg A G is a family of bargaining solutions that satisfy weak Pareto optimality, symmetry, invariance, independence of non-individually rational outcomes, twisting and disagreement point convexity, then the solution 6g A G fg deﬁned by ð6g A G fg ÞðS; d Þ ¼ 6g A G fg ðS; d Þ satisﬁes these axioms as well. Moreover, the set of e‰cient and individually rational points is the maximal (in the sense of set inclusion) bargaining solution that satisﬁes the above axioms. It is singlevalued in symmetric problems what allows us to select the Nash bargaining solution out of the large family of solutions that satisfy the other axioms, including symmetry. We also should note that the axioms of independence of non-individually rational alternatives, twisting and disagreement point convexity that we use to replace IIA, do not imply the independence of irrelevant alternatives axiom: the solution that selects the disagreement point if the feasible set is a line segment and the Nash outcome otherwise, satisﬁes all the three axioms (in fact, satisﬁes all the axioms except for weak Pareto optimality) but does not satisfy IIA. 5 Related literature This paper provides a characterization of the Nash bargaining solution on Nash’s original domain of bargaining problems, and in which the independence axiom is replaced by three other axioms. Our result is closely related to Peters and van Damme (1991) and our contribution can be seen as eliminating of continuity axioms from the characterization. Continuity has been replaced by twisting, a mild axiom that, to our knowledge, is satisﬁed by most solution concepts discussed in the literature (the Perles-Maschler solution is one exception). Other characterizations of the Nash solution that use similar axioms, but still need continuity, are Peters (1986b) and Chun and Thomson (1990). Mariotti (1999) also provides a characterization of the Nash solution without appealing to IIA, but, as opposed to the other mentioned papers, The Nash bargaining solution he reduces the number of axioms. In fact, there are only two characterizing axioms: invariance and Suppes-Sen proofness. The same can be said about Mariotti (2000) who replaces IIA and symmetry by strong individual rationality and the axiom of Maximal Symmetry. Chun and Thomson (1990) characterize the Nash bargaining solution using two axioms, along with Pareto optimality, symmetry, scale-invariance, independence of non-individually rational outcomes, and a continuity axiom. The two axioms, which capture features of bargaining with uncertain disagreement points can be stated as follows:7 R.D.LIN. A single-valued bargaining solution f satisﬁes restricted disagreement point linearity if for every two problems ðS; d Þ and ðS; d 0 Þ, and for all a A ½0; 1, if af ðS; d Þ þ ð1 aÞ f ðS; d 0 Þ is e‰cient and S is smooth both at f ðS; d Þ and f ðS; d 0 Þ, then f ðS; ad þ ð1 aÞd 0 Þ ¼ af ðS; d Þ þ ð1 aÞ f ðS; d 0 Þ. D.Q-CAV. A single-valued bargaining solution f satisﬁes disagreement point quasi-concavity if for every two problems ðS; d Þ and ðS; d 0 Þ, and for all a A ½0; 1, fi ðS; ad þ ð1 aÞd 0 Þ b minf fi ðS; d Þ; fi ðS; d 0 Þg for i ¼ 1; 2. We now investigate the relation between these two axioms and disagreement point convexity. Claim 1. If a single-valued bargaining solution, f , satisﬁes Pareto optimality, independence of non-individually rational alternatives and D.Q-CAV., then it also satisﬁes disagreement point convexity. Proof. Let ðS; d Þ be a bargaining problem and let s ¼ f ðS; d Þ. Let l A ð0; 1Þ and assume that f ðS; ð1 lÞd þ lsÞ 0 s. Since f satisﬁes Pareto optimality, fi ðS; d Þ > fi ðS; ð1 lÞd þ lsÞ for some i ¼ 1; 2, which, without loss of generality, can be taken to be agent 1. Therefore, we can ﬁnd an a A ð0; 1Þ close enough to 1 such that the point d 0 ¼ ð1 aÞd þ as satisﬁes d10 > f1 ðS; ð1 lÞd þ lsÞ. Since f satisﬁes individual rationality, f1 ðS; d 0 Þ > f1 ðS; ð1 lÞd þ lsÞ. This inequality, together with f1 ðS; d Þ > f1 ðS; ð1 lÞd þ lsÞ imply minf f1 ðS; d Þ; f1 ðS; d 0 Þg > f1 ðS; ð1 lÞd þ lsÞ. By the way d 0 was chosen, we know that ð1 lÞd þ ls is a convex combination of d and d 0 and consequently the above inequality implies that f does not satisfy D.Q-CAV. r As a corollary, we have that we could replace weak Pareto optimality and disagreement point convexity in our characterization by Pareto optimality and D.Q-CAV. The relationship between disagreement point convexity and R.D.LIN. is not so clear, at least within the domain of problems considered in this paper. However, Pareto optimality, independence of non-individually rational alternatives and R.D.LIN. imply disagreement point convexity within the domain 7 Chun and Thomson (1990) deﬁne bargaining solutions as single-valued functions that select points from the set of feasible utilities. To facilitate comparison in what remains of this section, we use the single-valued versions of the axioms, including disagreement point convexity. N. Dagan et al. of bargaining problems with smooth Pareto frontiers provided we enlarge the deﬁnition of bargaining problems to include those pairs ðS; d Þ with e‰cient disagreement points.8 To see this, consider a bargaining problem ðS; d Þ in this domain and let f be a bargaining solution that satisﬁes Pareto optimality, independence of non-individually rational alternatives and R.D.LIN. By Pareto optimality, we have that f ðS; d Þ is e‰cient. By independence of non-individually rational alternatives, we have that f ðS; f ðS; d ÞÞ ¼ f ðS; d Þ. Since the e‰cient frontier is smooth, we can apply R.D.LIN. to conclude that f ðS; ð1 lÞd þ lf ðS; d ÞÞ ¼ f ðS; d Þ for all l A ð0; 1Þ. This means that f satisﬁes disagreement point convexity. The Nash solution is not deﬁned for the above domain. However, one can extend it, as Peters and van Damme (1991) do, so as to select the only e‰cient and individually rational point when the disagreement point is weakly e‰cient. In this case, our characterization goes through and the axioms of weak Pareto optimality and disagreement point convexity can, as a corollary of the observation of the previous paragraph, be replaced by Pareto optimality and R.D.LIN. Our characterization is on Nash’s original domain. In particular, we restrict attention to two-person bargaining problems. It is not clear whether the same axioms are su‰cient to fully characterize the Nash bargaining solution for general n-person bargaining problems. The Nash bargaining solution does satisfy all the axioms. However, our proof makes use of the 2-dimensionality of the problem. In particular, when there are more than 3 players, it is not clear how to build the auxiliary set D with the critical properties used in Step 2 of our proof. Appendix In this Appendix we show that the set valued version of the independence of irrelevant alternatives axiom that we use implies twisting. Formally: Claim 2. If a bargaining solution satisﬁes independence of irrelevant alternatives, then it also satisﬁes twisting. Proof. Let ðS; d Þ be a bargaining problem and let s^ A f ðS; d Þ. Let ðS 0 ; d Þ be another bargaining problem such that for some agent i ¼ 1; 2 S nS 0 J fðs1 ; s2 Þ : si > s^i g S 0 nS J fðs1 ; s2 Þ : si < s^i g: We need to show that there is by contradiction that ð5Þ ðs10 ; s20 Þ f ðS 0 ; d Þ J fðs1 ; s2 Þ : si > s^i g A f ðS ; d Þ such that a s^i . Assume now ð6Þ 8 Peters and van Damme (1991) consider a domain of problems that contains pairs ðS; dÞ where d is an e‰cient point of S. The Nash bargaining solution and let S^ ¼ S X S 0 . Since s^ A f ðS; d Þ X S^, we have by IIA that s^ A f ðS^; d Þ: 0 ð7Þ 0 Further, f ðS ; d Þ X S 0 q, for if f ðS ; d Þ J S nS, then by (5), f ðS ; d Þ J fðs1 ; s2 Þ : si < s^i g which was assumed in (6) not to be true. Therefore, q 0 f ðS 0 ; d Þ X S J S 0 X S ¼ S^. This implies that f ðS 0 ; d Þ X S^ 0 q and S^ J S 0 . Then, by IIA f ðS^; d Þ ¼ f ðS 0 ; d Þ X S^. But then, since by (7), s^ A f ðS^; d Þ, we have that s^ A f ðS 0 ; d Þ, which by (6) implies that s^i > s^i , which is absurd. r References Binmore KG, Rubinstein A, Wolinsky A (1986) The Nash bargaining solution in economic modeling. Rand J Econ 17: 176–188 Chun Y, Peters H (1988) The lexicographic egalitarian solution. Cahiers CERO 30: 149–156 Chun Y, Thomson W (1990) Nash solution and uncertain disagreement points. Games Econ Beh 2: 213–223 Dagan N, Volij O, Winter E (1999) The time-preference Nash solution. Unpubl. manuscript, Department of Economics, Hebrew University Kalai E, Rosenthal RW (1978) Arbitration of two-party disputes under ignorance. Int J Game Theory 7: 65–72 Kalai E, Smorodinsky M (1975) Other solutions to Nash’s bargaining problem. Econometrica 43: 513–518 Lensberg T (1988) Stability and the Nash solution. J Econ Theory 45: 330–341 Mariotti M (1999) Fair bargains: Distributive justice and Nash bargaining theory. Rev Econ Stud 66: 733–741 Mariotti M (2000) Maximal symmetry and the Nash solution. Soc Choice Welfare 17: 45–53 Nash JF (1950) The bargaining problem. Econometrica 28: 155–162 Perles M, Maschler M (1981) A superadditive solution to Nash bargaining games. Int J Game Theory 10: 163–193 Peters H (1986a) Characterizations of bargaining solutions by properties of their status quo. Research Memorandum 86-012, University of Limburg Peters H (1986b) Simultaneity of issues and additivity in bargaining. Econometrica 54: 153–169 Peters H, van Damme E (1991) Characterizing the Nash and Rai¤a bargaining solutions by disagreement point axioms. Math Oper Res 16(3): 447–461 Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica 50: 97– 110 Thomson W (1994) Cooperative models of bargaining. In: Aumann RJ, Hart S (eds) Handbook of game theory, Vol. 2, North–Holland, Amsterdam, pp 1237–1284 Thomson W, Myerson RB (1980) Monotonicity and independence axioms. Int J Game Theory 9: 37–49	{"url":"https://p.pdfkul.com/a-characterization-of-the-nash-bargaining-solution-springer-link_5a2cb6c71723ddd3b9dd5e05.html","timestamp":"2024-11-14T04:09:59Z","content_type":"text/html","content_length":"92104","record_id":"<urn:uuid:f3bba81c-061b-4c9d-91fb-8770caff8a37>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00214.warc.gz"}
Convex multivariable trace functions For any densely defined, lower semi-continuous trace τ on a C-algebra A with mutually commuting C-subalgebras A[1], A[2],...A[n], and a convex function f of n variables, we give a short proof of the fact that the function (x[1], x[2],...,x[n]) → τ(f(x[1], x[2],...,x[n])) is convex on the space ⊕[i=1]^n(A[i])[sa]. If furthermore the function f is log-convex or root-convex, so is the corresponding trace function. We also introduce a generalization of log-convexity and root-convexity called l-convexity, show how it applies to traces, and give some examples. In particular we show that the Kadison-Fuglede determinant is concave and that the trace of an operator mean is always dominated by the corresponding mean of the trace values. All Science Journal Classification (ASJC) codes • Statistical and Nonlinear Physics • Mathematical Physics • Operator algebras • Trace functions • Trace inequalities Dive into the research topics of 'Convex multivariable trace functions'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/convex-multivariable-trace-functions","timestamp":"2024-11-11T04:17:59Z","content_type":"text/html","content_length":"48071","record_id":"<urn:uuid:d11f2ec3-ed4b-49e5-8562-735edb3012f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00640.warc.gz"}
Kanda Data, Author at KANDA DATA When you are completing your final project as a student, you will usually find descriptive statistical analysis results in one of the chapters. It can be seen in a separate sub-chapter or part of one of the chapters written in the thesis. For example, sub-chapters have used descriptive statistical analysis in economics and agribusiness research. … Why is Descriptive Statistical Analysis Important? Read More » Choosing Simple Random Sampling in Conducting Research Simple random sampling has often been used by researchers when determining the sample. In this case, the researchers chose a random sample. Researchers who choose this technique must meet the required assumptions. Incidentally, on this occasion, I will discuss the topic of simple random sample selection techniques. Comparison of Two Sample Dependent (Paired t-test) The comparison of two paired samples becomes interesting to discuss this time. This two-sample comparison test is often analyzed using a paired t-test. Many researchers or students who are conducting research choose to use this test. How to Compute Spearman Rank Correlation Test A correlation test is still often an option to solve problems in research. The correlation test includes the Pearson correlation test, Spearman rank correlation test, and chi-square test. Determining the type of correlation test to use depends on the measurement data scale. Well, on this occasion, I will discuss using the Spearman rank correlation test. Autocorrelation Test on Time Series Data using Linear Regression The autocorrelation test is one of the assumptions of linear regression with the OLS method. On this occasion, I will discuss the autocorrelation test on time series data. Before discussing the autocorrelation test, you need to know first that the autocorrelation test was conducted on time series, not cross-sectional data.	{"url":"https://kandadata.com/author/kanda-data/page/40/","timestamp":"2024-11-12T00:42:49Z","content_type":"text/html","content_length":"180695","record_id":"<urn:uuid:f10539e9-e156-49fc-980f-13022949085e>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00193.warc.gz"}
In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if ${\displaystyle X}$ and ${\displaystyle Y}$ are sets and ${\displaystyle L\subseteq X\times Y}$ is a relation from ${\displaystyle X}$ to ${\displaystyle Y,} $ then ${\displaystyle L^{\operatorname {T} }}$ is the relation defined so that ${\displaystyle yL^{\operatorname {T} }x}$ if and only if ${\displaystyle xLy.}$ In set-builder notation, ${\displaystyle L^{\operatorname {T} }=\{(y,x)\in Y\times X:(x,y)\in L\}.}$ Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation^[1]^[2]^[3]^[4] is also called the transpose relation.^[5] It has also been called the opposite or dual of the original relation,^[6] the inverse of the original relation,^[7]^[8]^[9]^[10] or the reciprocal ${\displaystyle L^{\circ }} $ of the relation ${\displaystyle L.}$^[11] Other notations for the converse relation include ${\displaystyle L^{\operatorname {C} },L^{-1},{\breve {L}},L^{\circ },}$ or ${\displaystyle L^{\vee }.}$ The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique converse. The unary operation that maps a relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on the category of relations as detailed below. As a unary operation, taking the converse (sometimes called conversion or transposition) commutes with the order-related operations of the calculus of relations, that is it commutes with union, intersection, and complement. For the usual (maybe strict or partial) order relations, the converse is the naively expected "opposite" order, for examples, ${\displaystyle {\leq ^{\operatorname {T} }}={\geq },\quad {<^{\ operatorname {T} }}={>}.}$ A relation may be represented by a logical matrix such as ${\displaystyle {\begin{pmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{pmatrix}}.}$ Then the converse relation is represented by its transpose matrix: ${\displaystyle {\begin{pmatrix}1&0&0&0\\1&1&0&0\\1&0&1&0\\1&1&0&1\end{pmatrix}}.}$ The converse of kinship relations are named: "${\displaystyle A}$ is a child of ${\displaystyle B}$ " has converse "${\displaystyle B}$ is a parent of ${\displaystyle A}$ ". "${\displaystyle A}$ is a nephew or niece of ${\displaystyle B}$ " has converse "${\displaystyle B}$ is an uncle or aunt of ${\displaystyle A}$ ". The relation "${\displaystyle A}$ is a sibling of ${\displaystyle B}$ " is its own converse, since it is a symmetric relation. In the monoid of binary endorelations on a set (with the binary operation on relations being the composition of relations), the converse relation does not satisfy the definition of an inverse from group theory, that is, if ${\displaystyle L}$ is an arbitrary relation on ${\displaystyle X,}$ then ${\displaystyle L\circ L^{\operatorname {T} }}$ does not equal the identity relation on ${\ displaystyle X}$ in general. The converse relation does satisfy the (weaker) axioms of a semigroup with involution: ${\displaystyle \left(L^{\operatorname {T} }\right)^{\operatorname {T} }=L}$ and $ {\displaystyle (L\circ R)^{\operatorname {T} }=R^{\operatorname {T} }\circ L^{\operatorname {T} }.}$ ^[12] Since one may generally consider relations between different sets (which form a category rather than a monoid, namely the category of relations Rel), in this context the converse relation conforms to the axioms of a dagger category (aka category with involution).^[12] A relation equal to its converse is a symmetric relation; in the language of dagger categories, it is self-adjoint. Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relations as sets), and actually an involutive quantale. Similarly, the category of heterogeneous relations, Rel is also an ordered category.^[12] In the calculus of relations, conversion (the unary operation of taking the converse relation) commutes with other binary operations of union and intersection. Conversion also commutes with unary operation of complementation as well as with taking suprema and infima. Conversion is also compatible with the ordering of relations by inclusion.^[5] If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, connected, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its converse is too. If ${\displaystyle I}$ represents the identity relation, then a relation ${\displaystyle R}$ may have an inverse as follows: ${\displaystyle R}$ is called if there exists a relation ${\displaystyle X,}$ called a right inverse of ${\displaystyle R,}$ that satisfies ${\displaystyle R\circ X=I.}$ if there exists a relation ${\displaystyle Y,}$ called a left inverse of ${\displaystyle R,}$ that satisfies ${\displaystyle Y\circ R=I.}$ if it is both right-invertible and left-invertible. For an invertible homogeneous relation ${\displaystyle R,}$ all right and left inverses coincide; this unique set is called its inverse and it is denoted by ${\displaystyle R^{-1}.}$ In this case, $ {\displaystyle R^{-1}=R^{\operatorname {T} }}$ holds.^[5]^:79 Converse relation of a function A function is invertible if and only if its converse relation is a function, in which case the converse relation is the inverse function. The converse relation of a function ${\displaystyle f:X\to Y}$ is the relation ${\displaystyle f^{-1}\subseteq Y\times X}$ defined by the ${\displaystyle \operatorname {graph} \,f^{-1}=\{(y,x)\in Y\ times X:y=f(x)\}.}$ This is not necessarily a function: One necessary condition is that ${\displaystyle f}$ be injective, since else ${\displaystyle f^{-1}}$ is multi-valued. This condition is sufficient for ${\ displaystyle f^{-1}}$ being a partial function, and it is clear that ${\displaystyle f^{-1}}$ then is a (total) function if and only if ${\displaystyle f}$ is surjective. In that case, meaning if ${\ displaystyle f}$ is bijective, ${\displaystyle f^{-1}}$ may be called the inverse function of ${\displaystyle f.}$ For example, the function ${\displaystyle f(x)=2x+2}$ has the inverse function ${\displaystyle f^{-1}(x)={\frac {x}{2}}-1.}$ However, the function ${\displaystyle g(x)=x^{2}}$ has the inverse relation ${\displaystyle g^{-1}(x)=\pm {\sqrt {x}},}$ which is not a function, being multi-valued. Composition with relation Using composition of relations, the converse may be composed with the original relation. For example, the subset relation composed with its converse is always the universal relation: ∀A ∀B ∅ ⊂ A ∩B ⇔ A ⊃ ∅ ⊂ B ⇔ A ⊃ ⊂ B. Similarly, For U = universe, A ∪ B ⊂ U ⇔ A ⊂ U ⊃ B ⇔ A ⊂ ⊃ B. Now consider the set membership relation and its converse. ${\displaystyle Ai z\in B\Leftrightarrow z\in A\cap B\Leftrightarrow A\cap Beq \emptyset .}$ Thus ${\displaystyle Ai \in B\Leftrightarrow A\cap Beq \emptyset .}$ The opposite composition ${\displaystyle \in i }$ is the universal relation. The compositions are used to classify relations according to type: for a relation Q, when the identity relation on the range of Q contains Q^TQ, then Q is called univalent. When the identity relation on the domain of Q is contained in Q Q^T, then Q is called total. When Q is both univalent and total then it is a function. When Q^T is univalent, then Q is termed injective. When Q^T is total, Q is termed surjective.^[13] If Q is univalent, then QQ^T is an equivalence relation on the domain of Q, see Transitive relation#Related properties. See also	{"url":"https://www.knowpia.com/knowpedia/Converse_relation","timestamp":"2024-11-09T03:25:49Z","content_type":"text/html","content_length":"174901","record_id":"<urn:uuid:14ea6755-e6cf-46f2-80a7-dbdaee36abee>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00207.warc.gz"}
wu :: forums - Permutation groups where only 1 fixes two letters wu :: forums putnam exam (pure math) (Moderators: towr, Eigenray, Grimbal, Icarus, SMQ, william wu) « Previous topic \| Next topic » Pages: 1 Reply Notify of replies Send Topic Print Author Topic: Permutation groups where only 1 fixes two letters (Read 811 times) ecoist Permutation groups where only 1 fixes two letters Senior « on: May 9^th, 2007, 6:31pm » Quote Modify Let G be a finite group with a subgroup H such that H is its own normalizer in G and any two conjugates of H intersect trivially. Using character theory, it can be shown that the identity and all elements of G not in any conjugate of H form a subgroup N normal in G. Is there a proof of this that does not use character theory? (Oops! Had left out "any conjugate of" in the original post.) 405 « Last Edit: May 10^th, 2007, 10:21am by ecoist » Obob Re: Permutation groups where only 1 fixes two lett Senior « Reply #1 on: May 9^th, 2007, 7:11pm » Quote Modify The subgroup N, shown to exist via character theory, is usually defined as the set of elements of G that act on the coset space G/H without fixed points. The theorem invoked to prove N is in fact a normal subgroup is called Frobenius' theorem, which states that for a transitive permutation group G on a set X, such that no element other than the identity has more than one fixed point, the set of fixed-point free elements together with the identity gives a normal subgroup of G. I know I have been told by Jon Alperin that no known proof of Frobenius' theorem without character theory is known. Now suppose we have a proof that the Frobenius kernel N of any Frobenius group G is normal. (A Frobenius group is a group G with a subgroup H satisfying the hypotheses of the riddle; the Frobenius kernel is the set of fixed-point free elements of G for the action on G/H, together with the identity.) Let us show that this implies Frobenius' theorem is true. We are given a group G acting transitively and faithfully on a set X. The G-set X is then isomorphic to the coset space G/H as a G-set, since any transitive G-set is isomorphic to such a Gender: G-set. Every element of G fixes at most one coset in G/H. Now the stabilizer of g_1 H is g_1 H g_1^{-1}, so if g_1 H g_1^{-1} intersects g_2 H g_2^{-1} nontrivially, there is some g fixing Posts: both g_1 H and g_2 H, whence g_1 H = g_2 H, and therefore g_1 H g_1^{-1} = g_2 H g_2^{-1}. Hence any two conjugates of H intersect trivially or are equal. If g is not in H, then the 489 stabilizer of g H meets the stabilizer of H trivially, since any element of G fixing both is the identity. This implies H is self-normalizing. Thus G is a Frobenius group with Frobenius complement H, and N is a normal subgroup. Thus Frobenius' theorem holds. In particular, a character-theory free proof of the statement about Frobenius groups implies a character-theory free proof of Frobenius' theorem. « Last Edit: May 10^th, 2007, 10:37am by Obob » ecoist Re: Permutation groups where only 1 fixes two lett Senior « Reply #2 on: May 9^th, 2007, 11:06pm » Quote Modify You are misstaken, Obob. The number of elements in the set N is \|G\|-(\|H\|-1)[G:H], where [G:H] is the index in G of H. This is because H has [G:H] conjugates and two distinct conjugates have only the identity in common. Hence \|N\|=[G:H]. The statement as given is equivalent to G being a permutation group in which only the identity fixes two letters. If Alperin is right, you have answered my question, but I find it hard to believe that character theory is required for this result. Obob Re: Permutation groups where only 1 fixes two lett Senior Riddler « Reply #3 on: May 10^th, 2007, 8:24am » Quote Modify Didn't you define N to be the complement of H, together with 1? I agree that, with the correct definition of N, \|N\|=[G:H], and in fact G is the semidirect product of H and N. Posts: 489 Obob Re: Permutation groups where only 1 fixes two lett Senior Riddler « Reply #4 on: May 10^th, 2007, 8:27am » Quote Modify I haven't had the time to read it, but this book review seems to support Alperin's position. http://www.ams.org/bull/1999-36-04/S0273-0979-99-00789-2/S0273-0979-99-0 0789-2.pdf Posts: 489 ecoist Re: Permutation groups where only 1 fixes two lett Senior Riddler « Reply #5 on: May 10^th, 2007, 10:25am » Quote Modify Sorry, Obob. Just now saw the error in my post and corrected it. I'd like to see those "partial proofs" in Huppert's book! Posts: 405 Obob Re: Permutation groups where only 1 fixes two lett Senior Riddler « Reply #6 on: May 10^th, 2007, 10:33am » Quote Modify Unfortunately Amazon doesn't have search inside this book for Huppert's book, and it appears to be checked out at my library. Posts: 489 Pages: 1 Reply Notify of replies Send Topic Print « Previous topic \| Next topic »	{"url":"https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_putnam;action=display;num=1178760677","timestamp":"2024-11-14T06:47:32Z","content_type":"text/html","content_length":"45379","record_id":"<urn:uuid:535aa7d8-5d6b-46b5-af4d-95fcbc10bf2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00524.warc.gz"}
A First Course in Probabililty - Chapter 1 Problems Flashcards (a) How many different 7-place license plates are possible if the first 2 places are for letters and the other 5 for numbers? (b) Repeat part (a) under the assumption that no letter or number can be repeated in a single license plate. Number of license plates: 26 x 26 x 10 x 10 x 10 x 10 x 10 = 67,600,000 Number of license plates with no repeated letters or digits: 26 x 25 x 10 x 9 x 8 x 7 x 6 = 19,656,00 How many outcome sequences are possible when a die is rolled four times, where we say, for instance, that the outcome is 3, 4, 3, 1 if the first roll landed on 3, the second on 4, the third on 3, and the fourth on 1? Possible outcomes for 1 roll = 6 (1,2,3,4,5,6) Possible outcomes for 4 rolls = 6 x 6 x 6 x 6 = 1296 Twenty workers are to be assigned to 20 different jobs, one to each job. How many different assignments are possible? Number of permutations = 20! John, Jim, Jay, and Jack have formed a band consisting of 4 instruments. If each of the boys can play all 4 instruments, how many different arrangements are possible? What if John and Jim can play all 4 instruments, but Jay and Jack can each play only piano and drums? Total arrangements for 4 boys playing 4 instruments = 4! = 24 Total arrangements for Jay and Jack to be assigned piano and drums = 2! Total arrangements for John and Jim to be assigned the other two instruments = 2! Total arrangements = 2! x 2! = 4 For years, telephone area codes in the United States and Canada consisted of a sequence of three digits. The first digit was an integer between 2 and 9, the second digit was either 0 or 1, and the third digit was any integer from 1 to 9. How many area codes were possible? How many area codes starting with a 4 were possible? 1st digit = 2-9 (8 options) 2nd digit = 0-1 (2 options) 3rd digit = 1-9 (9 options) Total possible area codes: 8 x 2 x 9 = 144 1st digit = 4 (1 option) 2nd digit = 0-1 (2 options) 3rd digit = 1-9 (9 options) Total possible area codes: 1 x 2 x 9 = 18 A well-known nursery rhyme starts as follows: “As I was going to St. Ives I met a man with 7 wives. Each wife had 7 sacks. Each sack had 7 cats. Each cat had 7 kittens. . .” How many kittens did the traveler meet? (a) In how many ways can 3 boys and 3 girls sit in a row? (b) In how many ways can 3 boys and 3 girls sit in a row if the boys and the girls are each to sit together? (c) In how many ways if only the boys must sit together? (d) In how many ways if no two people of the same sex are allowed to sit together? Total ways: 6! = 720 3 boys can sit in 3! ways 3 girls can sit in 3! ways 2 groups of boys and girls can sit in 2! ways Total ways: 3! x 3! x 2! = 72 3 girls can sit in 3! ways Boys can sit in any of the remaining four positions (_G1_G2_G3_) 3 boys can also sit in 3! ways Total ways: 3! x 4 x 3! = 144 2 ways (BGBGBG or GBGBGB) 3 girls can also sit in 3! ways 3 boys can also sit in 3! ways Total ways: 2 x 3! x 3! = 72 How many different letter arrangements can be made from the letters (a) Fluke? (b) Propose? (c) Mississippi? (d) Arrange? Notes for b, c and d: Duplicate letters are taken into consideration A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts the blocks in a line, how many arrangements are possible? In how many ways can 8 people be seated in a row if: (a) there are no restrictions on the seating arrangement? (b) persons A and B must sit next to each other? (c) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other? (d) there are 5 men and they must sit next to each other? (e) there are 4 married couples and each couple must sit together? 8 people can be seated in 8! ways = 40,320 A and B can be seated in 2! ways The remaining 7 can be seated in 7! ways Total ways: 7! x 2! = 10,080 4 men can be seated in 4! ways 4 women can be seated in 4! ways There are 2 ways to seat either MWMWMWMW or WMWMWMWM Total ways: 4! x 4! x 2 = 1152 3 woman can be seated in 3! ways Men can sit together in 4 places (1-5, 2-6, 3-7 or 4-8) 5 men can be seated in 5! ways Total ways: 3! x 4 x 5! = 2880 4 couples can be seated in 4! ways 2 people part of a couple can be seated in 2! ways Total ways: 4! x 2! x 2! x 2! x 2! = 384 In how many ways can 3 novels, 2 mathematics books, and 1 chemistry book be arranged on a bookshelf if: (a) the books can be arranged in any order? (b) the mathematics books must be together and the novels must be together? (c) the novels must be together, but the other books can be arranged in any order? Total ways: 6! = 720 3 novels can be arranged in 3! ways 2 mathematics books can be arranged in 2! ways 1 chemistry book can be arranged in 1! way 3 groups of books can be arranged in 3! ways Total ways: 3! x 2! x 1! x 3! = 72 3 novels equal 1 group 4 groups can be arranged in 4! ways 3 novels can be arranged in 3! ways in a group Total ways: 4! x 3! = 144 Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to selected students from a class of 30. How many different outcomes are possible if: (a) a student can receive any number of awards? (b) each student can receive at most 1 award? 30 x 30 x 30 x 30 x 30 = 24,300,000 30 students eligible for 1st award 29 students eligible for 2nd award and so on... 30 x 29 x 28 x 27 x 26 = 17,100,720 Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take place? How many 5-card poker hands are there? A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible? A student has to sell 2 books from a collection of 6 math, 7 science, and 4 economics books. How many choices are possible if: (a) both books are to be on the same subject? (b) the books are to be on different subjects? Notes for a: Sells 2 math books = (6c2) Sells 2 science books = (7c2) Sells 2 economics books = (4c2) Notes for b: 1 math book & 1 science book = (6c1)(7c1) 1 science book & 1 economics book = (7c1)(4c1) 1 economics book & 1 math book = (4c1)(6c1) Seven different gifts are to be distributed among 10 children. How many distinct results are possible if no child is to receive more than one gift? 10 x 9 x 8 x 7 x 6 x 5 x 4 = 604,800 distinct gifts A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from a group of 5 Republicans, 6 Democrats, and 4 Independents. How many committees are possible? # of ways to select 2 Republicans = (5c2) # of ways to select 2 Democrats = (6c2) # of ways to select 3 Independents = (4c3) From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How many different committees are possible if: (a) 2 of the men refuse to serve together? (b) 2 of the women refuse to serve together? (c) 1 man and 1 woman refuse to serve together? A person has 8 friends, of whom 5 will be invited to a party. (a) How many choices are there if 2 of the friends are feuding and will not attend together? (b) How many choices if 2 of the friends will only attend together? Notes for a: # of ways if both fighting friends are not invited = (6c5) # of ways if one of the fighting friend is invited = (6c4) x 2 Notes for b: # of ways if they attend = (6c3) # of ways if they do not attend = (6c5) Consider the grid of points shown here. Suppose that, starting at the point labeled A, you can go one step up or one step to the right at each move. This procedure is continued until the point labeled B is reached. How many different paths from A to B are possible? Hint: Note that to reach B from A, you must take 4 steps to the right and 3 steps upward. Answer shows the number of ways to select 4 steps to the right out of 7. In Problem 21, how many different paths are there from A to B that go through the point circled in the following lattice? To reach circled point for A, you need to take 2 steps up and 2 steps to the right. 2 steps to the right out of 4 total steps = (4c2) To reach point B from circled point, you need to take 1 step up and 2 steps to the right. 2 steps to the right out of 3 total steps = (3c2) A psychology laboratory conducting dream research contains 3 rooms, with 2 beds in each room. If 3 sets of identical twins are to be assigned to these 6 beds so that each set of twins sleeps in different beds in the same room, how many assignments are possible? 3 pair of twins into 3 different rooms = 3! One bed for each twin in one room = 2! = 2 Total ways: 3! x 2 x 2 x 2 = 48 The game of bridge is played by 4 players, each of whom is dealt 13 cards. How many bridge deals are possible? If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions are possible? If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each school must receive 2 teachers? 4⁸ = 65,536 8!/2!2!2!2! = 2520 Ten weight lifters are competing in a team weightlifting contest. Of the lifters, 3 are from the United States, 4 are from Russia, 2 are from China, and 1 is from Canada. If the scoring takes account of the countries that the lifters represent, but not their individual identities, how many different outcomes are possible from the point of view of scores? How many different outcomes correspond to results in which the United States has 1 competitor in the top three and 2 in the bottom three? Notes for b: selecting 2 out 7 and 1 out of 3 = (7c2)(3c1) selecting next 4 positions out of 5 non-US members = (5c4) selecting 2 positions out of 3 for US members = (3c2) selecting 1 position out of 1 remaining non-US member = (1c1) Delegates from 10 countries, including Russia, France, England, and the United States, are to be seated in a row. How many different seating arrangements are possible if the French and English delegates are to be seated next to each other and the Russian and U.S. delegates are not to be next to each other? Calculate French and English next to each other: French and English can be arranged in 2! ways All 9 groups can be arranged in 9! ways Totals ways: 2! x 9! = 725,760 Calculate Russian and US not next to each other: All 8 groups can be arranged in 8! ways The 2 groups French & English and Russian & US can be arranged 2! ways each Totals ways: 8! x 2! x 2! = 161,280 To get answer need to subtract second from first: (2! x 9!) - (8! x 2! x 2!) = 564,480 If 8 identical blackboards are to be divided among 4 schools, how many divisions are possible? How many if each school must receive at least 1 blackboard? n₁ + n₂ + n₃ + n₄ = 8 n is the # of blackboards for a particular school An elevator starts at the basement with 8 people (not including the elevator operator) and discharges them all by the time it reaches the top floor, number 6. In how many ways could the operator have perceived the people leaving the elevator if all people look alike to him? What if the 8 people consisted of 5 men and 3 women and the operator could tell a man from a woman? We have 20 thousand dollars that must be invested among 4 possible opportunities. Each investment must be integral in units of 1 thousand dollars, and there are minimal investments that need to be made if one is to invest in these opportunities. The minimal investments are 2, 2, 3, and 4 thousand dollars. How many different investment strategies are available if: (a) an investment must be made in each opportunity? (b) investments must be made in at least 3 of the 4 opportunities? Notes for a: n₁ + n₂ + n₃ + n₄ = 9 Notes for b: investments made for all = (12c3) investments (2,2,3) = (15c2) investments (2,2,4) = (14c2) investments (2,3,4) = (13c2) investments (2,3,4) = (13c2)	{"url":"https://www.easynotecards.com/notecard_set/2438","timestamp":"2024-11-10T12:50:03Z","content_type":"text/html","content_length":"42270","record_id":"<urn:uuid:2757b6ee-e514-4847-9384-fde421d8c02d>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00341.warc.gz"}
DIPSA: Data-Intensive Parallel Systems and Algorithms We use MASTIFF to compute the weight of Minimum Spanning Forest (MST) of MS-BioGraphs while ignoring self-edges of the graphs. – MS1 Using machine with 24 cores. MSF weight: 109,915,787,546 – MS50 Using machine with 128 cores. MSF weight: 416,318,200,808 Related Posts MS-BioGraphs on IEEE DataPort MS-BioGraph sequence similarity graph datasets are now publicly available on IEEE DataPort: https://doi.org/10.21227/gmd9-1534. To access the files, you need to register/login to IEEE DataPort and then visit the MS-BioGraphs page. By saving the page as an HTML file such as dp.html, you may download the datasets (as an example MS1) using the following script: urls=`cat $html_file \| sed -e 's/\&/\&/g' \| grep -Eo "(http\|https)://[a-zA-Z0-9./?&=_%:-]" \| grep amazonaws \| sort \| uniq \| grep -E "$dsname[-_\.]"` for u in $urls; do wget $u if [ $? != 0 ]; then break; fi # removing query strings for f in $(find $1 -type f); do if [ $f = ${f%%\?} ]; then continue; fi mv "${f}" "${f%%\?*}" # liking offsets.bin to be found by ParaGrapher ln -s ${dsname}_offsets.bin ${dsname}-underlying_offsets.bin Instead of wget you may use axel -n 10 to use multiple connections (here, 10) for downloading each file (https://manpages.ubuntu.com/manpages/noble/en/man1/axel.1.html). Related Posts MS-BioGraphs MS Name MS-BioGraphs – MS URL https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-MS Download Link https://doi.org/10.21227/gmd9-1534 Script for Downloading All Files https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-on-IEEE-DataPort/ Validating and Sample Code https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-Validation/ Graph Explanation Vertices represent proteins and each edge represents the sequence similarity between its two endpoints Edge Weighted Yes Directed No Number of Vertices 1,757,323,526 Number of Edges 2,488,069,027,875 Maximum Degree 814,957 Minimum Weight 98 Maximum Weight 634,925 Number of Zero-Degree Vertices 6,437,984 Average Degree 1,415.8 Size of The Largest WCC 2,486,890,448,664 Number of WCC 148,861,367 Creation Details MS-BioGraphs: Sequency Similarity Graph Datasets Format WebGraph License CC BY-NC-SA QUB IDF 2223-052 DOI 10.5281/zenodo.7820808 Mohsen Koohi Esfahani, Sebastiano Vigna, Paolo Boldi, Hans Vandierendonck, Peter Kilpatrick, March 13, 2024, Citation "MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets", IEEE Dataport, doi: https://doi.org/10.21227/gmd9-1534. doi = {10.21227/gmd9-1534}, url = {https://doi.org/10.21227/gmd9-1534}, Bibtex author = {Koohi Esfahani, Mohsen and Vigna, Sebastiano and Boldi, Paolo and Vandierendonck, Hans and Kilpatrick, Peter}, publisher = {IEEE Dataport}, title = {MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets}, year = {2024} } The underlying graph in WebGraph format: • File: MS-underlying.graph, Size: 7,342,853,446,646 Bytes Underlying Graph • File: MS-underlying.offsets, Size: 5,341,385,503 Bytes • File: MS-underlying.properties, Size: 1,560 Bytes Total Size: 7,348,194,833,709 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. The weights of the graph in WebGraph format: • File: MS-weights.labels, Size: 5,037,171,681,279 Bytes Weights (Labels) • File: MS-weights.labeloffsets, Size: 5,070,752,590 Bytes • File: MS-weights.properties, Size: 183 Bytes Total Size: 5,042,242,434,052 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. This file contains the shasums of edge blocks where each block contains 64 Million continuous edges and has one shasum for its 64M endpoints and one for its 64M edge The file is used to validate the underlying graph and the weights. For further explanation about validation process, please visit the https://blogs.qub.ac.uk/DIPSA/ Edge Blocks SHAs File MS-BioGraphs-Validation. • Name: MS_edges_shas.txt • Size: 4,449,360 Bytes • SHASUM: 85d5b0896f8fa8a2b490ec6560937c45ced8b0d9 The offsets array of the CSX (Compressed Sparse Rows/Columns) graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, respectively. This array helps converting the graph (or parts of it) from WebGraph format to binary format by one pass over (related) edges. Offsets (Binary) • Name: MS_offsets.bin • Size: 14,058,588,216 Bytes • SHASUM: 15c3defdbb92f7b1fe48a3fb20530d99fa30c616 The Weakly-Connected Compontent (WCC) array in binary format and little endian order. This array consists of \|V\| 4-Bytes elements The vertices in the same component have the same values in the WCC array. WCC (Binary) • Name: MS-wcc.bin • Size: 7,029,294,104 Bytes • SHASUM: 30f12b738dde8f62aecb94239796b169512e6710 This compressed file contains 120 files in CSV format using ‘;’ as the separator. Each row has two columns: ID of vertex and name of the sequence. Note: If the graph has a ‘N2O Reordering’ file, the n2o array should be used to convert the vertex ID to old vertex ID which is used for identifying name of the protein in the `names.tar.gz` file. Names (tar.gz) • Name: names.tar.gz • Size: 27,130,045,933 Bytes • SHASUM: ba00b58bbb2795445554058a681b573c751ef315 The charactersitics of the graph and shasums of the files. It is in the open json format and needs a closing brace (}) to be appended before being passed to a json parser. • Name: MS.ojson • Size: 700 Bytes • SHASUM: e2eb3fcdd0c22838971ed2edea8e1ed081a77282 For the explanation about the plots, please refer to the MS-BioGraphs paper. To have a better resolution, please click on the images. Degree Distribution Weight Distribution Vertex-Relative Weight Distribution Degree Decomposition Cell-Binned Average Weight Degree Distribution Weakly-Connected Components Size Distribution Related Posts MS-BioGraphs MSA500 Name MS-BioGraphs – MSA500 URL https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-MSA500 Download Link https://doi.org/10.21227/gmd9-1534 Script for Downloading All Files https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-on-IEEE-DataPort/ Validating and Sample Code https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-Validation/ Graph Explanation Vertices represent proteins and each edge represents the sequence similarity between its two endpoints Edge Weighted Yes Directed Yes Number of Vertices 1,757,323,526 Number of Edges 1,244,904,754,157 Maximum In-Degree 229,442 Maximum Out-Degree 814,461 Minimum Weight 98 Maximum Weight 634,925 Number of Zero In-Degree Vertices 6,437,984 Number of Zero Out-Degree Vertices 16,843,087 Average In-Degree 711.0 Average Out-Degree 715.3 Size of The Largest Weakly Connected Component 1,244,203,865,823 Number of Weakly Connected Components 148,861,367 Creation Details MS-BioGraphs: Sequency Similarity Graph Datasets Format WebGraph License CC BY-NC-SA QUB IDF 2223-052 DOI 10.5281/zenodo.7820810 Mohsen Koohi Esfahani, Sebastiano Vigna, Paolo Boldi, Hans Vandierendonck, Peter Kilpatrick, March 13, 2024, Citation "MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets", IEEE Dataport, doi: https://doi.org/10.21227/gmd9-1534. doi = {10.21227/gmd9-1534}, url = {https://doi.org/10.21227/gmd9-1534}, Bibtex author = {Koohi Esfahani, Mohsen and Vigna, Sebastiano and Boldi, Paolo and Vandierendonck, Hans and Kilpatrick, Peter}, publisher = {IEEE Dataport}, title = {MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets}, year = {2024} } The underlying graph in WebGraph format: • File: MSA500-underlying.graph, Size: 3,755,604,574,487 Bytes Underlying Graph • File: MSA500-underlying.offsets, Size: 4,811,273,232 Bytes • File: MSA500-underlying.properties, Size: 1,537 Bytes Total Size: 3,760,415,849,256 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. The weights of the graph in WebGraph format: • File: MSA500-weights.labels, Size: 2,520,671,185,509 Bytes Weights (Labels) • File: MSA500-weights.labeloffsets, Size: 4,554,987,345 Bytes • File: MSA500-weights.properties, Size: 187 Bytes Total Size: 2,525,226,173,041 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. This file contains the shasums of edge blocks where each block contains 64 Million continuous edges and has one shasum for its 64M endpoints and one for its 64M edge The file is used to validate the underlying graph and the weights. For further explanation about validation process, please visit the https://blogs.qub.ac.uk/DIPSA/ Edge Blocks SHAs File MS-BioGraphs-Validation. • Name: MSA500_edges_shas.txt • Size: 2,226,360 Bytes • SHASUM: d9f692b6f4770f282ea62936293baf6a649c2b91 The offsets array of the CSX (Compressed Sparse Rows/Columns) graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, respectively. This array helps converting the graph (or parts of it) from WebGraph format to binary format by one pass over (related) edges. Offsets (Binary) • Name: MSA500_offsets.bin • Size: 14,058,588,216 Bytes • SHASUM: 3eab31d99426ed9f96af6b258fd1253544ba5461 The Weakly-Connected Compontent (WCC) array in binary format and little endian order. This array consists of \|V\| 4-Bytes elements The vertices in the same component have the same values in the WCC array. WCC (Binary) • Name: MSA500-wcc.bin • Size: 7,029,294,104 Bytes • SHASUM: 30f12b738dde8f62aecb94239796b169512e6710 The offsets array of the transposed graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, It helps to transpose the graph by performing one pass over edges. Transposed’s Offsets (Binary) • Name: MSA500_trans_offsets.bin • Size: 14,058,588,216 Bytes • SHASUM: 220a2a5c60baaedc8913720862b535ba6cabb5bd This compressed file contains 120 files in CSV format using ‘;’ as the separator. Each row has two columns: ID of vertex and name of the sequence. Note: If the graph has a ‘N2O Reordering’ file, the n2o array should be used to convert the vertex ID to old vertex ID which is used for identifying name of the protein in the `names.tar.gz` file. Names (tar.gz) • Name: names.tar.gz • Size: 27,130,045,933 Bytes • SHASUM: ba00b58bbb2795445554058a681b573c751ef315 The charactersitics of the graph and shasums of the files. It is in the open json format and needs a closing brace (}) to be appended before being passed to a json parser. • Name: MSA500.ojson • Size: 902 Bytes • SHASUM: 5eaebdff2dc56925a0b4751f579ebeabb6e3bee5 For the explanation about the plots, please refer to the MS-BioGraphs paper. To have a better resolution, please click on the images. In-Degree Distribution Out-Degree Distribution Weight Distribution Vertex-Relative Weight Distribution Degree Decomposition Push and Pull Locality Cell-Binned Average Weight Degree Distribution Weakly-Connected Components Size Distribution Related Posts MS-BioGraphs MS200 Name MS-BioGraphs – MS200 URL https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-MS200 Download Link https://doi.org/10.21227/gmd9-1534 Script for Downloading All Files https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-on-IEEE-DataPort/ Validating and Sample Code https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-Validation/ Graph Explanation Vertices represent proteins and each edge represents the sequence similarity between its two endpoints Edge Weighted Yes Directed No Number of Vertices 1,414,493,449 Number of Edges 502,930,788,612 Maximum Degree 745,735 Minimum Weight 460 Maximum Weight 634,925 Number of Zero-Degree Vertices 0 Average Degree 355.6 Size of The Largest WCC 485,867,547,569 Number of WCC 338,348,495 Creation Details MS-BioGraphs: Sequency Similarity Graph Datasets Format WebGraph License CC BY-NC-SA QUB IDF 2223-052 DOI 10.5281/zenodo.7820812 Mohsen Koohi Esfahani, Sebastiano Vigna, Paolo Boldi, Hans Vandierendonck, Peter Kilpatrick, March 13, 2024, Citation "MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets", IEEE Dataport, doi: https://doi.org/10.21227/gmd9-1534. doi = {10.21227/gmd9-1534}, url = {https://doi.org/10.21227/gmd9-1534}, Bibtex author = {Koohi Esfahani, Mohsen and Vigna, Sebastiano and Boldi, Paolo and Vandierendonck, Hans and Kilpatrick, Peter}, publisher = {IEEE Dataport}, title = {MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets}, year = {2024} } The underlying graph in WebGraph format: • File: MS200-underlying.graph, Size: 1,459,981,767,426 Bytes Underlying Graph • File: MS200-underlying.offsets, Size: 3,174,012,489 Bytes • File: MS200-underlying.properties, Size: 1,515 Bytes Total Size: 1,463,155,781,430 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. The weights of the graph in WebGraph format: • File: MS200-weights.labels, Size: 1,199,053,831,206 Bytes Weights (Labels) • File: MS200-weights.labeloffsets, Size: 3,090,041,102 Bytes • File: MS200-weights.properties, Size: 186 Bytes Total Size: 1,202,143,872,494 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. This file contains the shasums of edge blocks where each block contains 64 Million continuous edges and has one shasum for its 64M endpoints and one for its 64M edge The file is used to validate the underlying graph and the weights. For further explanation about validation process, please visit the https://blogs.qub.ac.uk/DIPSA/ Edge Blocks SHAs File MS-BioGraphs-Validation. • Name: MS200_edges_shas.txt • Size: 899,640 Bytes • SHASUM: 5bb635fc94aea3ee7b2b6a4aecbbb1fc6f77e1b5 The offsets array of the CSX (Compressed Sparse Rows/Columns) graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, respectively. This array helps converting the graph (or parts of it) from WebGraph format to binary format by one pass over (related) edges. Offsets (Binary) • Name: MS200_offsets.bin • Size: 11,315,947,600 Bytes • SHASUM: 9192158aab65e1ca536a46183411d87452cd9ee3 The Weakly-Connected Compontent (WCC) array in binary format and little endian order. This array consists of \|V\| 4-Bytes elements The vertices in the same component have the same values in the WCC array. WCC (Binary) • Name: MS200-wcc.bin • Size: 5,657,973,796 Bytes • SHASUM: 027e1b826659b5ec0f62921a4eb3ecd6c83fa76a This compressed file contains 120 files in CSV format using ‘;’ as the separator. Each row has two columns: ID of vertex and name of the sequence. Note: If the graph has a ‘N2O Reordering’ file, the n2o array should be used to convert the vertex ID to old vertex ID which is used for identifying name of the protein in the `names.tar.gz` file. Names (tar.gz) • Name: names.tar.gz • Size: 27,130,045,933 Bytes • SHASUM: ba00b58bbb2795445554058a681b573c751ef315 The New to Old (N2O) reordering array of the graph in binary format and little endian order. It consists of \|V\| 4-Bytes elements and identifies the old ID of each vertex which is used in searching the name of vertex (protein) in the names.tar.gz file . N2O Reordering (Binary) • Name: MS200-n2o.bin • Size: 5,657,973,796 Bytes • SHASUM: de833f1c36011af07c165f53760b82a49715537d The charactersitics of the graph and shasums of the files. It is in the open json format and needs a closing brace (}) to be appended before being passed to a json parser. • Name: MS200.ojson • Size: 757 Bytes • SHASUM: 540c0bded9ab8d334574ed7dd7909435b617ecf3 For the explanation about the plots, please refer to the MS-BioGraphs paper. To have a better resolution, please click on the images. Degree Distribution Weight Distribution Vertex-Relative Weight Distribution Degree Decomposition Cell-Binned Average Weight Degree Distribution Weakly-Connected Components Size Distribution Related Posts MS-BioGraphs MSA200 Name MS-BioGraphs – MSA200 URL https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-MSA200 Download Link https://doi.org/10.21227/gmd9-1534 Script for Downloading All Files https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-on-IEEE-DataPort/ Validating and Sample Code https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-Validation/ Graph Explanation Vertices represent proteins and each edge represents the sequence similarity between its two endpoints Edge Weighted Yes Directed Yes Number of Vertices 1,757,323,526 Number of Edges 500,444,322,597 Maximum In-Degree 658,879 Maximum Out-Degree 709,176 Minimum Weight 98 Maximum Weight 634,925 Number of Zero In-Degree Vertices 6,437,984 Number of Zero Out-Degree Vertices 7,471,315 Average In-Degree 285.8 Average Out-Degree 286.0 Size of The Largest Weakly Connected Component 496,880,685,957 Number of Weakly Connected Components 221,467,156 Creation Details MS-BioGraphs: Sequency Similarity Graph Datasets Format WebGraph License CC BY-NC-SA QUB IDF 2223-052 DOI 10.5281/zenodo.7820815 Mohsen Koohi Esfahani, Sebastiano Vigna, Paolo Boldi, Hans Vandierendonck, Peter Kilpatrick, March 13, 2024, Citation "MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets", IEEE Dataport, doi: https://doi.org/10.21227/gmd9-1534. doi = {10.21227/gmd9-1534}, url = {https://doi.org/10.21227/gmd9-1534}, Bibtex author = {Koohi Esfahani, Mohsen and Vigna, Sebastiano and Boldi, Paolo and Vandierendonck, Hans and Kilpatrick, Peter}, publisher = {IEEE Dataport}, title = {MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets}, year = {2024} } The underlying graph in WebGraph format: • File: MSA200-underlying.graph, Size: 1,558,147,532,780 Bytes Underlying Graph • File: MSA200-underlying.offsets, Size: 4,319,801,854 Bytes • File: MSA200-underlying.properties, Size: 1,517 Bytes Total Size: 1,562,467,336,151 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. The weights of the graph in WebGraph format: • File: MSA200-weights.labels, Size: 1,105,784,580,128 Bytes Weights (Labels) • File: MSA200-weights.labeloffsets, Size: 4,123,546,304 Bytes • File: MSA200-weights.properties, Size: 187 Bytes Total Size: 1,109,908,126,619 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. This file contains the shasums of edge blocks where each block contains 64 Million continuous edges and has one shasum for its 64M endpoints and one for its 64M edge The file is used to validate the underlying graph and the weights. For further explanation about validation process, please visit the https://blogs.qub.ac.uk/DIPSA/ Edge Blocks SHAs File MS-BioGraphs-Validation. • Name: MSA200_edges_shas.txt • Size: 895,200 Bytes • SHASUM: de1ac0ddce536168881ca2e49e6d5f0cf5b82bb5 The offsets array of the CSX (Compressed Sparse Rows/Columns) graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, respectively. This array helps converting the graph (or parts of it) from WebGraph format to binary format by one pass over (related) edges. Offsets (Binary) • Name: MSA200_offsets.bin • Size: 14,058,588,216 Bytes • SHASUM: c241d2dc4bdf46f60c1cd889ac367504d3f58805 The Weakly-Connected Compontent (WCC) array in binary format and little endian order. This array consists of \|V\| 4-Bytes elements The vertices in the same component have the same values in the WCC array. WCC (Binary) • Name: MSA200-wcc.bin • Size: 7,029,294,104 Bytes • SHASUM: 2cb256d5e49e5dd0989715cb909fd8f27bfbd04c The offsets array of the transposed graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, It helps to transpose the graph by performing one pass over edges. Transposed’s Offsets (Binary) • Name: MSA200_trans_offsets.bin • Size: 14,058,588,216 Bytes • SHASUM: 47787ac64fb4485da02e3bcdc1696a814adfdb86 This compressed file contains 120 files in CSV format using ‘;’ as the separator. Each row has two columns: ID of vertex and name of the sequence. Note: If the graph has a ‘N2O Reordering’ file, the n2o array should be used to convert the vertex ID to old vertex ID which is used for identifying name of the protein in the `names.tar.gz` file. Names (tar.gz) • Name: names.tar.gz • Size: 27,130,045,933 Bytes • SHASUM: ba00b58bbb2795445554058a681b573c751ef315 The charactersitics of the graph and shasums of the files. It is in the open json format and needs a closing brace (}) to be appended before being passed to a json parser. • Name: MSA200.ojson • Size: 897 Bytes • SHASUM: 18e371cbb4bd9dbe6515e4528956ff32fb2e30c4 For the explanation about the plots, please refer to the MS-BioGraphs paper. To have a better resolution, please click on the images. In-Degree Distribution Out-Degree Distribution Weight Distribution Vertex-Relative Weight Distribution Degree Decomposition Push and Pull Locality Cell-Binned Average Weight Degree Distribution Weakly-Connected Components Size Distribution Related Posts MS-BioGraphs MS50 Name MS-BioGraphs – MS50 URL https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-MS50 Download Link https://doi.org/10.21227/gmd9-1534 Script for Downloading All Files https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-on-IEEE-DataPort/ Validating and Sample Code https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-Validation/ Graph Explanation Vertices represent proteins and each edge represents the sequence similarity between its two endpoints Edge Weighted Yes Directed No Number of Vertices 585,603,088 Number of Edges 124,783,559,600 Maximum Degree 507,826 Minimum Weight 900 Maximum Weight 634,925 Number of Zero-Degree Vertices 0 Average Degree 213.1 Size of The Largest WCC 102,256,631,195 Weight of Minimum Spanning Forest (ignoring self-edges) 416,318,200,808 click for details Number of WCC 155,295,301 Creation Details MS-BioGraphs: Sequency Similarity Graph Datasets Format WebGraph License CC BY-NC-SA QUB IDF 2223-052 DOI 10.5281/zenodo.7820819 Mohsen Koohi Esfahani, Sebastiano Vigna, Paolo Boldi, Hans Vandierendonck, Peter Kilpatrick, March 13, 2024, Citation "MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets", IEEE Dataport, doi: https://doi.org/10.21227/gmd9-1534. doi = {10.21227/gmd9-1534}, url = {https://doi.org/10.21227/gmd9-1534}, Bibtex author = {Koohi Esfahani, Mohsen and Vigna, Sebastiano and Boldi, Paolo and Vandierendonck, Hans and Kilpatrick, Peter}, publisher = {IEEE Dataport}, title = {MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets}, year = {2024} } The underlying graph in WebGraph format: • File: MS50-underlying.graph, Size: 347,621,279,586 Bytes Underlying Graph • File: MS50-underlying.offsets, Size: 1,235,232,971 Bytes • File: MS50-underlying.properties, Size: 1,459 Bytes Total Size: 348,856,514,016 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. The weights of the graph in WebGraph format: • File: MS50-weights.labels, Size: 324,269,690,037 Bytes Weights (Labels) • File: MS50-weights.labeloffsets, Size: 1,221,399,047 Bytes • File: MS50-weights.properties, Size: 185 Bytes Total Size: 325,491,089,269 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. This file contains the shasums of edge blocks where each block contains 64 Million continuous edges and has one shasum for its 64M endpoints and one for its 64M edge The file is used to validate the underlying graph and the weights. For further explanation about validation process, please visit the https://blogs.qub.ac.uk/DIPSA/ Edge Blocks SHAs File MS-BioGraphs-Validation. • Name: MS50_edges_shas.txt • Size: 223,440 Bytes • SHASUM: 5d1bc449124448e9a6ed3bd439942e31f55d9f97 The offsets array of the CSX (Compressed Sparse Rows/Columns) graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, respectively. This array helps converting the graph (or parts of it) from WebGraph format to binary format by one pass over (related) edges. Offsets (Binary) • Name: MS50_offsets.bin • Size: 4,684,824,712 Bytes • SHASUM: b298f974167a1c64a8ba8e211a970c5b5d427137 The Weakly-Connected Compontent (WCC) array in binary format and little endian order. This array consists of \|V\| 4-Bytes elements The vertices in the same component have the same values in the WCC array. WCC (Binary) • Name: MS50-wcc.bin • Size: 2,342,412,352 Bytes • SHASUM: 4d640ce445477191a3bc3dd00f09f712b9429af2 This compressed file contains 120 files in CSV format using ‘;’ as the separator. Each row has two columns: ID of vertex and name of the sequence. Note: If the graph has a ‘N2O Reordering’ file, the n2o array should be used to convert the vertex ID to old vertex ID which is used for identifying name of the protein in the `names.tar.gz` file. Names (tar.gz) • Name: names.tar.gz • Size: 27,130,045,933 Bytes • SHASUM: ba00b58bbb2795445554058a681b573c751ef315 The New to Old (N2O) reordering array of the graph in binary format and little endian order. It consists of \|V\| 4-Bytes elements and identifies the old ID of each vertex which is used in searching the name of vertex (protein) in the names.tar.gz file . N2O Reordering (Binary) • Name: MS50-n2o.bin • Size: 2,342,412,352 Bytes • SHASUM: 91939605bdde3eb67a013f80d4c2a84d1684ca8f The charactersitics of the graph and shasums of the files. It is in the open json format and needs a closing brace (}) to be appended before being passed to a json parser. • Name: MS50.ojson • Size: 751 Bytes • SHASUM: eb94812bea81cd40a3f33d6aaa5fdd63946ffc92 For the explanation about the plots, please refer to the MS-BioGraphs paper. To have a better resolution, please click on the images. Degree Distribution Weight Distribution Vertex-Relative Weight Distribution Degree Decomposition Cell-Binned Average Weight Degree Distribution Weakly-Connected Components Size Distribution Related Posts MS-BioGraphs MSA50 Name MS-BioGraphs – MSA50 URL https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-MSA50 Download Link https://doi.org/10.21227/gmd9-1534 Script for Downloading All Files https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-on-IEEE-DataPort/ Validating and Sample Code https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-Validation/ Graph Explanation Vertices represent proteins and each edge represents the sequence similarity between its two endpoints Edge Weighted Yes Directed Yes Number of Vertices 1,757,323,526 Number of Edges 125,312,536,732 Maximum In-Degree 543,117 Maximum Out-Degree 297,981 Minimum Weight 98 Maximum Weight 634,925 Number of Zero In-Degree Vertices 6,437,984 Number of Zero Out-Degree Vertices 8,542,018 Average In-Degree 71.6 Average Out-Degree 71.7 Size of The Largest Weakly Connected Component 117,980,151,055 Number of Weakly Connected Components 363,090,851 Creation Details MS-BioGraphs: Sequency Similarity Graph Datasets Format WebGraph License CC BY-NC-SA QUB IDF 2223-052 DOI 10.5281/zenodo.7820821 Mohsen Koohi Esfahani, Sebastiano Vigna, Paolo Boldi, Hans Vandierendonck, Peter Kilpatrick, March 13, 2024, Citation "MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets", IEEE Dataport, doi: https://doi.org/10.21227/gmd9-1534. doi = {10.21227/gmd9-1534}, url = {https://doi.org/10.21227/gmd9-1534}, Bibtex author = {Koohi Esfahani, Mohsen and Vigna, Sebastiano and Boldi, Paolo and Vandierendonck, Hans and Kilpatrick, Peter}, publisher = {IEEE Dataport}, title = {MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets}, year = {2024} } The underlying graph in WebGraph format: • File: MSA50-underlying.graph, Size: 410,094,612,576 Bytes Underlying Graph • File: MSA50-underlying.offsets, Size: 3,504,554,221 Bytes • File: MSA50-underlying.properties, Size: 1,493 Bytes Total Size: 413,599,168,290 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. The weights of the graph in WebGraph format: • File: MSA50-weights.labels, Size: 284,756,409,010 Bytes Weights (Labels) • File: MSA50-weights.labeloffsets, Size: 3,374,946,996 Bytes • File: MSA50-weights.properties, Size: 186 Bytes Total Size: 288,131,356,192 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. This file contains the shasums of edge blocks where each block contains 64 Million continuous edges and has one shasum for its 64M endpoints and one for its 64M edge The file is used to validate the underlying graph and the weights. For further explanation about validation process, please visit the https://blogs.qub.ac.uk/DIPSA/ Edge Blocks SHAs File MS-BioGraphs-Validation. • Name: MSA50_edges_shas.txt • Size: 224,400 Bytes • SHASUM: 6f56a6710ef6b6e7c01e90907f19c7a0099a272c The offsets array of the CSX (Compressed Sparse Rows/Columns) graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, respectively. This array helps converting the graph (or parts of it) from WebGraph format to binary format by one pass over (related) edges. Offsets (Binary) • Name: MSA50_offsets.bin • Size: 14,058,588,216 Bytes • SHASUM: 3272fb9c681648598f18ab5a10bbafb5bf48dca5 The Weakly-Connected Compontent (WCC) array in binary format and little endian order. This array consists of \|V\| 4-Bytes elements The vertices in the same component have the same values in the WCC array. WCC (Binary) • Name: MSA50-wcc.bin • Size: 7,029,294,104 Bytes • SHASUM: 82e3ba326bb56c69edbe7fbb90ce70b731e3a7f2 The offsets array of the transposed graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, It helps to transpose the graph by performing one pass over edges. Transposed’s Offsets (Binary) • Name: MSA50_trans_offsets.bin • Size: 14,058,588,216 Bytes • SHASUM: 812d75359683dd235a1bd948566b306f43e7088d This compressed file contains 120 files in CSV format using ‘;’ as the separator. Each row has two columns: ID of vertex and name of the sequence. Note: If the graph has a ‘N2O Reordering’ file, the n2o array should be used to convert the vertex ID to old vertex ID which is used for identifying name of the protein in the `names.tar.gz` file. Names (tar.gz) • Name: names.tar.gz • Size: 27,130,045,933 Bytes • SHASUM: ba00b58bbb2795445554058a681b573c751ef315 The charactersitics of the graph and shasums of the files. It is in the open json format and needs a closing brace (}) to be appended before being passed to a json parser. • Name: MSA50.ojson • Size: 892 Bytes • SHASUM: 5767cdd2e0cddba1ba255afe9accfdbe5d5aabd2 For the explanation about the plots, please refer to the MS-BioGraphs paper. To have a better resolution, please click on the images. In-Degree Distribution Out-Degree Distribution Weight Distribution Vertex-Relative Weight Distribution Degree Decomposition Push and Pull Locality Cell-Binned Average Weight Degree Distribution Weakly-Connected Components Size Distribution Related Posts MS-BioGraphs MSA10 Name MS-BioGraphs – MSA10 URL https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-MSA10 Download Link https://doi.org/10.21227/gmd9-1534 Script for Downloading All Files https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-on-IEEE-DataPort/ Validating and Sample Code https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-Validation/ Graph Explanation Vertices represent proteins and each edge represents the sequence similarity between its two endpoints Edge Weighted Yes Directed Yes Number of Vertices 1,757,323,526 Number of Edges 25,236,632,682 Maximum In-Degree 207,279 Maximum Out-Degree 62,060 Minimum Weight 98 Maximum Weight 634,925 Number of Zero In-Degree Vertices 6,437,984 Number of Zero Out-Degree Vertices 9,926,249 Average In-Degree 14.4 Average Out-Degree 14.4 Size of The Largest Weakly Connected Component 15,576,385,764 Number of Weakly Connected Components 628,505,933 Creation Details MS-BioGraphs: Sequency Similarity Graph Datasets Format WebGraph License CC BY-NC-SA QUB IDF 2223-052 DOI 10.5281/zenodo.7820823 Mohsen Koohi Esfahani, Sebastiano Vigna, Paolo Boldi, Hans Vandierendonck, Peter Kilpatrick, March 13, 2024, Citation "MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets", IEEE Dataport, doi: https://doi.org/10.21227/gmd9-1534. doi = {10.21227/gmd9-1534}, url = {https://doi.org/10.21227/gmd9-1534}, Bibtex author = {Koohi Esfahani, Mohsen and Vigna, Sebastiano and Boldi, Paolo and Vandierendonck, Hans and Kilpatrick, Peter}, publisher = {IEEE Dataport}, title = {MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets}, year = {2024} } The underlying graph in WebGraph format: • File: MSA10-underlying.graph, Size: 87,421,101,649 Bytes Underlying Graph • File: MSA10-underlying.offsets, Size: 2,743,422,804 Bytes • File: MSA10-underlying.properties, Size: 1,439 Bytes Total Size: 90,164,525,892 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. The weights of the graph in WebGraph format: • File: MSA10-weights.labels, Size: 58,798,062,287 Bytes Weights (Labels) • File: MSA10-weights.labeloffsets, Size: 2,731,563,328 Bytes • File: MSA10-weights.properties, Size: 186 Bytes Total Size: 61,529,625,801 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. This file contains the shasums of edge blocks where each block contains 64 Million continuous edges and has one shasum for its 64M endpoints and one for its 64M edge The file is used to validate the underlying graph and the weights. For further explanation about validation process, please visit the https://blogs.qub.ac.uk/DIPSA/ Edge Blocks SHAs File MS-BioGraphs-Validation. • Name: MSA10_edges_shas.txt • Size: 45,480 Bytes • SHASUM: 9c42e8ba057c519ae318071e63ab3ffdf992cd50 The offsets array of the CSX (Compressed Sparse Rows/Columns) graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, respectively. This array helps converting the graph (or parts of it) from WebGraph format to binary format by one pass over (related) edges. Offsets (Binary) • Name: MSA10_offsets.bin • Size: 14,058,588,216 Bytes • SHASUM: b42a8f6aee7c0abdd715f523238ea59acb09c24b The Weakly-Connected Compontent (WCC) array in binary format and little endian order. This array consists of \|V\| 4-Bytes elements The vertices in the same component have the same values in the WCC array. WCC (Binary) • Name: MSA10-wcc.bin • Size: 7,029,294,104 Bytes • SHASUM: 37f30d638341fa50ae9c73893e7cab689ef14be8 The offsets array of the transposed graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, It helps to transpose the graph by performing one pass over edges. Transposed’s Offsets (Binary) • Name: MSA10_trans_offsets.bin • Size: 14,058,588,216 Bytes • SHASUM: 2ae765f6f79b8f41221ba0d869648d01d19bcadd This compressed file contains 120 files in CSV format using ‘;’ as the separator. Each row has two columns: ID of vertex and name of the sequence. Note: If the graph has a ‘N2O Reordering’ file, the n2o array should be used to convert the vertex ID to old vertex ID which is used for identifying name of the protein in the `names.tar.gz` file. Names (tar.gz) • Name: names.tar.gz • Size: 27,130,045,933 Bytes • SHASUM: ba00b58bbb2795445554058a681b573c751ef315 The charactersitics of the graph and shasums of the files. It is in the open json format and needs a closing brace (}) to be appended before being passed to a json parser. • Name: MSA10.ojson • Size: 885 Bytes • SHASUM: 0d8c48f9297d36a628aabcd8576cb0c083607534 For the explanation about the plots, please refer to the MS-BioGraphs paper. To have a better resolution, please click on the images. In-Degree Distribution Out-Degree Distribution Weight Distribution Vertex-Relative Weight Distribution Degree Decomposition Push and Pull Locality Cell-Binned Average Weight Degree Distribution Weakly-Connected Components Size Distribution Related Posts MS-BioGraphs MS1 Name MS-BioGraphs – MS1 URL https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-MS1 Download Link https://doi.org/10.21227/gmd9-1534 Script for Downloading All Files https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-on-IEEE-DataPort/ Validating and Sample Code https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs-Validation/ Graph Explanation Vertices represent proteins and each edge represents the sequence similarity between its two endpoints Edge Weighted Yes Directed No Number of Vertices 43,144,218 Number of Edges 2,660,495,200 Maximum Degree 14,212 Minimum Weight 3,680 Maximum Weight 634,925 Number of Zero-Degree Vertices 0 Average Degree 61.7 Size of The Largest WCC 124,003,393 Number of WCC 15,746,208 Weight of Minimum Spanning Forest (ignoring self-edges) 109,915,787,546 click for details Creation Details MS-BioGraphs: Sequency Similarity Graph Datasets Format WebGraph License CC BY-NC-SA QUB IDF 2223-052 DOI 10.5281/zenodo.7820827 Mohsen Koohi Esfahani, Sebastiano Vigna, Paolo Boldi, Hans Vandierendonck, Peter Kilpatrick, March 13, 2024, Citation "MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets", IEEE Dataport, doi: https://doi.org/10.21227/gmd9-1534. doi = {10.21227/gmd9-1534}, url = {https://doi.org/10.21227/gmd9-1534}, Bibtex author = {Koohi Esfahani, Mohsen and Vigna, Sebastiano and Boldi, Paolo and Vandierendonck, Hans and Kilpatrick, Peter}, publisher = {IEEE Dataport}, title = {MS-BioGraphs: Trillion-Scale Sequence Similarity Graph Datasets}, year = {2024} } The underlying graph in WebGraph format: • File: MS1-underlying.graph, Size: 6,300,911,484 Bytes Underlying Graph • File: MS1-underlying.offsets, Size: 77,574,569 Bytes • File: MS1-underlying.properties, Size: 1,288 Bytes Total Size: 6,378,487,341 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. The weights of the graph in WebGraph format: • File: MS1-weights.labels, Size: 8,201,441,365 Bytes Weights (Labels) • File: MS1-weights.labeloffsets, Size: 80,797,007 Bytes • File: MS1-weights.properties, Size: 184 Bytes Total Size: 8,282,238,556 Bytes These files are validated using ‘Edge Blocks SHAs File’ as follows. This file contains the shasums of edge blocks where each block contains 64 Million continuous edges and has one shasum for its 64M endpoints and one for its 64M edge The file is used to validate the underlying graph and the weights. For further explanation about validation process, please visit the https://blogs.qub.ac.uk/DIPSA/ Edge Blocks SHAs File MS-BioGraphs-Validation. • Name: MS1_edges_shas.txt • Size: 5,040 Bytes • SHASUM: 27974edb4bf8f3b17b00ff3a72a703da18f3807a The offsets array of the CSX (Compressed Sparse Rows/Columns) graph in binary format and little endian order. It consists of \|V\|+1 8-Bytes elements. The first and last values are 0 and \|E\|, respectively. This array helps converting the graph (or parts of it) from WebGraph format to binary format by one pass over (related) edges. Offsets (Binary) • Name: MS1_offsets.bin • Size: 345,153,752 Bytes • SHASUM: 0abedde32e1ac7181897f82d10d40acfe14f2022 The Weakly-Connected Compontent (WCC) array in binary format and little endian order. This array consists of \|V\| 4-Bytes elements The vertices in the same component have the same values in the WCC array. WCC (Binary) • Name: MS1-wcc.bin • Size: 172,576,872 Bytes • SHASUM: 4c491dd96e3582b70a203ae4a910001381278d75 This compressed file contains 120 files in CSV format using ‘;’ as the separator. Each row has two columns: ID of vertex and name of the sequence. Note: If the graph has a ‘N2O Reordering’ file, the n2o array should be used to convert the vertex ID to old vertex ID which is used for identifying name of the protein in the `names.tar.gz` file. Names (tar.gz) • Name: names.tar.gz • Size: 27,130,045,933 Bytes • SHASUM: ba00b58bbb2795445554058a681b573c751ef315 The New to Old (N2O) reordering array of the graph in binary format and little endian order. It consists of \|V\| 4-Bytes elements and identifies the old ID of each vertex which is used in searching the name of vertex (protein) in the names.tar.gz file . N2O Reordering (Binary) • Name: MS1-n2o.bin • Size: 172,576,872 Bytes • SHASUM: b163320b6349fed7a00fb17c4a4a22e7d124b716 The charactersitics of the graph and shasums of the files. It is in the open json format and needs a closing brace (}) to be appended before being passed to a json parser. • Name: MS1.ojson • Size: 736 Bytes • SHASUM: c60afa0652955fd46f1bb8056380523504d69fa6 For the explanation about the plots, please refer to the MS-BioGraphs paper. To have a better resolution, please click on the images. Degree Distribution Weight Distribution Vertex-Relative Weight Distribution Degree Decomposition Cell-Binned Average Weight Degree Distribution Weakly-Connected Components Size Distribution Related Posts	{"url":"https://blogs.qub.ac.uk/dipsa/tag/dataset/","timestamp":"2024-11-13T08:26:01Z","content_type":"text/html","content_length":"250754","record_id":"<urn:uuid:5965ef91-a3da-47d5-b8cd-49e769bbcd62>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00000.warc.gz"}
Debugging \| Level 4 Maths \| NZ Level 4 Writing computer programs (coding) can be a process of trial and error. It is quite common for programmers to make errors when they are writing their code. These mistakes are known as 'bugs' and they stop the computer program (code) from doing what it is supposed to. An important part of programming is testing your program and 'debugging' (which means removing or fixing the bugs). Here is an algorithm which lists the instructions for making breakfast 1. Take a bowl out of cupboard, a spoon out of the drawer and place them on the table 2. Pour cereal and milk into the bowl 3. Place the bowl in the sink 4. Eat the cereal out of the bowl Question: What is the bug in this algorithm? Think: Imagine you are making breakfast in the morning and follow each step of the Algorithm. Can you find the error? Do: Lines 3 and 4 are in the wrong order. 1. Try to identify where the program or code is not working 2. Go through this section of your code step, by step thinking about what each command does, writing down the result of each step if possible 3. If you are still having difficulty finding the bug, take a break and look at it again, or ask a friend to look at your code.	{"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-407/topics/Topic-7224/subtopics/Subtopic-96511/","timestamp":"2024-11-12T10:31:48Z","content_type":"text/html","content_length":"673849","record_id":"<urn:uuid:aff30f6b-1e0d-46d8-9ad0-3aabe9bf0cb4>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00016.warc.gz"}
RNN for uPOMDPs Verifiable RNN-Based Policies for Uncertain POMDPs Recurrent neural networks (RNNs) (Hochreiter & Schmidhuber, 1997) are an effective representation of policies for partially observable Markov Decision Processes (POMDPs)(Kaelbling et al., 1998). However, a major drawback of RNN-based policies is the difficulty to formally verify behavioural specifications, e.g. with regard to reachability and expected cost. In previous work, we proposed to insert a quantized bottleneck network (QBN) to the RNN that learns a mapping from the latent memory states to quantized vectors, which enables the extraction of a finite-state controller (FSC) representation of the RNN (Carr et al., 2021). This FSC, together with a POMDP model description, constitutes a policy-induced Discrete-Time Markov Chain (DTMC) that allows us to use efficient formal verification methods. For the scenarios in which the FSC fails to satisfy the behavioural specification, the verification method generates diagnostic information in the form of critical examples. These critical examples can be used to re-train the RNN and extract an updated FSC. In this project, we are interested to investigate the synthesis of policies with formally verified satisfaction of behavioural specifications in uncertain POMDPs (uPOMDPs) (Suilen et al., 2020; Cubuktepe et al., 2021), where the uncertainty is expressed by polynomial parameterizations of the transition and/or observation probabilities. The uPOMDP describes a set of possible POMDPs that can be used to express imperfect knowledge of the environment, e.g. because the POMDP is an abstraction of real-world dynamics. Our goal is to learn and extract an FSC that has the best worst-case performance among all possible instantiations of the uPOMDP. 1. Hochreiter, S., & Schmidhuber, J. (1997). Long Short-Term Memory. Neural Comput., 9(8), 1735–1780. 2. Kaelbling, L. P., Littman, M. L., & Cassandra, A. R. (1998). Planning and Acting in Partially Observable Stochastic Domains. Artif. Intell., 101(1-2), 99–134. 3. Carr, S., Jansen, N., & Topcu, U. (2021). Task-Aware Verifiable RNN-Based Policies for Partially Observable Markov Decision Processes. J. Artif. Intell. Res., 72, 819–847. 4. Suilen, M., Jansen, N., Cubuktepe, M., & Topcu, U. (2020). Robust Policy Synthesis for Uncertain POMDPs via Convex Optimization. IJCAI, 4113–4120. 5. Cubuktepe, M., Jansen, N., Junges, S., Marandi, A., Suilen, M., & Topcu, U. (2021). Robust Finite-State Controllers for Uncertain POMDPs. AAAI, 11792–11800.	{"url":"https://ai-fm.org/projects/rnn_upomdp/","timestamp":"2024-11-10T06:31:20Z","content_type":"text/html","content_length":"11469","record_id":"<urn:uuid:e11b1a34-322e-4748-99f3-1528ad0cf5ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00513.warc.gz"}
Pearson Edexcel IAL – P3: Pure Mathematics 3 (WMA13 010) – Samir Sha’lan Course Type Full Course Program British Level Pearson Edexcel IAL (A2) Subject Mathematics • General • 1- Algebraic Methods □ 1.1 Arithmetic operations with algebraic fractions □ 1.2 Improper Fractions • 2- Functions and Graphs □ 2.1 The Modulus Function □ 2.2 Functions and Mapping □ 2.3 Composite Functions □ 2.4 Inverse Functions □ 2.5 y = \|f(x)\| and y = f(\|x\|) □ 2.6 Combining Transformations □ 2.7 Solving Modulus Problems • 3-Trigonometric Functions □ 3.1 Secant, Cosecant, and Cotangent □ 3.2 Graphs of sec(x), cosec(x), cot(x) □ 3.3 Using sec(x), cosec(x), and cot(x) □ 3.4 Trigonometric Identities □ 3.5 Inverse Trigonometric Functions • 4-Trigonometric Addition Functions □ 4.1 Addition Formulae □ 4.2 Using the Angle Addition Formulae □ 4.3 Double-Angle Formulae □ 4.4 Solving Trigonometric Equations □ 4.5 Simplifying a cos(x) ± b sin(x) □ 4.6 Proving Trigonometric Identities • 5- Exponentials and Logarithms □ 5.1 Exponential Functions □ 5.2 y = e^(ax+b) + c □ 5.3 Natural Logarithms □ 5.4 Logarithms and Non-Linear Data □ 5.5 Exponential Modelling • 6- Differentiation □ 6.1 Differentiating sin(x) and cos(x) □ 6.2 Differentiating Exponentials and Logarithms P3: Pure Mathematics 3 (WMA13 010) is a continuation of the foundational studies in mathematics, designed to deepen students’ understanding of advanced mathematical concepts. This course builds upon the principles introduced in Pure Mathematics 1 and 2, focusing on further development of algebra, calculus, and mathematical reasoning. It is tailored for students preparing for higher education in mathematics or related disciplines. – Simplification of rational expressions including factorizing and cancelling, and algebraic division definition of a function. Domain and range of functions. Composition of functions. Inverse functions and their graphs, the modulus function, and Combinations of the transformations – Knowledge of secant, cosecant and cotangent and of arc sin, arc cos and arctan. Their relationships to sine, cosine and tangent. Understanding of their graphs and appropriate restricted domains, Knowledge and use of sec2 θ = 1 + tan2 θ and cosec2 θ = 1 + cot2 θ, Knowledge and use of double angle formulae; use of formulae for sin (A ± B), cos(A ± B) and tan (A ± B) and of expressions for a cos θ + b sin θ in the equivalent forms of r cos(θ ± a) or rsin (θ ± a). – The function ex and its graph, The function ln x and its graph; ln x as the inverse function of e^x, and Use logarithmic graphs to estimate parameters in relationships of the form y = ax^n and y = kb^x – Differentiation of e^kx, ln kx, sin kx, cos kx, tan kx and their sums and differences – Differentiation using the product rule, the quotient rule and the chain rule, and understand and use exponential growth and decay – Integration of e^kx, 1 / x , sin kx, cos kx and their sums and differences, in addition to Integration by recognition of known – Location of roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x in which f(x) is continuous, and Approximate solution of equations using simple iterative methods, including recurrence relations of the form x^n + 1 = f(x^n) Join now and get amazing discount To get access to the free lessons, first you need to sign up, if you are an existing user, please login to your account and enjoy viewing free lessons.	{"url":"https://metastars.online/product/pearson-edexcel-ial-p3-pure-mathematics-3-samir-shalan/","timestamp":"2024-11-10T21:45:45Z","content_type":"text/html","content_length":"193158","record_id":"<urn:uuid:959ffda3-c952-401e-9596-47f9b01a7d84>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00480.warc.gz"}
Biplot.PLSR: Partial Least Squares Biplot in MultBiplotR: Multivariate Analysis Using Biplots in R Adds a Biplot to a Partial Lest Squares (plsr) object. Adds a Biplot to a Partial Lest Squares (plsr) object. The biplot is constructed with the matrix of predictors, the dependent variable is projected onto the biplot as a continuous supplementary An object of class ContinuousBiplot with the dependent variables as supplemntary. Oyedele, O. F., & Lubbe, S. (2015). The construction of a partial least-squares biplot. Journal of Applied Statistics, 42(11), 2449-2460. X=as.matrix(wine[,4:21]) y=as.numeric(wine[,2])-1 mifit=PLSR(y,X, Validation="None") mibip=Biplot.PLSR(mifit) plot(mibip, PlotVars=TRUE, IndLabels = y, ColorInd=y+1) For more information on customizing the embed code, read Embedding Snippets.	{"url":"https://rdrr.io/cran/MultBiplotR/man/Biplot.PLSR.html","timestamp":"2024-11-11T07:37:13Z","content_type":"text/html","content_length":"31689","record_id":"<urn:uuid:593ceeae-1f3d-4d4a-8ec7-a8398cdb1ea3>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00260.warc.gz"}
What is the VC dimension of a hyperplane of dimension D? What is the VC dimension of a hyperplane of dimension D? For hyperplanes in Rd, the VC-dimension is d+1. What does VC dimension measure? In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the capacity (complexity, expressive power, richness, or flexibility) of a set of functions that can be learned by a statistical binary classification algorithm. What is VC dimension in SVM? The VC dimension of {f(α)} is the maximum number of. training points that can be shattered by {f(α)} For example, the VC dimension of a set of oriented lines in R2 is three. In general, the VC dimension of a set of oriented hyperplanes in Rn is n+1. What is the VC dimension of a line? The VC dimension is the highest number so that there is a set of that cardinality that can be shattered. For the case of 3 distinct points S={x,y,z} (x What is the VC dimension of the class of circle? The VC dimension is the maximum number of points that can be shattered. {(5,2), (5,4), (5,6)} cannot be shattered by circles, but {(5,2), (5,4), (6,6)} can be shattered by circles, so the VC dimension is at least 3. How do you prove VC dimensions? under the definition of the VC dimension, in order to prove that VC(H) is at least d, we need to show only that there’s at least one set of size d that H can shatter. shattered by oriented hyperplanes if and only if the position vectors of the remaining points are linearly independent. hyperplanes in Rn is n+1. What is VC dimension of instances points on a real? The VC dimension of a classifier is defined by Vapnik and Chervonenkis to be the cardinality (size) of the largest set of points that the classification algorithm can shatter [1]. What is the VC dimension of a finite hypothesis space? The VC-dimension of a hypothesis space H is the cardinality of the largest set S that can be shattered by H. Fact: If H is finite, then VCdim H log \|H\|. If the VC-dimension is d, that means there exists a set of d points that can be shattered, but there is no set of d+1 points that can be shattered. What is the VC dimension of a convex classifier? Since a set with 2d + 1 points can be shattered, the VC dimension of the set of convex polygons with at most d vertices is at least 2d + 1. c1 = 1U – 1s1l \| U ∈ c,s1 ∈ Ul. What is shatter in VC dimension? 1 VC-dimension. A set system (x, S) consists of a set x along with a collection of subsets of x. A subset containing A ⊆ x is shattered by S if each subset of A can be expressed as the intersection of A with a subset in S. VC-dimension of a set system is the cardinality of the largest subset of A that can be shattered. What is the VC dimension of a triangle? Proof: The VC-dimension of a triangle is at least 7. All possible labelling of the seven points aligned on a circle can be separated using the triangles. See the figure below. Why is VC dimension of circle 3? Since some set of 3 points is shattered by the class of circles, and no set of 4 points is, the VC dimension of the class of circles is 3. What is the VC dimension of an origin centered circle? Origin-centered circles and spheres (:ans:) The VC dimension is 2. With any set of three points, they will be at some radii r1≤r2≤r3 r 1 ≤ r 2 ≤ r 3 from the origin, and no function f will be able to label the points at r1 and r3 with +1 while labeling the point at r2 with −1 . Why is VC dimension useful? VC dimension is useful in formal analysis of learnability, however. This is because VC dimension provides an upper bound on generalization error. The mathematics of this are quite complex. The basic idea is that reducing VC dimension has the effect of eliminating potential generalization errors. What is the VC dimension of the set of hypothesis? The VC dimension of a set of hypotheses H is the size of the largest set C ⊆ X such that C is shattered by H. If H can shatter arbitrarily sized sets, its VC dimension is infinite. We now study the VC dimension of some finite classes, more in particular: classes of boolean functions. Why is VC dimension important? VC dimension in mathematics The basic idea is that reducing VC dimension has the effect of eliminating potential generalization errors. So if we have some notion of how many generalization errors are possible, VC dimension gives an indication of how many could be made in any given context. What is the VC dimension of a hypothesis class? Definition 3 (VC Dimension). The VC-dimension of a hypothesis class H, denoted VCdim(H) is the size of the largest set C ⊂ X that can be shattered by H. If H can shatter sets of arbitrary size, then VCdim(H) = ∞. Can you have VC dimension of 0? Any non-empty class trivially shatters a set of size 0, thus the VC dimension is non-negative. Also, the VC dimension is equal to zero iff H has precisely one hypothesis – a constant function. Can 4 points be shattered? No set of 4 points can be shattered. Suppose we have four points arranged such that they define a rectangle. Now, suppose we want to select two points (A&C, in this case). The minimum enclosing square for A&C must contain either B or D – so we can’t capture just two points with a square. Can VC dimension of H be 3? The VC dimension of H here is 3 even though there may be sets of size 3 that it cannot shatter. under the definition of the VC dimension, in order to prove that VC(H) is at least d, we need to show only that there’s at least one set of size d that H can shatter.	{"url":"https://www.evanewyork.net/what-is-the-vc-dimension-of-a-hyperplane-of-dimension-d/","timestamp":"2024-11-04T10:45:42Z","content_type":"text/html","content_length":"46841","record_id":"<urn:uuid:301f668e-8c89-44e8-b1c5-0a666db9239c>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00642.warc.gz"}
Two problems on random analytic functions in Fock spaces Let (Formula Presented) be an entire function on the complex plane, and let (Formula Presented) be its randomization induced by a standard sequence of independent Bernoulli, Steinhaus, or Gaussian random variables. In this paper, we characterize those functions such that is almost surely in the Fock space for any. Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, also known as regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include: (a) a characterization of random analytic functions in the mixed-norm space, an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; and (c) a complete description of random multipliers between different Fock spaces. • Fock spaces • Random analytic functions • mixed norm space Dive into the research topics of 'Two problems on random analytic functions in Fock spaces'. Together they form a unique fingerprint.	{"url":"https://scholars.ncu.edu.tw/en/publications/two-problems-on-random-analytic-functions-in-fock-spaces","timestamp":"2024-11-09T10:13:07Z","content_type":"text/html","content_length":"62353","record_id":"<urn:uuid:a9cc6b93-6a6b-4978-967f-9b98d8dea51b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00518.warc.gz"}
Parallel Shear Sort This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations. This entry provides a formalisation of parallel shear sort, a comparison-based sorting algorithm intended for highly parallel systems. It sorts $n$ elements in $O(\log n)$ steps, each of which involves sorting $\sqrt{n}$ independent lists of $\sqrt{n}$ elements each. If these smaller sort operations are done in parallel with a conventional $O(n\log n)$ sorting algorithm, this leads to an overall work of $O(n \log^2(n))$ and a span of $O(\sqrt{n}\log^2(n))$ -- a considerable improvement over conventional non-parallel sorting. Session Parallel_Shear_Sort	{"url":"https://devel.isa-afp.org/entries/Parallel_Shear_Sort.html","timestamp":"2024-11-12T21:48:28Z","content_type":"text/html","content_length":"8503","record_id":"<urn:uuid:ad70f2d2-452a-414e-bd8e-c852b0dd7d18>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00206.warc.gz"}
Calculating the Force That Causes a Given Momentum Change Question Video: Calculating the Force That Causes a Given Momentum Change Physics • First Year of Secondary School An object that has a momentum of 12 kg ⋅ m/s is acted on by a constant force for 8 seconds, after which the object’s momentum is 6 kg ⋅ m/s. Determine the force that acted on the object. Video Transcript An object that has a momentum of 12 kilogram-meters per second is acted on by a constant force for eight seconds, after which the object’s momentum is six kilogram-meters per second. Determine the force that acted on the object. Okay, so we have an object that starts out with a momentum of 12 kilogram-meters per second. Let’s imagine that this pink blob here is our object. And we’ll label its initial momentum as 𝐩 one so that we have 𝐩 one is equal to 12 kilogram-meters per second. Now, momentum is a vector quantity, which means that it has a direction as well as a magnitude. Let’s imagine that the object’s momentum is in this direction. We are then told that the object is acted on by a constant force for eight seconds and that after this eight-second interval, the object’s momentum is six kilogram-meters per second. So we can draw our object again after a time interval of eight seconds, which we’ve labeled as Δ𝑡. We’ll label its new momentum as 𝐩 two so that we have 𝐩 two is equal to six kilogram-meters per second. Since the original momentum 𝐩 one was positive and the new momentum 𝐩 two is also positive, then 𝐩 two will be in the same direction as 𝐩 one. So we can represent that with an arrow like this. We are asked to work out the force which acted on the object to cause this change in momentum. We can recall that whenever we have a force which acts to change the momentum of an object, then if the object’s momentum changes by an amount of Δ𝐩 over a time interval of Δ𝑡, then the force 𝐹 which acts on that object is equal to Δ𝐩 divided by Δ𝑡. So we know the time interval Δ𝑡, and we know the momentum the object started with and the momentum it ended up with. We can use these values of 𝐩 one and 𝐩 two to calculate the change in the object’s momentum Δ𝐩. Δ𝐩 must be equal to the final momentum 𝐩 two minus the initial momentum 𝐩 one. If we now sub in the values for 𝐩 two and 𝐩 one, we find that the change in momentum is equal to negative six kilogram-meters per second. The value we’ve calculated for Δ𝐩 is negative because 𝐩 two is smaller than 𝐩 one. In other words, the momentum of the object has decreased, and so we have a negative change in the object’s momentum. We now have values for both Δ𝐩 and Δ𝑡. So it’s time to go ahead and sub those values into this equation to calculate the force 𝐹. When we do this, we get that 𝐹 is equal to negative six kilogram-meters per second divided by eight seconds. At this stage, it’s worth pointing out that the kilogram-meter per second is the SI base unit for momentum, and the second is the SI base unit for time. So Δ𝐩 and Δ𝑡 are both expressed in their SI base units. This means that the force 𝐹 that we calculate will be in its own SI base unit, which is the newton. When we evaluate this expression for 𝐹, we get a result of negative 0.75 newtons. The fact that this force is negative means that it acts in the opposite direction to the object’s momentum. So since in our diagram we had the momentum of the object directed to the right, then in this diagram, the force will be directed to the left. This force 𝐹 is exactly what we were asked to calculate. And so our answer to the question is that the force which acted on the object is equal to negative 0.75 newtons.	{"url":"https://www.nagwa.com/en/videos/743132387982/","timestamp":"2024-11-12T09:30:59Z","content_type":"text/html","content_length":"249232","record_id":"<urn:uuid:34cab4ea-d007-4170-ae1c-c3d648a02727>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00098.warc.gz"}
Adding Fractions Calculator • Enter fractions separated by commas (e.g., 1/2, 3/4, etc.). • Click "Calculate" to calculate the result. • Click "Clear" to clear the input and results. • Click "Copy" to copy the result to the clipboard. • Your calculation history will be displayed below. The Adding Fractions Calculator is a specialized tool designed to simplify the process of adding fractions. This guide provides an in-depth analysis of the calculator, covering its functionality, the mathematical concepts involved in adding fractions, and the benefits of using such a tool. Understanding Fractions Before delving into the calculator’s workings, it’s essential to understand what fractions are and why they are important. A fraction represents a part of a whole and is expressed as numerator/ denominator. The numerator indicates how many parts are taken, while the denominator shows the total number of equal parts that make up a whole. Adding Fractions: The Mathematical Concept Adding fractions involves a few steps: 1. Identifying Common Denominators: To add fractions, their denominators must be the same. If they are not, find the least common denominator (LCD), which is the smallest number that both denominators can divide into without a remainder. 2. Converting Fractions to Have Common Denominators: If the fractions have different denominators, convert them into equivalent fractions with the common denominator. 3. Adding the Numerators: Once the fractions have the same denominator, add their numerators. 4. Simplifying the Result: The resulting fraction is then simplified if possible, reducing it to its lowest terms. The Adding Fractions Calculator: How It Works The Adding Fractions Calculator simplifies this process. Here’s how to use it: 1. Inputting Fractions: Enter the fractions to be added, separated by commas (e.g., 1/2, 3/4). 2. Calculation: Click “Calculate” to process the addition. 3. Result: The calculator displays the sum of the fractions, handling the conversion to a common denominator and the addition internally. 4. Additional Features: The tool also offers options to clear inputs, copy results, and view calculation history. The Importance of the Least Common Denominator (LCD) Finding the LCD is crucial in adding fractions. The LCD is the smallest multiple that is common to the denominators of two or more fractions. For example, for fractions with denominators 3 and 4, the LCD is 12, as it’s the smallest number divisible by both 3 and 4. Advantages of Using the Adding Fractions Calculator 1. Accuracy: Ensures precise calculations, reducing the likelihood of errors. 2. Efficiency: Saves time compared to manual calculations, especially with complex fractions. 3. User-Friendly: Simplifies the process for those uncomfortable with mathematics. 4. Educational Value: Helps students understand the process of adding fractions and the concept of common denominators. Real-World Applications Understanding and being able to add fractions is crucial in various fields: • Cooking and Baking: Recipes require fraction addition for ingredient measurements. • Construction and Carpentry: Measurements frequently involve fractions. • Academic Education: Essential for students in mathematics and related subjects. The Adding Fractions Calculator is a valuable tool for anyone needing to add fractions, from students to professionals. It simplifies a process that can be complex, ensuring accuracy and efficiency. Understanding the mathematical principles behind adding fractions enhances its usefulness, allowing users to not only perform calculations quickly but also grasp the concepts involved. Last Updated : 03 October, 2024 One request? I’ve put so much effort writing this blog post to provide value to you. It’ll be very helpful for me, if you consider sharing it on social media or with your friends/family. SHARING IS ♥️ Sandeep Bhandari holds a Bachelor of Engineering in Computers from Thapar University (2006). He has 20 years of experience in the technology field. He has a keen interest in various technical fields, including database systems, computer networks, and programming. You can read more about him on his bio page. 23 thoughts on “Adding Fractions Calculator” 1. Brown Colin The guide clearly outlines the functionality and usage of the adding fractions calculator. It’s a highly efficient and user-friendly tool. 2. Tiffany61 The adding fractions calculator’s educational value is highlighted well. It’s a crucial tool for facilitating the understanding and accurate addition of fractions. 3. Ian Stewart The article provides comprehensive insights into the adding fractions calculator, highlighting its importance, functionality, and benefits. It’s indeed an invaluable tool for precise and efficient fraction addition. 4. Craig Palmer This article effectively highlights the educational value and practical applications of understanding and adding fractions, catering to a wide audience. 5. Mason Clark The article effectively highlights the advantages of using the adding fractions calculator. Its accuracy and efficiency are significant for various fields, making it a must-have tool. 6. Wood Gavin Indeed, its user-friendly features and real-world applications make it an essential resource. 7. Bennett Adam Absolutely, the calculator is indispensable for anyone dealing with fractions, providing both accuracy and efficiency. 8. Rmorris The guide thoroughly explains the steps involved in adding fractions and underscores the practical applications of this mathematical concept in real-world scenarios. 9. Jayden11 The guide effectively explains the mathematical concepts behind adding fractions, emphasizing the educational and real-world significance of the calculator. 10. Adam98 The guide provides a comprehensive understanding of fractions and the calculator’s significance. It’s a valuable resource for anyone dealing with fractions. 11. Lucas18 The explanation of the mathematical concepts behind adding fractions is incredibly informative. A necessary tool for individuals dealing with fractions regularly. 12. Graham Joanne Indeed, its applications in everyday life, from cooking to academic education are noteworthy. 13. Butler Carlie I agree, it simplifies the process and has educational benefits for students. 14. Jayden Baker Absolutely, this calculator is a game-changer for precise and efficient fraction addition. 15. Lloyd Elliott Certainly, the educational value and usability of the calculator are praised, making it a significant resource. 16. Megan Russell Absolutely, the calculator’s mathematical significance and user-friendly features make it an essential resource. 17. Edward07 I agree, the calculator’s importance and practical applications are well-elaborated in this guide. 18. Charlotte Khan The importance of finding the least common denominator is highlighted effectively. This article provides an excellent resource for understanding and utilizing the adding fractions calculator. 19. Finley12 Absolutely, the calculator’s real-world applications and educational value are commendable and applicable across different disciplines. 20. Kgriffiths Absolutely, this guide serves as a valuable tool for accurate and efficient fraction addition. 21. Tim46 Undoubtedly, the calculator simplifies the process and ensures accurate results, making it an invaluable tool. 22. Keith Johnson I agree, the article effectively emphasizes the advantages and real-world applications of the adding fractions calculator. 23. Jason Reid Definitely, the calculator’s functionality and educational value are commendable.	{"url":"https://calculatoruniverse.com/adding-fractions-calculator/","timestamp":"2024-11-01T18:57:00Z","content_type":"text/html","content_length":"260763","record_id":"<urn:uuid:5bfd2e91-fa59-4d20-8808-35481805f420>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00786.warc.gz"}
The Virginia Calculator: Thomas Fuller, A Gifted Mathematician Black History: Special Delivery!! “Image Ownership: Public Domain” Thomas Fuller (1710-1790) was known as the “Virginia Calculator”. Fuller was stolen from his native land and came to America at the age of 14 in 1724. He was considered “illiterate” because he could not read and write in English. However he demonstrated an amazing aptitude for mathematics. He was able to solve extremely complex math problems in his head in very short periods of time. His slave owners, Presley and Elizabeth Cox were both illiterate as well but quickly recognized Fuller’s unusual talent. He became a key asset in the management of their plantation in Virginia. It is believed that Fuller, acquired his mathematical skills as a boy in West Africa. He claimed that his knowledge came from experimental math functions such as counting the hairs in a cow’s tail or counting grains In a bushel of wheat. Some of the mathematical computations he solved are so complex that they are now done by computers. It is easy to see why his slave owners refused numerous offers to buy Fuller. Another example of his ability happened in 1780 when he was 70 years old. Several men from Pennsylvania had heard of Fuller’s amazing talents and traveled to Virginia to challenge his skills. Two of the questions they asked him were, “How many seconds were in a year and a half?” and, “How many seconds had a man lived who is 70 years, 17 days and 12 hours old?” He correctly answered, “47,304,000 and 2,210,500,800 in less than 2 minutes. He received an objection to one of his answers. One of the men felt that the answer was actually smaller. Fuller responded by telling the man, that he had forgotten to include the leap years in his calculation. The man then accepted that Fuller’s answer was correct. Fuller died in Virginia in 1790 at the age of 80. Despite his extraordinary math skills, he had never learned to read or write.	{"url":"https://dev.digitalconcrete.co/the-virginia-calculator-thomas-fuller-a-gifted-mathematician/","timestamp":"2024-11-12T10:32:48Z","content_type":"text/html","content_length":"50262","record_id":"<urn:uuid:b1a84e93-09eb-45f8-bd58-b775fde2a441>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00280.warc.gz"}
A Sound Type System for Physical Quantities, Units, and Measurements This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations. The present Isabelle theory builds a formal model for both the International System of Quantities (ISQ) and the International System of Units (SI), which are both fundamental for physics and engineering. Both the ISQ and the SI are deeply integrated into Isabelle's type system. Quantities are parameterised by dimension types, which correspond to base vectors, and thus only quantities of the same dimension can be equated. Since the underlying "algebra of quantities" induces congruences on quantity and SI types, specific tactic support is developed to capture these. Our construction is validated by a test-set of known equivalences between both quantities and SI units. Moreover, the presented theory can be used for type-safe conversions between the SI system and others, like the British Imperial System (BIS). Session Physical_Quantities	{"url":"https://devel.isa-afp.org/entries/Physical_Quantities.html","timestamp":"2024-11-01T23:06:52Z","content_type":"text/html","content_length":"12146","record_id":"<urn:uuid:fb450946-c90d-4a92-8e3d-8989210f74db>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00305.warc.gz"}
Frictional Forces in context of mass to velocity 27 Aug 2024 Journal of Physics and Engineering Volume 12, Issue 3, 2023 Frictional Forces: A Study on the Relationship between Mass and Velocity This article explores the concept of frictional forces and their impact on the motion of objects. We examine the relationship between mass and velocity in the context of frictional forces, providing a theoretical framework for understanding this phenomenon. Friction is a fundamental force that opposes the motion of an object, converting some of its kinetic energy into heat. The magnitude of frictional forces depends on various factors, including the surface roughness, normal force, and coefficient of friction. In this article, we focus on the relationship between mass (m) and velocity (v) in the context of frictional forces. When an object is moving with a certain velocity, it experiences a frictional force (F_f) that opposes its motion. The magnitude of F_f depends on the coefficient of friction (μ), normal force (N), and mass (m) of the object: F_f = μ \* N Since the normal force (N) is equal to the weight of the object (mg), we can rewrite the equation as: F_f = μ \* m \* g where g is the acceleration due to gravity. Relationship between Mass and Velocity As an object moves with a certain velocity, its kinetic energy (KE) increases. However, the frictional force opposes this motion, converting some of the kinetic energy into heat. As a result, the velocity of the object decreases over time. We can express this relationship as: v = v_0 - (F_f / m) \* t where v_0 is the initial velocity, F_f is the frictional force, m is the mass, and t is time. In conclusion, this article has explored the concept of frictional forces in the context of mass and velocity. We have provided a theoretical framework for understanding the relationship between these variables, highlighting the impact of friction on an object’s motion. Further research is needed to fully understand the complexities of frictional forces and their effects on various systems. • [1] Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics. • [2] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. Note: The references provided are for general knowledge and not specific to the article’s content. Related articles for ‘mass to velocity’ : • Reading: Frictional Forces in context of mass to velocity Calculators for ‘mass to velocity’	{"url":"https://blog.truegeometry.com/tutorials/education/df051bae46f4f2ecff8fd025a8dcb160/JSON_TO_ARTCL_Frictional_Forces_in_context_of_mass_to_velocity.html","timestamp":"2024-11-05T07:11:07Z","content_type":"text/html","content_length":"16537","record_id":"<urn:uuid:ea84b91a-9ad4-4c92-856f-51563fe60b26>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00876.warc.gz"}
g03fcf (multidimscal_ordinal) NAG FL Interface g03fcf (multidimscal_ordinal) FL Name Style: FL Specification Language: 1 Purpose g03fcf performs non-metric (ordinal) multidimensional scaling. 2 Specification Fortran Interface Subroutine g03fcf ( typ, n, ndim, d, x, ldx, stress, dfit, iter, iopt, wk, iwk, ifail) Integer, Intent (In) :: n, ndim, ldx, iter, iopt Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iwk(n(n-1)/2+nndim+5) Real (Kind=nag_wp), Intent (In) :: d(n(n-1)/2) Real (Kind=nag_wp), Intent (Inout) :: x(ldx,ndim) Real (Kind=nag_wp), Intent (Out) :: stress, dfit(2n(n-1)), wk(15nndim) Character (1), Intent (In) :: typ C Header Interface #include <nag.h> void g03fcf_ (const char typ, const Integer n, const Integer ndim, const double d[], double x[], const Integer ldx, double stress, double dfit[], const Integer iter, const Integer iopt, double wk[], Integer iwk[], Integer *ifail, const Charlen length_typ) The routine may be called by the names g03fcf or nagf_mv_multidimscal_ordinal. 3 Description For a set of objects, a distance or dissimilarity matrix can be calculated such that is a measure of how ‘far apart’ the objects are. If have been recorded for each observation this measure may be based on Euclidean distance, ${d}_{ij}=\sum _{k=1}^{p}{\left({x}_{ki}-{x}_{kj}\right)}^{2}$ , or some other calculation such as the number of variables for which ${x}_{kj}e {x}_{ki}$ . Alternatively, the distances may be the result of a subjective assessment. For a given distance matrix, multidimensional scaling produces a configuration of points in a chosen number of dimensions, , such that the distance between the points in some way best matches the distance matrix. For some distance measures, such as Euclidean distance, the size of distance is meaningful, for other measures of distance all that can be said is that one distance is greater or smaller than another. For the former metric scaling can be used, see , for the latter, a non-metric scaling is more appropriate. For non-metric multidimensional scaling, the criterion used to measure the closeness of the fitted distance matrix to the observed distance matrix is known as is given by, $∑i=1n∑j=1 i-1 (dij^-dij~) 2 ∑i=1n∑j=1 i-1dij^2$ is the Euclidean squared distance between points is the fitted distance obtained when is monotonically regressed on , that is is monotonic relative to and is obtained from with the smallest number of changes. So is a measure of by how much the set of points preserve the order of the distances in the original distance matrix. Non-metric multidimensional scaling seeks to find the set of points that minimize An alternate measure is $∑i=1n∑j=1 i-1 (dij^2-dij~2) 2 ∑i=1n∑j=1 i-1dij^4$ in which the distances in are replaced by squared distances. In order to perform a non-metric scaling, an initial configuration of points is required. This can be obtained from principal coordinate analysis, see . Given an initial configuration, uses the optimization routine to find the configuration of points that minimizes . The routine uses a conjugate gradient algorithm. will find an optimum that may only be a local optimum, to be more sure of finding a global optimum several different initial configurations should be used; these can be obtained by randomly perturbing the original initial configuration using routines from Chapter G05 4 References Chatfield C and Collins A J (1980) Introduction to Multivariate Analysis Chapman and Hall Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press 5 Arguments 1: $\mathbf{typ}$ – Character(1) Input On entry : indicates whether is to be used as the criterion. $\mathit{STRESS}$ is used. $\mathit{SSTRESS}$ is used. Constraint: ${\mathbf{typ}}=\text{"S'}$ or $\text{"T'}$. 2: $\mathbf{n}$ – Integer Input On entry: $n$, the number of objects in the distance matrix. Constraint: ${\mathbf{n}}>{\mathbf{ndim}}$. 3: $\mathbf{ndim}$ – Integer Input On entry: $m$, the number of dimensions used to represent the data. Constraint: ${\mathbf{ndim}}\ge 1$. 4: $\mathbf{d}\left({\mathbf{n}}×\left({\mathbf{n}}-1\right)/2\right)$ – Real (Kind=nag_wp) array Input On entry : the lower triangle of the distance matrix stored packed by rows. That is must contain , for $\mathit{i}=2,3,\dots ,n$ $\mathit{j}=1,2,\dots ,\mathit{i}-1$ . If is missing then set ; for further comments on missing values see Section 9 5: $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{ndim}}\right)$ – Real (Kind=nag_wp) array Input/Output On entry : the th row must contain an initial estimate of the coordinates for the th point, for $\mathit{i}=1,2,\dots ,n$ . One method of computing these is to use On exit: the $\mathit{i}$th row contains $m$ coordinates for the $\mathit{i}$th point, for $\mathit{i}=1,2,\dots ,n$. 6: $\mathbf{ldx}$ – Integer Input On entry : the first dimension of the array as declared in the (sub)program from which is called. Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$. 7: $\mathbf{stress}$ – Real (Kind=nag_wp) Output On exit: the value of $\mathit{STRESS}$ or $\mathit{SSTRESS}$ at the final iteration. 8: $\mathbf{dfit}\left(2×{\mathbf{n}}×\left({\mathbf{n}}-1\right)\right)$ – Real (Kind=nag_wp) array Output On exit : auxiliary outputs. , the first elements contain the distances, , for the points returned in , the second set of contains the distances ordered by the input distances, , the third set of elements contains the monotonic distances, , ordered by the input distances, and the final set of elements contains fitted monotonic distances, , for the points in . The corresponding to distances which are input as missing are set to zero. If ${\mathbf{typ}}=\text{"S'}$, the results are as above except that the squared distances are returned. Each distance matrix is stored in lower triangular packed form in the same way as the input matrix $D$. 9: $\mathbf{iter}$ – Integer Input On entry : the maximum number of iterations in the optimization process. A default value of $50$ is used. A default value of $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5nm\right)$ (the default for e04dgf/e04dga) is used. 10: $\mathbf{iopt}$ – Integer Input On entry : selects the options, other than the number of iterations, that control the optimization. The tolerance $\epsilon$ is set to $0.00001$ (Section 7). All other values are set as described in Section 9. The tolerance $\epsilon$ is set to ${10}^{-i}$ where $i={\mathbf{iopt}}$. All other values are set as described in Section 9. No values are changed, therefore, the default values of e04dgf/e04dga are used. 11: $\mathbf{wk}\left(15×{\mathbf{n}}×{\mathbf{ndim}}\right)$ – Real (Kind=nag_wp) array Workspace 12: $\mathbf{iwk}\left({\mathbf{n}}×\left({\mathbf{n}}-1\right)/2+{\mathbf{n}}×{\mathbf{ndim}}+5\right)$ – Integer array Workspace 13: $\mathbf{ifail}$ – Integer Input/Output On entry must be set to to set behaviour on detection of an error; these values have no effect when no error is detected. A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not. If halting is not appropriate, the value is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit. On exit unless the routine detects an error or a warning has been flagged (see Section 6 6 Error Indicators and Warnings If on entry , explanatory error messages are output on the current error message unit (as defined by Errors or warnings detected by the routine: On entry, ${\mathbf{ldx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$. On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ndim}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{n}}>{\mathbf{ndim}}$. On entry, ${\mathbf{ndim}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{ndim}}\ge 1$. On entry, ${\mathbf{typ}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{typ}}=\text{"S'}$ or $\text{"T'}$. On entry, all the elements of ${\mathbf{d}}\le 0.0$. The optimization has failed to converge in function iterations. Try either increasing the number of iterations using or increasing the value of , given by , used to determine convergence. Alternatively try a different starting configuration. The conditions for an acceptable solution have not been met but a lower point could not be found. Try using a larger value of , given by The optimization cannot begin from the initial configuration. Try a different set of points. The optimization has failed. This error is only likely if . It corresponds to An unexpected error has been triggered by this routine. Please contact Section 7 in the Introduction to the NAG Library FL Interface for further information. Your licence key may have expired or may not have been installed correctly. Section 8 in the Introduction to the NAG Library FL Interface for further information. Dynamic memory allocation failed. Section 9 in the Introduction to the NAG Library FL Interface for further information. 7 Accuracy After a successful optimization the relative accuracy of should be approximately , as specified by 8 Parallelism and Performance Background information to multithreading can be found in the g03fcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information. The optimization routine used by has a number of options to control the process. The options for the maximum number of iterations ( Iteration Limit ) and accuracy ( Optimality Tolerance ) can be controlled by respectively. The printing option ( Print Level ) is set to to give no printing. The other option set is to stop the checking of derivatives ( ) for efficiency. All other options are left at their default values. If however is used, only the maximum number of iterations is set. All other options can be controlled by the option setting mechanism of with the defaults as given by that routine. Missing values in the input distance matrix can be specified by a negative value and providing there are not more than about two thirds of the values missing the algorithm may still work. However the does not allow for missing values so an alternative method of obtaining an initial set of coordinates is required. It may be possible to estimate the missing values with some form of average and then to give an initial set of coordinates. 10 Example The data, given by Krzanowski (1990) , are dissimilarities between water vole populations in Europe. Initial estimates are provided by the first two principal coordinates computed. 10.1 Program Text 10.2 Program Data 10.3 Program Results	{"url":"https://support.nag.com/numeric/nl/nagdoc_latest/flhtml/g03/g03fcf.html","timestamp":"2024-11-06T15:13:39Z","content_type":"text/html","content_length":"75223","record_id":"<urn:uuid:6f03b49d-c7e3-41c4-ae9e-3b28ff3bbfb6>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00008.warc.gz"}
Sample space Archives Welcome to the first post from my new series. Here we’re going look at a famous probability question often called the birthday problem. This is actually a more general question related to the probability of at least one coincidence after a fixed number of draws from a discrete uniform distribution. This post is part of my series Probability Questions from the Real World. [Read more…]	{"url":"https://www.probabilisticworld.com/tag/sample-space/","timestamp":"2024-11-05T11:02:03Z","content_type":"text/html","content_length":"115941","record_id":"<urn:uuid:4a765672-47b3-4918-a128-b90025f32e97>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00005.warc.gz"}
Subtracting fractions Look at the denominators (bottom numbers) to see if they have a common denominator. As they are mixed numbers it is a good idea to write them as improper fractions (where the numerator is larger than the denominator): So the question is: 2 is the denominator of the first fraction and 5 is the denominator of the second fraction. These fractions do not have a common denominator. To be able to subtract the fractions they need to have a common denominator. 2 and 5 have a lowest common multiple of 10, so we change both fractions so that they have a common denominator of 10: We have converted the fractions so that they have a common denominator and can now be subtracted:	{"url":"https://thirdspacelearning.com/gcse-maths/number/subtracting-fractions/","timestamp":"2024-11-04T17:39:46Z","content_type":"text/html","content_length":"250186","record_id":"<urn:uuid:90f95985-6f1b-4ce4-b438-13fe36fbc3bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00111.warc.gz"}
Which sale gives the greatest percent discount? 20% off the original price. 75% of the original price. Original price uced by 30%. It depends on the original price. Answer:75% off the original priceNo it doesn't depend on the original price.Why, because when it says original price uced by 30%, that basically means 30% off the original price.Let's say the original price is $100, okay 20% off the original price is only $20 off, 75% off of the original price is $75 off, and to have the original price uced by 30% means the price goes from 100 to 70, but the price would be $25 in this case aer getting 75% off.	{"url":"https://thibaultlanxade.com/algebra/which-sale-gives-the-greatest-percent-discount-20-off-the-original-price-75-of-the-original-price-original-price-uced-by-30-it-depends-on-the-original-price","timestamp":"2024-11-04T13:54:00Z","content_type":"text/html","content_length":"29987","record_id":"<urn:uuid:fbcfd922-ca2b-46e1-84e1-e7a8e177dde6>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00507.warc.gz"}
Study of a plot \| R-bloggersStudy of a plot [This article was first published on » R , and kindly contributed to ]. (You can report issue about the content on this page ) Want to share your content on R-bloggers? if you have a blog, or if you don't. I began to think on a nice way of plotting campaign expenditures in a paper I’m working on. I thought this would be something like the following–simple but meaningful even when there are outliers in both tails. Though I like the seniors Tukey’s boxplot and scatter plots, I had already used them the last time I published about this topic, so I’d like to oxygenate my figures; I thought a type of Manhattan plot could do the job. The very idea is to have types of elections, districts or parties along the X-axis, with the negative logarithm of the association (p-value) between a candidate’s spending and votes displayed on the Y-axis. Thus, each dot on the plot indicates a candidate’s position on this metric. Because stronger associations have the smallest p-values (a log of 0.05 = -1.30103), their negative logarithms will be positivie and higher (e.g., 1.3), while those with p-values not statistically significant (whatever that means these days, maybe nothing ) will stay below this line. The positive thing of this version is that it draws our attention to the upper outliers instead to the average association, which tends to be left-skewed because Brazilian elections typically attract many sacrificial lamb candidates who expend nearly nothing in their campaigns.	{"url":"https://www.r-bloggers.com/2014/12/study-of-a-plot/","timestamp":"2024-11-05T00:32:28Z","content_type":"text/html","content_length":"84821","record_id":"<urn:uuid:f9047533-6e72-4efb-97e5-a5f7023318a0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00013.warc.gz"}
Problem set 8 Yevdokimov, Oleksiy. 2011. "Problem set 8." Australian Senior Mathematics Journal. 25 (2), pp. 63-63. Article Title Problem set 8 ERA Journal ID 40314 Article Category Article Author Yevdokimov, Oleksiy Journal Title Australian Senior Mathematics Journal Journal Citation 25 (2), pp. 63-63 Number of Pages 1 Year 2011 Place of Adelaide, South Australia ISSN 0819-4564 Web Address (URL) http://www.aamt.edu.au This article introduces a regular problem section in the Australian Senior Mathematics Journal. It notes that the section aims to give readers an opportunity to exchange Abstract interesting mathematical problems and solutions. It adds that the set in each issue will consist of up to five problems. Keywords mathematics; problem solving ANZSRC Field of 490303. Numerical solution of differential and integral equations Research 2020 490401. Algebra and number theory 390109. Mathematics and numeracy curriculum and pedagogy Public Notes File reproduced in accordance with the copyright policy of the publisher/author. Byline Affiliations Department of Mathematics and Computing Institution of University of Southern Queensland Permalink - • 1741 total views • 111 total downloads • 0 views this month • 0 downloads this month Related outputs Yevdokimov, Oleksiy. 2021. "Minimising mathematical anxiety in teaching mathematics and assessing studentâ€™s work." Kollosche, David (ed.) 11th International Mathematics Education and Society Conference: Exploring new ways to connect (2021). Virtual Germany. Yevdokimov, Oleksiy. 2010. "Links between problems and their role in students' preparation for mathematical contests." Avotina, Maruta, Bonca, Dace, Falk de Losada, Maria, Ramana, Liga and Soifer, Alexander (ed.) WFNMC6: 6th Congress of the World Federation of National Mathematics Competitions. Riga, Latvia 25 - 31 Jul 2010 Australia. Yevdokimov, Oleksiy. 2013. 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"Notes about teaching mathematics as relationships between structures: a short journey from early childhood to higher mathematics." The De Morgan Journal. 2 (1), pp. 69-83. Tall, David, Yevdokimov, Oleksiy, Koichu, Boris, Whiteley, Walter, Kondratieva, Margo and Cheng, Ying-Hao. 2012. "Cognitive development of proof." Hanna, Gila and de Villiers, Michael (ed.) Proof and proving in mathematics education . Dordrecht, Germany. Springer. pp. 13-49 Yevdokimov, Oleksiy. 2011. "On creating a 'free of anxiety' assessment environment for students with special needs ." NORSMA6: New Trends in Special Needs Education in Mathematics: Problems and Possibilities. Kristiansand, Norway 02 - 04 Nov 2011 Kristiansand, Norway. Yevdokimov, Oleksiy. 2009. "Problem set 4." Australian Senior Mathematics Journal. 23 (2), pp. 63-63. Yevdokimov, Oleksiy. 2009. "Problem set 3." Australian Senior Mathematics Journal. 23 (1), pp. 63-63. Yevdokimov, Oleksiy. 2008. 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Bonds and Yields - Working and Calculation Bonds and Yields – Working and Calculation By sm-blogger / December 6, 2023 When discussions about bonds revolve, we often hear bond yields going up and down. The bond yield represents what a bondholder will receive by investing in a bond. In this blog, you will get a clear understanding of a bond yield in India. What Is a Bond Yield? A bond yield is what you expect to get back from your investment in bonds. It indicates the total return that you will receive over a bond’s lifespan, including both interest and principal. When you purchase a bond from the primary market, the price you pay depends on various factors. These factors include a bond’s term, rates of comparable bonds in the market, and the promised interest payments. All these elements put together help to calculate bond yields. What is Bond Equivalent Yield (BEY)? A bond equivalent yield serves as a metric used by investors to determine the annual percentage yield for fixed-income securities, especially those with discounted short-term terms that make periodic payouts on a monthly, quarterly, or semi-annually basis. It helps you assess the annual yield of a bond or any fixed-income security that does not provide an annual payout. Let’s understand the concept of bond equivalent yield with the help of its formula and an example. The following is the formula to calculate BEY. BEY = [(Face value – Purchase price) / Purchase price] * (365/d) Here, d = number of days left for bond maturity. Suppose, you purchase a bond for ₹100, which would offer ₹150 at maturity. This bond will mature after 300 days. Thus, the BEY calculation will show – BEY = [(₹150 – ₹100) / ₹100] * 365/300 = 0.608 or 60.8% Thus, you will make a yield of 60.8% by investing in these bonds. Formula and Calculation of a Bond Yield The calculation of bond yields includes two factors: the annual coupon payment and the current bond price. An annual coupon payment is determined by multiplying the coupon rate by the bond’s par value. Usually, this annual coupon payment remains constant over the bond’s life, while the bond price might fluctuate. To calculate a bond yield, you can use the following formula. Bond yield = Annual coupon payment/ Price of the bond To understand this calculation better, consider an example. Suppose a bond has a face value of ₹2,000 and has an annual interest rate of 10%. The current price of the bond will be ₹2,400. Annual coupon payment = ₹2,000*10%= ₹200 Bond yield = ₹200/₹2,400 = ₹0.0833 or 8.33% Working of Bond Yields A bond yield in the secondary market depends upon the demand and supply equilibrium. Suppose you have a bond of 5-year maturity with a 5% coupon rate and a face value of ₹20,000. Each year you will earn an interest of ₹1000 from the bond. Now, if the interest rate on this bond rises above 5%, the investors will not buy these bonds from you. They will buy bonds having interest rates of more than 5%. In such cases, you will have to reduce the price of your bond to increase the bond yield. When your bond price decreases, the annual coupon rate will increase because of the lower face value of the bond. As a result, it will increase the bond yield. This is how bond yield works based on the prevailing interest rate in the market. Bond Yield vs. Bond Price A bond yield and bond price have an inverse relationship. It implies when the price of a bond rises, its yield decreases and vice versa. This relationship is crucial for investors to understand when making decisions in the bond market. • When the bond price rises, the percentage return or yield decreases. • Similarly, when the bond price falls, the bond yield increases. Investors closely monitor these changes to assess the profitability and risk associated with their bond investments. When it comes to investment in bonds, understanding the concept of bond yields and using its formula accurately will help you assess investment opportunities and potential risks while navigating the financial markets in the long run. Moreover, it will also help you to identify in advance when to trade your bond and make necessary adjustments in your portfolio as per your investment horizon and market conditions. How do bond yields affect the economy? A rise in bond yields indicates that the issuing entity will have to pay more as a yield since there will be a rise in the cost of borrowing. In case the bond is issued by a government body, its yields will have an overall impact on the economy as it will put excessive pressure on the interest rate of the banking system. What causes bond yields to rise? Inflation, interest rates, yield curve and economic growth causes a bond yield to rise. Furthermore, bond yields like corporate bond yields are influenced by a company’s credit rating in the industry Is it good if bond yields are high? Yes, higher bond yields result in investors receiving large interest payments from that particular bond. However, bonds with higher yields also come with increased risk. What is 10-year bond yield? A 10-year bond yield in India is the rate of return on a 10-year Treasury bond. It is an ideal investment option for risk-averse individuals looking for a fixed income. 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W. T. Tutte In the not-too-distant future, people are going to be celebrating the 100th birthday of Bill Tutte. He’s not quite as well-known as Alan Turing, which is a shame since they were both equally invaluable in the cryptanalysis at Bletchley Park. You may also know him from the problem of squaring the square, which he accomplished with three of his friends whilst they were students at Trinity. The story of how they met was rather interesting and extremely serendipitous, and is recounted at the beginning of this article on squaring.net. The catalysis for this was a complete chance encounter in London between Trinity mathmos Arthur Stone and Cedric Smith, who happened to both reside in the area. The scope of the conversation was as vacuous as the empty set, and not even their names were exchanged at the time. Much more interesting was the next edge of the acquaintance graph, which was created in an equally improbable manner: As first-year undergraduates, Cedric Smith met Rowland Brooks in a lecture on almost periodic functions. This was the equivalent of a Part III lecture, and their attendance was the result of an error in a timetable. This administrative mistake became apparent from the incomprehensibility of the lecture! Nevertheless, it was very fortunate that the timetables were erroneous, since it ultimately led to a very productive union of four mathematicians. The fourth, of course, was Bill Tutte, who was reading chemistry (which has since been engulfed into the Natural Sciences Tripos) as well as mathematics. He knew Rowland Brooks, and later met Smith and Stone by extension. Tutte will be the primary focus of the majority of this article, so it seems appropriate to include a picture. Even better, here is a photograph of a bronze bust by the accomplished sculptor, artist, painter and former musician, singer and actress, Gabriella Bollobás, who happens to be married to the eminent mathematician Professor Bollobás: Squaring the square The four of them met on a regular basis to discuss mathematical problems. One of these was the problem of finding a squared square, that is to say a dissection of a square into a finite number of smaller squares, no two of which are the same size. William Tutte wrote a detailed account of their effort to solve this mathematical problem, which spanned the years 1936 through to 1938. The first major insight was that a squared rectangle could be represented by a network of one-ohm resistors, where the Thevenin equivalent resistance is equal to the aspect ratio of the squared rectangle. Specifically, imagine that the rectangle is made from a material of uniform resistivity, where a square of the material has a resistance of one ohm between opposite edges. Then, given a dissection of the rectangle into squares, we can replace each horizontal line with a perfect conductor, and each vertical line with a perfect insulator, and since all current flows vertically this has no effect on the total resistance. A corollary of this is that the resulting squared rectangle must have sides in a ratio corresponding to the resistance of a circuit composed of one-ohm resistors; this forces it to be rational. Tutte et al also showed that the dimensions of the constituent squares must also be commensurate with the side lengths of the rectangle. The next important discovery was by Smith’s mother, who assembled the squares into a perfect rectangle in a different arrangement from the original configuration. Eventually, it was noticed that this corresponded to a resistance-preserving transformation of the corresponding electrical circuit. They reasoned that it might be possible to tile the same rectangle with two completely different sets of squares, which would trivially result in a squared square: Problematically, the squared rectangle produced by Smith’s mother featured precisely the same set of squares as the original, so the resulting dissection of the square would be replete with pairs of identical squares. Hence, further ingenuity was required to discover a perfect squared square. The structure of Smith’s circuit was observed to naturally partition into a ‘rotor’ with threefold rotational symmetry and a ‘stator’. With a one-wire stator, it was realised that it could be possible to produce two rectangles sharing only one common square, which could be overlapped to yield a configuration trivially completable into a squared square. It transpires that Smith and Stone collaboratively found one, at almost precisely the same time that Brooks did so. These order-69 squares were the first to be discovered, but far from being optimal. The Trinity Mathematical Society logo is an order-21 squared square, which has been proved to be minimal by exhaustive search. Cryptanalysis of the Lorenz cipher A year later, Britain and Germany plunged into the Second World War. (I’m not going to mention the war again, I promise!) This involved quite a lot of secret communication on both sides, and much of the Allied codebreaking efforts were concentrated in Bletchley Park. Bletchley Park defeated two great ciphers. The first of these, which is probably more well known, was Arthur Scherbius’ Enigma cipher used by the Luftwaffe and Kriegsmarine. The effort began with the Polish mathematician Marian Rejewski. He knew roughly how the Enigma worked from models stolen by intelligence agencies, but had to embrace the immense challenge of determining both the internal wiring of the rotors and a method for cryptanalysing the variable key (which changes with each day). Rejewski built a machine, the bomba, for brute-forcing the key settings. Eventually, the Polish intelligence passed their discoveries to Bletchley Park, where people such as Alan Turing (King’s College, Cambridge) and Gordon Welchman (Trinity, yet again) found alternative and more flexible methods of cryptanalysing the Enigma based on Rejewski’s insights. However, the cryptanalysis of the more difficult Lorenz cipher was an infinitely more impressive achievement, mainly because it was accomplished without anyone having ever seen one of the Lorenz encryption machines. I shall briefly describe its modus operandi. The Lorenz cipher used the Baudot code, where each character is represented by five binary digits (either dots or crosses). The enciphering machine (or Schlüsselzusatz) produced a pseudorandom stream of information which was XOR’d with the plaintext to yield a ciphertext. The same stream of information, when XOR’d with the incoming ciphertext, would result in perfectly comprehensible plaintext. This Vernam cipher was a known form of encryption, but cryptanalysists at Bletchley Park had no idea how the pseudorandom bit generator operated. What would be helpful is if the output of the pseudorandom bit generator could actually be isolated. This happened when a lazy operator was asked to repeat a message with the same key, and abbreviated it slightly, giving a huge vulnerability. Specifically, XORing the two encrypted messages would neutralise the contribution of the Schlüsselzusatz, giving the symmetric difference of the two messages. John Tiltman was able to carefully separate this into the two original messages, treating it as a basic autokey cipher. Moreover, he could XOR the encrypted and decrypted messages to isolate the output of the Schlüsselzusatz. Enter Tutte. Tutte was given the output of the Schlüsselzusatz, and asked to deduce the internal workings of the machine. This, of course, was an incredibly difficult task. He started by looking for approximate periodicity (perhaps the lecture on almost periodic functions helped!) in the bitstream. He found a period-41 pattern, suggesting the presence of a rotor χ1 with 41 teeth. Moreover, he determined that another more complicated rotor (ψ1) driven by a motor wheel μ contributed to the encryption. Proceeding in this manner, cryptanalysts determined how each of the other four bits were generated (there were five independent mechanisms with coprime periods). Armed with Tutte’s deduction, Alan Turing et al were able to work on algorithmically cracking this cipher. This began with the electromechanical Heath Robinson machines, which were replaced by the much more efficient electronic Colossus computers, designed and built by engineer Tommy Flowers. Gordon Welchman was somewhat disapproving of this shift from good old electromechanical relays to those new-fangled thermionic valves; fortunately, Turing was more forgiving. Nowadays, of course, relays and valves have both been rendered obsolete by microscopic transistors on silicon chips, and may eventually be further antiquated by advances in carbon nanotechnology. The intercepted and decrypted communications gave the Allies a massive advantage. I seem to recall that they even sent a Lorenz-encrypted message to sabotage the Nazi military by issuing unhelpful orders, although I might be confusing this with the Sherlock Holmes mystery, The Adventure of the Dancing Men. Graph theory Squaring the square, whilst initially appearing to be combinatorial geometry, eventually succumbed to a graph-theoretic approach. Many of Bill Tutte’s mathematical accomplishments were in graph theory, including the disproof of Tait’s conjecture that every 3-regular polyhedral graph admits a Hamiltonian cycle. He has a few graphs named after him, one of which is the Tutte 8-cage. This is a 3-regular graph with girth 8, and indeed is the unique smallest such graph. It possesses a large amount of symmetry, a small amount of which is apparent in the following picture: Its full symmetry group is PΛL(2,9), the automorphism group of S6. This is apparent from the standard construction of the 8-cage: take the 15 unordered pairs of letters {a,b,c,d,e,f}, together with the 15 unordered triples of unordered pairs (such as (ac,bf,de)), and connect a triple to a pair when they are incident. This has an obvious action of S6 (permuting the letters), but also features an outer automorphism interchanging the two halves of this bipartite graph. Indeed, this is one of the simplest ways to see the outer automorphism of S6. Tutte was also responsible for the snark conjecture, that every snark contains the Petersen graph as a minor. When proved much later by Robertson, Seymour et al, this gave an indirect proof of the four colour theorem as an immediate corollary. Other theorems include the BEST theorem (where the ‘T’ in the acronym stands for ‘Tutte’), Tutte-Berge formula and Tutte’s homotopy theorem. William Tutte passed away in 2002, after a long life (by contrast with Turing’s untimely cyanide-laced Rosacaean death) and a very fruitful and productive mathematical career. There is an obituary in the Guardian. 0 Responses to W. T. Tutte 1. My names is Alison Hayes and I am chairman of the Bill Tutte Memorial Working group based in Newmarket, in Suffolk, UK, the town where Bill was born. For the past two years we have been raising funds for a commemorative sculpture called The Codebreaker, which is will be unveiled this September to mark Bill’s wartime codebreaking achievements, which as you recognised, do not get anything like as much recognition as those of Alan Turing. Go to http://www.billtuttememorial.org.uk to see more. We are also collected funds for a Bill Tutte scholarship under which we plan to award a grant of £1000 per year for three years to a local student going to study maths and/or computer science at university. Could you tell me where the bust featured in your article is ? Someone told me it was in the Microsoft headquarters ? Best regards □ Hello, Alison. This is excellent news, thank you; I am very pleased that Tutte’s revolutionary work will be adequately commemorated. The punched-tape panels are brilliant, and I may even visit the sculpture when it is unveiled (to coincide with the centenary?). I approve of the scholarship as well — is it specific to Trinity College, Cambridge (where Tutte matriculated), or applicable to any I’m not sure where the bust is located, but I’ll ask Gabriella Bollobás directly (serendipitously, she has e-mailed me to confirm my booking of a seat at a performance of Schubert’s Winterreise, so I’ll enquire when I reply to the e-mail) and notify you accordingly. Adam P. Goucher □ I spoke to Gabriella at the Winterreise. Apparently, she is going to be sculpting another W. T. Tutte bust. 2. Thanks for your job. A greeting. This entry was posted in Uncategorized. 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Fung-Hsieh 7 Factor Model - Breaking Down Finance Fung-Hsieh 7 Factor Model The Fung-Hsieh 7 factor model is a risk factor model commonly used to evaluate hedge funds’ performance. The seven factors are risk factors that explain a large proportion of the returns of hedge funds. The model was proposed door David Hsieh and William Fung in 2001 in a paper titled Hedge Fund Benchmarks: A Risk-Based Approach”. The aim of the factors is to capture the returns to a well-diversified portfolio of hedge funds. On this page, we discuss the Fung-Hsieh 7 factor model definition. In particular, we discuss the risk factors that are included in the model. We also briefly introduce the Fung-Hsieh 8 factor model. Fung-Hsieh 7 factor model definition The 7 factor model for hedge funds is actually a simple linear factor model, similar to the Fama and French 3 factor model for individual stocks. In this case, however, the model is used to explain hedge fund returns. The model uses the following 7 factors: • Bond Trend-Following Factor • Currency Trend-Following Factor • Commodity Trend-Following Factor • Equity Market Factor • The Equity Size Spread Factor • The Bond Market Factor • The Bond Size Spread Factor The first three factors are trend-following factors proposed by Fung and Hsieh in a different paper called “The Risk in Hedge Fund Strategies: Theory and Evidence from Trend Followers“. These factors are available here. Next, the equity market factor is captured using the S&P 500, the size factor is the difference between the Russell 2000 index monthly total return – Standard & Poor’s 500 monthly total return. The bond market factor is proxied using the the monthly change in the 10-year treasury constant maturity yield Finally, the size spread factor is measured using the monthly change in the Moody’s Baa yield less 10-year treasury constant maturity yield (month end-to-month end). Fung-Hsieh 7 factor model formula Once we have the factors, the model looks as follows where PTFS are the trend-following factors for Bonds, Currencies, and Commodities, EQ is the equity factor, ES is the Equity Size factor, BM is the bond market, and BS is the Bond Size factor. Fung-Hsieh 8 factor model A few years after the introduction of the 7 factor model, Fung and Hsieh added an eighth factor. This model is referred to as the Fung-Hsieh 8-factor model. The additional factor that is added to the model is the MSCI Emerging Market index. We discussed the Fung-Hsieh 7 factor model. This is the workhorse model when researchers try to explain hedge funds’ return and analyze whether hedge funds generate alpha.	{"url":"https://breakingdownfinance.com/finance-topics/performance-measurement/fung-hsieh-7-factor-model/","timestamp":"2024-11-09T14:23:06Z","content_type":"text/html","content_length":"235733","record_id":"<urn:uuid:cb6e63e8-068d-43d1-8929-8238fe987d5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00750.warc.gz"}
Computer Graphics and Computer Animation: A Retrospective Overview Chapter 19: Quest for Visual Realism 19.4 Noise functions and Fractals Perlin Noise Function Vase Ken Perlin received a B.A. in theoretical mathematics from Harvard University in 1979 and his Ph.D. in Computer Science from New York University in 1986. Perlin started his graphics career as the System Architect for computer generated animation at MAGI (Mathematical Applications Group, Inc.) in Elmsford, NY. While at MAGI he worked on the movie TRON and began thinking about what would be his noise functions and how they could be used to efficiently create textures for use in complex images. Between 1984 and 1987 Perlin was Head of Software Development at R/Greenberg Associates in New York. He then became a professor in the Media Research Laboratory in the Department of Computer Science at New York University and served as the co-Director of the NYU Center for Advanced Technology. According to his bio on the NYU site, he has served on the Board of Directors of the New York chapter of ACM/SIGGRAPH, and on the Board of Directors of the New York Software Industry Association. His research interests include graphics, animation, and multimedia. In 2002 he received the NYC Mayor’s award for excellence in Science and Technology and the Sokol award for outstanding Science faculty at NYU. In 1991 he received a Presidential Young Investigator Award from the National Science Foundation. Perlin Noise Functions In 1997 Perlin won an Academy Award for Technical Achievement from the Academy of Motion Picture Arts and Sciences for his noise and turbulence procedural texturing techniques, which are widely used in feature films and television. As Perlin said in a lecture at the Game Developers Conference HardCore seminars titled Making Noise ^[1] in 1999, “The first thing I did in 1983 was to create a primitive space-filling signal that would give an impression of randomness. It needed to have variation that looked random, and yet it needed to be controllable, so it could be used to design various looks. I set about designing a primitive that would be “random” but with all its visual features roughly the same size (no high or low spatial Perlin Noise Functions I ended up developing a simple pseudo-random “noise” function that fills all of three dimensional space (a slice of the 3D is shown here). In order to make it controllable, the important thing is that all the apparently random variations be the same size and roughly isotropic. Ideally, you want to be able to do arbitrary translations and rotations without changing its appearance too Perlin modified his noise functions so that he could make naturally looking textures using controllable mathematical expressions, which he integrated into shaders. He later expanded the noise to generate 3D models, a process he dubbed Hypertexture in a 1989 paper. A tutorial about Perlin noise, with code, can be found on the scratchapixel.com tutorial site. Benoit Mandelbrot Benoit Mandelbrot is largely responsible for the present interest in fractal geometry, which has found its way into the fibre of computer graphics. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature. In 1958, after a stint at the Institute for Advanced Study, he came to the United States permanently and began his long standing collaboration with IBM as an IBM Fellow at their Thomas J. Watson Research Laboratory in Yorktown Heights. The IBM Watson laboratory provided Mandelbrot the opportunity to research many different concepts and approaches to looking at nature, in his words “an opportunity which no university post could have given me.” After retiring from IBM, became a professor at Yale University, where he received tenure and served as the Sterling Professor of Mathematical Sciences. He had been introduced at an early age to the mathematical concepts of the mathematicians Gaston Julia and his rival Pierre Fatou, whose contributions to the mathematics discipline were important but pretty much forgotten because of their rivalry until Mandelbrot revived a discussion of them. Mandelbrot had been working in other areas of scientific concepts, not in math, until he started rethinking about some areas of geometry and how they could provide a lens for visualizing science. This interest in geometric concepts brought him back to some of the ideas of Julia, in particular the one unifying aspect of certain geometries that was the concept of self-similarity. In the mid-1970s he coined the word “fractal” as a label for the underlying objects, since he observed that they had fractional dimensions. A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale, that is they have similar properties at all levels of magnification or across all times. Just as the sphere is a concept that unites physical objects that can be described in terms of that shape, so fractals are a concept that unites plants, clouds, mountains, turbulence, and coastlines, that do not correspond to simple geometric shapes. According to Mandelbrot, “I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means “to break” or to create irregular fragments. It is therefore sensible – and how appropriate for our needs – that, in addition to “fragmented” (as in fraction or refraction), fractus should also mean “irregular,” both meanings being preserved in fragment.” (The Fractal Geometry of Nature, page 4.) He gives a mathematical definition of a fractal as a set for which the Hausdorff-Besicovich dimension strictly exceeds the topological dimension. With the aid of computer graphics, Mandelbrot was able to show how Julia’s work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics. An example fractal is the Mandelbrot set (others include the Lyapunov fractal, Julia set, Cantor set, Sierpinski carpet and triangle, Peano curve and the Koch snowflake). To graph the Mandelbrot set a test determines if a given number in the complex number domain is inside the set or outside the set. The test is based on the equation Z = Z^2 + C where C represents a constant number, meaning that it does not change during the testing process. Z starts out as zero, but it changes as the equation is repeatedly iterated. With each iteration a new Z is created that is equal to the old Z squared plus the constant C. The actual value of Z as it changes is not of interest per se, only its magnitude. As the equation is iterated, the magnitude of Z changes and will either stay equal to or below 2 forever (and will be part of the Mandelbrot set), or it will eventually surpass 2 (and will be excluded from the set). To create the visual representation a color is assigned to a number if it is not part of the Mandelbrot set. The actual color value is determined by how many iterations it took for the number to surpass 2. Mandelbrot’s work was first described in his book Les Objets Fractals, Forn, Hasard et Dimension (1975) and later in The Fractal Geometry of Nature (1982). Loren Carpenter was employed at Boeing in Seattle when he decided he wanted to pursue a career in the evolving graphics film production industry. As an engineer at Boeing, Carpenter worked on problems related to the creation of high-quality renderings of free-form surfaces. He was also responsible for the development of algorithms for the use of fractal geometry as a tool for creating complex scenes for graphic display. Scene from Vol Libre In 1980 Carpenter used the fractal concept to create mountains for his film Vol Libre, which generated widespread interest in the possibilities that this approach promised. His technical contributions, along with the seminal work with fractals, resulted in a position with Lucasfilm’s Computer Division in 1981 (see the sidebar at the end of this section). He recreated the fractal mountains used in Vol Libre as part of the Genesis Demo for Star Trek II: The Wrath of Khan. Another of his contributions, the A-buffer hidden surface algorithm, was central to the systems used by many production companies to create images for television and motion pictures. He is the author of numerous fundamental papers in computer image synthesis and display algorithms. His research contributions include motion blur, fractal rendering, scan-line patch rendering, the A-buffer, distributed ray tracing and many other algorithmic approaches to image making. He holds several patents both personally and through Pixar, and his technical awards include the third SIGGRAPH Technical Achievement Award in 1985. In 1986, when Lucasfilm’s Computer Division spun off to form Pixar, Loren became Chief Scientist for the company. In 1993, Loren received a Scientific and Technical Academy Award for his fundamental contributions to the motion picture industry through the invention and development of the RenderMan image synthesis software system. RenderMan has been used by many computer-generated films, including use by Lucasfilm’s Industrial Light and Magic to render the dinosaurs in Jurassic Park. Cinematrix Wand Carpenter also patented an interactive entertainment system which, through the use of simple retroreflectors, allows large audiences to play a variety of games together either as competing teams or unified toward a common goal, such as flying a plane. Enthusiastic audiences have shown that many types of people find this new method of communicating fun and exciting. Concurrently with his leadership of Pixar, Loren and his wife Rachel founded Cinematrix to explore the intersection of computers and art. Cinematrix’s Interactive Entertainment Systems division is focusing on the development of an interactive audience participation technology that enables thousands of people to simultaneously communicate with a computer, making possible an entire new class of human-computer Reflectors used by audience to control motion Other people involved in using fractals as a basis for their work in image-making include Richard Voss, Ken Musgrave, Michael Barnsley, Melvin Prueitt (and high school and college students all over the world!) Loren Carpenter, Rob Cook and Ed Catmull – And the Oscar Goes to… (IEEE Spectrum, April 2001) Movie 19.12 Vol Libre 1980 film by Loren Carpenter, created while he was at Boeing, and shown to attendees of the 1980 SIGGRAPH conference The following account is from the book Droidmaker- George Lucas and the Digital Revolution, by Michael Rubin. Fournier gave his talk on fractal math, and Loren gave his talk on all the different algorithms there were for generating fractals, and how some were better than others for making lightning bolts or boundaries. “All pretty technical stuff,” recalled Carpenter. “Then I showed the film.” He stood before the thousand engineers crammed into the conference hall, all of whom had seen the image on the cover of the conference proceedings, many of whom had a hunch something cool was going to happen. He introduced his little film that would demonstrate that these algorithms were real. The hall darkened. And the Beatles began. Vol Libre soared over rocky mountains with snowy peaks, banking and diving like a glider. It was utterly realistic, certainly more so than anything ever before created by a computer. After a minute there was a small interlude demonstrating some surrealistic floating objects, spheres with lightning bolts electrifying their insides. And then it ended with a climatic zooming flight through the landscape, finally coming to rest on a tiny teapot, Martin Newell’s infamous creation, sitting on the mountainside. The audience erupted. The entire hall was on their feet and hollering. They wanted to see it again. “There had never been anything like it,” recalled Ed Catmull. Loren was beaming. “There was strategy in this,” said Loren, “because I knew that Ed and Alvy were going to be in the front row of the room when I was giving this talk.” Everyone at SIGGRAPH knew about Ed and Alvy and the aggregation at Lucasfilm. They were already rock stars. Ed and Alvy walked up to Loren Carpenter after the film and asked if he could start in October. (Available as an eBook from the iTunes store) Ken Perlin, An image synthesizer, SIGGRAPH 1985, pp. 287 – 296. Ken Perlin and Eric Hoffert, Hypertexture, SIGGRAPH 1989, pp. 253 – 262. Mandelbrot, B. Fractals: Form, Chance and Dimension, W. H. Freeman and Company, 1977, xviii + 265 pp. Mandelbrot, B. The Fractal Geometry of Nature, W. H. Freeman and Company, 1982, xii + 461 + xvi pp. (twenty-two reprints by 2005) Mandelbrot, B. Fractals and Chaos: The Mandelbrot Set and Beyond, New York NY: Springer, 2004, xii + 308 pp. Lane, J., L. Carpenter, T. Whitted, and J. Blinn, “Scan Line Methods for Displaying Parametrically Defined Surfaces,” CACM, 23(1), January 1980. Gallery 19.3 Fractal Images	{"url":"https://ohiostate.pressbooks.pub/graphicshistory/chapter/19-4-noise-functions-and-fractals/","timestamp":"2024-11-08T03:08:31Z","content_type":"text/html","content_length":"127872","record_id":"<urn:uuid:eaa6505e-bc72-4fb7-9169-3f5f175bd44e>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00537.warc.gz"}
The Operational Amplifier, August 1967 Electronics World August 1967 Electronics World Table of Contents Wax nostalgic about and learn from the history of early electronics. See articles from Electronics World, published May 1959 - December 1971. All copyrights hereby acknowledged. There is no such thing as too many introductory articles on operational amplifiers (opamps). Of course, when this story was written for Electronics World back in 1967, opamps were relatively new to the scene. Prior to the advent of opamps, circuit design for controllers, filter, comparators, isolators, and just plain old amplification was much more involved. Opamps suddenly allowed designers to not worry as much about biasing, variations in power supply voltages, and other annoyances, and instead focus on function. Even from the very beginning with the μa741 operational amplifier, the parameters came close to those of an ideal device: infinite input impedance, zero output impedance, perfect isolation between ports, and infinite bandwidth. OK, the bandwidth spec was more constrained compared to the other three, but still, with frequencies being what they were compared to today, it was close enough. Opamps allowed engineers to design with the simplicity of LaPlace The Operational Amplifier - Circuits & Applications By Donald E. Lancaster These highly versatile controllable-gain modular or integrated-circuit packages have been used in computer and military circuits. New price and size reductions have opened commercial and consumer markets. Here are complete details on what is available and how the devices are used. Once exclusively the mainstay of the analog-computer field, operational amplifiers are now finding diverse uses throughout the rest of the electronics industry. An operational amplifier is basically a high-gain, d.c.-coupled bipolar amplifier, usually featuring a high input impedance and a low output impedance. Its inherent utility lies in its ability to have its gain and response precisely controlled by external resistors and capacitors. Since resistors and capacitors are passive elements, there is very little problem keeping the gain and circuit response stable and independent of temperature, supply variations, or changes in gain of the op amp itself. Just how these resistors and capacitors are arranged determines exactly what the operational amplifier will do. In essence, an op amp provides "instant gain" that may be used for practically any circuit from a.c., d.c., and r.f. amplifiers, to precision waveform generators, to high- "Q" inductorless filters, to mathematical problem solvers. Op amps used to be quite expensive, but many of today's integrated circuit versions now range from $6 to $20 each and less in quantity. Due to price breaks that have occurred very recently, the same benefits now available to the analog computer, industrial, and military markets are now extended to commercial and consumer circuits. One obvious application will be in hi-fi preamps where a single integrated circuit can replace the bulk of the low-level transistor circuitry normally used. Fig. 1A shows the op-amp symbol. An op amp has two high-impedance inputs, the inverting input and the non-inverting input, as indicated by a "-" or a "+" on the input side of the amplifier. The inverting input is out-of-phase with the output, while the non-inverting input is in-phase with the output. The amplifier has an open-loop gain A, which may range from several thousand to several On closer inspection, we see three distinct parts to any operational amplifier's internal circuitry, as shown in Fig. 1B. A high-input-impedance differential amplifier forms the first stage, with the inverting input going to one side and the non-inverting input the other. The purpose of this stage is to allow the inputs to differentially drive the circuit and also to provide a high input There are several possibilities for this input stage. If an ordinary matched pair of transistors (or the integrated circuit equivalent) is used, an input impedance from 10,000 to 100,000 ohms will result, combined with low drift, low cost, and wide bandwidth. By using four transistors in a differential Darlington configuration, the input impedance may be nearly one megohm. Drift and circuit cost are traded for this benefit. Field-effect transistors are sometimes used, yielding input impedances of 100 megohms, but often with limited bandwidths. FET integrated-circuit operational amplifiers are not yet available, limiting this technique to the modular-style package at present. One or two novel techniques allow extreme input impedances, but presently at very high cost. One approach is to use MOS transistors with their 10^13-ohm input impedance; a second is to use a varactor diode parametric amplifier arrangement on the input. The input differential amplifier is followed by ordinary voltage-gain stages, designed to bring the total voltage gain up to a very high value. Terminals are usually brought out of the voltage-gain stage to allow the frequency and phase response of the op amp to be tailored for special applications. This is usually done by adding external resistors and capacitors to these terminals. Since an operational amplifier is bipolar, the output can swing either positive or negative with respect to ground. A dual power-supply system, one negative and one positive, is required. The final op-amp stage is a low-impedance power-output stage, which may take the form of a single emitter-follower, a push-pull emitter-follower, or a class-B power stage. This final circuit serves to make the output loading and the over-all gain and frequency response independent. It also provides a useful level of output power. The gain of an operational-amplifier circuit is always chosen be much less than the open-loop gain of the amplifier itself. This allows the circuit response to be precisely determined by the external feedback and input network impedances. Feedback is almost ways applied to the inverting (-) input. This is negative feedback for any change in output tries to produce an opposing change in the input. The feedback and input network impedances are normally chosen such that they are much larger than the op amp's output impedance, much smaller than the op amp's input impedance, and such that the gain they require for proper operation is much less than the amp's gain. If these assumptions are met, the ratio of input to output voltage (the gain of the circuit) will be given by: Circuit Gain = For instance, the op-amp circuit of Fig. 5B has an input impedance of 1000 ohms and a feedback impedance of 10,000 ohms Its gain will be - 10k/1k = - 10. Any of the op amps of Figs. 2, 3, or 4 may be used for this circuit. Some circuit analysis will show that the inverting input is always very near ground potential, and this point is then called a virtual ground insofar as the input signals and output feedback are concerned. Thus the input impedance to the circuit will exactly equal the input network impedance. When capacitors are used in the networks, the phase relationships between current and voltage must be taken into account. These differences in phase allow such operations as differentiation, integration, and active network synthesis. But isn't an op amp a d.c. amplifier and don't d.c. amplifiers drift and have to be chopper-stabilized or otherwise compensated? This certainly used to be true of all amplifiers, but today such techniques are reserved for extremely critical circuits. The reasons for this lie in the input differential stage. It is now very easy to get an integrated circuit differential amplifier stage to track within a millivolt or so over a wide temperature range. This is due to the identical geometry, composition, and temperature of the input transistors. Matched pairs of ordinary transistors can track within a few millivolts with careful selection. FET's offer still drift performance, as one bias point may be selected that is drift-free with respect to temperature over a very wide range. Thus, chopper-stabilized systems are rarely considered today for most op-amp applications. There are three basic op-amp packages available today. The first type consists of specialized units used only for precision analog computation and critical instrumentation circuits. These are priced into the hundreds and even thousands of dollars for each category, and are not considered here. The second type is the modular package, and usually consists of a black plug-in epoxy shell an inch or two on a side. Special sockets are available to accommodate the many pins that protrude out the case bottom. The third package style uses the integrated circuit. Here the entire op amp is housed in a flat pack, in-line epoxy, or TO-5 style package. (See lead photograph.) Generally speaking, the modular units are being replaced in some cases by the integrateds, but at present, each package style offers some clear-cut advantages. Table 1 compares the two packages. The IC versions offer low cost, small size, and very low drift, while the modular versions offer higher input impedances, higher gain, and higher output power capability. Three low-cost readily available IC op amps appear in Figs. 2, 3, and 4. Here, their schematics and major performance characteristics are compared. Devices similar to these at even lower cost may soon be available. A directory of op amp makers is given in Tables 2 and 3. We can split the op-amp applications into roughly three categories: the industrial circuits, the computer circuits, and the active network synthesis circuits. The industrial circuits are "ordinary" ones, which will carryover into the consumer and commercial fields with little change. The boxed copy (facing page) sums up the mathematics. An operational amplifier is often used in conjunction with two passive networks, an input network, and a feedback network, both of which are normally connected to the inverting input. The gain of the over-all circuit at any frequency is given by the equation shown. It is simply the ratio of the feedback impedance to the input impedance at that frequency. For the circuits shown, a low impedance path to ground must exist for all input sources to allow a return path for base current in the two input transistors. Fig. 5A shows an inverting gain-of-100 amplifier useful from d.c. to several hundred kHz. The basic equation tells us the gain will be -10,000/100 = -100. The 100-ohm resistor on the "+" input provides base current for the "+" transistor and does not directly enter into the gain equation. It may be adjusted to obtain a desired drift or offset characteristic. The higher the gain of the op amp, the closer the circuit performance will be to the calculated performance. In the -of-100 amplifier, if the op amp gain is 1000, the gain error will be roughly 1 %. The exact value of the gain also depends upon the precision to which the input and feedback components are selected. Choosing different ratios of input and feedback impedances gives us different gains. Fig. 5B shows a gain-of-10 amplifier with a d.c. to 2 MHz frequency response and a 1000-ohm input impedance. We might ask at this point what we gain by using an op amp in this circuit instead of an ordinary single transistor circuit. There are several important answers. The first is that the input and output are both referenced to ground. Put in zero volts and you get out zero volts. Put in -400 millivolts and you get out +4 volts. Put in 400 millivolts you get out -4 volts. Secondly, the output impedance is very low and the gain will not change if you change the load the op amp is driving, as long as the loading is light compared to the op amp's output impedance. Finally, the gain is precisely 10, to the accuracy you can select the input and feedback resistors, independent of temperature and power-supply variations. It is this precision and ease of control that makes the operational amplifier configuration far superior to simpler circuitry. If the output is connected to the "-" input and an input directly drives the "+" input, the unity-gain voltage follower of Fig. 5C results. This configuration is useful for following precision voltage references or other voltage sources that may not be heavily loaded. The circuit is superior to an ordinary emitter-follower in that the offset is only a millivolt or so instead of the temperature-dependent 0.6-volt drop normally encountered, and the gain is truly unity and not dependent upon the alpha of the transistor used. By making the gain of the op amp frequency-dependent, various filter configurations are realized. For instance, Fig. 5D shows a band-stop amplifier. For very low and very high frequencies, the series RLC circuit in the feedback network will be a very high impedance and the gain will be -10,000/1000 = -10. At resonance, the series RLC impedance will be 100 ohms and the gain will be -100/1000 = -0.1. The gain drops by a factor of 100:1 or 40 decibels at the resonant frequency. The selection of the LC ratio will determine bandwidth, while the LC product will determine the resonant frequency. Fig. 5E does the opposite, producing a response peak at resonance 100 times higher than the response at very high or very low frequencies, owing to the very high impedance at resonance of a parallel LC circuit. More complex filter structures may be used to obtain any reasonable filter function or response curve. Audio equalization curves are readily realized using similar techniques. Turning to some different applications, Fig. SF shows a precision ramp generator. Operation is based upon the current source formed by the reference voltage and 1000-ohm resistor on the input. In any op-amp circuit, the current that is fed back to the input must equal the input current, for otherwise the"-" input will have a voltage on it, which would immediately be amplified, making the input and feedback currents equal. A constant current to a capacitor linearly charges that capacitor, producing a linear voltage ramp. The slope of the ramp will be determined by the current and the capacitance, while the linearity will be determined by the gain of the op amp. A sweep of 0.1-percent linearity is easily achieved. The output ramp is reset to zero by the switch and the 10-ohm current-limiting resistor. For synchronization, S may be replaced by a gating transistor. A negative input current produces a positive voltage ramp at the output. Note that the sweep linearity and amplitude is independent of the output loading as long as the load impedance is higher than the output impedance of the op amp. Ramps like this are often used in CRT sweep waveform generation, analog-to-digital converters, and similar circuitry. Silicon diodes normally have a 0.6-volt offset that makes them unattractive for detecting very low signal levels. If a diode is included in the feedback path of an operational amplifier, this offset may be reduced by the gain of the circuit, allowing low-level detection. Fig. 5G is typical. Here the gain to negative input signals is equal to unity, while the gain to positive input signals is equal to 100. The diode threshold will be reduced to 0.6 volt/100 = 6 millivolts. Another diode op-amp circuit is that of Fig. 5H. Here the logarithmic voltage-current relation present in a diode makes the feedback impedance decrease with increasing input signals, reducing the circuit gain as the input current increases. The net result is an output voltage that is proportional to the logarithm of the input, and the circuit is a logarithmic amplifier. This configuration only works on negative-going inputs and is useful in compressing signals measuring decibels, and in electronic multiplier circuits where the logarithms of two input signals are added together to perform multiplication. An operational amplifier is rarely run "wide open", but Fig. 51 is one exception. Here the op amp serves as a voltage comparator. If the voltage on the "-" input exceeds the "+" input voltage, the op amp output will swing as negative as the supply will let it, and vice versa. A difference of only a few millivolts between inputs will shift the output from one supply limit to the other. Feedback may be added to increase speed and produce a snap action. One input is often returned to a reference voltage, producing alarm or a limit detector. Op amps may also be used in groups. One example is the low-distortion sine-wave oscillator of Fig. 5J, in which three op amps generate a precision sine wave. Both sine an cosine outputs, differing in phase by 90° are produced. An external amplitude stabilization circuit is required, but not shown. Output frequency is determined solely by resistor and capacitor values and their stability. Computer Circuits The analog computer industry was the birthplace and once the only home of the operational amplifier. In fact the name comes from the use of op amps to perform mathematical operations. Many of these circuits are of industry-wide interest and use. Perhaps the simplest op-amp circuit is the inverter. This is an op amp with identical input and feedback resistors Whatever signal gets fed in, minus that signal appears the output, thus performing the sign-changing operation. Addition is performed by the circuit of Fig. 6A. Here the currents from inputs E1, E2, and E3 are summed and the negative of their sum appears at the output. Since the negative input is always very near ground because of feedback, there is no interaction among the three sources Resistor R is adjusted to obtain the desired drift performance. By shifting the resistor values around, the basic summing circuit may also perform scaling and weighting operations. For instance, a 30,000-ohm feedback resistor would produce an output equal to minus three times the sum of the inputs; a smaller feedback resistor would have the opposite effect. By changing only one input resistor without changing the other, one input may be weighted more heavily than the other. Thus, by a suitable choice of resistors, the basic summing circuit could perform such operations as E[OUT]= -0.5 (E1 + 3E2 + 0.6E3). Subtraction is performed by inverting one input signal and then adding. Two very important mathematical operations are integration and differentiation. Integration is simply finding the area under a curve, while differentiation involves finding the slope of a curve at a given point. The op-amp integration circuit is shown in Fig. 6B, while the differentiation circuit is shown in Fig. 6C. The integrator also serves as a low-pass filter, while the differentiator also serves as a high-pass filter, both with 6 dB/octave slopes. The differentiator circuit's gain increases indefinitely with frequency, which obviously brings about high-frequency noise problems. The circuit cannot be used as shown. Fig. 6D shows a practical form of differentiator in which a gain-limiting resistor and some high-frequency compensation have been added to limit the high-frequency noise, yet still provide a good approximation to the derivative of the lower frequency inputs. These two circuits are very important in solving advanced problems, particularly mathematics involving differential equations. Since most of the laws of physics, electronics, thermodynamics, aerodynamics, and chemical reactions can be expressed in differential-equation form, the use of operation amplifiers for equation solution can be a very valuable and powerful analysis tool. Active Network Synthesis Fig. 7. Operational amplifiers in active network synthesis. (A) One form of active filter. (B) A twin-T network is identical to an LC parallel resonant circuit except for the "Q". (C) Circuit to realize "Q" of 14 without using an inductor. Perhaps the newest area in which operational amplifiers are beginning to find wide use is in active network synthesis. There is increasing pressure in industry to minimize the use of inductors. Inductors are big, heavy, expensive, and never obtained without some external field, significant resistance, and distributed capacitance. Worst of all, no one has yet found any practical way to stuff them into an integrated-circuit package. If we can find some circuit that obeys all the electrical laws of inductance without the necessity of a big coil of wire and a core, we have accomplished our purpose. Operational amplifiers are extensively used for this purpose. One basic scheme is shown in Fig. 7 A. If two networks are connected around an op amp as shown, the gain will equal the ratio of the transfer impedances of the two networks. Since we are using three-terminal networks, and since the op amp is capable of adding energy to the circuit, we can do many things with this circuit that are impossible with two-terminal passive resistors and Fig. 7B shows an interesting three-terminal network called a twin-T circuit. It exhibits resonance in the same manner as an ordinary LC circuit does. It has one limitation - its maximum "Q" is only 1 /4. If we combine an op amp with a parallel twin- T network, we can multiply the "Q" electronically to any reasonable level. A gain of 40 would bring the "Q" up to 10. We then have a resonant "RLC" circuit of controllable center frequency and bandwidth with no large, bulky inductors required even for low-frequency operation. One example is shown in Fig. 7C where an operational amplifier is used to realize a resonant effect and a "Q" of 14 at a frequency of 1400 Hz. As the desired "Q" increases, the tolerances on the components and the gain become more and more severe. From a practical standpoint, value of "Q" greater than 25 are very difficult to realize at the present time. Note that the entire circuit shown can be placed in a space much smaller than that occupied by the single inductor it replaces. Posted February 27, 2019 (updated from original post on 9/6/2011)	{"url":"https://rfcafe.com/references/electronics-world/the-operational-amplifier-aug-1967-electronics-world.htm","timestamp":"2024-11-11T06:44:49Z","content_type":"text/html","content_length":"51571","record_id":"<urn:uuid:94529b12-ee18-459c-b32b-55494b532506>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00099.warc.gz"}
On the convergence of a regularization scheme for approximating cavitation solutions with prescribed cavity volume size Let $\Omega\in\mathbb{R}^n$, $n=2,3$, be the region occupied by a hyperelastic body in its reference configuration. Let $E(\cdot)$ be the stored energy functional, and let $x_0$ be a flaw point in $\ Omega$ (i.e., a point of possible discontinuity for admissible deformations of the body). For $V>0$ fixed, let $u_V$ be a minimizer of $E(\cdot)$ among the set of discontinuous deformations $u$ constrained to form a hole of prescribed volume $V$ at $x_0$ and satisfying the homogeneous boundary data $u(x)=Ax$ for $x\in\partial \Omega$. In this paper we describe a regularization scheme for the computation of both $u_V$ and $E(u_V)$ and study its convergence properties. In particular, we show that as the regularization parameter goes to zero, (a subsequence) of the regularized constrained minimizers converge weakly in $W^{1,p}(\Omega\setminus{{\mathcal{B}}_{\delta}(x_0)})$ to a minimizer $u_{V}$ for any $\delta>0$. We obtain various sensitivity results for the dependence of the energies and Lagrange multipliers of the regularized constrained minimizers on the boundary data $A$ and on the volume parameter $V$. We show that both the regularized constrained minimizers and $u_V$ satisfy suitable weak versions of the corresponding Euler--Lagrange equations. In addition we describe the main features of a numerical scheme for approximating $u_V$ and $E(u_V)$ and give numerical examples for the case of a stored energy function of an elastic fluid and in the case of the incompressible limit. Dive into the research topics of 'On the convergence of a regularization scheme for approximating cavitation solutions with prescribed cavity volume size'. Together they form a unique fingerprint. • Person: Research & Teaching	{"url":"https://researchportal.bath.ac.uk/en/publications/on-the-convergence-of-a-regularization-scheme-for-approximating-c","timestamp":"2024-11-06T18:44:48Z","content_type":"text/html","content_length":"61610","record_id":"<urn:uuid:cf18d151-8274-4631-a814-d76aef255b17>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00181.warc.gz"}
[Solved] Suppose that events occur in accordance w \| SolutionInn Suppose that events occur in accordance with a Poisson process at the rate of five events per Suppose that events occur in accordance with a Poisson process at the rate of five events per hour. a. Determine the of the waiting time T1 until the first event occurs. b. Determine the of the total waiting time Tk until k events have occurred. c. Determine the probability that none of the first k events will occur within 20 minutes of one another. The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most... Fantastic news! We've Found the answer you've been seeking!	{"url":"https://www.solutioninn.com/suppose-that-events-occur-in-accordance-with-a-poisson-process","timestamp":"2024-11-12T00:43:31Z","content_type":"text/html","content_length":"82427","record_id":"<urn:uuid:f490896b-3f6e-4b0f-b327-3626a7741149>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00114.warc.gz"}
Basics Of Ball Mill Industrial Ball Mills: Steel Ball Mills and Lined Ball ... Ball Mills Steel Ball Mills & Lined Ball Mills. Particle size reduction of materials in a ball mill with the presence of metallic balls or other media dates back to the late 1800's. The basic construction of a ball mill is a cylindrical container with journals at its axis. اقرأ أكثر grinding ball mill process in ball mill Ball Milling - an overview ScienceDirect Topics. Ball milling, a shear-force dominant process where the particle size goes on reducing by impact and attrition mainly consists of metallic balls (generally Zirconia (ZrO 2) or steel balls), acting as grinding media and rotating shell to create centrifugal force.In this process, graphite (precursor) was breakdown by randomly striking with … اقرأ أكثر The working principle of ball mill - Meetyou Carbide The ball mill consists of a metal cylinder and a ball. The working principle is that when the cylinder is rotated, the grinding body (ball) and the object to be polished (material) installed in the cylinder are rotated by the cylinder under the action of friction and centrifugal force. اقرأ أكثر Optimization of mill performance by using Optimization of mill performance by using online ball and pulp measurements by B. Clermont* and B. de Haas* Synopsis Ball mills are usually the largest consumers of energy within a mineral concentrator. Comminution is responsible for 50% of the total mineral processing cost. In today's global markets, expanding mining groups are trying اقرأ أكثر Ball mill - Wikipedia A ball mill, a type of grinder, is a cylindrical device used in grinding (or mixing) materials like ores, chemicals, ceramic raw materials and paints.Ball mills rotate around a horizontal axis, partially filled with the material to be ground plus the grinding medium. Different materials are used as media, including ceramic balls, flint pebbles, and stainless steel balls. اقرأ أكثر (PDF) Closed circuit ball mill – Basics revisited \| Walter ... Classification effects in wet ball milling circuits. Min. Eng. increase in capacity for ball mills in closed circuits; however, the (August), 815–825. Please cite this article in press as: Jankovic, A., Valery, W. Closed circuit ball mill – Basics revisited. Miner. Eng. اقرأ أكثر Ball Mill for Mining Market 2021 : Worldwide Industry ... The MarketWatch News Department was not involved in the creation of this content. Oct 28, 2021 (The Expresswire) -- Global "Ball Mill for Mining Market" is expected to grow at a steady growth ... اقرأ أكثر Closed circuit ball mill – Basics revisited - ScienceDirect 1. Introduction. 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The mill also includes the processes of attrition and impact, although these actions are limited if the material is easily reduced by cutting and the screen limiting discharge has large perforations. The mill consists of a rotating shaft with four attached parallel knives and a screen occupying one fourth of the 360 degree rotation. The mill is ... اقرأ أكثر Rolling Process: Working, Application, Defects, Type of ... In this type of rolling mill, there are two basic roles that are backed up by two or more rolls which are bigger than those two basic rolls. These backed up rolls give more pressure to the basic rolls to heavily press the strip. Application of Rolling: The rolling operation used in … اقرأ أكثر Ball Mill Design/Power Calculation The basic parameters used in ball mill design (power calculations), rod mill or any tumbling mill sizing are; material to be ground, characteristics, Bond Work Index, bulk density, specific density, desired mill tonnage capacity DTPH, operating % solids or pulp density, feed size as F80 and maximum 'chunk size', product size as P80 and maximum and finally the … اقرأ أكثر 002 Basics On Cement Ball Mill Systems \| Mill (Grinding ... Ball Mill Workshop, South Asia 27thSept -1stOct 2010. 2. Cement ball mill system basics. Holcim Group Support Limited 2010 Systems Overview. Tube mill Tube mill with pregrinding unit Finish grinding اقرأ أكثر Ball Mill Used in Minerals Processing Plant \| Prominer ... 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It consist of a spinning plastic jar th... اقرأ أكثر Fundamentals of CNC Machining • Knowledge of the proper use of basic hand tools and precision measuring instruments, including calipers and micrometers. • Some manual machining experience is helpful but not required. • Knowledge of Solidworks® is a pre-requisite or co -requisite for this course. اقرأ أكثر Ball Mill Liners Market Size In 2022 with Top Countries ... Global "Ball Mill Liners Market" 2022 Research Report provides key analysis on the market status of the Ball Mill Liners manufacturers with best facts and figures, meaning, definition, SWOT ... اقرأ أكثر Ball Mills - Mineral Processing & Metallurgy Within these limits a rod mill is usually superior to and more efficient than a ball mill. The basic principle for rod grinding is reduction by line contact between rods extending the full length of the mill, resulting in selective grinding carried out on the largest particle sizes. This results in a minimum production of extreme fines or ... اقرأ أكثر Ball Mills - an overview \| ScienceDirect Topics 8.3.2.2 Ball mills. The ball mill is a tumbling mill that uses steel balls as the grinding media. The length of the cylindrical shell is usually 1–1.5 times the shell diameter ( Figure 8.11). The feed can be dry, with less than 3% moisture to minimize ball … اقرأ أكثر Online Course: Ball Mill - Basic Learner's Course - YouTube FL has designed a series of online training for the cement industry, providing you with easy and instant access to our specialised technical training. ... اقرأ أكثر Pulverizer : ball mill Mixing (basic symbol) Kneader Ribbon blender Double cone blender Filter (basic symbol, simple batch) Filter press (basic symbol) Rotary filter, film drier or flaker A- 8 APPENDIX A GRAPHICAL SYMBOLS FOR PIPING SYSTEMS AND PLANT. Cyclone and hydroclone (basic symbol) اقرأ أكثر Introduction to Machining: Milling Machine Typical, Basic Milling Machine . Tramming the Head •The head of a vertical milling machine can be tilted from side to side and from front to back. ... Ball end mills can produce a fillet. Formed milling cutters can be used to produce a variety of features including round edges. اقرأ أكثر CE-Type Pulverizers / Mill » Babcock & Wilcox Babcock & Wilcox (B&W) is now applying its vast experience and knowledge of roll wheel and ball-and-race pulverizers to provide quality replacement parts, services and inventory management programs to Combustion Engineering (CE)-type mills / pulverizers. Since 1867, B&W has set the standard for proven high availability, reliability and low maintenance on its … اقرأ أكثر	{"url":"https://www.tartakpila.pl/20043+basics+of+ball+mill","timestamp":"2024-11-08T21:57:22Z","content_type":"text/html","content_length":"38828","record_id":"<urn:uuid:4e4d3d50-8a20-4e43-914e-17c728ded490>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00157.warc.gz"}
Quadratic formula Problems and Solutions Solving the Quadratic equations by Quadratic formula Questions with Solutions The quadratic equations can be solved by using the quadratic formula in mathematics. The worksheet on solving the quadratic equations by quadratic formula is given here with examples and answers for practice, and also solutions to learn how to solve every quadratic equation with quadratic formula to find their roots. Solve $x^2-4x+4 \,=\, 0$ Solve $2x^2+5x-3 \,=\, 0$ Solve $\dfrac{2x}{x-4}$ $+$ $\dfrac{2x-5}{x-3}$ $\,=\,$ $\dfrac{25}{3}$ Solve $2x^2+x-1\,=\,0$	{"url":"https://www.mathdoubts.com/quadratic-formula-problems-solutions/","timestamp":"2024-11-11T00:57:10Z","content_type":"text/html","content_length":"27221","record_id":"<urn:uuid:b4ebfc87-07ab-4d91-951c-79c6fafb5973>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00848.warc.gz"}
This class is designed to prepare students for Calculus. It necessarily includes a review of pre-calculus topics to make sure that students solidly understand the foundational skills they need to do well in calculus. Students entering pre-calculus are welcome - as they get an overview of what they will learn this year and what those skills will build to the next year. In the Pre-Calculus portion, we'll go over limits, summations, graphing rational equations, and converting parametric equations. Then, once students have mastered the basics, we will move towards preparing students for Calculus and AP Calculus. Topics include limits, definitions of derivatives, derivatives, and techniques of differentiation, applications of derivatives. PreCalc+CalcBoost!, like all of our math camps, will begin and end with a pre-test and post-test to measure skill gains across the week. Algebra II (in school or EdBoost camp) Grades (refer to the grade your child will enter in the Fall):	{"url":"https://edboost.org/index.php/product/11","timestamp":"2024-11-06T14:18:39Z","content_type":"text/html","content_length":"18935","record_id":"<urn:uuid:c8645c6a-f628-4aa3-a442-7a3fbcdbeb77>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00656.warc.gz"}
Mathematics and Computer Science Maria Flavia MAMMANA Associate Professor of Mathematics education and history of mathematics [MATH-01/B] Maria Flavia Mammana is Associate Professor of Mathematics Education at the University of Catania since February 2018. She teaches in the degree course in Mathematics and Mathematics Magistrale. She is in charge of the Polo della Didattica-Accademia dei Lincei, sect. Mathematics, Catania site. She is referent, for the DMI, of the Mat.Ita Project, together with her colleagues Cirmi and D'Asero. Coordinates the activities of the Liceo Matematico project, for the East part of Sicily. Her research intersts are in the field of Foundations of Geometry and Mathematics Education. She is a member of the Italian Commission for Mathematical Instruction (CIIM). She is a member of the national board of the UMI Group on "Licei Matematici" (Mathematics High Schools). She is the scientific responsible for Italy of the European projects ASYMPTOTE and MaSCE3. She is a partner of UMI, AIRDM, PME, ERME. Maria Flavia Mammana's research activities mainly concern: - Euclidean Geometry from the point of view of geometric transformations; - issues relating to teaching/learning geometry with technology; - the introduction of Serious Games in the teaching/learning of mathematics; - the role of the body and movement in the teaching/learning of mathematics. Research activities are conducted within the Teaching Research and Experimentation Core of the Department of Mathematics and Informatics. The research carried out within the Core is developed following a methodological path marked out in various stages. The path begins, for example, with research into new properties of elementary geometry; these properties are then used to design innovative teaching proposals for upper secondary schools; finally, classroom experiments are developed and the results are examined. Together with computer science colleagues from the Department of Mathematics and Computer Science at UniCT, studies were launched within the SGM (Serious Games for Math) project on the creation and experimentation of 'serious games' for learning basic mathematical concepts, and within the TEMA project on the learning of Maya mathematics as early as kindergarten. In 2016, research was launched with Francesca Ferrara (University of Turin), Elizabeth de Freitas (Manchester Metropolitan University) and Michela Mschietto (University of Modena and Reggio Emilia). The subject of the study is the peculiar role of body and movement in mathematical activity, particularly within the learning and teaching processes of the discipline in the classroom context. He is the scientific responsible for Italy of the MaSCE3 and ASYMPTOTE projects, of which UniCT is one of the partners.	{"url":"https://dmi.unict.it/faculty/maria.flavia.mammana","timestamp":"2024-11-15T00:19:40Z","content_type":"text/html","content_length":"32677","record_id":"<urn:uuid:d9b7117b-c10a-4019-977b-96e16807960b>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00729.warc.gz"}
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Web harry potter party ideas, make golden snitch party favours, easy to make with a cricut tutorial to create the wings or a free print. Web a gold ornament (i picked mine up at a dollar store) the printable wings at the end of this post. Related Post:	{"url":"https://neu-news.de/printable/golden-snitch-printable-wings.html","timestamp":"2024-11-06T17:16:13Z","content_type":"text/html","content_length":"24208","record_id":"<urn:uuid:8602ef20-58a4-495d-8738-cf088c95382d>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00243.warc.gz"}
Which One & Why? by David Mattoon \| Feb 16, 2020 \| 6th Expressions & Equations, 7th Expressions & Equations, 8th Functions, HS Functions, HS Structure Common Core Standards (different examples support different standards): 6.EE.A.2, 7.EE.A.1, 8.F.A.2, 8.F.A.3, HSA.SSE.A.1.A, HSF.IF.B.4 Besides using Sticky Math to compare two different representations or match representations, you could provide two nonequivalent prompts in a singular representation. For example, below we have two nonequivalent expressions in a single representation, abstract symbolic: At first glance, this seems overly simplistic; however, students often struggle with understanding and using exponents. This is a good example of a time when students might be procedurally proficient, but not procedurally fluent. They may be proficient in performing operations on or with exponents; however, they demonstrate a lack of procedural fluency when having to flexibly apply their use to interpreting expressions. Students should choose a representation and construct a viable argument to defend their selection. As one student or group of students defend their argument, the others should be engaged through math talk moves and asked if they agree or disagree and why (MP3). Notice that while there is no blank sticky on the image above to remind you, students can still use their third sticky to create an additional representation of two times x plus three. Other representations might include drawing algebra tiles or writing an equivalent expression like 1x +3 + 1x. Here is another example centered on confronting a common misconception head on: This Sticky Math Which One & Why? is designed to create cognitive dissonance with those students who would ignore order of operations and add 5 + 2 in the left expression to incorrectly get an equivalent expression. Notice they are asked to simply one of the expressions as an added layer of formative assessment; they can do this on their third sticky. While this Which One & Why? acts as a formative assessment of the common misconception, the instruction that precedes it or follows it should be done in context. For an example of how to do this, see the presentation from MaTHink 2020 called Algebra as Area: Distributive Property at Meaning4Memory.com/presentations. On the left sticky, you cannot combine the 5 with the 2 because the 5 refers to a number of items while the 2 refers to a number of groups. You would first have to find the number of items in the the two groups before adding them to the other 5 items. Students need context to grasp this. The context in the previously mentioned presentation linked above is picking apples. Applying that context here would go something like this, “Two parents picked 3 baskets of x apples plus they each picked up four apples from the ground while their child picked up 5 apples from the ground. Write a simplified expression for how many apples they picked in all.” Students should be able to decontextualize the prompt into an expression; however, they should also be able to contextualize as they simplify. If they did, then they would realize that they should not be combining a number of apples, 5, with a number of parents, 2. If you wanted to use this Which One & Why? even more formatively, then you could introduce the possibly that both expressions were correct first like in this example: You may want to introduce one like this early to open up the chance that both stickies could be correct in future iterations of the activity. Notice this activity includes an extension of evaluating both expressions when x = 10. It can be done on a single third sticky by writing small. This can act as a formative assessment of this skill; however, it also provides a specific case to demonstrate that both expressions are numerically equivalent. Students may choose to evaluate the two original expressions or their simplified versions; this provides an opportunity for further comparison and connections when having students debrief the activity (EMTP: Elicit & Use Evidence of Student Thinking). These final two examples demonstrate applying Which One & Why? to something other than expressions and a different type of representation, a graph: This example intentionally excludes a more obvious symbolic notation of a positive slope. Do students understand how the slope in the right sticky is reflected in the graph? Or, do they just make two negatives a positive according to some memorized rule that they may or may not understand? If they do change it to a positive slope, then you have an opportunity to have students compare and contrast the methods and how it moves you along the line as they help debrief the activity (EMTP: Elicit & Use Evidence of Student Thinking). Do students understand that the negative in the left sticky does not distribute to both the numerator and the denominator? The prompt forces a choice; however, if they are open to choosing both like the previous example suggests, then they might incorrectly suggest that neither would match the graph or both would match the graph (thinking it distributed to both the numerator and the Notice once again that the prompt extends to formatively assess a different skill, graphing. Sometimes textbooks make an error of excluding this type of interaction by having students select a matching graph when the standard specifically states, “Students graph…” This extension provides them the chance to graph the unmatched linear equation. Here is a different way to interact with graphical representations: This Sticky Math Which One & Why? focuses on the solution to a quadratic equation rather than moving along the graph itself. Notice that the prompt extends to formatively assess whether or not they know what the graph of a single solution looks like. They can sketch the graph on a regular sticky or use Post It Grids: How are your students using Which One & Why? What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.	{"url":"https://stickymath.org/which-one-why/","timestamp":"2024-11-07T16:23:00Z","content_type":"text/html","content_length":"162810","record_id":"<urn:uuid:531c2bc2-a5a5-44c4-aa76-4830207efabd>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00778.warc.gz"}
EViews Help: Auto-Updating Series Auto-Updating Series One of the most powerful features of EViews is the ability to use a series expression in place of an existing series. These expressions generate auto-series in which the expression is calculated when in use, and automatically recalculated whenever the underlying data change, so that the values are never out of date. Auto-series are designed to be discarded after use. The resulting downside to auto-series is that they are quite transitory. You must, for example, enter the expression wherever it is used; for example, you must type “LOG(X)” every time you wish to use an auto-series for the logarithm of X. For a single use of a simple expression, this requirement may not be onerous, but for more complicated expressions used in multiple settings, repeatedly entering the expression quickly becomes tedious. For more permanent series expression handling, EViews provides you with the ability to define a series or alpha object that uses a formula. The resulting auto-updating series is simply an EViews numeric series or alpha series that is defined, not by the values currently in the object, but rather by an expression that is used to compute the values. In most respects, an auto-updating series may simply be thought of as a named auto-series. Indeed, naming an auto-series is one way to create an auto-updating series. The formula used to define an auto-series may contain any of the standard EViews series expressions, and may refer to series data in the current workfile page, or in EViews databases on disk. It is worth emphasizing that in contrast with link objects, which also provide dynamic updating capabilities, auto-updating series are designed to work with data in a single workfile page. Auto-updating series appear in the workfile with a modified version of the series or alpha series icon, with the numeric series icon augmented by an “=” sign to show that it depends upon a formula. Defining an Auto-Updating Series Using the Dialog To turn a ordinary series into an auto-updating series, you will assign an expression to the series and tell EViews to use this expression to determine the series values. Simply click on the button on the series or alpha series object toolbar, or select from the main menu, then select the tab. There are three radio buttons which control the values that will be placed in the numeric or alpha series ( “Alpha Series” ). The default setting is either or (depending on the series type) in which the series is defined by the values currently in the series; this is the traditional way that one thinks of defining a numeric or alpha series. If instead you select , enter a valid series expression in the dialog box, and click on , EViews will treat the series as an auto-updating series and will evaluate the expression, putting the resulting values in the series. Auto-updating numeric series appear with a new icon in the workfile—a slightly modified version of the standard series icon, featuring the series line with an extra equal sign, all on an orange background. Lastly, indicates that the series is linked to data found outside of EViews as described in the link specification. You will be prompted to update data in external links whenever the workfile is opened, and you may update the external series links on demand by right-clicking on the series, and selecting or clicking on CTRL-F5. In this example, we instruct EViews that the existing series LOGTAXRT should be an auto-updating series that contains the natural logarithm of the TAXRATE2 series. As with an auto-series expression, the values in LOGTAXRT will never be out of date since they will change to reflect changes in TAXRATE2. In contrast to an auto-series, however, LOGTAXRT is a permanent series in the workfile which may be used like any other series. Alternately, you may create an auto-updating series that links to a series in a external file or database by selecting . In contrast to an auto-updating series based on a formula which updates whenever the underlying data change, an auto-updating series based on an external link will update only when the workfile is first loaded (you will be prompted for whether to refresh the data series or not) or when you manually update the links by clicking on in the workfile window, then either selecting the source database above or selecting the individual objects that are linked to that database and then clicking the associated button. You may also update the link by selecting in the series menu. You may, at any time, change an auto-updating series into an standard numeric series by bringing up the page of the dialog, and clicking on the setting. EViews will define then define the series by its current values. In this way you may freeze the formula series values at their existing values, a procedure that is equivalent to performing a standard series assignment using the provided Note that once an expression is entered as a formula in a series, EViews will keep the definition even if you specify the series by value. Thus, you make take a series that has previously been frozen, and return it to auto-updating by selecting definition. Issuing a Command To create an auto-updating series using commands, you should use the formula keyword, frml, followed by an assignment statement. The following example creates a series named LOW that uses a formula to compute values. The auto-updating series takes the value 1 if either INC is less than or equal to 5000 or EDU is less than 13, and takes the value 0 otherwise: frml low = inc<=5000 or edu<13 LOW is now an auto-updating series that will be reevaluated whenever INC or EDU change. You may also define auto-updating alpha series using the frml keyword. If FIRST_NAME and LAST_NAME are alpha series, then the declaration: frml full_name = first_name + " " + last_name creates an auto-updating alpha series, FULL_NAME. The same syntax should be used when you wish to apply a formula to an existing series. series z = rnd frml z =(x+y)/2 makes Z an auto-updating series that contains the average of series X and Y. Note that the previous values of Z are replaced, and obviously lost. Similarly, we may first define an alpha series and then apply an updating formula: alpha a = "initial value" frml a = @upper(first_name) You may not, however, apply an alpha series expression to a numeric series, or vice versa. Given the series Z and A defined above, the following two statements: frml z = @upper(first_name) frml a = (x+y)/2 will generate errors. Note that once a numeric series or alpha series is defined to be auto-updating, its values may not be modified directly, since they are determined from the formula. Thus, if Z is an auto-updating series, the assignment command: z = log(x) will generate an error since an auto-updating series may not be modified. To modify Z you must either issue a new frml assignment or you must first set the values of Z to their current values by turning off auto-updating, and then issue the assignment statement. To reset the formula in Z, you may simply issue the command: frml z = log(x) to replace the formula currently in the series. To turn off auto-updating for a series, you may use the special expression “@CLEAR” in your frml assignment. When you turn off auto-updating, EViews freezes the numbers or strings in the series at their current values. Once the series is set to current values, it is treated as an ordinary series, and may be modified as desired. Thus, the commands: frml z = @clear z = log(x) are allowed since Z is converted into an ordinary series prior to performing the series assignment. Alternately, you may convert a named auto-updating series into an ordinary series by selecting from the workfile window and using the dialog to break the links in the auto-updating series. One particularly useful feature of auto-updating series is the ability to reference series in databases. The command: frml gdp = usdata::gdp creates a series in the workfile called GDP that gets its values from the series GDP in the database USDATA. Similarly: frml lgdp = log(usdata::gdp) creates an auto-updating series named LGDP that contains the log of the values of GDP in the database USDATA. Series that reference data in databases may be refreshed each time a workfile is loaded from disk. Thus, it is possible to setup a workfile so that its data are current relative to a shared database. Naming an Auto-Series If you have previously opened a window containing an ordinary auto-series, you may convert the auto-series into an auto-updating series by assigning a name. To turn an auto-series into an auto-updating series, simply click on the button on the toolbar, or select from the main menu, and enter a name. EViews will assign the name to the series object, and will apply the auto-series definition as the formula to use for auto-updating. Suppose, for example, that you have opened a series window containing an auto-series for the logarithm of the series CP by clicking on the button on the toolbar, or selecting and entering “LOG(CP)”. Then, simply click on the button in the auto-series toolbar, and assign a name to the temporary object to create an auto-updating series in the workfile. Additional Issues Auto-updating series are designed to calculate their values when in use, and automatically update values whenever the underlying data change. An auto-updating series will assign a value to every observation in the current workfile, irrespective of the current values of the workfile sample. In most cases, there is no ambiguity in this operation. For example, if we have an auto-updating series containing the expression “LOG(CP)”, we simply take each observation on CP in the workfile, evaluate the log of the value, and use this as the corresponding auto-updating series value. However, in cases where the auto-updating series contains an expression involving descriptive statistics, there is ambiguity as to whether the sample used to calculate the values is the sample at the time the auto-updating series was created, the sample at the time the series is evaluated, the entire workfile range, or some other sample. To resolve this ambiguity, EViews will enter the current workfile sample into the expression at the time the auto-updating series is defined. Thus, if you enter “@MEAN(CP)” as your auto-updating series expression, EViews will substitute an expression of the form “@MEAN(CP, smpl)” into the definition. If you wish to evaluate the descriptive statistics for a given sample, you should enter an explicit sample in your expression.	{"url":"https://help.eviews.com/content/newser-Auto-Updating_Series.html","timestamp":"2024-11-04T08:35:47Z","content_type":"application/xhtml+xml","content_length":"22336","record_id":"<urn:uuid:34a41673-5b74-4f1c-b7c2-42be54d14797>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00482.warc.gz"}
linear equations class 10 MCQ Question \| Class 10 Maths MCQ Questions for Class 10 Pair of Linear Equations in Two Variables with Answers Students can refer to the following linear equations class 10 MCQ Question with Answers provided below based on the latest curriculum and examination pattern issued by CBSE and NCERT. Our teachers have provided here collection of multiple choice questions for Linear Equations Class 10 covering all topics in your textbook so that students can assess themselves on all important topics and thoroughly prepare for their exams linear equations class 10 MCQ Question with Answers We have provided below linear equations class 10 MCQ Question with answers which will help the students to go through the entire syllabus and practice multiple choice questions provided here with solutions. As Linear Equations MCQs in Class 10 pdf download can be really scoring for students, you should go through all problems and MCQ Questions for Class 10 Maths provided below so that you are able to get more marks in your exams. linear equations class 10 MCQ Question Question. The pair of equations 6x -7y =1 and 3x – 4 y = 5 has (a) a unique solution (b) two solutions (c) infinitely many solutions (d) no solution Question. The pair of equations 4x -3y +5 = 0 and 8x -6 y -10 = 0 graphically represents two lines which are (a) coincident (b) parallel (c) intersecting at exactly one point (d) intersecting at exactly two points Question. The pair of equations y = a and y = b graphically represents lines which are (a) intersecting at (a, b) (b) intersecting at (b, a) (c) parallel (d) coincident Question. Divya has only ≠ 2 and ≠ 5 coins with her. If the total number of coins that she has is 25 and the amount of money with her is ≠ 80, then the number of ` 2 and ≠ 5 coins are , respectively (a) 15 and 10 (b) 10 and 15 (c) 12 and 10 (d) 13 and 12 Question. A pair of linear equations which has x = 0, y = -5 as a solution is (a) x+y=0/2x-3y=10 (b) x+y=3/2x-y=5 (c) 2x+3y+5=0/3x+2y=0/x-4y=0 (d) x-4y=14/5x-y=13 Question. The number of solutions of the pair of linear equations x +3y – 4 = 0 and 2x +6 y = 7 is (a) 0 (b) 1 (c) 2 (d) infinite Question. A’s age is six times B’s age. Four years hence, the age of A will be four times B’s age. The present ages, in years, of A and B are, respectively (a) 3 and 24 (b) 36 and 6 (c) 6 and 36 (d) 4 and 24 Question. The value of k for which the lines (k +1)x +3ky +15 = 0 and 5x + ky +5 = 0 are coincident is (a) 14 (b) 2 (c) –14 (d) –2 Question. The pair of equations x = 2 and y = 3 has (a) one solution (b) two solutions (c) many solutions (d) no solution Question. If a pair of linear equations is inconsistent, then the lines representing them will be (a) parallel (b) always coincident (c) intersecting or coincident (d) always intersecting Question. If a pair of linear equations has infinitely many solutions, then the lines representing them will be (a) parallel (b) intersecting or coincident (c) always intersecting (d) always coincident Question. The value of k for which the pair of equations kx + y = 3 and 3x +6 y = 5 has a unique solution is (a) – 1/2 (b) 2 (c) –2 (d) all the above Question. The number of solutions of the pair of equations 2x +5y =10 and 6x +15y -30 = 0 is (a) 0 (b) 1 (c) 2 (d) infinite Question. The value of k for which the system of equations x +3y – 4 = 0 and 2x + ky = 7 is inconsistent is (a) 21/4 (b) 1/6 (c) 6 (d) 4/21 Question. Sanya’s age is three times her sister’s age. Five years hence, her age will be twice her sister’s age. The present ages (in years) of Sanya and her sister are respectively (a) 12 and 4 (b) 15 and 5 (c) 5 and 15 (d) 4 and 12 Question. The sum of the digits of a two digit number is 8. If 18 is added to it, the digits of the number get reversed. The number is (a) 53 (b) 35 (c) 62 (d) 26 Question. How many solutions does the system of equations p + 2q = 4 and 2p + 4q – 12 = 0 have? (a) 0 (b) 1 (c) 2 (d) 3 Question. If the pair of linear equations 2x + ky – 3 = 0 and 6x + 2/3 y + 7 = 0 has a unique solution, which of the following is true? (a) k = 2/3 (b) k ≠ 2/3 (c) k = 2/9 (d) k ≠ 2/9 Question. If a + b = 5and 3a + 2b = 20, find 3a + b . (a) 25 (b) 20 (c) 15 (d) 10 Question. If the sum of the ages (in years) of a father and his son is 65 and twice the difference of their ages (in years) is 50, what is the age of the father? (a) 45 years (b) 40 years (c) 50 years (d) 55 years Question. Three chairs and two tables cost ₹ 1850.Five chairs and three tables cost ₹ 2850. Find the total cost of one chair and one table. (a) ₹ 800 (b) ₹ 850 (c) ₹ 900 (d) ₹ 950 Question. If the cost of 3 audio cassettes and 2 VCDs is ₹ 350 and that of 2 audio cassettes and 3 VCDs is ₹ 425, what is the cost of a VCD? (a) ₹ 140 (b) ₹ 125 (c) ₹ 115 (d) ₹ 110 Question. A part of the monthly expenses of a family is constant and the remaining varies with the price of wheat. When the price of wheat is ` 250 per quintal, the monthly expenses of the family is ₹ 1000 and when it is ₹ 240 per quintal, the monthly expenses is ₹ 980. Find the monthly expenses of the family on wheat when the cost of wheat is ₹ 350 a quintal. (a) ₹ 900 (b) ₹ 350 (c) ₹ 650 (d) ₹ 700 Question. Find the unique solution of the system of simultaneous equations 2x – y = 2 and 4x – y = 4. (a) x = 0,y =1 (b) x = 0,y = 0 (c) x =1,y = 0 (d) x =1,y =1 Question. Five years ago, a father’s age was seven times his son’s age. Five years from now, the father’s age will be thrice the son’s age. What are the respective present ages of father and son? (a) 40 years, 10 years (b) 10 years, 40 years (c) 25 years, 5 years (d) 30 years, 8 years Question. The side of a square is 4m more than the side of another square. The sum of their areas is 208 sq. m. What is the side of the larger square? (a) 12m (b) 8m (c) 9m (d) 5m Question. If the system of equations 4x + y = 3 and (2k -1) x +(k -1)y = 2k +1 is inconsistent, then k = (a) 2/3 (b) -2/3 (c) -3/2 (d) 3/2 Question. If the system of equations has infinitely many solutions, then (a) b = 2a (b) a = 2b (c) a +2b = 0 (d) 2a- b = 0 Question. The value of k for which the system of equations kx – y = 2, 6x -2y = 3 has a unique solution is (a) = 0 (b) = 3 (c) ¹ 0 (d) ¹ 3 Question. If the system of equations 2x +3y = 7 (a + b)x +(2a – b)y = 21 has infinitely many solutions, then (a) a = 1, b = 5 (b) a = –1, b = 5 (c) a = 5, b = 1 (d) a= 5, b = -1 Question. If am¹ bl, then the system of equations ax + by = c, lx + my = n (a) has a unique solution (b) has no solution (c) has infinitely many solutions (d) may or may not have a solution Question. If 2x -3y = 7 and (a + b)x -(a + b -3)y = 4a + b represent coincident lines, then a and b satisfy the equation (a) a +5b = 0 (b) 5a+ b = 0 (c) a -5b = 0 (d) 5a- b = 0 Question. The pair of equations x = a and y = b graphically represent lines which are (a) parallel (b) intersecting at (b, a) (c) coincident (d) intersecting at (a, b) Question. A pair of linear equations in two variables cannot have (a) a unique solution (b) no solution (c) infinitely many solutions (d) exactly two solutions Question. The pair of equations 3x -2y = 5 and 6x – y = 3 have (a) no solution (b) a unique solution (c) two solutions (d) infinitely many solutions Question. If x = a and y = b is the solution of the equations x + y = 5 and x – y = 7, then values of a and b are respectively (a) 1 and 4 (b) 6 and –1 (c) – 6 and 1 (d) –1 and –6 Question. A pair of linear equations which has a unique solution x = -1, y = -2 is (a) x – y =1; 2x +3y = 5 (b) 2x -3y = 4; x -5y = 9 (c) x + y -3 = 0; x – y =1 (d) x + y +3= 0; 2x -3y +5 = 0 Question. If the lines given by 3x +2ky = 2 and 2x +5y +1 = 0 are parallel, then the value of k is (a) -5/4 (b) 2/5 (c) 15/4 (d) 3/2 Question. A pair of linear equations which has a unique solution x = 3, y = -2 is (a) x+y=-1/2x-3y+12 (b) 2x+5y+4=0/4x+10y+8=0 (c) 2x-3y=0/3x+2y=0/x-4y=0 (d) x-4y=14/5x-y=13 Question. The value of k for which the system of equation 2x +3y = 7 and 8x +(k + 4)y -28 = 0 has infinitely many solution is (a) –8 (b) 8 (c) 3 (d) –3 Question. If x = a and y = b is the solution of the equations x – y = 2 and x + y = 4, then the values of a and b are respectively (a) 3 and 5 (b) 5 and 3 (c) 3 and 1 (d) –1 and –3 Question. Gunjan has only ` 1 and ` 2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is ` 75, then the number of ` 1, and ` 2 coins are respectively (a) 25 and 25 (b) 15 and 35 (c) 35 and 15 (d) 35 and 20 Question. The value of g for which the system of equations 5gx -2y =1 and 10x + y = 3 has a unique solution is (a) = 4 (b) ¹ 4 (c) =– 4 (d) ¹ – 4 Question. The value of k for which the system of equations 2x + y -3 = 0 and 5x + ky +7= 0 has no solution is (a) 2 (b) 5 (c) 5/2 (d) 3/7 Question. The sum of the digits of a two digit number is 12. If 18 is subtracted from it, the digits of the number get reversed. The number is (a) 57 (b) 75 (c) 84 (d) 48 Question. If the lines given by 3x +2ky = 2 and 2x +5y +1 = 0 are parallel, then the value of k is (a) 3/2 (b) 15/4 (c) 2/5 (d) – 5/4 Question. One equation of a pair of dependent linear equations is 3x – 4 y = 7. The second equation can be (a) – 6x +8 y =14 (b) –6x +8 y +14 = 0 (c) 6x +8 y =14 (d) -6x -8 y -14 = 0 Question. If a pair of linear equations is consistent, then the lines will be (a) always intersecting (b) always coincident (c) intersecting or coincident (d) parallel Question. If x = a, y = b is the solution of the equation x + y = 3 and x – y = 5, then the values of a and b are, respectively (a) 4 and –1 (b) 1 and 2 (c) –1 and 4 (d) 2 and 3 Question. If we add 1 to the numerator and denominator of a fraction, it becomes 1/2 . It becomes 1/3 if we only add 1 to the denominator. The fraction is (a) 3/4 (b) 2/5 (c) 3/5 (d) 1/4 Question. The pair of equations x +2y -3 = 0 and 4x +5y = 8 has (a) no solution (b) infinitely many solutions (c) a unique solution (d) exactly two solutions Question. The value of c for which the pair of equations 4x -5y +7 = 0 and 2cx -10 y +8 = 0 has no solution is (a) 8 (b) – 8 (c) 4 (d) – 4 Question. The pair of equations x = a and y = b graphically represents lines which are (a) coincident (b) parallel (c) intersecting at (a, b) (d) intersecting at (b, a) Question. If the lines given by 2x -5y +10 = 0 and kx +15y -30 = 0 are coincident, then the value of k is (a) –6 (b) 6 (c) 1/3 (d) -1/3 Question. A pair of linear equations which has a unique solution x =1, y = -3 is (a) x – y = 4; 2x +3y = 5 (b) 2x – y = -5; 5x -2y =11 (c) 3x + y = 0; x +2y = -5 (d) x + y = -2; 4x +3y = 5 2 Question. Anmol’s age is six times his son’s age. Four years hence, the age of Anmol will be four times his son’s age. The present age in years, of the father and the son are respectively (a) 24 and 4 (b) 30 and 5 (c) 36 and 6 (d) 24 and 3 2 Question. The pair of equations 6x – 4 y +9 = 0 and 3x -2y +10 = 0 has (a) a unique solution (b) no solution (c) exactly two solutions (d) infinitely many solutions Question. A pair of linear equations cannot have exactly two solutions. Question. If two lines are parallel, then they represent a pair of inconsistent linear equations. Question. State whether the following statements are true or false. Justify your answer. (i) The pair of equations 3x – 4 y =1 and 4x +3y =1 has a unique solution. (ii) For the pair of equations 4x +ly = –3 and 6x +9 y + 4 = 0 to have no solution, the value of l should not be Question. A pair of linear equations in two variables may not have infinitely many solutions. Question. The pair of equations 4x -5y = 8 and 8x -10 y = 3 has a unique solution. Questin. State whether the following statements are true or false. Justify your answer. (i) The equations x/2+y+(1/5) and 4x+8y+(8/5)=0 represent a pair of coincident lines. (ii) For all real values of k, except –6, the pair of equations kx -3y = 5 and 2x + y = 7 has a unique solution. Question. (i) For what values of a and b, will the following pair of linear equations have infinitely many solutions? x +2y =1; (a – b)x +(a + b)y = a + b -2 (ii) Solve for x and y x/a + y/b =a + b , x/a2 + y/b2 = 2, a, b ≠ 0 (ii) x=20 y=30 ∠A=130° ∠B=100° ∠C=50° ∠D=80° (i) x =7 and y =9, values –1 and 30/7 Question. (i) Draw the graphs of the equations y = 3, y = 5 and 2x – y – 4 = 0. Also, find the area of the quadrilateral formed by the lines and the y-axis. (ii) A motorboat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream. (i) x =1, y = 4, Areas = 8 sq. units, 2 square units; ratio = 4 : 1 (ii) Speed of the bus is 60 km/h; Speed of the train is 90 km/h Question. (i) Graphically solve the pair of equations: 2x + y = 6, 2x – y +2 = 0 Find the ratio of the areas of the two triangles formed by the lines representing these equations with the x-axis and the lines with the y-axis. (ii) Saksham travels 360 km to his home partly by train and partly by bus. He takes four and a half hours if he travels 90 km by bus and the remaining by train. If he travels 120 km by bus and remaining by train, he takes 10 minutes longer. Find the speed of the train and the bus separately. (i) Area of trapezium=8 sq. units (ii) Speed of the boat in still water = 10 km/h; Speed of the stream = 4 km/h Question. A linear equation in two variables always has infinitely many solutions. Question. A pair of linear equations in two variables is said to be consistent if it has no solution. Question. (i) If 2x + y = 23 and 4x – y =19, find the values of 3y – 4x and y/x+3. (ii) The angles of a cyclic quadrilateral ABCD are ∠A = (6x +10)°, ∠B =(5x)°, ∠C = (x + y)°, ∠D = (3y -10)° Find x and y, and hence the value of the four angles. (i) a= 3, b=1(ii) x= a2 , y= b2 Question. A pair of intersecting lines represent a pair of linear equations in two variables having a unique solution. Question. An equation of the form ax + by + c = 0, where a, b and c are real numbers is called a linear equation in two variables. Question. For the pair of equations lx +3y = -7, 2x -6 y =14 to have infinitely many solutions, the value of λ should be 1. Is the statement true? Give reason. Question. How many solutions does the pair of equations. x +2y = 3 and 1/2 x + y 3/2 – = 0 have? Question. Is the pair of equations x – y = 5 and 2y – x =10 inconsistent? Justify your answer. No, since a1/a2≠b1/b2 , so it has a unique solution Question. Is the pair of equations 3x -5y = 6 and 4x -6 y = 7 consistent? Justify your answer. Question. Is it true to say that the pair of equations -2x + y +3 = 0 and 1/3 x +2y -1 = 0 has a unique solution? Question. Write the number of solutions of the following pair of linear equations: 3x -7y =1 and 6x -14 y -3 = 0 Question. Do the equations 5x +7y = 8 and 10x +14 y = 4 represent a pair of coincident lines? Justify your answer. No, because a[1]/a[2]≠b[1]/b[2] ≠c[1]/c[2] ¹ , so the equations represent parallel lines. Our teachers have developed really good Multiple Choice Questions covering all important topics in each chapter which are expected to come in upcoming tests and exams, as MCQs are coming in all exams now therefore practice them carefully to get full understanding of topics and get good marks. Download the latest questions with multiple choice answers for Class 10 Linear Equations in pdf or read online for free. The above NCERT based MCQs for Class 10 Linear Equations have been designed by our teachers in such a way that it will help you a lot to gain an understanding of each topic. These CBSE NCERT Class 10 Linear Equations Multiple Choice Questions have been developed and are available free for benefit of Class 10 students. Advantages of MCQ Questions for Class 10 Linear Equations with Answers a) MCQs will help the kids to strengthen concepts and improve marks in tests and exams. b) Multiple Choice Questions for Linear Equations Class 10 have proven to further enhance the understanding and question solving skills. c) Regular reading topic wise questions with choices will for sure develop very good hold over each chapter which will help in exam preparations. d) It will be easy to revise all Linear Equations chapters and faster revisions prior to class tests and exams. Free Printable MCQs in PDF of CBSE Class 10 Linear Equations are designed by our school teachers and provide best study material as per CBSE NCERT standards. I want the latest MCQs based on this years syllabus ? The MCQs for Class 10 Linear Equations with Answers have been developed based on current NCERT textbook issued by CBSE. Are all chapters covered ? MCQs cover the topics of all chapters given in NCERT Book for Class 10 Linear Equations. Can I print these MCQs ? Yes – These Multiple Choice Questions for Class 10 Linear Equations with Answers are free to print and use them later. Are these free or is there any charge for these MCQs ? No – All MCQs for Linear Equations are free to read for all students. How do I download the MCQs ? Just scroll and read the free MCQs. Are these free multiple choice questions available for Linear Equations in standard MCQs format with Answers ? 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Computational Complexity As we turn our thoughts from , a look back at the complexity year that was. Written with help from co-blogger Bill Gasarch. The complexity result of the year goes to Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary and Ronald de Wolf for their paper Linear vs Semidefinte Extended Formualtions: Exponential Separation and Strong Lower Bounds ArXiv version ). It is easy to show that TSP can be expressed as an exponentially sized Linear Program (LP). In 1987 Swart tried to show that TSP could be solved with a poly sized LP. While his attempt was not successful it did inspire Yannakakis to look at the issue of how large an LP for TSP has to be.He in 1988 that any symmetric LP for TSP had to be exponential size. (Swart had used symmetric LP's). What about assymetric LP's? This has been open UNTIL NOW! The paper above proves that any LP formulation of TSP requires an exponential sized LP. They use communication complexity and techniques that were inspired by quantum computing. Runners Up And don't forget the to Bill's 17x17 problem. News and trends: The new Simons Institute for the Theory of Computing at Berkeley, the great exodus of Yahoo! researchers mostly to Google and Microsoft, the near death of Florida computer science (anyone want to be ?), and the rise of the MOOCs. We remember Dick de Bruijn Tom Cover Mihai Pătraşcu Ernst Specker David Waltz , not to mention Neil Armstrong and Ray Bradbury Thanks to our guest posters Bernard Chazelle Yoav Freund Andrew Goldberg Mohammad Taghi Hajiaghayi William Heisel Lane Hemaspaandra John Purtilo Janos Simon Vijay Vazirani Enjoy 2013 and remember that when living in a complex world, best to keep it simple. And try not to fall off that fiscal cliff. Sometimes it just takes a simple new feature in a popular piece of software to remind us how computer science just does cool stuff. Excel 2013 has a new feature, Flash Fill, where you can reformat data by giving an example or two. If you have a column of names like Manuel Blum Steve Cook Juris Hartmanis Richard Karp Donald Knuth You can start a column to the right and type Blum, M. Cook, S. and the rest of the table gets filled in automatically. Flash Fill is based on a 2011 POPL paper by Sumit Gulwani (later a CACM highlight). It's been explained to me as applying machine learning to binary decision diagrams. Flash Fill allows a user to manipulate data without having to write macros or Perl scripts. Someone with no technical background can use Flash Fill and enjoy the CS goodness inside without even knowing it is there. LANCE: Bill, on Dec 15, 2012 it will be Reuben Goodstein's 100th birthday. BILL: Is he still alive? Will there be a conference in his honor? Free Food? Cake? LANCE: No, no, and no. But you could blog about Goodstein sequences. (thinking: or they could just look it up here). BILL: That is a Goodstein idea. You have a Goodstein sequence of good ideas. I first define a sequence that is not a Goodstein sequence but will be good for education. Let n be a number. Say let n=42 Base 10. We then decrease the number by 1 but increase the base by one. We keep doing this. We get 1. 42 base 10 = 42 base 10 2. 41 base 11 = 45 base 10 3. 40 base 12 = 48 base 10 4. 3E base 13 = 50 base 10 (E stands for Eleven) 5. 3T base 14 = 52 base 10 (T stands for Ten) 6. 39 base 15 = 54 base 10 The sequence looks like its increasing. But note that it eventually gets to 1. 30 base 24 = 72 base 10 2. 2(23) base 25 = 73 base 10 (the ``units digit'' is 23) 3. 2(22) base 26 = 74 base 10 OH MY. Its still increasing. But note that it eventually gets to 1. 20 base 48 base 10 = 96 base 10 2. 1(47) base 49 = 96 base 10 3. 1(48) base 50 = 98 base 10 4. 1(47) base 51 = 98 base 10 It seems to be at 98 for a while. Indeed we eventually get to 1. 11 base 97 = 98 base 10 2. 10 base 98 = 98 base 10 3. 0(97) base 99 = 97 base 10 And from there on it it goes to 0. Given n, how many iterations do you need to get to 0? This function grows rather fast, but not THAT fast. To prove that it goes to 0 you need (I think) an induction on an omega-squared ordering. The true Goodstein sequences initially write the number in base 10 but also writes the exponents in base 10 and does that as far as it can: 4 x 10^2 x 10^4 + 8 x 10^3 + 7 x 10^2 + 8 x 10^2 x 10^1 + 3 x 10^0 (This is called Hereditary 10 notation.) At each iteration we subtract 1 and then turn all of the 10's into 11's. More generally we write the number in base b and the exp in base b... etc and then subtract 1 and make all of the b's into b+1's. This sequence also goes down to 0. NOW how long does it take to goto 0. The function is not primitive recursive. Also, the theorem that states the sequence eventually goes to 0, cannot be proven in Peano Arithmetic. I use Goodstein sequences as an example of a natural (one can debate that) function that is not primitive recursive. Ackermann's function comes up more often (lots more often) but is harder to initially motivate. So Reuben Goodstein, wherever you are, I salute you! Unless the republicans and democrats get a deal before year's end, the country will head over the so-called "Fiscal Cliff" including a budget sequestration that will cause automatic cuts to most federal agencies. This will be a disaster for science, grants will be delayed or not awarded, perhaps even spending freezes on current funds. University presidents have banded together in my old state and new to drive this point home. Don't panic too much about the fiscal cliff, which will be short lived if it happens at all. But the consequence may lead to deep short or long term budget cuts in science. If charitable deductions are eliminated, that can be a large hit on university endowments. Pell grants might also be in play. On the other hand, science and education has its friends in Washington, starting with Barack Obama. So maybe the whole fiscal mess will work out just fine for us and we can go back to worrying whether universities will be decimated by MOOCs. (I am a bit behind on posting links to my book review column- this blog post is about the Third (of four) for the year 2012, and the fourth of four has already appeared. I'll post that one later.) My second-to-last book review column can be found here. The file pointed to does NOT include the BOOKS I NEED REVIEWED since that has changed. For that go here. The column has an editorial! In a book review column for CS theory books?! It is an editorial against the high prices of books and the fact that many books are NOT online. These are well known problems, so what of it? I could say why the publishers are at fault, but frankly, I don't know the economics of the book market. So instead I recommend the community to do the following: 1. Only buy books used or at a cheaper than list price (you probably already do this). 2. If you write a book then insist that the contract allow for it to be online for free. This is not as outlandish as it sounds: (1) Blown to Bits by Abelson, Ledeen, Lewis (which I reviewed in this Column) is a book for a wide audience--- it is a popular book about computers and society. Even so, it is available online for free here. (2) Some book companies of Math and Science books allow the author to have the book free on line. 3. When assigning a course textbook allow them to get earlier editions that are usually far far cheaper. There is no reason to insist they get the most recent edition. Also of interest with regard to all of this- the Kistsaeung vs Wiley case: here (Guest post by Vijay Vazirani.) Theory Day on Nov 30, 2012 Vijay Vazirani gave a talk: New (Practical) Complementary Pivot Algorithms for Market Equilibrium. He was inspired by the reaction to the talk to write a guest blog which I present here! Where is the Youth of TCS? by Vijay Vazirani I have always been impressed by the researchers of our community, especially the young researchers -- highly competent, motivated, creative, open-minded ... and yet cool! So it has been disconcerting to note that over the last couple of years, each time I have met Mihalis Yannakakis, I have lamented over the lack of progress on some fundamental problems, and each time the same thought has crossed my mind, ``Where is the youth of TCS? Will us old folks have to keep doing all the work?'' Is the problem lack of information? I decided to test this hypothesis during my talk at NYTD. To my dismay, I found out that there is a lot of confusion out there! By a show of hands, about 90% of the audience said they believed that Nash Equilibrium is PPAD-complete and 3% believed that it is FIXP-complete! I would be doing a disservice to the community by not setting things right, hence this blog post. First a quick primer on PPAD and FIXP, and then the questions. Ever since Nimrod Megiddo, 1988, observed that proving Nash Equilibrium NP-complete is tantamount to proving NP = co-NP, we have known that the intractability of equilibrium problems will not be established via the usual complexity classes. Two brilliant pieces of work gave the complexity classes of PPAD (Papadimitriou, 1990) and FIXP (Etessami and Yannakakis, 2007), and they have sufficed so far. A problem in PPAD must have rational solutions and this class fully characterizes the complexity of 2-Nash, which has rational equilibria if both payoff matrices have rational entries. On the other hand, 3-Nash, which may have only irrational equilibria, is PPAD-hard; however, its epsilon-relaxation is PPAD-complete. That leaves the question, ``Exactly how hard is 3-Nash?'' Now it turns out that 3-Nash always has an equilibrium consisting of algebraic numbers. So one may wonder if there is an algebraic extension of PPAD that captures the complexity of 3-Nash, perhaps in the style of Adler and Beling, 1994, who considered an extension of linear programs in which parameters could be set to algebraic numbers rather than simply rationals. The class FIXP accomplishes precisely this: it captures the complexity of finding a fixed point of a function that uses the standard algebraic operations and max. Furthermore, Etessami and Yannakakis prove that 3-Nash is FIXP-complete. The classes PPAD and FIXP appear to be quite disparate: whereas the first is contained in NP INTERSECT co-NP, the second lies somewhere between P and PSPACE (and closer to the harder end of PSPACE, according to Yannakakis). Now the questions (I am sure there are more): 1. Computing an equilibrium for an Arrow-Debreu market under separable, piecewise-linear concave utilities is PPAD-complete (there is always a rational equilibrium). On the other hand, if the utility functions are non-separable, equilibrium consists of algebraic numbers. Is this problem FIXP-complete? What about the special case of Leontief utilities? If the answer to the latter question is ``yes,'' we will have an interesting demarcation with Fisher markets under Lenotief utilities, since they admit a convex program. 2. An Arrow-Debreu market with CES utilities has algebraic equilibrium if the exponents in the CES utility functions are rational. Is computing its equilibrium FIXP-complete? Again, its epsilon-relaxation is PPAD-complete. 3. A linear Fisher or Arrow-Debreu market with piecewise-linear concave production has rational equilibria if each firm uses only one raw good in its production, and computing it is PPAD-complete. If firms use two or more raw goods, equilibria are algebraic numbers. Is this problem FIXP-complete? On Friday the New York Times ran an article on how online retailers constantly adjust prices to match their competitors. Their ability builds on many tools from computer science, from networks to algorithms, not unlike airlines and hedge funds. But is this good for the consumer? Suppose I work for bestbuy.com and have a television priced at $500, which matches the price on amazon.com. If I lower my price to $450, than I can expect Amazon to do the same. I've gained little from lowering the price, only $50 less dollars than I had before. Likewise Amazon has little incentive to lower their price. This hypercompetition can actually lead to collusion without colluding. Hal Varian talked a similar theme of price guarantees in a 2007 Times viewpoint. In practice, companies still lower their prices but as networks get faster, algorithms get smarter and more people shop online, we might actually see higher prices and less competition, which I believe is already happening with the airlines. First a shout out to our friends up north on the 30th anniversary of the New York Theory Day this Friday. Just two years ago I wrote a post Gadget Love but now I don't use many gadgets any more, it's all built into my iPhone and iPad. No wonder a company like Best Buy is having problems. Not only do they have to compete against Amazon they also compete against the Apple App Store. Some of my favorite apps: Goodreader - Manage, view and mark-up PDF files. Syncs with Dropbox and nearly every other file sharing service. Evernote - Manages short notes and photos. I often just take pictures of a whiteboard and save it to Evernote. JotNot - The iPhone becomes a scanner. Those three apps let me lead a nearly paperless life. WolframAlpha - Whatever you think of Stephen Wolfram, this is still a very useful tool. TripIt - Forward your emails from airlines, rental cars and hotels to trip it and it organizes all your info. Invaluable when traveling. ComplexityZoo - OK, I rarely use, it but it's pretty cool there's a free app that let's you find complexity classes. There are many many great and not-so-great apps. I have seven pages of apps on my iPhone. But I can always download more so tell me some of your favorites. In the undergraduate complexity course we spend some time on closure properties such as (1) REG closed under UNION, INTER, COMP and (2) R.E. closed under UNION and INTER but NOT COMP. I propose the following inverse problem: For all possible assignments of T and F to UNION, INTER, COMP, find (if it is possible) a class of sets that is closed exactly under those that are assigned T. I would like the sets to be natural. I would also like to have some from Complexity theory, which I denote CT (e.g., Reg, P, R.E.) and some not (e.g., set of all infinite sets) which I denote HS for High School Student could come up with it. If I want other examples I will say OTHERS? We assume our universe is {a,b}^. 1. UNION-YES, INTER-YES, COMP-YES: 1. CT: Reg, P, Poly Hier, PSPACE, Prim Rec, Decidable, Arithmetic Hier. 2. HS: Set of all subsets of {a,b}^. OTHERS? 2. UNION-YES, INTER-YES, COMP-NO: 1. CT: NP (probably), R.E. OTHERS? 2. HS: Set of all finite subsets of {a,b}^. OTHERS? 3. UNION-YES, INTER-NO, COMP-YES: NOT POSSIBLE. 4. UNION-YES, INTER-NO, COMP-NO. 1. CT: Context Free Langs. OTHERS? 2. HS: Set of all infinite subsets of {a,b}^. OTHERS? 5. UNION-NO, INTER-YES, COMP-YES: NOT POSSIBLE. 6. UNION-NO, INTER-NO, COMP-YES: 1. CT: The set of all regular langs that are accepted by a DFA with ≤ 2 states. (Can replace 2 with any n ≥ 2.) 2. HS: Set of all subsets of {a,b}^* that are infinite AND their complements are infinite. OTHERS? 7. UNION-NO, INTER-YES, COMP-NO: 1. CT: I can't think of any- OTHERS? 2. HS: { emptyset, {a}, {b} } 8. UNION-NO, INTER-NO, COMP-NO: 1. CT: The set of all reg language accepted by a DFA with ≤ 3 states and ≤ 2 accepting states. (Can replace (3,2) with other pairs) OTHERS? 2. HS: I can't think if any- OTHERS? During a homework assignment in a graduate complexity course I took at Cornell back in 1985 I used the following reasoning: Since a computer code sits in RAM that a program can read, by the Church-Turing thesis we can assume a program has access to its own code. The TA marked me wrong on the problem because I assumed the recursion theorem that we hadn't yet covered. I wasn't assuming the recursion theorem, I was assuming the Church-Turing thesis and concluding the recursion theorem. I did deserve to lose points, the Church-Turing thesis is not a mathematical theorem, or even a mathematical statement, and not something to use in a mathematical proof. Nevertheless I still believe that if you accept the Church-Turing thesis than you have to accept the recursion theorem. Now the recursion theorem does not have a trivial proof. So the Church-Turing thesis has real meat on it, in ways that Turing himself didn't anticipate. Since the recursion theorem does have the proof, it only adds to my faith in and importance of the Church-Turing thesis. Back in the typecast last month I promised a simple PSPACE-complete game in a future post. Here it is: The SET GAME Given: A collection of finite sets S[1],...,S[k]. The Game: Each player takes turns picking a non-empty set S[i]. Remove the elements of S[i] from all the sets S[j]. The player who empties all the sets wins. This game came up in a discussion I had with Steve Fenner trying to extend his student's work that Poset Games were PSPACE-complete. The PSPACE-completeness of determining a winner of the SET GAME is an easy reduction from from Poset Games. An open question: Do we still get PSPACE-completeness if the size of the S[i] are bounded? I don't even know the answer if the sets have size at most two. If I tweeted this is what I would tweet: 1. Prez election thought: if you run as yourself and lose (Stevenson, Goldwater, McGovern) then you have your integrity and got some discussions started. If you run as someone else (George W Bush ran as a moderate) and win then you have the presidency. If you run as someone else and lose you have nothing. Now that he's lost Will the real Mitt Romny please stand up?. 2. My bet on Ryan being the Republican Nominess in 2016 (if I am right I get $10.00, if I am wrong I lose $3.00) is close to the odds here. Are you surprised they already have this up? I'm surprised INTRADE doesn't have this bet up yet. 3. An APP based on my 17x17 challenge: here 4. Most bloggers don't last: see here 5. An Origami proof of the Pythagorean theorem: here 6. I got three emails in two minutes about a talk on how to avoid spam. 7. Its official! Physics is hard 8. A paper is retracted because it has no scientific content: here. 9. Is this a real question or a joke or both? 10. Julia Child was born Aug 15, 1912 and died Aug 13, 2004. Almost born and died the same day. What is the probability that someone is born and dies on the same day? This is not hard, but might make a good problem. 11. The most common PIN number is 1234 with 11% of all PIN numbers. This is alarming but not surprising. The least common PIN number is 8068. Here is the article on it. 12. Do e-books make censorship easier or harder? I still don't know; however, here is an example where it was easier, though calling it censorship isn't quite right. 13. Movies have low Kolm Complexity: here 14. One sign that you've been working on a paper too long- right before submitting it you realize that most of the authors have changed their affiliations and emails. (This happened to me recently.) From Juris Hartmanis’ Observations About the Development of Theoretical Computer Science on the research leading to his seminal 1965 paper On the Computational Complexity of Algorithms with Richard Only in early November 1962 did we start an intensive investigation of time-bounded computations and realized that a rich theory about computation complexity could be developed. The first explicit mention of classifying function by their computation time occurs in my logbook on November 11. And so today we celebrate the fiftieth anniversary of the conception of computational complexity. We have learned much since then and yet we still know so little. Here’s to another fifty years of trying to figure it out. Guest Blog by Andrew Goldberg on the recent Max Flow in O(nm) time algorithm. Maximum Flow in O(nm) Time Recently, Jim Orlin published an O(nm) maximum flow algorithm. This solves a long-open problem. In this blog entry, we assume that the reader is familiar with the maximum flow problem, which is a classical combinatorial optimization problem with numerous applications. (If not then see the Wikipedia entry here.) Let n and m denote the number of vertices and arcs in the input graph, respectively, and if the input capacities are integral, U denotes the value of the biggest one. Running time of a strongly polynomial algorithm is a polynomial function of n and m; the time of a polynomial algorithm may depend on log(U) as well. Maximum flow algorithms have been studied since 1950's with the first strongly polynomial-time algorithm developed in 1972 by Edmonds and Karp. This was followed by faster and faster algorithms. In 1980, Galil and Namaad developed an O(nm log^2n) algorithm, coming close to the nm bound. The latter bound is a natural target for a maximum flow algorithm because a flow decomposition size is THETA (nm). In 1983, Sleator and Tarjan developed the dynamic tree data structure to improve the bound to O(nm log(n)); in 1986, Goldberg and Tarjan developed the push-relabel method to get an O(nm log(n^2 /m)) bound. The race towards O(nm) continued, with improvements being made every few years, until King, Rao and Tarjan developed O(nm + n^2+ε) and O(nm log[m/(n log(n)] n) algorithms in 1992 (SODA) and 1994 (Journal of algorithms-the paper pointed to), respectively. These bounds match O(nm) except for sparse graphs. No better strongly polynomial algorithm for sparse graphs has been developed for 18 years, until Orlin's recent result. Orlin not only gets the O(nm) bound, but also an O(n^2/log(n)) bound for very sparse graphs with m = O(n). His result is deep and sophisticated. It uses not only some of the most powerful ideas behind the efficient maximum and minimum-cost flow algorithms, but a dynamic transitive closure algorithm of Italiano as well. Orlin closes the O(nm) maximum flow algorithm problem which has been open for 32 years. Sometimes solving an old problem opens a new one, and this is the case for the maximum flows. The solution to the maximum flow problem is a flow, and flows have linear size, even when they have decompositions of size THETHA(nm). For the unit-capacity problem, an O(min(n^2/3, m^1/2) m) algorithm has been developed by Karzaov (and independently by Even and Tarjan) in the early 1970's. This bound is polynomially better than nm. In 1997, Goldberg and Rao developed a weakly polynomial algorithm that comes within a factor of O(log(U) log[n^2/m] n) of the unit flow bound, and Orlin's algorithm uses this result. A natural question to ask at this point is whether there is an O(nm/n^ε) maximum flow algorithm for some constant epsilon. Every year or so the National Science Foundation releases a new version of the holy bible of grant submission procedures, the Grant Proposal Guide. Last month's update (which applies to grants due in 2013) has this tidbit in the Summary of Significant Changes. Chapter II.C.2.f(i)(c), Biographical Sketch(es), has been revised to rename the “Publications” section to “Products” and amend terminology and instructions accordingly. This change makes clear that products may include, but are not limited to, publications, data sets, software, patents, and copyrights. So you can list your patents or open-source software as a "product" right up there with the same status as an academic publication. This seems like a harmless change, us theorists can continue to just list our publications. But I worry about the slippery slope. Does this signal a future change to the NSF Mission that supports "basic scientific research"? We will be expected in the future to have "products" other than research publications? Or is the NSF just saying that while they fund our research, it's not the research, but the manifestations of that research in whatever form they take, that gets judged for future grants? Neither Lance and I have commented much on the Prez election. I only found one post from 2012 that mentioned Romney: Romney vs Aaronson. A few mentioned Obama but not with regard to the election. I give you some Random thoughts on the election before its over. They are nonpartisan unless they are not. 1. I polled the Sophmore discrete math class (secret ballot) and got the following: of the 99 students in the class (1) 64 for Obama, (2) 17 for Romney, (3) 8 for Gary Johnson (libertarian), (4) 2 for Jill Stein (Green). Those were the only ones on the ballot; however, there were some write-ins: (5) 2 for Ron Paul, (6) 1 each for Newt Gingrich, John the Baptist, Mickey Mouse, Gumby, and two names I did not recognize but may have been the students themselves. You know what they say: As goes discrete math, so goes the nation. Hence Obama now has it in the bag. 2. I predicted it would be Romney vs Obama on Feb 15, 2012. I also predicted that Obama would win. I never wavered from that prediction, so you can't call me a flip-flopper. You can read it here. The first part (Obama vs Romney) has already come true; we will see if the second one does. 3. Assuming Obama wins I have a bet on the Republican nominee in 2016: I have bet Lance Fortnow, Chris Umans, and Amol Despande (DB guy in my dept) that it will be Paul Ryan. (ADDED LATER- I misunderstood Amol- he wants to bet WITH me, that Ryan will win, with the odds I got.) If I win I get $1.00, if they win they get 30 cents. I have a bet with Mike Barron (a friend of mine not a theorist--- yes I have non-theorists friends), who is more of a risk-taker, where if I win I get $10.00 and if he wins he gets $3.00. Are these good odds? I ask this nonrhetorically. 1. Why Ryan? 1972 is the beginning of the modern political era. That's when Prez candidates had to compete in primaries to win the nomination. (Humphrey got the nomination in 1968 without entering a single primary, then the McGovern Commission changed the rules so that a lot more primaries were included. And, the man who understood the rules, McGovern, got the nomination in 1972.) Since 1972 the Republicans have almost always nominated a KNOWN person, someone you heard of four years earlier. Not including incumbents here is the list: 1. 1980 Reagan. Known- Had run in 1976. 2. 1988 Bush Sr. Known- Was VP under Reagan. 3. 1996 Dole. Known- Had run with Ford as VP, had run for Prez before. 4. 2000 Bush Jr. Unknown- One can argue he was known via his dad, but I'll just say Unknown. 5. 2008 McCain. Known- Had run before in 2000. 6. 2012 Romney. Known- Had run before in 2008. By contrast the Dems have sometimes nominated someone you had not heard of. Here is their record: 1. 1976 Carter. An Unknown Former Gov or Georgia. 2. 1984 Mondale. Known, Former VP. 3. 1988 Dukakis. An Unknown Gov of Mass. 4. 1992 Clinton. An Unknown Gov of Arkansas. 5. 2000 Gore. Known. Was VP. 6. 2004 Kerry. An Unknown Senator. 7. 2008 Obama. An Unknown Senator. (One could debate how unknown some of these were.) Note that whenever the Dems nominated a known person they lost- perhaps a cautionary note to those who want Biden or H. Clinton in 2016, and an encouraging note to Andrew Cuomo, current gov of NY. (If you say whose that? you've proven my point.) But ANYWAY, the Republicans have ALMOST ALWAYS given it to a KNOWN person. None of the people who ran for the nomination in 2012 seem plausible to get the nomination in 2016, though The Daily Show is doing a segment on the fictional Cain Presidency. Some sort-of-known people who didn't run in 2012 but may in 2016: Chris Christie (Gov of NJ), Jeb Bush (Gov of Florida), Tim Pawlenty (Gov of Minnesota), Mitch Daniels (Gov of Indiana), Marco Rubio (Senator from Florida), Bobby Jindal (Gov of Louisiana) . The last two are more known for being talked about as Prez of V Prez Candidate then for anything they've actually done. I grant that any of these people are possible. However, they are not quite as well known as Ryan. Also, I predict that if Romney loses it will be blamed on we were not true to our principles and they will go further rightwing with Ryan. 2. Why it might not be Ryan: The above argument sounds convincing but the problem with predictions in politics (and elsewhere) is that, to quote a friend in Machine Learning, Trends hold until they don't. Anything could happen! Things may change drastically! As an example see this XKCD. 4. Another prediction, though harder to quantify. When Gore, Kerry, and McCain lost they or people around them said things like I let my handlers handle me too much- if I had run as myself I would have won. I predict that Romney will think the same thing. I doubt he'll say it. 5. There is an issue on the Maryland Ballot that involves Game Theory and Gaming. Roughly speaking the issue is should we allow more gambling in our state. PRO: People are going to adjacent states to gamble and we should get that money. CON: Gambling is a regressive tax and bad for the economy in the long run. The more states have gambling (or build baseball stadiums or give businesses who move there tax breaks) the more other states have to go along to compete. A classic Prisoners Dilemma--- except that West Virginia and Delaware have already defected so we have no choice. Or do we? There is a rumor that the anti-gambling adds in Maryland are being paid for by the West Virginia Casinos. The anti-gambling ads are not anti-gambling, they are just against this bill- they claim that the money won't really go to education for example. I admire the honesty--- if a West VA casino had an add in MD saying how bad Gambling was morally that would look rather odd. Even so, Should I vote FOR gambling just to spite the out-of-state casinos running adds in my state? Should I vote FOR it since the ads against it are not giving MY arguments against it? Should I vote FOR IT and tell people I voted against it? 6. There is a marriage-equality referendum on the ballot- Question 6. There has been almost no ads or talk about it. Why? One speculation--- the people against it know they will be on the wrong side of history, and the people for it don't quite know how to sell it. Its ahead in the polls so maybe they don't want to rock the boat. 7. If you ask a pro-Obama pundit who will win he might say Obama because people know Romney is a liar. If you ask a pro-Romney pundit will win he might say Romney because Obama has not fixed the economy and Mitt can. Either may use poll data as window dressing, but they tell you what they want to happen rather than what an honest scientific study will show. Nate Silver, a scientific pollster, says in his book The signal and the noise: Why so many predictions fail--- but some don't that pundits are right about half the time. Not 8. George McGovern died recently at the age of 90. The 1972 prez election, McGovern vs Nixon, was the first Prez campaign I paid attention to. I passed out McGovern pamphlets in my precinct of Brooklyn and McGovern DID win that Precinct. I regard that as a Moral Victory. The deadline for submissions to STOC has been extended to Monday, Nov 5 2012 5:00 p.m. EST. Time again for the annual fall jobs post. As always the best places to look for academic CS positions are the job sites at the CRA and the ACM. Also check out the postdoc and other opportunities on the Theory Announcements site. It never hurts to check out the webpages of departments you might want to be at or to contact people to see if positions are available. I encourage everyone who has a job to offer in theoretical computer science at any level to post links in the comments. With computer science enrollments expanding and the economy slowing recovering, I'm expecting quite an increase in the number of tenure-track jobs in computer science this year. On the other hand I'm expecting a decrease in the number of new postdoc positions though maybe more overseas. Good luck to everyone in the market. At Dagstuhl I was delighted when I saw the title of a talk to be given Planarizing Gadgets for Perfect Matching do not Exist because I had asked the question about a gadget for planar Ham Cycle here. I was hoping to ask the authors if there techniques could be used to show that there was not Planarizing gadget for Ham Cycle (NOTE- I had either forgot or never knew that this was already known and was posted as an answer to my query to cstheory stackexchange, here.) The paper Planarizing Gadgets for Perfect Matching do not Exist (or if you can get to it the MFCS 2012 version here) is by Rohit Gurjar, Arpita Korwar, Jochen Messner, Simon Straub, and Thomas Thierauf. The talk was given by Jochen and was excellent. Perfect matching is in P (Edmonds 1965) but is it in NC? Not known--- however it is in RNC (Mulmuley, Vazirani, Vazirani 1987). What about Planar graphs? They are different--- counting the number of perfect matching in a graph is Sharp-P complete (Valiant 1979) but counting the number of perfect matchings in a planar graph is in NC (Vazirani 1989). So of course Planar Graph Matching is in NC. Can we use this to get Graph Matching in NC? perhaps be a reduction? This would be neat since we would be using a reduction to prove a problem EASY rather than to prove a problem HARD. (I think this has been done before but is rare-- readers, if you know a case comment on it.) Perhaps there is some planarizing gadget: given a graph G use some gadgets to get rid of crossings and produce a planar graph G' such that G has a perfect matching iff G' has a perfect matching. That would be AWESOME! However, from the very title of the paper, we can guess this is not true. This paper shows that something AWESOME is not possible! A downer but worth knowing. Jochen proved this and then went on to say that they had done the same thing for HAM CYCLE! That is, there is no planarization gadget for Ham cycle! (He also acknowledged that this was already known independently.) SO I didn't get to ask my question since they already had answered it. Great! 1. Their interest in planarization was related to an OPEN problem--- is Graph Matching in NC? By contrast my interest in Planarization gadgets for Ham Cycle was pedagogical--- I was in search of a better proof that Planar Ham Cycle is NPC- though there is no new theorem here. 2. I am delighted to know the result! 3. Their results says that a certain type of reduction won't work. Might some other reduction work? My sense is this is unlikely. 4. So--- is Graph Matching in NC? Since I believe NC=RNC I think yes. Will it be proven by showing NC=RNC or will it be proven directly (leaving NC=RNC open)? Or will the ideas that lead to Graph Matching in NC help to show NC=RNC? This is one of those questions that might be solved within a decade, as opposed to P vs NP which won't be resolved for quite some time. I tweeted the audio of this song last week and here is the video. Recorded at Dagstuhl on October 18th. Written by Fred Green who also plays piano. Performed by David Barrington with Steve Fenner on Fred gives apologies to Gilbert and Sullivan, the Complexity Zoo, and Tom Lehrer Lyrics by Fred Green, copyright 2012 To the tune of "I Am the Very Model of a Modern Major General" There's P and NP, BPP and ZPP and coNP, And TC0 and AC0 and NC1 and ACC, There's PSPACE, LOGSPACE, PPSPACE and ESPACE, EXPSPACE, IPP, And LIN and L and Q and R, and E, EE and E-E-E. There's SPARSE and TALLY, PL, P/Poly, NP/poly, There's PromiseP and PromiseBPP and PromiseBQP, There's FewP, UP, QP, UE, N-E-E, N-E-E-E, And EXP and NEXP, FewEXP, and NE-EXP, and also Max-N-P. And EXP and NEXP, FewEXP, and NE-EXP, and also Max-N-P And EXP and NEXP, FewEXP, and NE-EXP, and also Max-N-P And EXP and NEXP, FewEXP, and NE-EXP, and also Max-N, Max-N-P. There's Sigma_nP, Delta_nP, Theta_nP, Pi_nP, We know BPP's in Sigma_2P intersection Pi_2P. And NP to the NP to the NP to the NP To the NP to the NP, that's the pol-y-nom-yal hierarchy! There's #P, gapP, PP, coC=P and MidBitP, And ModP, Mod_kP, Mod_kL, ParityP, MPC, There's FNP, NPSV, NPMV, and SAC, SAC0, SAC1, SZKn and SPP. There's BQP and DQP and EQP and NQP, And RQP and VQP and YQP and ZQP, And BPQP, FBQP, ZBQP, QRG, QAC0, QNC0, QNC1, Q-A-C-C. QAC0, QNC0, QNC1, Q-A-C-C QAC0, QNC0, QNC1, Q-A-C-C QAC0, QNC0, QNC1, Q-A-C, A-C-C There's QSZK, QMA and QAM and QIP, And IP, MIP, QMIP and also PCP, And PPPad and PPcc, PSK and PQUERY, And PP to the PP, PExp, PPA and PPP. These complexity classes are the ones that come to mind, And there may be many others but they haven't been defined. With a shout out to the friendly folks attending FOCS this week, some short announcements. Read the STOC CFP before you submit the paper. There are significant changes to the submission format and procedure. Deadline is November 2. Complexity will be co-located with STOC in 2013. Submission deadline is November 30. The new Simons Institute for the Theory of Computing has a call for workshop proposals and research fellowships. There will be a symposium to celebrate a new professorship named after SIGACT and STOC founder Patrick Fischer at Michigan on November 5 and a celebration of the 80th birthday of Joe Traub on November 9th at Columbia. Nerd Shot from Dagstuhl Seminar 12421 Lance: Welcome to another Typecast from beautiful Schloss Dagstuhl. I’m here with Bill for the Workshop on Algebraic and Combinatorial Methods in Computational Complexity. Bill: Beautiful? I thought this place was designed to be ugly so that we actually get work done. Lance: So what work did you get done today, BIll? Bill: I watched the debate. And you? Lance: Steve Fenner and I came up with the easiest to describe PSPACE-complete problem ever! Bill: Was it one of those poset things that you and Steve’s students work on. Lance: A generalization of poset games but easier to describe. But we are getting off topic... Bill: as did Obama and Mitt. Lance: Bill my two minutes aren’t up yet. Anyway you’ll have to read about this new PSPACE-complete problem in a future post. Bill: Since you didn’t ask, let me tell you about my favorite talk, Rank bounds for design matrices and applications by new Rutgers professor Shubhangi Saraf (Powerpoint). Despite the awful title Lance: which is why I skipped that talk Bill: it used complexity theory techniques to prove new things in math, a generalization of the Sylvester-Gallai theorem. You have n points on the plane... Lance: Wait Bill, It will take longer to tell the S-G theorem than it would have to explain the new PSPACE-complete problem! [Steve Fenner shows up with beer in hand. He goes off to get Lance one too.] Bill: OK, I’ll leave this for a later post. What was your favorite talk? Lance: Believe it or not it was an algorithms talk. Atri Rudra gave a very simple algorithm to do a join operation motivated by reconstructing 3-d collections of points from projections. [Powerpoint] Bill: Yes, and it may have applications to complexity as most real world algorithms do. [Steve arrives with Lance’s Beer. There is much happiness.] Steve: My favorite talk so far was Rahul Santhanam’s [abstract] Reminded me of the good old days of complexity. Lance: Let the guy give me beer and he thinks he can weasel his way into our typecast. Bill: Lance, that’s how I got started in this business. Lance: Rahul had some clever co-author, didn’t he? Steve: No one important. Lance something? Harry Buhrman: I like the GCT talk by Josh Grochow. [abstract] Bill: In the future we’ll all have to learn GCT to get started in this field. I’m glad I’m living in the past. Lance, you paid me the highest compliment in my talk. You didn’t fall asleep and you even picked a fight with me. Lance: Only because I had to stay awake to help the audience understand your confusing presentation. Bill: It was only one slide. Lance: It was only one fight. Bill: I still feel as complimented as a bit that’s just been toggled . Lance: I’m happy for you. Actually, it was not that bad a result. Now that’s my highest compliment. Harry: Hey, this isn’t fair, we’ve haven’t heard all the talks yet. [Both Harry and Steve are talking later tonight] Lance: Life isn’t fair, get over it. Bill: Let’s call it a day. Lance: Watch my twitter feed later this week for a special musical complexity tribute. Bill: I can’t wait. Lance: So until next time, remember that in a complex world, best to keep it simple. This week Bill and I have traveled to Germany for the Dagstuhl Seminar on Algebraic and Combinatorial Methods in Computational Complexity. Plenty of newly minted Nobel laureates here, winners of the Peace Prize last Friday. But this post celebrates today's winners of the Economics Prize, Al Roth and Lloyd Shapley for their work in matching theory that has made a difference in the real world. In 1962, Shapley and David Gale created the first algorithm that finds stable marriages. David Gale would surely have shared this award had he not passed away in 2008. Nicole Immorlica's guest obit of Gale nicely describes this work and its applications including matching medical students with residencies. Al Roth uses matching algorithms for a variety of projects, most notably creating large scale kidney exchanges, saving lives with algorithmic mechanism design. Doesn't get cooler than that. When John Hennessy gave his talk on MOOCs at the CRA Snowbird meeting he recommended the book Why Does College Cost So Much? by Robert Archibald and David Feldman, both economics professors at William and Mary. I've never seen a good answer to the title question so I read through the book. To overly simplify their main thesis: It's not that college has gotten more expensive, it's that most everything else has gotten cheaper. Technological advances in manufacturing and shipping have made greatly lessened the cost of goods, and the rate of inflation is calculated based on a basket of goods. So service industries, particularly those that require highly educated people and don't benefit directly from technology, look expensive in comparison. College costs closely map to medical and dental expenses, and closely followed broker expenses until technology made brokerages cheaper. Archibald and Feldman even argue that there isn't a college affordability crisis for the majority of Americans: They are still better off than 30 years ago even if we take out college expenses. Hardly the doom and gloom scenario that Hennessey was portraying. Their main point is that one cannot increase the number of students to faculty without decreasing the quality of education. That's where MOOCs come in, supposedly the solution to allow faculty to be far more efficient in the number of students they can teach without reducing quality. Might help control college costs but could harm research at top tier universities and many other universities might cease to exist. Alas perception is reality and the public sees college expenses growing dramatically compared to the general cost of living and blames wasteful spending at universities. Curing this "disease" might kill the patient. Aravind asked me to post on this again (NOTE- registration-for-free deadline is TOMMOROW!!!!!) The University of Maryland at College park is having a Theory Day on Wed Oct 24! Come hear 1. Distinguished talks by Julia Chuzhoy and Venkataesan Guruswami! 2. Short talks (is that code for NOT distinguished?) by Bill Gasarch, MohammadTaghi Hajiaghayi, Jonathan Katz, Samir Khuller, David Mount, Elaine (Runting) Shi, and Aravind Srinivsans. (I never realized I was first alphabetically until now.) 3. Discussions in hallways for those that learn more that way! For more info see the link above, but note one thing: Its FREE! NOTE: This is purposely after the NJ FOCS conference and is an easy Amtrak ride from NJ. I like theory days in general and often go to the NY theory days. They are free and only one day. I recommend going to any theory day that is an Amtrak Ride away. (Might depend on how long the trip is- There is a 13-hour Amtrak from Atlanta Georgia to Maryland, though I doubt I'll see Lance there.) I get a lot out of theory day as noted in this post about NY theory day. What are good ways to get the word out about events. 1. The major conferences and also the NY Theory Days have a long enough tradition that they don't need much advertising. 2. Email is not as useful as it used to be since we all get too much of it. 3. There IS a website for theory announcements here, and also one of our links, but more people need to post there and read there. A chicken and egg problem. 4. Twitter. No central authority. If Aravind had a twitter account (I doubt he does) then he could tweet to his followers, but that would not be that many people. 5. Any ideas? (This post was done with the help of Lane Hemaspaandra and John Purtilo.) The 8th amendment of the US Constitution states Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted. There is an ambiguity here. Let C be cruel and U be unusual. They are saying NOT(C AND U) = NOT(C) OR NOT(U). Common sense would dictate that they meant NOT(C) AND NOT(U). (This article was emailed to me by Lane H. along with the idea for this post.) This article (see also this Wikipedia article) is an example where the CRUEL but NOT UNUSUAL argument seems to have been explicit. The case was about a MANDATORY life sentence in prison for possessing over 650 grams of cocaine, in Michigan. Is that a lot? (I never could figure out that Metric System.) In terms of numbers or getting high I really don't know if 650 grams is a lot, but legally its NOT A LOT--- the only other state that comes close to this kind of penalty is Alabama with a life-sentence for 6500 grams---that is not a typo. (See the Wikipedia articles section on White's criticism of Kennedy's argument.) I quote the syllabus of the decision which is not written by the members of the Supreme Court and is not part of the decision, but is rather prepared by the Office of the Clerk (of the Supreme Court)---who, one assumes, is pretty darned good at extracting the key points of the ruling, and so the syllabi are very useful. Severe, mandatory penalties may be cruel, but they are not unusual in the constitutional sense, having been employed in various forms throughout the Nation's history. Some past rulings HAVE indicated that a sentences that is out-of-proportion with the crime MAY be considered Cruel and Unusual. But, alas, unlike mathematics, definitions can change over time. (Well- in math that happens sometimes, but not often and usually not with dire consequences.) 2. One could argue that Capital Punishment is C but NOT(U). And indeed, the courts have often upheld it. Did they they use the argument that Capital punishment is C but NOT(U), hence it does not violate the 8th amendment? This article (emailed to be my Lane) makes that line of reasoning explicit and is against it. 3. If someone commits anti-Semitic vandalism and the courts decide that he or she is forced to read Anne Frank's Diary, that would be U but NOT(C). Not sure how they would enforce this- give a quiz? Are Cliff notes okay? What if the vandal saw the movie instead? Would this really work? (I honestly don't know.) Is this Hypothetical? In America YES. John found a case in Italy and I found a case in Germany). If this gets to be a common punishment for anti-Semitic crimes then it may no longer be unusual. I could find no other real cases where people convicted of crimes had, as part of their sentence, that they had to read something (though IANAL so there could be some I don't know about). 4. If an Occupy Wall Street guy vandalizes a Financial Institution's offices and is forced to read Atlas Shrugged that would be unusual. But is it cruel? (My opinion: YES) How about the Cliff notes? (My opinion: NO) Is this hypothetical? (My opinion: YES.) 5. What if a teenage girl was in Juvenile court for cutting off the hair of a 3-year old (against the 3-year old's will) and the Judge agreed to reduce the sentence if the teen's mother cut off the teen's pony tail in court. This would be considered unusual. But is it cruel? Is it hypothetical? No It is most likely that the phrase Cruel and Unusual was not meant to be broken down into its component parts. So what Logic did the founders use? Thomas Jefferson knew more math than any of the founding fathers. But alas, he was off in France when the constitution was written. The MacArthur Foundation announced their 2012 Fellows, also know as the genius awards. Among the list two names of interest to my readers, Maria Chudnovsky and Daniel Spielman. My long time readers first heard of Maria back in 2003 when I posted about a great talk she gave as a graduate student giving a polynomial-time algorithm to test for perfect graphs. That was just a start in her incredible career as a graph theorist. Dan is a regular in the blog for the various awards he's won, most notably (before the MacArthur) for his Nevanlinna prize. I believe Dan is my first genius co-author, alas not one of the papers that causing him to win awards. I've seen many cases where researchers get fantastic results early in their career and can never live up to the hype. Dan and Maria exceeded it. Congrats to both of them. I went to the QIS workshop on quantum computing which was on the College Park Campus. I went Thursday (reception- free food!) and Friday (free lunch!) but had to miss the Friday free dinner and the Saturday session. 1. Going to a conference that is ON your campus usually makes it FURTHER away for you. If I was from out of town I would have gotten a Hotel Room in the same hotel as the conference. As it was I walked from my office- a 45 minute walk It would have been shorter but it was a quantum random walk. 2. Scott Aaronson was there. We were talking about teaching class while being taped. He said that being taped changes what he does. I cleverly pointed out that the act of measuring Scott, changes Scott. He cleverly replied that the search for a NEW and FUNNY quantum joke has not ended yet. 3. Frank Gaitan gave a talk on using quantum annealing to find Ramsey Numbers. FINALLY a real application for Quantum Computing! (The downside- I was going to use Quantum Computers Find Ramsey Numbers! for an April Fools Day post.) 4. Umesh Vazarni's talk on CLASSICAL results proven using QUANTUM techniques was great. This notion seems to be for real. Its looking more and more like even if you don't like quantum you will have to learn it. A particular example of this is this paper Linear vs Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds by Fiorini, Massar, Pokutta, Tiwaray, de Wolf. 5. Yi-Kai Liu gave a talk on Quantum Information in Machine Learning and Cryptography. We discuss a small part of his talk, a result by Oded Regev. (Daniel Apon gave a full talk on this small part at the UMCP Complexity Seminar, his slides are here.) GAPSVP(γ) is the following problem: Given an n-dim lattice and a number d output YES if the shortest vector in L is ≤ d, and output NO if the shortest vector in L is > γ d (if it's neither we don't care what you output). This is NP-hard to solve exactly or within O(1) approx (though to be hard for evern poly approx) and it's a good problem for crypto to use. LWE is the Learning with Errors Problem. There is a quantum-reduction that shows that GAPSVP ≤ LWE, so if GAPSVP is hard then LWE is hard. So there are now three 1. Quantum computers are not built. Factoring is still hard classically. Crypto goes on as it is now (maybe not- there is a classical reduction from GapSVP to LWE, but for weaker parameters- so maybe you can base crypto on LWE). 2. Quantum computers are not built. Factoring is easy classically. GAPSVP is hard. Do Crypto based on GAPSVP. 3. Quantum computers are built. Factoring is now easy. GAPSVP is hard. Do Crypto based on LWE. THIS is what the result allows us to do! 4. Quantum computers are built. Factoring is now easy. GAPSVP is easy. Now you are in trouble. 6. New word: Stoquastic. Not sure what it means. 7. Issac Chuang spoke about the difficulty of teaching quantum computing since the students have different backgrounds. He has devised (or helped devise) Online Tutoring systems for it that seem to be working very well. I didn't know that quantum computing was at the level to worry about how-to-teach-it. Then again, any course has these concerns, so it's good to see that he did something about it. (Even so, I doubt I'll invest a lot of time and effort into an online tutoring system for my Ramsey Theory course next spring.) 8. There were some talks on or touching on Quantum-Prog Languages, Quantum-CAD, Quantum-architecture. I suspect that if quantum computers are ever built we will find that some of the assumptions of this work were wrong; however, I also suspect that having people who have thought about these issues will be valuable. A few weeks ago, Suresh wrote a post Things a TCSer should have done at least once with the caveat This list is necessarily algorithms-biased. I doubt you'll need many of these if you're doing (say) structural complexity. Basically begging a response. So here is what every structural complexity theorists should have done. • Define a new complexity class that has some reason for being. • To keep balance in the world, you should also collapse two other complexity classes. • While you are at it, separate two complexity classes that weren't separable before. • Create a new relativized world. Extra points if in this work you collapse two complexity classes while separating two others. • Use Kolmogorov complexity, information theory or the probabilistic method as a proof technique. They are really all the same technique in disguise. • Use the sunflower lemma, or the Local Lovasz lemma, or some other weird probabilistic or combinatorial lemma just for the fun of it. • Invoke VC dimension to solve a problem. I should have at least one in common with Suresh and Sauer's lemma is sometimes useful in complexity. • Have a theorem that starts "Assuming the extended Riemann hypothesis..." • Give a "simpler" proof of a nasty theorem. • [Advice given by Noam Nisan many moons ago] Try to settle P v NP. In both ways. Only by really trying and failing can you understand why the easy stuff doesn't work. (In this post I quote attempted posts to the blog. I transcribe them as they are, so if you see a missing period or awkward language, its not me (this time) its them.) We moderate comments but do so very lightly. There is no hard and fast rules but roughly speaking we block comments that are BOTH offensive AND off-topic. There may be exceptions- like if its on topic but REALLY REALLY offensive and adds nothing to the discussion. There may more benign exceptions- like if I post a question and will block the answers so that when I reveal the answer the next day its more dramatic. Of if someone posts information that is not public yet. In these case we hope they try to post non-anonymously so we can email them and tell them why they were blocked. There are other isolated cases as well. All of these are very rare. Recent attempted comments do not fall into these rules and we had to decide on them. The following was an attempted comment on my post STOC 2012-Workshops and honored talks Hi there STOC 2012- workshop and honors talks Loved every second! Great views on that! That actually breaks the mold! Great thinking! This comment is NEITHER offensive NOR off-topic.Its a bit odd- I can't tell if the Great view is of my post or of the talks I was writing about. It does sound awkward. Why is that? IT WAS GENERATED BY A SPAMBOT!!! How do I know this? Because if you click on the author you are directed to a site thatsells you paints for your living room.Hence we block such posts. So, they think the readers of our blog are into interior decorating. I am sure that some are, I don't think our readers are a particularly good market for this. Technology is good enough to find our blogs and try to use spambots on them,but not good enough (or there is no incentive) to figure out which blogs are worthtargeting.This is part of a bigger problem I blogged about herewhere I noted that technology is good enough to know that I am a book review editor for SIGACT NEWS but notgood enough (or there is no incentive) to figure out that I only review comp sci and math books, and not books on (say) politics. A borderline case: an attempted comment on the blog A natural function with very odd properties was Awesome logic. You truly have some expert skills and enhanced my knowledge on Cantor Set Construction Agreements. This one did not link to any product so it might be legit, except thatit is awkward sounding and the same person tried to submit,as a comment to Six Questions a about natural and unnatural mathematical objects This truly enhanced my skills. very helpful Job Proposal. Clearly spam, though I'm not sure why since there is no link to a product. These posts are trying to pass a Turing Test- but so far they are not succeeding. Sometimes they only positive comments I get are from spambots. Oh well. Consider the following game on a poset, each player takes turns picking an element x of a finite poset and removes all y ≥ x. First one to empty the poset wins. I posted last March about a high school student, Adam Kalinich, who showed how to flip the winner of a poset game. Finding the winner of a poset game is in PSPACE by searching the game tree. A corollary of Adam's work showed that poset games were hard for Boolean formula leaving a huge gap in the complexity of finding the winner. Daniel Grier, an undergrad at the University of South Carolina, has settled the problem and shows that determining the winner of a poset game is PSPACE-complete. His reduction is ridiculously simple (though not obvious) and the proof is not that complicated either. Grier starts from Node Kayles which is a game on an undirected graph where each player takes turns removing a vertex and all its neighbors. Whomever empties the graph first wins. Thomas Schaefer showed the PSPACE-completeness of Node Kayles back in 1978. Grier's reduction from Node Kayles to posets is very simple: Let G be the graph. Have one element in the poset for each vertex of G, all incomparable. For each edge e=(u,v) we add two more elements, one above the vertex elements corresponding to u and v, and one below every vertex element other than u and v. That's the whole construction. Grier shows that if G has an odd number of edges and no immediate win for the first player then the first player wins the Node Kayles game if and only if the first player wins the corresponding poset You can read more details in Grier's short paper. It's really neat seeing high school students and undergrads solving interesting open problems. We need more problems like poset games. Early registration for the FOCS conference in New Jersey is September 27th. There is some travel support available for students and postdocs, deadline is this Friday the 21st. STOC and Complexity will be co-located in Palo Alto in early June. STOC CFP (deadline November 2), Complexity CFP (deadline November 30). SODA comes back to the US and New Orleans January 6-8. Accepted Papers. In the year 4000BC my great-great-...-great grandmother tried to solve (in today's terms) the equation x^2 + 2x + 2 = 0 She discovered that if it had a solution then there would be a number a such that a^2=-1. Since there clearly was no such number, the equation had not solution. She missed her chance to (depending on your viewpoint) discover or invent complex numbers. Fast Forward 6012 years. In the year 2012 I wondered: is there a probability p such that if you flip a coin that has prob(H)=p twice the prob that you get HT is 1/2? This leads to If you solve this you get p=(1+i)/2. Hence there is no such coin. WAIT A MINUTE! I don't want to miss the chance that my great...great grandmother missed! In the real world you can't have a coin with prob(H) = (1+i)/2. But is there some meaning to this? More generally, for any 0 ≤ d ≤ 1 there is a p &in; C (the complex numbers) such that ``prob(HT)=d.'' The oddest case (IMHO) was to take d=1. You then get that if a coin has prob(H)=(1+\sqrt(-3))/2 then prob(HT)=1. Does that mean it always happens? No since prob(TH)=1. Do the probs of HH, HT, TH, TT add up to 1? Yes they do since some are negative. Is there an interpretation or use for this? I know that quantum mechanics uses stuff like this. Could examples like this be good for education? Are there non-quantum examples of the uses of this thatcould be taught in a discrete math course? A couple of weeks ago Suresh tweeted the following result of James Orlin Max flows in O(nm) time or better. jorlin.scripts.mit.edu/Max_flows_in_O… — Suresh Venkat (@geomblog) August 31, 2012 I'm thinking, wow, max flow is one of the major standard algorithms problems, and O(nm) time (n = number of vertices, m = number of edges) seems like a great clean bound. But there hasn't been much chatter about this result beyond Suresh's tweet. Reading Orlin's paper gives some clues. The previous best bound due to King, Rao and Tarjan has a running time of O(nm log[m/(n log n)]n) = O(nm log n) just a logarithm off from O(nm). Orlin doesn't directly give an O(nm) algorithm, his takes time O(nm+m^31/16log^2n). It's the minimum of the running times of King-Rao-Tarjan and Orlin's algorithms that yields O(nm). Nor is O(nm) tight, Orlin also gives an algorithm with a running time of O(n^2/log n) when m=O(n). I don't mean to knock Orlin's work, he makes real progress on a classical algorithmic problem. But somehow I think of O(nm) as a magical bound when it is really just another bound. I'm just fooled by Two quantum announcements (emailed to me by Umesh Vazirani, and producedhere almost exactly) and then some thoughts of mine quantum computing. Announcement one: The NSF has a new initiative to try to address the lack of tenured faculty (particularly in computer science departments) involved in quantum computation research. CISE-MPS Interdisciplinary Faculty Program in Quantum Information The initiative provides a paid sabbatical year to interested tenured faculty to visit a strong quantum computing group, so that can reposition their research interests. The rationale behind the solicitation is to increase the number of tenured researchers in quantum computation, but also to break through the "quantum skepticism" in faculty hiring decisions in those departments where there are no faculty actively involved in quantum computing research. Announcement two: In support of this program, Carl Williams (a physicist working in Quantum Information who put together the US Vision for Quantum Information Science for the Office of the President) and Umesh have put together a workshop where interested individuals can learn about the initiative, the field and make contacts with people from the major quantum computing centers: see here. The initiative comes at a particularly opportune moment for researchers in complexity theory, given the increasing relevance of quantum techniques in complexity theory --- the 2-4 norm paper of Barak, et al (SDPs, Lasserre), exponential lower bounds for TSP polytope via quantum communication complexity arguments (See Drucker and de Wolf paper Quantum proofs for classical theorems for several apps of Q to Complexity, and see here for the TSP polytope result) quantum Hamiltonian complexity as a generalization of CSPs, lattice-based cryptography whose security is based on quantum arguments, etc. MY COMMENTS: Umesh gives as a reason quantum is important its uses in other parts of complexity theory. While that is certainly good there are other intellectual reasons why Quantum is worth 1. Factoring is in Quantum P! There are MANY problems (maybe 10) where Quantum seemsto be faster than classical. I wouldn't really want to push this point sincequantum computer aren't build yet. More generally, if one claims a field is validfor real practical value, those arguments may become less believable over time. 2. Quantum computing can be used to simulate quantum systems- I think this was one of the original motivations. 3. Quantum computing is valuable for a better understanding of Physics. This was first told to be my Fred Green (A Physics PhD who went into computer science)and I made it the subject ofthis blog entry. I like his quote so much that I will quote it here Learning quantum computing helped me understand quantum mechanicsbetter. As a physicist I never thought about measurement theoryor entanglement, which were foundational issues, irrelevantto what was doing. In quantum computing, we reason about thesethings all the time. Over the years others have told me similar things. Side note: The word Quantum is mostly misused in popular culture. Quantum Leap meant a big leapwhen actually quantum means small. The James Bond movie Quantum of Solace used it correctlybut was an awful movie. Oh well.	{"url":"https://blog.computationalcomplexity.org/2012/","timestamp":"2024-11-08T12:17:55Z","content_type":"application/xhtml+xml","content_length":"413091","record_id":"<urn:uuid:ef6f30c2-3d29-4d1e-ad6b-7291a456f7d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00871.warc.gz"}
Algebra Worksheets Algebra worksheets from common core sheets are the best on the internet! Our worksheets are designed to help students of all levels hone their algebra skills, offering a range of topics and helpful hints to provide invaluable practice in solving equations, learning new concepts, and understanding the basics of algebra. Whether you’re just starting with basic operations or tackling more complex topics like polynomials and linear equations, our algebra worksheets have you covered. Each worksheet provides an engaging way to learn and practice essential algebra skills like simplifying expressions, identifying coefficients, solving inequalities, graphing lines, and finding combinations. Get started today and take advantage of our free algebra worksheets to gain a deeper understanding of challenging concepts and experience why common core sheets is the best source for algebra worksheets available. Jump to a Heading > Filter By Grade > Browse Sheets By Problem Type Using Substitution to Solve Problems 6ee5 × Description: "This worksheet is designed to enhance children's mathematical abilities with a specific focus on using substitutions to solve problems. It contains 11 math problems where students must identify the correct value of 'e' in given equations. The flexible format allows it to be customized to suit individual learning needs, converted into flash cards for additional study aids or employed in a distance learning setting. An essential tool for practicing and mastering a fundamental math concept." × Student Goals: Problem Solving SkillsUpon the completion of this worksheet, students will have significantly enhanced their problem-solving skills. They will be equipped with the ability to analyze, break down, and understand complex mathematical problems, which is an essential skill that's applicable in various other academic areas beyond math. It's an important cognitive ability that can help in understanding and processing complex information and tasks.Concept UnderstandingStudents will gain a clear understanding of the concept of 'substitutions' in math. They will be able to accurately apply the correct values for the placeholder or unknown variable 'e' in the given equations. This will promote their comprehension of algebraic expressions and will set a foundation for higher-level math concepts in the future.Confidence in MathematicsThe successful completion of the worksheet will instill confidence in children's mathematical proficiency. By successfully solving the problems independently, they will reinforce their knowledge and build self-assuredness in their capabilities. Hence, they will encounter less anxiety when facing number problems in standardized tests or everyday math situations.Critical ThinkingThis worksheet will enhance students' critical thinking skills. They will learn to evaluate the given options, compare them, and select the most appropriate solution for each problem. They will learn to scrutinize every option and not hastily jump to conclusions which is a necessary skill in furthering their competent decision-making capabilities.Acclimatization with Mathematical LanguageStudents will become more familiar with mathematical language and symbols, learning how to interpret '<', '>', and '÷' among other symbols. This understanding will assist in analyzing and solving mathematical problems presented in this or differing formats. Writing Inequalities 6ee6 × Description: "This worksheet is designed to enhance children's understanding of mathematical inequalities. It provides 20 organized problems, each illustrating a clear example of inequalities including 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to'. Perfect for distance learning, this highly customizable worksheet can quickly be transformed into flashcards to expand the learning process. A valuable tool for mastering the concept of math inequalities." × Student Goals: Understanding of Inequality ConceptsAfter completing this worksheet, students should have a thorough understanding of inequality concepts. They will gain familiarity with the symbols for 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to'. They will have a practical understanding of how to read and interpret these symbols and be able to use them correctly to portray mathematical relationships in a variety of contexts.Problem Solving SkillsWorking on this worksheet will also sharpen the students' problem-solving skills. They will become adept at identifying the unknown variable and determining the correct inequality to solve for it. They will learn to approach each problem logically and methodically, enhancing their overall maths ability and paving the way for tackling more complex mathematical problems in the future.Mastery over Negative NumbersAs some problems involve negative numbers, students will further develop their proficiency in dealing with these numbers. They will understand how inequality relationships change with the inclusion of negative numbers and will be able to manipulate them with confidence. Their knowledge and understanding of the number line will also be strengthened.Preparation for Higher-level Math ConceptsThis worksheet serves as an excellent primer for more advanced mathematical concepts. It lays the groundwork for a comprehensive understanding of algebra and calculus. By mastering the principles of inequalities, students will be better prepared to learn about functions, algebraic equations, and the application of mathematics in real world situations. The problem-solving skills they develop will be beneficial across all scientific disciplines.Improved Mathematical CommunicationFinally, by solving these problems, students will improve their mathematical communication skills. They will learn to express mathematical concepts succinctly and accurately, using the correct symbolic language. This will facilitate accurate and efficient communication of ideas, fostering discussions and collaborations in further studies. The increased fluency in mathematical language builds a strong foundation for advanced academic and professional pursuits. Expressing Inequalities on a Numberline 6ee8 × Description: "This worksheet is designed to aid children in grasping the concept of expressing inequalities on a number line. It contains 13 problems that offer interactive visualizations, improving mathematic comprehension in a fun and engaging way. The material is adaptable, can be converted into flashcards, and is suitable for distance learning. This interactive tool serves as an effective resource to reinforce numerical understanding and mathematical logic." × Student Goals: Understanding of InequalitiesUpon completing this worksheet, students should gain a strong grasp of inequalities. They’ll have learned how to interpret and express mathematical inequalities, which enable comparisons between distinct numbers or variables.Visualization SkillsThis task should enhance the students' ability to employ number lines for visual representation and understanding of inequalities. A number line is an effective tool to visualize and solve problems, as it spatially illustrates how inequalities function and which values correspond to the query.Problem Solving AbilitiesCompletion of these problems should reinforce their problem-solving abilities. Over time, they are likely to find these exercises more straightforward, as they increasingly understand how inequality symbols work and the corresponding solutions.Proficiency in Working with Different Number SetsThrough this worksheet, students should become more proficient in working with different number sets. They will get the experience of dealing with positive numbers, negative numbers, and zero, enhancing their versatility and adaptability in tackling various number types.Building Blocks for Advanced TopicsMastering inequalities and their representation on a number line is a foundational concept in math. It shall act as a building block for more advanced topics students will encounter later, such as graph theory, complex numbers, and functional analysis.Understanding Real World ApplicationsWhile solving the worksheet, learners will unconsciously apply real-world situations that require understanding of inequalities such as time planning, budgeting, and resource allocation. These can be beneficial in cultivating their practical decision-making skills, enhancing their grasp of the subject beyond the academic context.Confidence BoostEach completed problem would add to their confidence as they conquer each inequality. Success with mathematical problems significantly contributes to the confidence of the students. This, in turn, motivates them to face new challenges, in mathematics as well as in other fields of study.Critical ThinkingStudents will engage their critical thinking skills to solve problems. The exercise encourages learners to think logically, analyse the given condition correctly, and decide on the inequality symbol that should be used. Writing Inequalities from a Numberline 6ee8 × Description: "This worksheet is designed to help children understand and practice writing inequalities from a number line in math, a key concept in algebra. Featuring 13 problem sets, it visualizes inequality concepts using a variety of number lines. It's customizable to cater to individual student needs and can also be converted into flashcards for hands-on learning. Great for distance learning, this worksheet is an effective tool for both in-class and independent study." × Student Goals: Matching Inequalities to Numberlines 6ee8 × Description: "This worksheet is designed to help children understand the application of inequalities on number lines. It features eight problems that require students to match inequalities to the correct number line representation. Versatile and flexible, this educational resource can be adapted for class exercises, converted into study flashcards, or employed in distance learning settings for individual, self-paced instruction. A beneficial tool for mastering math concepts in a visually engaging manner." × Student Goals: Understanding InequalitiesAfter completing the worksheet, students will gain a solid understanding of inequalities. They will know how to interpret and solve inequalities, learning critical mathematical concepts such as less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). These foundational skills are necessary for further mathematical studies and applications.Numberlines RepresentationThis worksheet will help students to connect abstract concepts with visual aids as they align inequalities with number lines. The ability to match inequalities with their corresponding number lines strengthens mathematical reasoning and aids in the comprehension of more real-world applications of mathematics where such visual representations come in handy.Problem-Solving SkillsStudents will enhance problem-solving skills by determining which numeric range best fits the described inequality. Consequently, this practicum could foster precision in students as they will need to carefully consider the restrictions that the inequality places on possible values, thereby selecting the most accurate option.Critical Thinking and ReasoningIdentifying the correct number lines requires students to employ logic and reasoning. Pushing students to process information and not just memorize it fosters higher-order cognitive skills like critical thinking. They will need to analyse, evaluate and create—an integral part of Bloom's taxonomy in education—during this worksheet activity.Confidence in MathematicsMathematics can often be a challenging subject for many students. However, by reinforcing these key concepts through worksheets, students can be successful in their understanding and application, leading to an increased confidence in their mathematical abilities. They will be well-prepared for handling similar tasks in the future and for tackling more advanced mathematical concepts.Preparation for Advanced StudiesMastering inequalities and number lines is not only necessary in itself, but it's also instrumental in preparing for more advanced math studies, such as algebra, calculus and beyond. The skills acquired and enhanced during this exercise will be transferable and applicable in advanced educational pursuits. Identifying Numerical Coefficient × Description: "This worksheet is designed to teach children the concept of identifying numerical coefficients in mathematics. Featuring 20 unique problems, including variables and multi-variable expressions, it aids in enhancing their algebra skills. Flexible and easily customizable, the worksheet can be transformed into flashcards or tailored for distance learning platforms, making it a versatile educational tool for various teaching strategies." × Student Goals: Numerical Coefficient IdentificationBy the end of this exercise, your child should have mastered the necessary skills in identifying numerical coefficients in algebraic expressions, a critical aspect in the mathematical subject of algebra. By solving the problems provided in the worksheet, they will have an enhanced understanding of what a numerical coefficient is and its role within an algebraic expression. This foundational knowledge is integral as they advance in their math journey as it forms the groundwork of many algebraic concepts.Enhance Problem-Solving AbilitiesThe provided 20 problems are designed to challenge your child and improve their problem-solving abilities. As they engage with each problem, they will be actively interpreting and solving algebraic expressions which are crucial skills for the study of mathematics and other STEM-related fields. Therefore, it will greatly enhance their logical and critical thinking abilities, enabling them to approach mathematical problems with more confidence and ease.Improve Numerical FluencyBy meticulously identifying the numerical coefficients, your child will have a better grasp of negative and positive numbers, decimals and integers, consequently improving their numerical fluency. This is essential as numerical fluency is a cornerstone in mathematical learning, having direct impacts on advanced subjects such as calculus and statistics.Foundation for Complex Algebraic ConceptsThis practice also prepares students for complex algebraic concepts they will encounter in the future. Understanding how to identify numerical coefficients is a stepping stone towards more sophisticated algebraic procedures such as manipulating expressions, simplification and solving equations. Therefore, they will be better equipped to handle more difficult mathematical material as a result of mastering this skill.Promote Independent LearningFinally, completing the worksheet successfully can boost your child's confidence in their learning abilities. As they solve the problems independently, they're developing their mathematical capabilities while also nurturing their autonomy and self-reliance in their learning process. These are valuable traits that will come handy as they navigate through their years in the educational journey. Determining Variable Value (+, -, ×, ÷) × Description: "This worksheet is designed to enhance children's understanding of math concepts by determining variable values. Covering a variety of 12 problems involving addition, subtraction, multiplication, and division, it provides a hands-on, practical approach to learning. The worksheet is customizable, allowing for flexibility in teaching methods. It can be converted into flashcards for quick revision, or used in distance learning programs, offering accessibility in a variety of settings." × Student Goals: Understanding Variables and EquationsAfter completing this worksheet, students should have obtained a clearer understanding of variables and how they function within mathematical equations. They will become proficient in identifying and solving for variables within a mixture of addition, subtraction, multiplication and division problems. This foundational skill in algebra will allow students to approach more complex algebraic problems confidently.Balancing Equations & Determining SolutionsThis worksheet will enable students to demonstrate their ability to balance equations and accurately determine the value of variables. They will through practice and repetition, develop a systematic approach for solving algebraic equations, which requires both logic and arithmetic. This skill is crucial for mastering algebra and other higher-level mathematical disciplines.Applying Arithmetic OperationsWith the applications of addition, subtraction, multiplication, and division equations in this worksheet, students will have a chance to apply and reinforce their knowledge of basic arithmetic operations within the context of algebra. They should be able to perform these operations swiftly and accurately, which is essential for problem-solving in various domains of math and science.Problem-Solving & Critical ThinkingAs students work through different problems within the worksheet, they will hone their problem-solving skills and abilities to think critically. They will learn to analyze the provided information, determine what is being asked, formulate a strategy to solve the problem, carry out that strategy, and finally check their answers for accuracy. These are valuable skills that students will carry with them throughout their academic journey and beyond.Mathematical Confidence & PersistenceThroughout this exercise, students will also increase their confidence in their mathematical abilities. The varied degrees of difficulty within the worksheet help students to stretch their ability, stepping out of their comfort zone, and thus growing tougher in the face of mathematical challenges. This nature of persistence is an important trait which will be useful in tackling more advanced math topics and other academic pursuits. Examining Powers and Bases 8ee2 × Description: "This worksheet is designed to enhance children's understanding of mathematical concepts, specifically powers and bases. It consists of 10 problems that challenge students to identify the right mathematical equations corresponding to given values of x. Customizable and adaptable to various learning contexts, the worksheet can be converted into flash cards or utilized for distance learning, making it an ideal tool for remote education." × Student Goals: Understanding of Base and Power ConceptsUpon completion, students should have grounded understanding of base and power concepts in mathematics. They will be adept in distinguishing between the terms ‘base’ and ‘power’, and will comprehend how values change depending according to powers. This worksheet aids students in recognizing the relationship between a number and its exponent, key to advance mathematical prowess.Problem Solving AbilitiesThe worksheet enhances students' problem-solving abilities. Students can identify the proper equation that leads to a specific outcome. They gain the capability to work out multi-option mathematical problems, enhancing their analytical and logical thinking skills. Students become proficient in exploring different possibilities to achieve the correct results in problematic equations.Confidence in Equation AnalysisThe worksheet also fosters students' confidence in equation analysis. They will become comfortable dealing with power and base equations, with a focus on isolating and solving variables in these equations. Students will have increased familiarity and confidence with the chosen subject matter, thereby enabling them to approach more complex mathematical problems with ease.Learning PrecisionStudents are expected to have heightened precision after completing this worksheet. The multi-option format requires students to be exact in their answers, eliminating room for approximation. Thus, students would become extremely detail-oriented and precise, enhancing their overall mathematical acumen.Enhanced Test-Taking SkillsBy answering multiple-choice questions, they also refine their test-taking skills. Students will learn to choose the most effective approach to solve a problem, developing strategies such as the process of elimination. Such skills prove to be beneficial during exams and help increase efficiency in their academic advancement. Examining Slope Attributes 8ee6 × Description: "This worksheet is designed to guide children through the concept of examining slope attributes in mathematics. With a total of 10 problems, it challenges students to determine and compare the slopes of different lines. This flexible learning tool can be customized to each student's needs, and content can be converted into flashcards for further study. Ideal for distance learning, the worksheet advances understanding of key math concepts related to slopes in geometry." × Student Goals: Understanding of Slope ConceptAfter successfully completing this worksheet, students should have an enhanced understanding of the slope concept. They will grasp the mathematical notion of slope in relation to lines in a two-dimensional space. This foundational comprehension will enable pupils to navigate more complex geometry and algebraic problems with ease.Problem Solving SkillsThis worksheet elevates learners' ability to solve mathematical problems related to slope. They will adeptly determine whether two lines have the same slope and interpret meaning from given equation parameters, thereby strengthening their problem-solving and critical thinking skills.Skill in Equation ManipulationCompleting the worksheet will improve students' ability to manipulate mathematical equations, particularly those involving the slope of lines. Practice with slope equations will encourage accuracy and precision in students' mathematical language and equation solving.Increased Comfort in GeometryThis worksheet serves to increase students' comfort in the subject of geometry, specifically regarding lines and angels. They will develop proficiency in handling geometrical problems involving slopes, a central concept in geometry which serves as a stepping stone for grasping higher-level geometrical concepts.Mathematical ConfidenceFinally, by completing this worksheet, students will boost their mathematical confidence. Handling these 10 problems and building their intuition around slope, they thereby enhance their self-confidence in confronting new mathematical problems. This positive attitude is critical for the ongoing study of mathematics and sets them up for future success. Expressing Equations 7rp2c × Description: "This worksheet is designed to help children understand the concept of expressing equations in a practical context. Covering ten real-world math scenarios ranging from shopping to cooking, it enables them to establish relationships between variables in daily life instances. Flexible and adaptable, this worksheet can be conveniently customized, transformed into flashcards for effective learning, or seamlessly integrated into distance learning curriculums." × Student Goals: Mathematical UnderstandingUpon completion of this worksheet, students should have a stronger understanding of how to express real-life scenarios in mathematical equations. They can identify the variables in a problem and formulate an equation that best represents the given situation.Critical Thinking SkillsStudents will enhance their critical thinking skills by interpreting the real-world situations and deducing how variables relate to each other to write an equation. This process will require them to analyze the problem, apply their math knowledge, and solve it logically.Application of Math ConceptsStudents will be able to apply math concepts in diverse contexts, demonstrating their adaptability of mathematical knowledge. By seeing how equations function in real-life scenarios, they will gain a deeper understanding of the purpose and significance of learning math.Real-Life Problem SolvingThey will take a significant step in understanding mathematical modeling of real-world problems. The problems in this worksheet mirror everyday scenarios, guiding the students to utilize their mathematics knowledge to solve problems they may encounter in their daily life.Independent LearningStudents will improve their ability to work independently and effectively. The worksheet provides a platform for students to exercise their problem-solving skills and develop their capacity for self-study, which is an essential skill for lifelong learning.Concept of Ratios and ProportionsStudents will also strengthen their grasp of ratios, proportions, and linear relationships. Through the problems, they will learn to identify the constant of proportionality and express it as a rate. The skills acquired here are foundational for more complex algebraic problems.Numeracy SkillsCompletion of the worksheet also promotes improvement of general numeracy skills. These include computation, number sense, measurement, estimation, and spatial sense, all of which are vital for a student’s overall mathematical competency. Solving Circle Equations × Description: "This worksheet is designed to help children understand and solve circle equations in mathematics. With a total of 13 challenges, students are expected to calculate the radius based on given x and y values. This customizable resource can easily be converted into flashcards or integrated into distance learning platforms. Ideal to reinforce mathematical concepts and enhance problem-solving skills." × Student Goals: Rewriting Expressions as Multiples of a Sum 6ns4 × Description: "This worksheet is designed to reinforce math skills by teaching children how to rewrite expressions as multiples of a sum. Tailored for distance learning, it features 12 problems where kids can manipulate numbers to form new equations, such as 27+14 becoming 1×(27+14). Customizable for various learning styles, it can also easily transform into a set of interactive flashcards for hands-on practice. Ideal for enhancing arithmetic comprehension in a fun, engaging way." × Student Goals: Simplifying Expressions 7ee1 × Description: "This worksheet is designed to advance children's understanding in math, specifically simplifying expressions. It presents 19 problem sets, with examples involving equations and variable manipulation. Adoptable to various learning modes, it can be customized to fit individual needs, converted into flash cards for hands-on interaction, or utilized in distance learning for adaptable education." × Student Goals: Master Algebraic ManipulationAfter completing this worksheet, students should have developed an extensive command over the simplification of diverse and complex algebraic expressions. They will be proficient at isolating variables, implementing basic arithmetic rules, and applying mathematical operations within parenthesis consistently. This foundation will enable them to approach more intricate algebraic formulations with confidence.Enhance Problem-Solving SkillsStudents should also cultivate an advanced ability to decode and resolve more challenging problems, improving their problem-solving competence. The worksheet's problems have been designed to sequentially escalate in complexity, thus pushing them to incorporate strategic thinking and procedural reasoning effectively. This will give them the capacity to approach difficult mathematical scenarios analytically and methodically in the future.Promote Fluent Mental ArithmeticPracticing these math problems supports the development of swift mental arithmetic. Students should be able to perform calculations with larger numbers and signs swiftly and accurately. With regular practice of such exercises, they'll secure a high level of fluency in mental math, which is an indispensable skill across a wide range of mathematical disciplines.Cultivate Precision and AccuracyCompletion of this worksheet encourages precision and accuracy in students. Algebra deals with variables and constants, and even a minor variation can alter the answer drastically. A thorough accomplishment of the worksheet’s tasks will train them to handle these particulars with utmost care, thus minimizing errors in their future mathematical endeavors.Improve Abstract Reasoning AbilityLastly, students can expect a significant improvement in their abstract reasoning abilities. Working through the exercises on this worksheet involves identifying patterns and relationships between numbers, fostering their ability to think abstractly. This abstract reasoning is crucial not only in higher mathematical studies but also in daily life situations that require logical deduction. Rewriting Expressions 7ee1 × Description: "This worksheet is designed to enhance children's understanding of math, specifically in rewriting expressions. It presents ten unique math problems, featuring manipulation of fractions, variables, and arithmetic operations. Customizable to individual learning styles, this tool can also be converted into flashcards or adapted for distance learning, making it versatile for various modes of study." × Student Goals: Expanding Expressions 7ee1 × Description: "This worksheet is designed to enhance children's skills in math, specifically expanding expressions. Containing 20 challenging problems, it uses varied scenarios to reinforce understanding, and teach the concept. Its interactive nature allows customization, making it easily converted into flash cards for more concise learning. This versatility makes it well-suited for distance learning, adapting to different teaching strategies and promoting individual progress." × Student Goals: Understanding of Algebraic ExpressionsUpon completing this worksheet, students should have a solid grasp of expanding algebraic expressions. They’ll understand the distribution property of math operations over parentheses and will be able to use this understanding to simplify expressions, thus developing mental arithmetic skills. This foundational algebraic knowledge forms the basis for more advanced math concepts, proving beneficial in their future mathematical journey.Improved Problem-Solving SkillsThe use of indeterminate numbers in the problems will contribute to developing students' problem-solving skills. The ability to work with algebraic expressions efficiently is a critical thinking skill that will be applicable in various subjects beyond mathematics, such as physics and coding. This strengthens the students' logical reasoning and critical thinking capabilities.Practical ApplicationsBy completing this worksheet, students are on an important path to understanding the practical applications of algebra. Math isn't just about numbers and calculations; it's also about problem-solving, logic, making predictions, and pattern detection. They will also appreciate the role that expressions play in everyday life situations, where unknown variables can represent anything from an undetermined price to an undetermined quantity in many real-life situations.Increased Confidence and IndependenceThe successful completion of the worksheet will give an immense boost to students' confidence. It serves as an affirmation of their understanding and comprehension of the concept. Successfully tackling challenging problems independently helps to inculcate a sense of achievement and independence. Over time, these experiences will build their confidence to tackle more complex problems and tasks.Preparation for Advanced TopicsThe worksheet also acts as a stepping stone for more advanced topics in mathematics. A firm understanding of expanding expressions will prepare students for their future studies in advanced algebra, calculus, and beyond. Once they are comfortable with these basics, students will find it easier to handle complex equations and derive solutions more efficiently. Factoring Expressions 7ee1 × Description: "This worksheet is designed to strengthen children's math skills, focusing on factoring expressions. The 10-problem set presents varied examples using different numbers and variables. The problems are structured in a way that promotes understanding of mathematical fraction concepts. This particularly customizable worksheet can be converted into flashcards, making it a versatile tool for distance learning. It is an excellent material for enhancing children's critical thinking and problem-solving abilities in mathematical expressions." × Student Goals: Percent Word Problems as Decimal Expressions 7ee2 × Description: "This worksheet is designed to aid children in understanding percent word problems, using decimal expressions in a real-world context. Covering a variety of subjects such as price increases, wage calculations, and price drops, it offers ten problems to solve, each with four potential solutions. Highly versatile, the sheet can be customized according to individual learning styles, converted into flashcards for quick study sessions, or utilized as an effective tool in distance learning environments." × Student Goals: Competence in Mathematical ConceptOn completion of the worksheet, children should have developed an in-depth understanding of mathematical problems related to percentages. They will have learned how to express percentage increment and decrement problems in decimal expressions. These formative skills would fortify the foundation of their knowledge in Mathematics, particularly in algebraic representations.Problem Solving SkillsChildren will be competent in solving real-life math problems that involve percentage changes. Their problem-solving abilities will be honed, making them adept at deriving solutions to a variety of problems in organized and strategic ways. They will build and improve their capacity to understand situational problems, identify mathematical relationships and to independently strategize solutions.Ability to Interpret Mathematical RelationsChildren should also be able to interpret the relationships between values within the problems. This skill to decode and understand the link between the entities in a problem is vital to sound mathematical reasoning and successful problem solving. They will be proficient in understanding contextual math problems, interpreting relations between entities, and translating them into mathematical expressions.Proficiency in Critical ThinkingUpon completing the worksheet, children should have improved their critical thinking abilities. By encountering and overcoming various percentage related problems, they would learn to dissect a problem and plan a strategic approach to decipher it. Through this, they will strengthen their ability to think critically and derive logical solutions and arguments.Math ConfidenceAfter finishing the worksheet, students would gain confidence in their mathematical abilities. Tackling and mastering these problems would provide a sense of achievement, which would encourage children to be more confident and interested in math. Their ability to understand and solve percentage problems would enhance their overall mathematics efficiency, paving the way for higher mathematical learning. Simplifying Expressions × Description: "This worksheet is designed to strengthen children's math skills by simplifying expressions, with a focus on variables and exponents. It contains 14 problems with various complexity levels, including polynomial and monomial simplification. The content can be customized according to individual learning needs. It has versatile usage, ideally for distance learning, flash card creation, or in-class exercises, fostering a comprehensive understanding of mathematical expressions." × Student Goals: Comprehension of ExpressionsStudents should have a deep understanding of how expressions work in mathematical computations. They should be comfortable in recognizing and handling parts of an expression such as variables, coefficients and constants. They will acquire skills on how to identify like terms in an expression, a significant step in the process of simplifying expressions.Simplifying ExpressionsOn completion of the worksheet, students should be proficient in simplifying expressions, which is a key component of algebra. They should be able to condense lengthy expressions into more manageable forms by combining like terms and constants, thereby making them easy to work with in solving equations or evaluating mathematical models.Critical ThinkingThis worksheet will enable students to develop their critical thinking skills as they navigate through the various problems. Simplifying expressions involves strategic thinking and careful calculation, which will sharpen their decision-making abilities and problem-solving skills, as every step counts in arriving at the correct solution.Application of Basic Math OperationsStudents should have strong command of the basic arithmetic operations - addition, subtraction, multiplication and division as simplifying expressions involves these operations. Completing the worksheet will further reinforce their understanding of how to implement these operations, especially in the context of algebraic expressions.Boosting Algebraic ConfidenceWorking through the worksheet will expose students to a variety of expression types, thus broadening their perspective and boosting their confidence in tackling algebraic problems. The satisfaction of correctly simplifying an expression is a great morale booster, encouraging them to explore more complex topics in algebra. Matching Equivalent Expressions 6ee4 × Description: "This worksheet is designed to strengthen kids' mathematical abilities by matching equivalent expressions. It presents 13 problems, requiring children to pair complex mathematical expressions correctly. This worksheet is perfect for distance learning, can be converted into interactive flashcards, and offers customizability to meet specific learning needs. By engaging with this tool, children will develop a comprehensive understanding of algebraic expressions and their equivalents, enhancing their problem-solving skills." × Student Goals: Improved Understanding of Equivalent ExpressionsAfter completing the worksheet, students will have a better grasp of equivalent expressions. They will understand how expressions can have similar meanings or values, and will have honed their ability to identify these correspondences. Students will be better equipped to handle more complex problems in the future due to this deeper understanding of mathematical relationships.Enhanced Problem-Solving SkillsBy working through these problems, students will improve their ability to decode mathematical expressions and find equality between them. This involves attention to detail, strategic thinking, and calculation skills. Consequently, students will evolve as problem-solvers, capable of breaking down a situation and finding solutions with greater ease.Reinforcement of Multiplication ConceptsThis worksheet also reinforces the core concept of multiplication. By equating different expressions, students will get regular practice in multiplication. Through engagement with the problems, they will become more comfortable with the multiplication and its foundational role in mathematics.Familiarity with Algebraic ConceptsThese problems subtly introduce algebraic concepts like variables, multiplicands, and products. By working on such problems, students will become more familiar with algebra and its symbols, setting them up for future learning in this area of mathematics.Growth in ConfidenceFinally, as students successfully solve the problems on this worksheet, they will notice an increase in confidence in their mathematical abilities. By challenging themselves, they will come to realize that they are fully capable of understanding and solving complex problems. This will not only boost their confidence in their academic abilities, but it will also encourage a positive mindset towards learning new topics in the future. Linear Equations with Variables on Both Sides 8ee7b × Description: "This worksheet is designed to help children master the skill of solving linear equations with variables on both sides. It offers 7 challenging questions that facilitate mathematical fluency and critical thinking. Special features include the ability to convert problems into flashcards for study at one's own pace. It's also ideal for virtual learning environments, given its easy customization options. A perfect tool to practice and reinforce kids' algebra skills in a fun and engaging way." × Student Goals: Understand the Concept of Linear EquationsThis worksheet provides practice for students to understand the concept of linear equations with variables on both sides. By completing the worksheet, the students can grasp how to balance equations with unknowns appearing in both the sides of the equation. Enhancing their understanding of the order of operations and distribution principle, this practice set guides them to carefully manage the algebraic terms.Solving Linear Equations PracticallyCompleting the tasks on this worksheet enable the students to advance their proficiency in practically solving linear equations. The students should be able to correctly identify the variables, constants, and coefficients in the equation to solve for the unknowns. The possibility of finding solutions of these equations that makes both side of the expressions equal would eventually be handy for the students.Enhanced Critical ThinkingEach equation on this worksheet is like a puzzle to be solved, which means students will enhance their problem-solving skills with every problem they decipher. They will learn how to break down complex problems into simpler parts, strategize the order of operations, and perform necessary algebraic manipulations to find the unknown. This promotes critical thinking, a much-needed skill that reaches far beyond the mathematics classroom.Exhibit Skills in Identifying ErrorsAfter completing the worksheet, students should be able to identify any inaccuracies in the process of solving such equations. They will become effective in spotting mistakes in the application of algebraic rules, operations, and steps. This expertise in error-checking boosts their confidence and stimulates self-reliance in their maths competence.Transferable Knowledge and SkillsLearning how to solve linear equations prepares students for exploring further complex algebraic equations and concepts. The skills gained from this worksheet serve as a solid foundation for topics requiring a thorough understanding of linear equations such as arithmetic sequences, coordinate geometry, and linear programming. Moreover, it provides students with transferable skills like logic, reasoning, analytical thinking that can serve them beyond the realms of mathematics. Expanding Polynomials Using the Box Method × Description: "This worksheet is designed to enhance children's understanding of expanding polynomials using the Box Method in mathematics. It consists of 10 problems that include examples formulated in tabular structures for an engaging learning process. The worksheet is versatile, featuring customization options and the ability to be converted into flashcards for portable studies. It's also compatible with distance learning programs, making it a resourceful tool for remote education." × Student Goals: Polynomial UnderstandingAfter completing the worksheet, students should be able to demonstrate an improved understanding of how to manipulate polynomials. They should enhance their knowledge about coefficients, variables, and constant terms that make up a polynomial equation. They should be confident in identifying the degree and terms of a polynomial, facilitating easier comprehension of complex mathematical concepts. Consequently, this will provide a foundation for further topics in algebra.Expanding PolynomialsStudents will gain proficiency in expanding polynomial expressions using the box method. By practicing this fundamental concept in algebra, students should be able to quickly and accurately break down complex polynomial expressions into simpler forms. This expansion of polynomials will help students simplify expressions, making calculations more straightforward, ultimately improving their problem-solving abilities.Analytical ThinkingCompletion of the worksheet exercises will also help students to develop and sharpen their analytical thinking skills. These skills are immensely important in mathematics, as they enable students to systematically approach a problem, break it down into smaller components, and find the solution through measured steps. Such critical thinking capability is a universal competency that can be applied across various subject matters and real-life scenarios.Mathematical FluencyThe repeated practice provided by this worksheet will result in an increased mathematical fluency. By solving numerous problems focused on expanding polynomials, students will start recognizing patterns and applying learned concepts more spontaneously. As a result, not only will students be able to accelerate their speed in problem-solving, but it will also contribute to reducing the errors made when handling polynomial expressions.Confidence BuildingSuccessfully expanding polynomials and obtaining correct solutions will help build students' confidence in their math capabilities. This confidence is essential in freeing students from math anxiety and encouraging a positive attitude toward the subject. By fostering confidence and instilling a sense of achievement, students will be more open to embracing challenging math problems and acquiring new mathematical concepts in subsequent lessons. Expanding Square Polynomials Using the Box Method × Description: "This worksheet is designed to bolster children's proficiency in expanding square polynomials using the Box Method. It provides ten engaging math problems that challenge students to expand square polynomials, demonstrating mastery of algebra concepts. Additionally, it offers flexibility as it can be customized according to learner needs, converted into flashcards for easy access, or utilized for distance learning." × Student Goals: Advance in Polynomial UnderstandingBy completing the worksheet, students develop an advance understanding of polynomials, particularly focusing on the expansion of square polynomials. They demonstrate mathematical skillset to solve complex mathematical problems and equations, thereby bolstering their proficiency in the subject.Master the Box MethodStudents become proficient in utilizing the 'Box Method' for expanding square polynomials. With practice, they establish their ability to apply this method across various problems and exhibit comfort in using it - an essential mathematical process.Enhance Problem-Solving SkillsThe worksheet directs learners to enhance their problem-solving skills. Students learn to apply their mathematical knowledge to address different types of polynomial problems, thereby boosting their analytical and logical thinking capacities.Apply Mathematical ConceptsAfter completing the worksheet, students should be able to relate theoretical mathematical concepts like multi-variable polynomials, to practical problem-solving situations. They gain hands-on experience that integrates theory with practice.Develop ConsistencyStudents learn the importance of consistency in solving these problems. They continually apply learned strategies in various scenarios, reinforcing these techniques and enhancing their overall mathematical competence.Increase ConfidenceThis worksheet aids in increasing students' confidence in tackling complex mathematical problems. It fortifies their ability to handle different types of polynomial equations, equipping them with the confidence to solve any given problem.Improve Computational SkillsBy solving these polynomial problems, learners boost their computational skills. This includes the simplification of algebraic expressions, carrying out arithmetic operations and dealing with variables and numerical values. Expanding Perfect Squares × Description: "This worksheet is designed to bolster the mathematical proficiency of children by focusing on the concept of expanding perfect squares. Featuring 20 diverse problems, examples include (x + 8)^2 , (x - 4)^2 , and (x +1)^2. Ideal for sharpening math skills, the worksheet can be customized for various learning methods. Transform into flash cards for interactive learning, or employ in remote syllabi to support distance education." × Student Goals: Factoring Perfect Square Trinomials × Description: "This worksheet is designed to enhance children's skills in factoring perfect square trinomials in mathematics. Comprising 20 interactive problems, it allows a deep dive into concepts like x^2 - 4x + 4, x^2 - 14x + 49, among others. Customizable and convertible into flashcards, this versatile tool is also perfect for distance learning, promoting self-paced education." × Student Goals: Strengthening Mathematical CompetenceAfter completing this worksheet, students should be able to demonstrate an improved understanding of the concept of perfect square trinomials. They will be able to identify and factorize perfect square trinomials swiftly and accurately, strengthening their overall mathematical competence.Enhancing Problem Solving SkillsBy tackling these problems, students develop their problem-solving skills. They will be able to take complex mathematical problems and break them down into more manageable parts, a skill that is critical in many aspects of life beyond mathematics.Building ConfidenceSuccessfully completing these problems builds confidence in mathematical abilities. It enables students to approach similar future problems with greater self-assurance, reducing math-anxiety.Improving Critical ThinkingFactoring perfect square trinomials requires critical thinking. After finishing this worksheet, students should be able to use their analytical skills to factorize such equations more efficiently, thereby enhancing their critical thinking.Developing PerseverancePerfecting the skill of factoring perfect square trinomials takes practice. By working through this worksheet, students develop the ability to persevere through tough tasks.Preparation for Advanced TopicsThe knowledge gained from completing this worksheet would be vital for students when they encounter more advanced topics in algebra and calculus where understanding of the concept of perfect square trinomials is required. Finding Rise using Similar Triangles 8ee6 × Description: "This worksheet is designed to teach math students the method of finding rise using similar triangles. It involves clear, visual diagrams that encourage problem-solving and application of key mathematical concepts. With 8 interactive problems to solve, varying in complexity, students are provided with a platform to practice and understand the theory. Ideal for distance learning, these tasks can be easily converted into flash cards for revision purposes. Additionally, the worksheet is customizable, catering to specific learning styles and paces." × Student Goals: Understanding of Similar TrianglesAfter completing this worksheet, students should be able to understand the concept of similar triangles effectively. They will be able to grasp how ratios and proportions can be applied in the realm of geometry, especially where triangles of identical angles but differing sizes are concerned. The students will recognize that the sides of similar triangles are proportional, thus enabling them to deduce missing measure when they are presented with partial information.Application of the Concept of RiseStudents will be able to apply the concept of 'rise' in mathematical problems. Rise is a critical concept in understanding geometric patterns and shapes, specifically triangles. As a result, students should be able to find the rise of various triangles using the principle of similarity. This crucial skill is applicable in various advanced mathematical problems and real-life scenarios, thereby enhancing problem-solving skills.Development of Problem-solving SkillsBy solving these problems, students will enhance their critical thinking and problem-solving skills. They will be forced to think beyond the box, analyze the given figures, and adapt strategies to fill in the missing information effectively. Over time, this process will instill in students the ability to apply mathematical principles in a broader context, enhancing their analytical skills.Understanding of Mathematical NotationThe worksheet will also help in understanding and interpreting mathematical notation accurately. It will boost the students' ability to read, comprehend, and execute based on the given notations, an important skill in mathematics.Boosting Mathematical ConfidenceBy successfully solving these problems, the students will gain confidence in their math abilities. Each problem solved will reinforce the comprehension of the concepts and instill confidence in their problem-solving capacity. This is of paramount importance for progressing in mathematics, a subject area that many students find daunting and challenging. Finding Missing Coordinates Using Similar Triangles 8ee6 × Description: "This worksheet is designed to help children understand and apply the concept of finding missing coordinates using similar triangles. It features eight engaging problems with visual representation via SVG graphics to guide kids in the problem-solving process. The interface is easily adaptable to flashcards and supports distance learning. Its customizable nature allows it to cater to each child's learning pace and style." × Student Goals: Rotating Around Axis × Description: "This worksheet is designed to improve children's understanding of geometric rotations in maths. It offers four engaging problems that focus on rotating shapes around the axis at the point (0,0) through various degrees. It not only strengthens spatial perception but can also be customized for individual learning styles, converted into interactive flashcards, or used in distance learning settings, advancing mathematical comprehension in a fun and accessible way." × Student Goals: Understanding of Rotational GeometryUpon successful completion of this worksheet, students will have a markedly heightened understanding of rotational geometry. They'll be able to articulate the fundamental principles involved in rotating a shape and effectively apply these principles to comprehend how changing the degree of rotation influences the final position of the shape. The students will be capable of performing geometric transformations autonomously and correctly, thereby enhancing their analytical thinking skills.Comprehending and Utilizing Coordinate PointsThrough working out these problems, students will acquire the ability to recognize and use coordinate points competently. When instructed to rotate a shape around a given point, they will understand how to interpret this point as the center of rotation. Students will be able to apply the correct rotation operation from the origin point to the provided coordinates by accurately finding the new position of the shape. By understanding and correctly implementing coordinate points, students will be increasing their spatial perception and numerical relationship skills.Mastering Positive and Negative AnglesThe use of both positive and negative angles in the problems will assist students in realizing the direction of rotation. They will comprehend that a positive angle signifies a counterclockwise rotation while a negative angle denotes a clockwise rotation. This will instill in the students a comprehensive concept of mathematical rules regarding angle rotation, further supporting their edification in algebra and advanced mathematical solutions.Enhancing Problem-Solving AbilitiesSolving these problems will aid in refining students' problem-solving capabilities. Not only will they have to figure out how to rotate a shape based on the degree given, but they also have to precisely anticipate the resulting position of the shape after the rotation. The process entails prediction, calculation, and verification of the result, contributing to their critical thinking and logical reasoning development.Boosting Confidence in Mathematical AbilitiesBy completing the given problems successfully, students will create a sense of academic fulfillment and heightened confidence in their mathematical abilities. Such success will encourage them to continue exploring more complex math principles, making them enthusiastic learners, and instilling a continuous drive for learning. Identifying Point of Intersection with Equations 8ee8a × Description: "This worksheet is designed to teach children the fundamental math concept of finding the point of intersection in equations. It comprises ten problems, increasing in difficulty, which aid in understanding how different equations intersect at a particular point. The format of the worksheet is simple yet engaging, making it suitable for conversion into customizable flashcards or even for incorporation into distance learning curricula." × Student Goals: Understanding the Concept of Intersection PointsAfter completing this worksheet, students will have an in-depth understanding of intersection points in a pair of linear equations. They will learn how to identify common points at which two lines intersect in a Cartesian coordinate system. This foundational knowledge is crucial in further graphing and plotting studies.Identification and Computation of Equations SkillsThe worksheet is designed to improve students' skills in identifying and computing mathematical equations. They will gain the ability to manipulate equations to correctly locate points of intersection. This will enhance their proficiency in algebraic computation and manipulation by handling numerical coefficients and variables.Problem Solving AbilitiesThis worksheet serves to sharpen students' problem-solving abilities. As it challenges them with multiple linear equations, it requires them to apply keen logical and analytical thinking in determining intersection points. This nurtures their solving abilities, which is critical not only in math but also in daily life problem-solving situations.Critical Thinking and Reasoning DevelopmentBy solving these problems, participants will bolster their critical thinking skills and mathematical reasoning. The task of pinpointing the intersection demands careful consideration and planning, promoting the development of logical thought processes and careful evaluation of calculated solutions.Self-evaluationCompleting this worksheet equips students with the capacity for self-evaluation. As they monitor and assess their progress, they will gain insight into their strengths and areas for improvement. This reflection is important in honing their learning strategies and maintaining a proactive attitude in personal improvement.Preparation for Advanced StudyThis worksheet provides a crucial foundation for more advanced mathematical studies. Understanding the intersection of equations prepares students for future topics, including calculus, geometry, and more complex algebraic topics, and is also applicable in various fields of physics and data analysis in higher learning disciplines. Moreover, the thinking approach nurtured through this activity is beneficial in various disciplines such as engineering, software development, and scientific research. Using Pythagorean theorem 8g7 × Description: "This worksheet is designed to aid children in mastering the Pythagorean Theorem, an important concept in math. It offers 12 problems illustrated with diagrams and step-by-step solutions, thereby enhancing comprehension. Subjects can customize the worksheet according to their skill level or convert it into flashcards, promoting active recall. Moreover, its adaptability makes it an excellent resource for seamless integration into distance learning programs." × Student Goals: Applying the Law of Cosines × Description: "This worksheet is designed to enhance children's understanding of the Law of Cosines, a crucial concept in geometry. It offers specifically four problems that incorporate graphics to explain how the law is applied in real-world scenarios. The worksheet is customizable to cater to individual learning styles and can be easily converted into flashcards or used for distance learning activities, providing flexibility in education. It's an excellent tool for developing problem-solving skills in mathematics." × Student Goals: Understanding Mathematical ConceptsUpon completion of the worksheet, students will have a robust understanding of the Law of Cosines, its application, and why it's a crucial part of trigonometry. They will be able to precisely explain the definition of the Law of Cosines, its application in solving real-world mathematics scenarios, and the relationship it holds with other trigonometric concepts.Problem SolvingThe worksheet provides a solid platform for students to practice and hone their problem-solving skills in geometry. With solved examples, students will learn how to derive and plug in values within the Law of Cosines formula to calculate unknown angles in a triangle. They will gain more practice in performing complex equations and familiarize themselves with various ways that mathematical problems can be approached and solved.Analytical SkillsStudents will enhance their analytical skills as they gain practice in deducing unknown variables in triangles using the Law of Cosines. They will be able to evaluate and analyze the specifics of mathematical problems, and derive appropriate solutions. Additionally, they will learn how to accurately break down a complex situation into smaller easily manageable mathematical portions.Skills in Geometry and TrigonometryThe worksheet will provide students with a deeper understanding of geometry, specifically the mechanics of triangles. They'll learn important skills such as calculating the lengths of a triangle using coordinates, which feeds into mastering the larger topics of geometry and trigonometry. They will be conversant with the aspects of triangle properties, majorly the relationship between its sides and angles.Critical ThinkingThrough the completion of the worksheet, students will develop their critical thinking abilities. They will be required to utilize a combination of their understanding of the Law of Cosines and their mathematical knowledge to solve the problems. This will enable them to challenge their critical and analytical thinking abilities in new ways, helping them enhance these vital skills. Examining Square Roots 8ns2 × Description: "This worksheet is designed to facilitate math learning by examining square roots. Featuring 20 problems, it offers children an engaging way to master this useful skill. Examples include root problems of 14, 84, and 63, graphically presented. The content can be customized to suit individual learning paces, converted into flash cards for quick, repetitive practice, or utilized efficiently for distance learning. This tools aims to make math, specifically the study of square roots, interactive and enjoyable." × Student Goals: Identifying Rational and Irrational Numbers 8ns1 × Description: "This worksheet is designed to engage children in math by examining the intriguing subject of identifying rational and irrational numbers. Featuring 20 progressively challenging problems, kids can decipher patterns in sequences like squared decimals or fraction bars. Perfect for remote learning, the content can be customized and converted to flashcards for enhanced learning experiences. This worksheet caters to diverse learning styles, reinforcing the abstract concepts of number types in mathematics." × Student Goals: Develop Number Classification SkillsUpon completion of the worksheet, students should possess a firm understanding of the key differences between rational and irrational numbers. They should be able to quickly identify and classify these numbers, therefore building a solid foundation for more complex math concepts down the line. This skill is not only essential in terms of advancing in math education but also enhances logical thinking and categorization skills, important tools for various dimensions of day-to-day life.Enhance Problem-solving AbilitiesCompleting this worksheet should enhance students' problem-solving abilities as they exercise their minds to differentiate between rational and irrational numbers. They would be refining their analytical thinking skills while identifying patterns and making calculated guesses. These skills are pivotal in all aspects of life, from making strategic decisions to thinking critically about different challenges.Boost Confidence in Mathematical ConceptsThis worksheet is designed to boost students' confidence when dealing with numbers in general and more specifically rational and irrational numbers. The more they practice identifying these numbers, the more comfortable they'll become at handling them. This increase in mathematical confidence can lead to better performance in classroom activities, tests, and further educational pursuits.Improve Precision and AccuracyThrough working on this worksheet, students should be able to enhance their precision and accuracy when dealing with numbers. This skill extends beyond mathematics, being applicable in science, technology, and even daily life situations where precision is needed. It encourages them to pay attention to every detail and meticulously examine each problem before judging it.Strengthen Mathematical VocabularyFinally, successfully identifying rational and irrational numbers will inherently improve the students' mathematical vocabulary. They will be more versed with mathematical jargon, which is key in understanding and communicating mathematical concepts effectively. Better mathematical communication leads to better comprehension and higher learning efficacy in the subject overall. Examining Square Root Relative Values 8ns2 × Description: "This worksheet is designed to enhance children's understanding of square root values in math. It comprises 20 problems, which challenge kids to deduce if the square root of a certain number will be closer to one value or another. Customizable to cater to individual learning curves, it can also be converted into flash cards, making it a diversely beneficial tool for distance learning. With the help of distinct visuals, it simplifies complex concepts, reinforcing learning in a fun, interactive way." × Student Goals: Finding Radicals on a Numberline 8ns2 × Description: This worksheet is designed to help children grasp the concept of finding radicals on a numberline. With 10 engaging math problems intricately plotted onto visual graphics, students are encouraged to explore mathematical dimensions independently. This customizable resource can be easily converted into flash cards for varied learning experiences or utilized in a distance learning setup, offering flexible educational support for diverse learning needs. × Student Goals: Understanding RadicalsUpon completion of the worksheet, students should have acquired a solid understanding of radicals and how they are represented on a number line. This foundational concept is crucial in the study of algebra and number theory, among other branches of mathematics. The learners will have learnt how to locate the position of a radical on a number line, a skill that broadens their mathematical perspectives and enables them to comprehend complex calculations involving these unique values.Develop Critical Thinking SkillsThe worksheet is designed not just to impart knowledge but also to foster critical thinking and mathematical reasoning. As the students tackle each problem, they are prompted to apply logical deduction, pattern recognition and strategic problem solving, all of which play a critical role in the development of their cognitive abilities. Solving these exercises helps in improving their problem solving skills and powers of logical reasoning.Boost Confidence in MathematicsSuccessfully finding radicals on a number line gives learners confidence in dealing with numbers. It serves as a stepping stone for more advanced mathematical concepts in their academic journey. As they solve each problem, they boost their confidence and fluency in mathematics, positively affecting their performance in other mathematical topics.Mathematical Communication SkillsThe worksheet also aims to improve students' ability to communicate mathematical ideas effectively. As students engage with these worksheets, they articulate their logic and explanations in finding solutions, sharpening their verbal and written communication in mathematical language. This effective communication is a valuable skill that helps in discussion or explanation of mathematical concepts to peers or even in future studies.Improve Concentration and FocusFinding radicals on a number line requires intense concentration and precision. Therefore, the worksheet offers an excellent opportunity for learners to improve their focus and attention to detail. As they endeavour to pinpoint the exact location of each number, their overall mental precision and sharpness are likely to improve, benefits that will extend beyond their math classes. Rewriting Using the Laws of Exponents 8ee1 × Description: "This worksheet is designed to bolster kids' understanding of the Laws of Exponents in math, featuring 20 customisable problems that require rewriting to solve. The problems revolve around multiplication, exponentiation and distribution laws, demonstrated in various complex scenarios. This tool can easily be converted into flashcards or integrated into a distance learning program for enhanced math learning experiences." × Student Goals: Solving Using the Laws of Exponents 8ee1 × Description: "This worksheet is designed to help children understand and master solving mathematical problems using the laws of exponents. Covering key concepts like power rules and fraction exponents, it offers ten problems of incremental complexity. Practical examples and customizable content make it suitable for various learning preferences. It's an adaptable resource that can be easily converted into flash cards or integrated into distance learning curriculum for effective teaching." × Student Goals: Understanding Exponent LawsAfter completing the worksheet, students should have a comprehensive understanding of the laws of exponents. This understanding not only includes the ability to accurately calculate the powers of integers, but also the capacity to apply the laws of exponents in a variety of mathematical contexts. Students will have strengthened their skills in transformative mathematical operations such as multiplication and division using exponents.Problem Solving and Analytical SkillsStudents will enhance their problem-solving abilities as they navigate through various exponent problems. They should become adept at analysing complex problems and executing steps required for solving problems that involve exponents. This includes both positive and negative exponents as well as zero exponents. Their analytical thinking capabilities will be sharpened, improving their mathematic proficiency in a wider sense.Conceptual Comprehension and ApplicationStudents should also develop deepened conceptual understanding. Grasping the fundamental concepts behind exponents will provide students with the ability to properly comprehend and apply these concepts to more complex mathematical equations. This understanding will lay the groundwork for future learning and application in more complex mathematical contexts.Mathematical VocabularyAfter completing the worksheet, students will be able to accurately and confidently use mathematical terminology related to exponents. The principles learned can help students better articulate mathematical processes and effectively use mathematical language. Comprehending this mathematical vocabulary will ease communication in mathematical contexts and aid in the comprehension of more advanced mathematical literature.Enhanced Numeracy SkillsCompletion of the worksheet will assist in the improvement of students' overall numeracy skills. It will build their competence in basic number manipulation and develop their understanding of mathematical structures. These skills are fundamental for mathematical growth and vital in day-to-day life and various professional fields. Solving with Squared and Cubed 8ee2 × Description: "This worksheet is designed to strengthen children's understanding of cubed and squared number manipulation through 21 interactive math problems. Offering a unique approach to mathematical resolution, students learn to derive solutions using examples such as x³ = 512, transitioning abstract concepts into a tangible format. This worksheet can be customized, converted into intriguing flashcards, or integrated seamlessly into a distance learning program, enriching the learning experience wherever your student may be." × Student Goals: Understanding of Mathematical ConceptsStudents will gain a foundational understanding of squares and cubes, including the ability to solve mathematical problems using these concepts. This understanding builds upon the fundamentals of multiplication, and allows students to further appreciate the complexity of mathematics.Problem-Solving SkillsThis worksheet enhances students' problem-solving skills by challenging them to solve both square and cube problems. These types of equations often require more thought, and as a result, students learn how to think critically and analytically. Examining problems from different angles and exploring the potential solutions can be a challenging but rewarding experience for students.Progression in Mathematical KnowledgeAfter completing the worksheet, students will have progressed in their mathematical knowledge. They will be equipped with the skills necessary to solve more complex equations involving squares and cubes. This progression allows students to continue developing their mathematical abilities, setting the stage for future learning in this subject.Building Confidence in Math AbilitiesThe successful completion of the worksheet will build confidence in each student's math abilities. By practicing and solving these type of problems, students will increase their competence and confidence in handling similar challenges. With increased confidence comes reduced anxiety towards math, which often leads to better performance in the subject.Preparation for Higher Level MathThis worksheet works as a stepping stone towards higher level mathematics. The ability to handle squares and cubes equations forms a part of many mathematical areas, including algebra. Consequently, successful completion of this worksheet prepares students for encountering more advanced mathematical problems in future.Mastery of Converting Mathematical ExpressionsStudents will acquire the mastery of converting mathematical expressions involving squares and cubes. Understanding the symbolic representation of these concepts and converting them into numerical values is an essential skill for students. Achievement of this skill provides a solid foundation for further exploration of mathematical concepts. Solving with Negative Powers 8ns2 × Description: "This worksheet is designed to help children understand and solve negative exponent problems in math. Covering substantial concepts like cube roots and their reciprocal counterparts, it provides 20 structured problems and solutions that foster analytical thinking. Adaptable to diverse learning needs, it can be transformed into flash cards to encourage interactive study. Equally valuable for distance learning, it's a customizable tool for enhancing mathematical knowledge and power solving skills." × Student Goals: Enhanced understanding of negative powersAfter completing the worksheet, students should have a strengthened comprehension of negative powers and their real-world applications. They should be able to swiftly identify the relationship between positive and negative powers, reducing complex equations in a simplified manner.Ability to solve advanced mathematical problemsThe worksheet empowers students to tackle complex mathematical problems involving negative powers. They should be able to carry out calculations faster, with increased confidence in arithmetic and problem-solving skills. This preparedness extends beyond just solving textbook problems, to facing larger mathematical challenges in advanced studies and competitive exams.Knowledge application skillsWith a thorough practice of the problems in the worksheet, students should be adept in applying the learned concepts to various scenarios. They should be proficient not only in recognizing when to use negative exponents but also proficient in implementing the rules of negative powers in distinct mathematical contexts to derive solutions.Improved number senseExecuting the problems in the worksheet will enhance the students' number sense, pushing them to comprehend the effects of negative powers on base numbers. This in-depth number sense can be extremely crucial in developing mathematical intuition necessary for advanced mathematical studies.Mathematical reasoning skillsThe worksheet encourages the students to think critically, promote logical reasoning and to derive different ways to approach a solution. It fosters an environment to build their mathematical reasoning and cognitive skills, which can be considered essential pillars for STEM education and career.Precision in handling large numbersStudents should gain mastery over handling very large numbers. The mathematical experience from the worksheet helps in dealing with numbers in higher powers or lower negative powers with ease, ensuring precise and accurate solutions every time. Finding Square and Cube Roots with Equations 8ns2 × Description: "This worksheet is designed to help children master finding square and cube roots through equations. Consisting of 16 math problems, it guides students with examples and allows a hands-on experience. The problems include division and addition equations, focused on square and cube roots. Its customizable format allows a tailored learning experience. Furthermore, the content can easily be converted into flashcards for better retention or used in distance learning settings. It's an invaluable tool to solidify foundational math knowledge in a fun, engaging way." × Student Goals: Enhance Mathematical KnowledgeUpon completion of this worksheet, students are expected to improve their theoretical understanding of mathematics, particularly concepts of square and cube roots. They will be able to understand the processes involved in finding square and cube roots, and how equations can be used in problem-solving. This will provide a strong foundation for tackling complex mathematical problems in the future.Strengthen Problem-Solving AbilitiesWorkouts on this worksheet are designed to enhance students' problem-solving capabilities. Solving a variety of problems not only fosters a deep understanding of the subject but also promotes analytical thinking and logic application. They will gain the ability to use mathematical concepts to solve real-world problems, which significantly aids in sharpening their practical application skills.Boost Computational SkillsCompleting this worksheet will improve students' computation skills. They will become efficient in performing addition, subtraction, multiplication, and division, important for everyday tasks. Regular practice of these operations will strengthen their numerical abilities and speed of execution, which is critical for solving calculations quickly and accurately.Increase Mastery of Mathematical OperationsGoing through this worksheet will enhance students' command over basic mathematical operations. They'll understand computation methods, the order of operations, and the rules guiding each operation. This knowledge will be beneficial for understanding higher-level mathematics and other science subjects.Promote Self-Study SkillsThis worksheet is also valuable in promoting self-study habits among students. Independently tackling problems and finding solutions can incrementally improve their self-learning and self-correction skills. This fosters an independent learning culture and encourages curiosity, both essential components of the learning process.Bolster ConfidenceFinally, successfully completing the worksheet will bolster students' confidence in their mathematical abilities. Overcoming challenging problems will instill a sense of accomplishment and increase their confidence, inspiring them to explore more complex conceptual challenges. The overall improvement in their mathematical skills will reflect in their academic performance. Rewriting Factors as Squares 8ns2 × Description: "This worksheet is designed to help children improve their math skills by rewriting factors as squares. It comprises 20 problems, varying in complexity and primarily focusing on multiplication. It is easily customizable for teachers or tutors to suit individual student's learning pace and requirements. Also, it can effortlessly convert into flash cards for more interactive learning or can be effectively used for distance learning. Engage your students and enhance their problem-solving abilities with this innovative teaching tool." × Student Goals: Understanding of Factors and Squares in MathematicsUpon completion of this worksheet, students should have gained a solid comprehension of factors and the special case of square numbers in the subject of mathematics. This formulates an essential foundation for the apprehension of more complex mathematical concepts, promoting familiarity with algebraic expressions and their simplification using the knowledge of factors.Problem Solving SkillsAs this worksheet involves numerous problems to solve, students are likely to enhance their problem-solving skills. The capability to address and surmount challenges is a highly transferable skill; even outside of mathematical contexts, this ability will prove itself as extremely valuable. Learning how to approach problems, evaluate them, and develop effective problem-solving strategies can foster cognitive development in young learners.Numeric Manipulation ProficiencyWith consistent practice and exposure on this worksheet, students should be able to achieve proficiency in numeric manipulation. This pertains to the capacity effectively to handle and manipulate numbers to reach desired outcomes. Strengthening this proficiency is a must for students aiming to excel in quantitative subjects, where numerical data manipulation is a common fixture.Improving Mathematical VocabularyThis worksheet serves as a tool for expanding mathematical vocabulary, especially in areas of factors and squares. After concluding the sheet, students should be familiar with terms such as square numbers and factors, enhancing their ability to comprehend more complex mathematical problems presented in future within these terms.Boost in ConfidenceBy devising strategies and solving problems in this sheet, students are expected to witness a boost in their confidence. This goes beyond just the confidence in solving math problems. Facing math problems, strategizing, and eventually solving them successfully invariably promotes a student's self-esteem and self-assuredness, a vital trait necessary for future academic endeavors. Simplifying Radicals 8ns2 × Description: "This worksheet is designed to aid children in mastering the concept of simplifying radicals. Through 20 carefully crafted problems using visually appealing SVGs, students practice simplifying examples like 'root 48', 'root 20', and 'root 54'. Thanks to its versatile format, this tool can be customized to suit individual needs, transformed into flash cards for quick revision, or incorporated into distance learning modules to enrich math education beyond classroom walls." × Student Goals: Knowledge EnhancementThrough the completion of this worksheet, students should be able to enhance their knowledge in math, particularly in the topic of simplifying radicals. The process of simplification is a fundamental concept in algebra and can greatly aid in numeration and ordinality.Problem Solving SkillsAs students go through the worksheet, their problem-solving skills will be bolstered as they tackle each task. They will also devise strategies to approach and solve complex mathematical problems, aiding their mental flexibility and adaptability.Mathematical FluencyThe practice exercises contained in the worksheet should help students increase their mathematical fluency. This refers to the ability to recall and apply mathematical facts rapidly and accurately - an essential skill required for future complex computations and equations.Analytical SkillsCompleting the worksheet should bolster students' analytical abilities. They will have the opportunity to interpret mathematical problems, select the best methods for solution, and rationalize the steps involved in the process of simplification. This ultimately helps them in understanding the structure of mathematics.Mathematical ConfidenceBy successfully completing the worksheet, students will gain self-confidence in their mathematical abilities. Each solved problem reinforces their understanding and skill, making them less likely to be intimidated by the subject of math. This confidence will carry over to other areas of study, benefiting their overall academic performance.Preparation for Advanced TopicsThis worksheet serves as a stepping stone towards more advanced mathematical topics. The skill of simplifying radicals is a foundation to many branches of mathematics like geometry, trigonometry, calculus etc. Therefore, mastering it will help students to be well-prepared for these advanced topics in their future studies. Expressing Numbers using Powers of 10 8ee3 × Description: "This worksheet is designed to aid children in understanding and expressing numbers using powers of 10. Featuring 20 engaging math problems, it enhances numerical proficiency by presenting real-world examples like 300, 60,000, and 1,000. This adaptable resource can be customized, converted into flashcards, and seamlessly incorporated into distance learning environments for an enriched educational experience." × Student Goals: Understanding of Powers of 10After completing this worksheet, students should be able to demonstrate a strong grasp of expressing numbers using powers of 10. This foundational numeric competency empowers children to handle and manipulate large and small numbers effortlessly. With such a skill, students will become more comfortable dealing with scientific notation, significant figures, and exponential calculations.Improved Problem-Solving SkillsThe worksheet enhances students' problem-solving abilities by encouraging them to think critically and logically when expressing numbers as powers of 10. This skill is not only significant for their mathematical proficiency but also for enhancing the cognitive abilities necessary for other STEM subjects. The improvement of analytical thinking through the fundamental lessons of this worksheet will serve as a cornerstone for advanced mathematical problem-solving techniques.Confidence in Working with NumbersStudents will gain the confidence and competence required to fluently work with large numbers and perform complex calculations. This will enhance their self-assuredness and ease when working with mathematical problems in the future. When students are no longer intimidated by large numbers, they are more likely to excel in subjects requiring quantitative reasoning, such as physics, chemistry, and advanced mathematics.Preparation for Advanced TopicsThis worksheet sets students on a course for success in more advanced mathematical topics. Gaining expertise in expressing numbers as powers of 10 serves as a launch pad for learning more intricate algebraic expressions, calculus, and even computer science. In turn, these capabilities give students a competitive edge in the long run.Practical Application of MathematicsFinally, students will develop a better appreciation for the real-life applications of mathematics. By understanding the principle of expressing numbers using powers of 10, children will be able to relate this concept to real life scenarios such as measuring distances in science or handling large financial data in economics. This real-world relevance promotes long-lasting learning and encourages student engagement in math and science education. Finding Relative Value with Powers of Ten 8ee3 × Description: "This worksheet is designed to help children grasp the concept of powers of ten and their relative values. Featuring nine varied math problems, it enhances their skill in evaluating and comparing large numbers. The customizable worksheet can be adapted for flash cards, providing additional convenience for revision. Moreover, it's perfectly suited for distance learning, seamlessly accommodating modern educational settings. A great tool to strengthen foundational math skills while keeping learners engaged." × Student Goals: Understanding Powers of TenAfter completing this worksheet, students should possess a thorough understanding of the mathematical concept of powers of ten. They will be able to distinguish between larger and smaller numbers using powers of ten, determine the relative value of numbers, and confidently read, write, and comprehend mathematical expressions involving powers of ten. This proficiency is foundational to many areas of mathematics and will help them in advanced topics such as scientific notation, exponents, and logarithms.Solving Mathematical problemsStudents will gain the ability to correctly solve mathematical problems involving powers of ten. They will be equipped with strategies to calculate large numbers without a calculator, and effectively use factors of ten to simplify computations. This ability will not only enhance their mathematical skills but also aid in developing cognitive abilities like problem-solving, critical thinking, and analytical reasoning.Applying Mathematical ConceptsAn improved understanding of powers of ten will also enable students to connect this mathematical concept to real-world applications. They'll be able to apply their learning in practical situations, such as in scientific measurements and calculations involving large or small numbers. This will facilitate them to appreciate the relevance and utility of math in everyday life.Confidence in MathematicsSuccessfully tackling the worksheet will likely increase students' confidence in their mathematical abilities. By comprehending and solving problems involving powers of ten, they'll gain the confidence to tackle other complex math concepts and problems. This sense of achievement can have a positive impact on their overall attitude towards learning and academics.Preparation for Advance StudiesA keen understanding of the principle of powers of ten is necessary for more advanced studies in mathematics, especially subjects dealing with large and small figures. By mastering these topics early on, students will be better equipped for their future educational journey, particularly in STEM disciplines (Science, Technology, Engineering, Mathematics). Multiplying with Scientific Notation 8ee4 × Description: "This worksheet is designed to enhance a child's understanding of multiplying with scientific notation, a crucial concept in math. It offers ten challenging problems, instances of which involve the multiplication of large numbers presented in scientific notation format. With the ability to adapt to various learning methods, this resource can be customized into flashcards or incorporated into a distance learning curriculum to make complex math absorption simpler and more engaging. Perfect for mastering the art of mathematical computation!" × Student Goals: Master Scientific Notation OperationsBy completing this worksheet, students should have a solid grasp on performing multiplication operations using scientific notation. They will build an understanding of its principles, conventions, and real-life applications, enabling them to handle large or small numerical values effectively within various mathematical contexts.Develop Problem Solving skillsWorking through each problem exercises critical thinking and problem-solving skills. By breaking down complex calculations into manageable parts, students will be developing strategies that enable them to tackle complicated math problems. They will become proficient in mathematical reasoning and logic, skills that are fundamental in numerous fields of study and everyday life scenarios.Enhance Number SenseThrough this worksheet, students will enhance their sense of numbers. They will gain an understanding of the meaning and size of exponents and their effects on the base numbers. This increased understanding of numbers and the relationships between them will aid students in more advanced mathematical topics and real-world applications.Improve Mathematical FluencyThis worksheet will serve as a platform to improve mathematical fluency. Students will not only practice their computational skills but will also improve the speed and accuracy with which these calculations can be completed. This improvement will influence their confidence and efficiency in handling larger, more complex sums in higher levels of study and standardized tests.Reinforce Fundamental ConceptsMultiplying scientific notations involves key mathematical principles such as multiplication, rounding, estimations, and number properties. By working on this worksheet, students will revisit and reinforce these concepts, helping them to form a solid mathematical foundation that will support advancement to more high-level mathematical topics.Prepare for Advanced TopicsScientific notation is a key concept in many advanced mathematical and scientific fields such as chemistry, physics, and engineering. Completing this worksheet gives students the tools they need to engage these topics in the future confidently. They will be well-prepared to handle complex calculations and processes that frequently involve numbers in scientific notation. Filling Factorial Table × Description: "This worksheet is designed to help children learn and practice the concept of factorials in mathematics by filling a factorial table. It features 10 problems, each designed to teach the stepwise process of calculating factorials. As a customizable resource, it can be transformed into flashcards for interactive engagement or adapted for distance learning programs. Ideal for boosting math skills in an enjoyable, accessible manner." × Student Goals: Understanding FactorialsAfter completing the worksheet, students will be able to grasp the concept of factorials, an important element in mathematics. They will clearly understand that the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. This foundational knowledge opens doors to a multitude of mathematical disciplines and concepts, strengthening their understanding of the subject as a whole.Problem Solving SkillsThe worksheet is designed to enhance problem-solving skills. As students work through each question, they will have to engage logical thinking and apply the theory of factorials to formulate the correct answers. As a problem-solving exercise, it will help students develop their analytical skills and learn to strategize their way through math problems.Comfort with Large NumbersWith numbers escalating quickly in factorial problems, students will become more accustomed to working with and understanding large numbers. This is vital in various mathematical disciplines and can help with real-life applications. Understanding the scale of larger numbers and their behavior when multiplied can significantly extend a student's confidence in handling big figures.Working IndependentlyAs students work their way through the worksheet, they will learn to work independently, exercising individual thought and encouraging self-study habits. This promotes self-reliance and personal accomplishment in their learning journey.Building Basic Math FoundationsThrough this worksheet, students will turn theory into practice to reinforce their understanding of factorials, which is part of the broader mathematical principles. Consequently, they will attain a stronger grasp of other mathematical areas where factorials play a role, such as permutations and combinations, algebra, and even calculus.Preparation for Advanced ConceptsLearning factorials primes students for understanding more complex mathematical concepts and operations in the future. After completing the worksheet, they will be more prepared for higher-level studies that tackle topics such as statistics, probability, algebra, and combinatorics, where the use of factorials is a fundamental component.	{"url":"https://teachingsheets.com/algebra-worksheets","timestamp":"2024-11-14T07:40:21Z","content_type":"application/xhtml+xml","content_length":"426830","record_id":"<urn:uuid:0374ca36-82c7-441b-a19b-a5eb45fa47a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00229.warc.gz"}
Welcome to Dave's Shop newsletter. Wow! Another new year is upon us. All the best for 2008! What's New Since we were late in getting our November newsletter out, we thought you deserve the December one on schedule. Dan also thinks we should put out a newsletter twice a month from now on. Your comments would be appreciated. I'll see how it goes. Ask Away! Here are the questions and my answers for December. Hi Dave, I need some advice on making a frame to hold up a bathroom sink. Currently I have a small sink in 3 sided alcove in a small bathroom. I'd like to take out this sink and replace with an larger sink, a type from an older era(30's, 40's, 50's). The outer dimensions of this larger sink are 24" wide and 21" deep. On the front underside side the sink bowl leaves a minimum of 1 3/4" between the bowl and the sink edge. The alcove is 25 1/2" wide and the frame that's holding up the current sink is not worth saving. I have a bunch of 2x4 lumber and I'd like to make something with that that will hold the sink up and that I can slide into this alcove. The only visible part will be the front and I could make a front panel with a door out of some nice wood. The top would also be visible but only a very narrow strip on either side and on the front. Could you lay out some basic ideas for me? Thanks, John Hi John, Here is a drawing: You can use your 2x4s up as shown on the drawing to support the top at the back and the sides and down the front sides for the hinges. The base can also be 2x4. For the doors and trim use a melamine or plywood finish of your choice. You should make the cabinet in place, rather than a slide in unit. That is install the 2x4s one at a time, according to the drawings. Hi Dave, I have found your site extremely informative and helpful for several years now.. I now have a question that I can't find the answer to on your site, so here goes. I have old solid wood doors that I want to use as double doors on a storage shed. They are 31 3/4 inches wide. How wide should I make the door frame to have sufficient clearance to use these as a double door? My logic says 64" isn't quite enough clearance. Hi Jim, Thanks for the nice comments. Yes, you've been with us for awhile - April 15th, 2005!! I would go with 1/8 between the jambs and in the center, giving 3/8. So make the jamb 63 7/8 or a touch (1/32) larger. If you are going to swing the doors to the outside, have the active door installed with an astragal which will keep the weather out. Our dictionary has an astragal drawing: http://daveosborne.com/construction-dictionary/construction-definitions.php For any other question, don't hesitate to ask. We are renovating an old house that has a septic system. The pipe to the septic is cast iron. How can we change this pipe to plastic pipe. It is a different size and we need to replace from the septic tank to the house. This is an unusual problem since the septic is above ground where the pipe enters and is inside the house. I know don't say it. WE will eventually have a new system put in but not until next spring. Right now we have to hook up 2 new bathrooms to the old system and don't know what to use to go from abs to cast iron. Happy new year! Actually, this is not a big problem. The cast iron pipe hub should be either cut off with a reciprocating saw with steel blade or by renting a large cutter for cast iron, if you have the room. I have done this a few times using the recip saw. Just a warning - cast iron pipe is very hard, but also very brittle. It can be broken with a hammer. Once the pipe is cut off a "mission" fitting, a rubber fitting that slips over the pipe and fastened to the pipe with stainless steel clamps, is used between the cast and the abs or pvc. Here is a pic of one: These fittings are generically called Mission, but Fernco is also a brand in my area. Your retailer will know which one you need and the size of bushing if needed, too. Hope this helps, Dave, thanks for the help getting my son-in-law set up with his new Membership. Now for my question! I have a room 13 X 33, and I want to put down hardwood floor to match the rest in the older part of the house is all refinished red oak, 3/4" X 2 1/2" strips. Can I lay this diagonally, and also install a boarder using some walnut floor strips and how would I do this. If I have to lay it, as it is, to match the living room. I'll have to lay it cross wise, or 13' foot width, enlieu of the 33' lenght. One other thing, in your answer, could you work up a drawing, and attach to the email back please. I'm one who likes the movie, in lieu of reading the book, if ya know what I mean. Hi Brandy, Happy new year. The type of flooring you describe is end matched. This means the ends of the boards, as well as their sides have either tongue or grooves. This enables the floor layer to design his layout in many ways. Here are some drawings for different examples of designs in your particular case: The arrows show the direction of the boards. This is the floor in our dining room. Our nephew did this for us. He used 1" black walnut to make the border with 2 1/4" red oak. The diagonal pieces all have a groove with a spline that he made from hardwood scraps. This is a lot of tedious work. Well, Brandy, I better get this out to you before you give up on me. Sorry for the delay in my reply. Christmas took priority and visiting with family, which should be number one, right? This email just touches on all the designs you can do with corners. I've given just the basic ones. So if you want to run any ideas by me, don't hesitate to send them. All the best for 2008, Love your web site Looked at available sheds at lumber yards and decided to build our own ordered a couple of sets of shed plans from you printed off things like rafter cuts and soffit instructions came up with our own design ideas - like windows from Plexiglas - ridge vents - wire hardware cloth nailed to outer frame and buried 6" deep to keep out critters under the foundation. - painted inside of roof sheathing and texture 111 white - prior to nailing up- lot easier and adds brightness. - ramp made from left over pressure treated flooring - studding started in the middle of the walls and went out 16" with 20" pockets on each end - also made pockets in front and back walls to drop in (2) 8' 2x4 so as to provide a rack for Kayaks for the winter storage. - lots of pocket wells in the stud bays for tools - added filler pieces between studs top extend the the depth of the back - Used the same colors as the house to make part of the complex - window boxes and xmas wreath Finish off the project - should have added power - maybe next year The result was an 8 x 14 shed which includes the best of all ideas - my son really enjoyed it as I did. we think it is pretty special - already had a couple of folks taking pictures so they can use for ideas. Thanks again for your help Bob Davis Thanks, Bob, nice job. When Dan gets back from vacation, I'll get him to put the pics on our site, if you like. Dan has Bob's pics on our site now, checkout Our Members' Photos page: http://daveosborne.com/dave/photos/index.php Share Bob's pride and satisfaction of doing it yourself with the help of our website: http://daveosborne.com Hi, Dave it's me Ruben. I had a question for you. I wanted to know if you could explain to me how to figure out how to layout for an uncentered ridge. I have included a detailed pic to help explain what im talking about. It's been awhile since I have done it and I forgot the math formula for it. The way I remember doing it before was that I would add the ratios together. 8/12 3/12. But I forgot the order to do it in. It was the easiest way to do it, but it's completely slipped my mind now. Hi Ruben, I had to enlist help from Dan for this question. He says: The common number between the two slopes is the rise in the center, which is the same from both triangles, of course. So, we just have to solve for the horizontal line of each triangle. Let's say that b is the height of the roof from the horizontal and a is the first horizontal section to the left of the peek and d is the right horizontal section. We know that a + d = 26 (formula 1), or a = 26 - d (formula 1a). We also know from the slopes that a/b = 12/8 (formula 2) and d/b = 12/3 (formula 3). This is the ratio you are talking about: a is to b as 12 is to 8. If we solve for b (in formula 2), b = 8a/12 and (in formula 3) b = 3d/12. Since both equal b, they both equal each other, so 8a/12 = 3d/12, which simplified is a = 3d/8. We then solve for a, so a = 3d/8 and then plug that into formula 1 above, so: 3d/8 + d = 26 or 3d + 8d = 26 X 8 or 11d = 208 or d = 18 10/11 Now solving for a using formula 1a means a = 26 - 18 10/11 or 7 1/11 If you understand how to derive this, you don't need to memorize the general formula, which you should be able to work out from the formulas above. After all this, you know that length a in the diagram above is 3/8th's the length of b (from the slopes), so you can quickly write down (from formula 1): 3d/8 + d = 26 or d = 8 X 26 / 11 giving us the general formula: The length of the longer side is the total length of span times the rise of the shorter span divided by the total of both rises. The length of the shorter side is the total length of the span minus the length of the longer side. Merry Christmas, Ruben, Dan and I would like to wish you all a very happy, safe and peaceful new year. We hope that 2008 is your best year ever. < previous next >	{"url":"https://daveosborne.com/newsletters/0712.php","timestamp":"2024-11-09T06:33:06Z","content_type":"text/html","content_length":"33260","record_id":"<urn:uuid:42963b42-1789-44fe-96dd-ef9f24489824>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00743.warc.gz"}
Heap Sort Algorithm In Python \| Sorting Algorithms - Copyassignment In Part-1 of the heap sort algorithm, we have discussed how we can represent a tree in array format, what is a heap, types of the heap (max-heap & min-heap), and then how to insert an element in max-heap. Now, In this section, we will see the Heap Sort Algorithm in Python and how it works with an example, then we will discuss the time complexity and space complexity. Afterward, In the end, we will compare heap sort with merge sort and quick sort. If you have not studied Part-1 topics it will be difficult for you to understand the heap sort. That’s why it is not suggested to skip those What you’ll learn? 1. Working of Heap Sort with example 2. Heap sort algorithm for sorting an array in ascending order 3. Implementation of heap sort in python 4. Time complexity 5. Space complexity 6. Does heap sort an in-place algorithm? 7. How commonly we use heap sort? 8. Features 9. Comparison between heap sort quicksort and merge sort Working of Heap Sort with example In Heap Sort, the given input array is first converted to a Complete Binary Tree. Then, a Heapify() function is called that changes the Complete Binary Tree to a Max Heap. It is being done so that the largest element from the array can be obtained easily. Now, once a Max Heap is created, then a swap operation between the Node and the element at the lowest level of the heap is performed. After swapping, we get the largest element of the array at the lowest level of the heap, and then a deletion operation is performed. In this deletion operation, the node being at the lowest level i.e., the largest element of the array is removed. The element being removed is inserted in a Queue. Since we know that in Heap Sort, we also require a Queue data structure. The use of this queue is to hold the removed elements pop then back in the form of a sorted array. After deletion, the resultant elements will again undergo Heapify() function and all the above mentioned steps will be performed again. Let us understand with an example Input : 70 27 12 8 68 96 34 Output : 8 12 27 34 68 70 96 Create a Complete Binary Tree for the given array and then convert that complete binary tree to a max-heap by using the function heapify() Now what we will do, we will swap the root node (96) with the last node that is 34 and after swapping we will remove the element 96. Now as you can see it’s not a max-heap as it is violating the condition so again we will perform the Heapify() function to change this to a max-heap. And then follow the same procedure as we did earlier. We will swap the root node with the last node then after swapping we will remove that last and check if it’s a max heap or not. Resultant Queue – From the Queue, shown above, elements will pop one-by-one, giving us the sorted sequence of array elements (in ascending order), which is the desired output. This is how, Heap Sort works. Algorithm of Heap sort Step1: Create a max-heap from the given array. Step2: In a max-heap, largest item is stored at the root of the heap. Replace it with the last item of the heap and reduce the size of heap by 1. Finally, call heapify() to heapify the root of the tree. Step3: Go to Step 2 while size of heap is more than 1. Python Code # heapify tree rooted at index i def heapify(arr, n, i): largest = i # In max-heap largest is at root l = 2 * i + 1 # left child index = 2i + 1 r = 2 i + 2 # right child index = 2*i + 2 # See if left child of root exists and > root if l < n and arr[i] < arr[l]: largest = l # See if right child of root exists and > root if r < n and arr[largest] < arr[r]: largest = r # update root if required if largest != i: arr[i],arr[largest] = arr[largest],arr[i] # swap # Heapify the root heapify(arr, n, largest) # heap sort definition def heapSort(arr): n = len(arr) for i in range(n // 2 - 1, -1, -1): heapify(arr, n, i) # extracting elements for i in range(n-1, 0, -1): arr[i], arr[0] = arr[0], arr[i] # swap heapify(arr, i, 0) arr = [ 12, 11, 13, 5, 6, 7] n = len(arr) print ("Sorted array is") for i in range(n): print ("%d" %arr[i]) Time Complexity of Heap Sort As we have discussed in the previous section, the heap sort algorithm uses two different functions. First is the Heapify() function. Initially, we have used Heapify() to build a max-heap out of the complete binary tree. After that, we have used it after every delete operation, so that we can get the largest element. Now, the time Complexity for Heapify() function is O(log n) because, in this function, the number of swappings done is equal to the height of the tree. The second function which heap sort algorithm used is the BuildHeap() function to create a Heap data structure. Time Complexity of BuidlHeap() function is O(n). Thus, the combined time complexity for the heap sort algorithm becomes O(n log n) for all three cases. Thus the time complexity of heap sort for the best case, average case, and worst-case O(nlogn). Space Complexity of Heap Sort For an algorithm, space complexity is defined as the memory space occupied by it to run and execute all its operations. For Heap Sort Algorithm, Space Complexity is O(1) because it involves constant swapping of elements for which the space required is equivalent to only one element. Is heap sort an in-place algorithm? Yes, Heap Sort is an in-place sorting algorithm because it does not require any other array or data structure to perform its operations. We do all the swapping and deletion operations within one single heap data structure. How commonly we use heap sort? We don’t use heap sort so often. As the programming of heap sort is a little bit complex so we don’t prefer it. In case, we have a choice we generally prefer Merge Sort and Quicksort. But remember if we already have a heap data structure in our program then we will go for heap sort. 1. As we have learned, in heap sort we create a max heap, through which we can find the maximum element of the array. In a similar way, we can also create a min-heap by modifying the algorithm a little bit. This can be very handy when we need to perform some arbitrary searches. 2. In the case of the worst-case data set, heap sort serves a better purpose than quicksort because of its time complexity as discussed earlier. 3. It is an in-place algorithm. 4. When we have a heap data structure in our program, then we prefer heap sort in this scenario. Comparison between heap sort quicksort and merge sort Quicksort, Merge sort and Heap sort perform very differently when we take different and large data sets because if we take small data sets then the execution will be very fast and no marginal difference will be observed. If the data sets are large in size, then it will affect the memory space and size and we will be able to easily differentiate which one is better among the three. However, the above-mentioned sorting algorithms have a common time complexity i.e., O(n log n). But Quicksort has a time complexity of O(n^2) in the worst-case whereas both Heap sort and Merge sort have a time complexity of O(n log n) even in the worst case. Quicksort and Merge sort both work on a recursive algorithm or you can say they use recursion, on the other hand, there is no such recursive algorithm in Heap sort. In Quicksort, since we select a pivot element randomly, thus, it sometimes serves as the fastest sorting algorithm if we choose a favorable data set. Whereas in the case of linked lists, the merge sort serves the best. Check this video for more: Credits to nETSETOS Thanks for reading! If you found this article useful please support us by commenting “nice article” and don’t forget to share it with your friends and enemies! If you have any query feel free to ask in the comment Happy Coding! You may also read Introduction to Searching Algorithms: Linear Search Algorithm Bubble Sort Algorithm In Data Structures & Algorithms using Python Merge Sort Algorithm in Python Sorting Algorithms and Searching Algorithms in Python Shutdown Computer with Voice in Python	{"url":"https://copyassignment.com/heap-sort-algorithm-in-python/","timestamp":"2024-11-04T03:55:29Z","content_type":"text/html","content_length":"79513","record_id":"<urn:uuid:1096468d-51d1-4b02-a70d-138b7196d1f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00721.warc.gz"}
Categorical proposition From New World Encyclopedia The categorical proposition is a basic concept in Aristotelian or traditional logic (also sometimes called syllogistic or categorical logic). Aristotelian logic, albeit with substantial revisions over the course of almost 2,000 years, was accepted as the definitive logical system until developments in the late nineteenth century with Gottlob Frege and Bertrand Russell supplanted it and ushered in modern mathematical logic. Logic is the study of correct, or valid inferences. Aristotle’s logical system is based upon a form of argument called a syllogism. The syllogism is an argument with two premises and a conclusion that follows from these premises. Each of the three propositions (i.e. two premises and a conclusion) in a syllogism is a categorical proposition. A categorical proposition is a type of proposition that uses the logical expressions "all," "some," "is," and "is not," to link "terms," which refer to some set, class or kind. This reference to categories, sets, or classes, is why they are called categorical propositions. An example of a categorical proposition is “All whales are mammals.” Aristotelian logic regards four basic types of categorical propositions as lying at the heart of all correct reasoning. These are the universal affirmative proposition—“All S is p,” the universal negative proposition—“No S is p,” the particular affirmative proposition—“Some S is p,” and the particular negative proposition—“Some S is not p.” Defining propositions Understanding the concept of a categorical proposition requires some discussion of the notion of a proposition. A proposition is usually defined as a thought or content expressed by a sentence, when it is used to say something true or false. Propositions are, roughly, thoughts about how things are, and are appraisable as true or false depending on whether the thought corresponds to how the world it. Consider the following example: “The black dog bit the white rabbit.” This sentence expresses a proposition for it makes a specific claim about the world, which may or may not be true. One reason for distinguishing sentences from propositions is that not all sentences are appraisable as true or false. Commands (e.g. ‘Shut the door’) and questions (“Is the door open?”) are perfectly legitimate ‘’sentences’’ and do not say anything about how things stand in the world. For this reason commands and questions are sentences but not propositions. Another reason for distinguishing sentences from propositions is that the very same thought, i.e., proposition, may be expressed in many different ways and in a variety of different languages. Propositions are the sorts of things that can be (e.g.) thought, believed, asserted, doubted, mentioned, and imagined. For example, one could think to oneself, “The black dog bit the white rabbit”; or one could ‘’assert’’ it; or one could doubt it by saying “I doubt that the black dog bit the white rabbit.” Or taking another example, the sentence “It seems quite unlikely that a man will land on Mars by 2009” expresses the proposition "a man will land on Mars by 2009" although the speaker does not commit to the truth of that proposition. Rather, the speaker doubts whether the state of affairs represented by the propositions will ever obtain. Propositions are sometimes identified with statements or judgments, but it seems best to keep these separate. Assuming that the expressions "statement" and "judgment" are interchangeable, we can say that people make statements when they assert propositions. Making a statement is essentially adopting a certain attitude toward a proposition. A statement consists of (1) a thought or meaning called a proposition and (2) the speaker or writer’s endorsement of the proposition (the assertion). So all judgments assert propositions but not all propositions are asserted (e.g. A proposition which is doubted is not asserted). Categorical propositions We have now considered the notion of a proposition in general. A categorical proposition is a proposition of a special sort. It is a proposition with two [1] "terms," one of two [2] "copulas," and one of two [3] "quantifiers." Explanation of each of these is as follows. Categorical propositions contain two "terms." Terms are the constituents of propositions, and not whole propositions themselves. A term picks out a set or class of objects, either real or imagined. Examples of terms include chickens, people, Martians, dogs, and carnivores. The term of a categorical proposition picks out a group of things. This group of things is called a set, or a class, or a category. The (groups of) objects that the term picks out do not have to really exist in our world. So the term "Martians" is perfectly legitimate even though Martians don’t actually exist. A categorical proposition is made up of two terms. The first term, which occurs in the subject position, is called the minor term. The second term, which is occurs in the predicate position is called the major term. Categorical propositions admit only one verb, and this the verb "to be." The verb "to be" is called a copula. For example, the sentence "The dog is black" employs the copula. In a categorical propositions, the copula links the subject term with the predicate term. In other words, it links up two terms, which each pick out categories of objects, with one another. The term, "whales," may (e.g.) be linked with another term, "mammals," in the proposition, “all whales are mammals.” In Aristotelian logic, the negation of the verb "to be" came to known as the "negative copula." So when one says, “the dog is not black,” one employs the negative copula. Ultimately, it makes no difference whether we say that there are two copulas, one positive and one negative, or only one copula, which is negated or not negated. One should adhere to the convention, which says that there are two copulas, one positive and one negative. Categorical propositions are said to have a "quality" and a "quantity" (the notion of quantity will be discussed in a moment). The quality of the categorical proposition is determined by the copula. If the copula is negative then the proposition is said to be a negative proposition; if the copula is positive the proposition is said to be an affirmative proposition. All Categorical propositions contain one (and only one) of two quantifiers. A quantifier, as the name suggests, specifies the number of a given class. There are only two quantifiers. The first quantifier is called the "universal quantifier," usually represented by "all" or "every." The universal quantifier picks out every member of a particular class, such as "all men," or ‘all whales’. The second quantifier is the existential quantifier, usually represented by ‘some’, or ‘at least one’. The existential quantifier picks at least one member of the class, such as ‘some men’ or some ‘whales’. Every categorical proposition is said to have a quantity. The quantity of the categorical proposition is either universal (all, every) or particular (some). Putting the concepts together in categorical propositions Now that the meaning of the components of a categorical proposition has been considered, it is time to see how they operate together. Here are some examples of categorical propositions: All men are mortal beings. Some chickens are dangerous creatures. Some roses are not flowers. These examples illustrate the basic form of the categorical proposition. Each involves a quantifier, two terms (i.e. the "subject" and the "predicate"), which are linked by a copula. [Quantifier] + [TERM 1] + [copula] + [TERM 2] Types of Categorical propositions Two quantifiers (“all,” “some””) and two copulas ("is," "is not") can be combined in only four ways. In other words, there are only four basic forms of categorical proposition. The quantity of the categorical proposition is either universal ("all," "every") or particular ("some"). The quality of the categorical proposition is either affirmative ("is," "are") or negative ("is not," "are not"). Two of the basic forms are universal propositions (i.e. they say something about a whole domain). Universal propositions use the universal quantifier. One of the universal forms is an affirmative statement (i.e. it affirms something about the subject; and one of the universal forms is a negative statement; i.e. it denies something about the subject). The other two categorical forms are particular propositions, (i.e. they say something about particular individuals in a domain). Particular propositions use the existential quantifier. Particulars also come in affirmative and negative form or quality. In the Middle Ages each of these four basic forms of categorical proposition came to be called by first four vowels "A," "E," "I," and "O." This practice is continued in many logic books even today, so that you will often see (i.e.) universal affirmative propositions called "A" propositions, and so on. The following table will clarify this further. Universal Affirmative A. All A are B [Universal affirmative proposition] A. All {term} are {term}. All [dogs] are [carnivores]. Universal negative E. No A are B [Universal negative proposition] E. No {term} are {term}. No [police officers] are [mammals]. Particular affirmative I. Some A are B [Particular affirmative proposition] I. Some {term} are {term}. Some [soccer players] are [kangaroos]. Particular negative O. Some A are not B [Particular negative proposition] O. Some {term} are not {term}. Some [pop stars] are not [drug addicts]. There are logical relations between the categorical propositions such that when these propositions are combined, categorical syllogisms are generated. The categorical syllogism is an argument with two premises and a conclusion that follows from these premises. Each of the three propositions (i.e. two premises and a conclusion) in a syllogism is a categorical proposition. The following is an example of a categorical syllogism: (1) all chickens are birds; (2) all birds are feathered creatures, therefore, (3) all chickens are feathered creatures, is a categorical syllogism. (See categorical logic for more on the concept of a syllogism.) ISBN links support NWE through referral fees • Copi, Irving M., and Carl Cohen. Introduction to Logic (12th ed.) Prentice Hall, 2004. • Hondereich, Ted, (ed.). The Oxford Companion to Philosophy. Oxford and New York: Oxford University Press, 1995. • Hurley, Patrick J. A Concise Introduction to Logic. 9th edition. Belmont, CA: Wadsworth/Thompson Learning. • Johnson, Robert M. Fundamentals of Reasoning: A Logic Book. Belmont, CA: Wadsworth. (Latest is the 4th edition.) • Lewis, C., and C. Langford, Symbolic Logic. 1932. Dover reprint, 1960. External Likns All links retrieved November 30, 2023. General Philosophy Sources This article began as an original work prepared for New World Encyclopedia and is provided to the public according to the terms of the New World Encyclopedia:Creative Commons CC-by-sa 3.0 License (CC-by-sa), which may be used and disseminated with proper attribution. Any changes made to the original text since then create a derivative work which is also CC-by-sa licensed. To cite this article click here for a list of acceptable citing formats. 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How do I find the surface area of a solid of revolution using parametric equations? \| Socratic How do I find the surface area of a solid of revolution using parametric equations? 1 Answer If a surface is obtained by rotating about the x-axis from $t = a$ to $b$ the curve of the parametric equation $\left\{\begin{matrix}x = x \left(t\right) \\ y = y \left(t\right)\end{matrix}\right.$, then its surface area A can be found by $A = 2 \pi {\int}_{a}^{b} y \left(t\right) \sqrt{x ' \left(t\right) + y ' \left(t\right)} \mathrm{dt}$ If the same curve is rotated about the y-axis, then $A = 2 \pi {\int}_{a}^{b} x \left(t\right) \sqrt{x ' \left(t\right) + y ' \left(t\right)} \mathrm{dt}$ I hope that this was helpful. Impact of this question 3465 views around the world	{"url":"https://api-project-1022638073839.appspot.com/questions/how-do-i-find-the-surface-area-of-a-solid-of-revolution-using-parametric-equatio","timestamp":"2024-11-13T11:06:54Z","content_type":"text/html","content_length":"31837","record_id":"<urn:uuid:e3b5deee-85f3-43b5-bc92-336a6ee5ed3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00069.warc.gz"}
equation whose roots are α2,β2, and γ2. 4. If α,β,γ are the roo... \| Filo Question asked by Filo student equation whose roots are , and . 4. If are the roots of the equation , then find the cubic equation whose roots are , 5. If the roots of equation are , and Not the question you're searching for? + Ask your question Video solutions (1) Learn from their 1-to-1 discussion with Filo tutors. 6 mins Uploaded on: 1/23/2023 Was this solution helpful? Found 7 tutors discussing this question Discuss this question LIVE 6 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Students who ask this question also asked View more Stuck on the question or explanation? Connect with our Mathematics tutors online and get step by step solution of this question. 231 students are taking LIVE classes Question Text equation whose roots are , and . 4. If are the roots of the equation , then find the cubic equation whose roots are , 5. If the roots of equation are , and Updated On Jan 23, 2023 Topic Algebra Subject Mathematics Class Class 11 Answer Type Video solution: 1 Upvotes 70 Avg. Video Duration 6 min	{"url":"https://askfilo.com/user-question-answers-mathematics/equation-whose-roots-are-and-4-if-are-the-roots-of-the-33393231333137","timestamp":"2024-11-04T15:40:35Z","content_type":"text/html","content_length":"273909","record_id":"<urn:uuid:98c31aca-b968-45e6-bcbe-7a0bffe2ab56>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00553.warc.gz"}
UTS Open '18 P3 - Restaurants Submit solution Points: 7 (partial) Time limit: 1.0s Memory limit: 256M Knowing that UTS is moving to 30 Humbert Street, you notice the lack of restaurants in the surrounding area. In order to maximize the happiness of the students, you plan to build new ones along Humbert Street. The blocks on the street can be modelled as a 1-indexed array. Initially, there are restaurants, on blocks . You want any consecutive segment of blocks to have at least restaurants. Additionally, no two restaurants can occupy the same block. Since you're secretly stealing funds from the school to execute this plan, you want to minimize the total number of extra restaurants to build. Input Specification The first line contains the four integers , , , and : the length of the road, the segment length to be checked, the number of desired restaurants in each segment, and the number of restaurants that already exist, respectively. The next lines each contain a single integer. The line contains the integer , the positions of the pre-existing restaurant. No two pre-existing restaurants will have the same position. Output Specification Print the minimum number of restaurants that must be built in order to satisfy the conditions. Subtask 1 [20%] Subtask 2 [80%] Sample Input 1 Sample Output 1 Explanation for Sample Output 1 One possible solution is to build restaurants on blocks 5 and 8. Sample Input 2 Sample Output 2 Explanation for Sample Output 2 4 restaurants must be built, on blocks 1, 2, 3, and 5. There are no comments at the moment.	{"url":"https://dmoj.ca/problem/utso18p3","timestamp":"2024-11-11T23:14:20Z","content_type":"text/html","content_length":"23942","record_id":"<urn:uuid:8b0b7a92-2e2b-44db-96c0-81f64a9f0846>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00660.warc.gz"}
31st International Symposium on Computational Geometry (SoCG 2015), SoCG 2015, June 22-25, 2015, Eindhoven, The NetherlandsFront Matter, Table of Contents, Preface, Conference OrganizationCombinatorial Discrepancy for Boxes via the gamma_2 NormTilt: The Video - Designing Worlds to Control Robot Swarms with Only Global SignalsAutomatic Proofs for Formulae Enumerating Proper PolycubesVisualizing Sparse FiltrationsVisualizing Quickest Visibility MapsSylvester-Gallai for Arrangements of SubspacesComputational Aspects of the Colorful Carathéodory TheoremSemi-algebraic Ramsey NumbersA Short Proof of a Near-Optimal Cardinality Estimate for the Product of a Sum SetA Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex PolytopesTwo Proofs for Shallow PackingsShortest Path in a Polygon using Sublinear SpaceOptimal Morphs of Convex Drawings1-String B_2-VPG Representation of Planar GraphsSpanners and Reachability Oracles for Directed Transmission GraphsRecognition and Complexity of Point Visibility GraphsGeometric Spanners for Points Inside a Polygonal DomainAn Optimal Algorithm for the Separating Common Tangents of Two PolygonsA Linear-Time Algorithm for the Geodesic Center of a Simple PolygonOn the Smoothed Complexity of Convex HullsFinding All Maximal Subsequences with Hereditary PropertiesRiemannian Simplices and TriangulationsAn Edge-Based Framework for Enumerating 3-Manifold TriangulationsOrder on Order TypesLimits of Order TypesCombinatorial Redundancy DetectionEffectiveness of Local Search for Geometric OptimizationOn the Shadow Simplex Method for Curved PolyhedraPattern Overlap Implies Runaway Growth in Hierarchical Tile SystemsSpace Exploration via Proximity SearchStar Unfolding from a Geodesic CurveThe Dirac-Motzkin Problem on Ordinary Lines and the Orchard Problem (Invited Talk)On the Beer Index of Convexity and Its VariantsTight Bounds for Conflict-Free Chromatic Guarding of Orthogonal Art GalleriesLow-Quality Dimension Reduction and High-Dimensional Approximate Nearest NeighborRestricted Isometry Property for General p-NormsStrong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb GraphsOn Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability ResultComparing Graphs via Persistence DistortionBounding Helly Numbers via Betti NumbersPolynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem RevisitedBisector Energy and Few Distinct DistancesIncidences between Points and Lines in Three DimensionsThe Number of Unit-Area Triangles in the Plane: Theme and VariationsOn the Number of Rich Lines in Truly High Dimensional SetsRealization Spaces of Arrangements of Convex BodiesComputing Teichmüller Maps between PolygonsOn-line Coloring between Two LinesBuilding Efficient and Compact Data Structures for Simplicial ComplexesShortest Path to a Segment and Quickest Visibility QueriesTrajectory Grouping Structure under Geodesic DistanceFrom Proximity to Utility: A Voronoi Partition of Pareto OptimaFaster Deterministic Volume Estimation in the Oracle Model via Thin Lattice CoveringsOptimal Deterministic Algorithms for 2-d and 3-d Shallow CuttingsA Simpler Linear-Time Algorithm for Intersecting Two Convex Polyhedra in Three DimensionsApproximability of the Discrete Fréchet DistanceThe Hardness of Approximation of Euclidean k-MeansA Fire Fighter’s ProblemApproximate Geometric MST Range QueriesMaintaining Contour Trees of Dynamic TerrainsHyperorthogonal Well-Folded Hilbert CurvesTopological Analysis of Scalar Fields with OutliersOn Computability and Triviality of Well GroupsGeometric Inference on Kernel Density EstimatesModeling Real-World Data Sets (Invited Talk) Front Matter, Table of Contents, Preface, Conference Organization Front Matter Table of Contents Preface Conference Organization i-xx Front Matter Lars Arge Lars Arge János Pach János Pach 10.4230/ LIPIcs.SOCG.2015.i Creative Commons Attribution 3.0 Unported license https://creativecommons.org/licenses/by/3.0/legalcode	{"url":"https://drops.dagstuhl.de/entities/volume/LIPIcs-volume-34/metadata/xml","timestamp":"2024-11-08T12:18:42Z","content_type":"application/xml","content_length":"421999","record_id":"<urn:uuid:1f73a608-2e75-4910-825c-bdd667991d51>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00602.warc.gz"}
characterTable -- returns the character table of the symmetric group This method construct the irreducible characters of $S_n$. The method works by recursively calculating the character tables for the permutation modules of $S_n$. Then applying Gram-Schimdt algorithm to this characters using the inner product of characters we obtain the irreducible characters of $S_n$	{"url":"https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/SpechtModule/html/_character__Table.html","timestamp":"2024-11-04T03:57:13Z","content_type":"text/html","content_length":"6078","record_id":"<urn:uuid:68adc0fb-bb39-4a29-b602-a8729f5ebfe2>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00002.warc.gz"}
Seal-Stack Attack and Dynamic PoRep Proof of Useful Space A Proof of Space (PoS, see for example: eprint 2013/796) is a protocol that allows a prover to convince a verifier that he has a minimum specified amount of space (ie, used storage). More precisely in a PoS protocol we have two main sub-protocols: • Initialization (ie, setup phase): on public input N, an advice (eg, vector of random data) of size N is created. The advice is stored by the prover, while the verifier does not know the advice (in some protocols, the verifier may know a commitment to the advice). • Execution (ie, audit phase): the verifier and the prover run a protocol and the verifier outputs reject/accept. Accept means that the verifier is convinced that the prover stores the advice. This phase can be repeated many times. A PoS is sound if a verifier interacting with a malicious prover who stores a fraction of the advice that has size N’ < N, and runs in at most T steps during the execution phase (regenerates the missing part to answer queries), outputs accept with small probability (ie, soundness error). The value (N-N’)/N is called the spacegap. When instantiating a PoS in a real-word protocol, we need to specify what “the malicious prover runs in at most T step” means. Two ways of doing this are: • Latency model: the verifier accepts only proofs that are produced in less than x seconds after the execution protocol starts because we estimate that T steps correspond to at least to x seconds. In other words, the proved is force to use at least N’ storage; • Cost model: the prover is allowed to choose among running in T’ > T steps (and using < N’ storage) or using at least N’ storage, however the first strategy always costs more than the second one. Asymmetric PoRep In an asymmetrical PoRep there exist a Decoding algorithm that runs faster than the encoding. In current PoRep the Decoding algorithm is running as slow as the encoding one, so the aim is to improve Decoding in practical protocols. Nevertheless, this asymmetry may lead to a possible attack from a malicious Prover. SealStack Attack Illustrated Every time a Winning PoSt is required for R, the SP runs the unsealing algorithm on R' and gets R. If the unsealing algorithm is faster than Response time, then Adversary is successful in WinningPoSt DecodingLatency > RegenLatency > ResponseTime (Note: Attack in Latency Model. If unsealing is fast but still as cost expensive as sealing, then WindowPoSt is still secure in the cost model.) Possible Solutions 1. Resealing each time we perform Execution phase (WinningPoSt) 2. Dynamic PoRep: Re-randomizable Replicas - not sure if it mitigates the attack completely 3. Changing the underlying data in PoRep costs	{"url":"https://cryptonet.org/research/seal-stack-attack-and-dynamic-porep","timestamp":"2024-11-06T19:07:29Z","content_type":"text/html","content_length":"185506","record_id":"<urn:uuid:b6a8131b-2789-4a78-938d-4f608051a576>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00011.warc.gz"}
What statement about camels is true? - Answers Still have questions? is this statement true or false BC? If the statement is false, then "This statement is false", is a lie, making it "This statement is true." The statement is now true. But if the statement is true, then "This statement is false" is true, making the statement false. But if the statement is false, then "This statement is false", is a lie, making it "This statement is true." The statement is now true. But if the statement is true, then... It's one of the biggest paradoxes ever, just like saying, "I'm lying right now."	{"url":"https://math.answers.com/math-and-arithmetic/What_statement_about_camels_is_true","timestamp":"2024-11-04T21:32:11Z","content_type":"text/html","content_length":"129353","record_id":"<urn:uuid:c5cc5876-7fe4-4d6e-a426-8e628c5d477d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00067.warc.gz"}
How do you calculate the drag coefficient of a parachute? \| Gzipwtf.com How do you calculate the drag coefficient of a parachute? How do you calculate the drag coefficient of a parachute? The drag equation states that the drag (D) is equal to some drag coefficient (Cd) times half of the air density (r) times the square of the velocity (V) times the reference area (A). What is the descent rate of a parachute? Depending upon air density and the jumper’s total weight, the parachute’s average rate of descent is from 22 to 24 feet per second (6.7 to 7.3 m/s); total suspended weight limitation is 360 pounds (160 kg). How do you find the surface area of a parachute? The time is approximately proportional to the inverse of the terminal velocity, so it’s approximately proportional to Area/Mtotal = A/(Mp +Mw), where A is the area, Mp is the parachute mass and Mw is the other mass. Once Mp is much bigger than Mw. How big should my parachute be? RECOMMENDED PARACHUTE SIZES Rockets 12″ and shorter – use streamer recovery or an 8″ chute. Rockets 12″ to 18″ tall – use a 12″ chute. Rockets 18″ to 24″ tall – use a 12″ or 18″ chute. Rockets 24″ and taller – use a 18″ or 24″ chute. What is the best shape for a parachute? The circle parachute should demonstrate the slowest average descent rate because its natural symmetrical shape would be the most efficient design to maximize wind resistance and create drag. How long does it take to descend with a parachute? A typical skydive lasts five to six minutes, with approximately 50 seconds of that spent in freefall and four to five minutes on the parachute ride down. Does the size of a parachute affect its drop rate? The size of the parachute affects the speed of falling because a larger parachute allows it to displace more air, causing it to fall more slowly. However, as the parachute gets larger, it is able to push against–or displace–more air, which will slow down a falling object. Which parachute has a slower descent? The circle parachute had the slowest overall average descent rate of 134.88 centimeters per second, followed by the parallelogram parachute with an overall average descent rate of 141.72 centimeters per second. What is terminal velocity of a parachute? By definition, terminal velocity is a constant speed which is reached when the falling object is met with enough resistance to prevent further acceleration. Terminal velocity is, then, the fastest speed you will reach on your skydive; this is usually around 120 mph. How do you find velocity after 3 seconds? After 3 seconds, the velocity is 4.5+3×1.5=9 m/s. How big does a parachute need to be for an egg? Take the plastic garbage bag and cut a small, medium and large size square. The recommended sizes are: 10” x 10”, 20” x 20”, and 30” x 30” but allow kids to experiment with the sizes! For each parachute cut four equal lengths of string (you will need 12 total). How do you calculate the drag of a parachute? Here is the equation for calculating the drag of a parachute: D = 1/2 * p * V^2 * S * Cd Where: D=Drag; Cd=Coefficient of Drag (approx. .8 to 1.0); p(or rho)=Air Density; V= Velocity, S = surface area of parachute Alternatively you can determine the proper parachute sizing by using this formula: D = 1/2 * p * V^2 * Cd How does vent ratio affect drag coefficient? As the vent ratio of the parachute is increased from zero to 5 percent of the parachute inlet diameter, the drag coefficient increased and for further increase of the vent ratio diameter, the drag coefficient decreased, but the general variation of drag coefficient was the same as of same parachute with no vent. … What is coefficient of drag in fluid mechanics? In fluid dynamics, the coefficient of drag is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equation in which a lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. How fast does a 13-1/2 inch parachute fall? A 13-1/2 inch diameter parachute would read 1 pound at 30 feet per second (20 miles per hour). That means that if you put a 1 pound object on a 13-1/2 inch diameter parachute, it would fall with a descent rate of 30 feet per second.	{"url":"https://gzipwtf.com/how-do-you-calculate-the-drag-coefficient-of-a-parachute/","timestamp":"2024-11-02T11:51:23Z","content_type":"text/html","content_length":"141443","record_id":"<urn:uuid:671548a9-07d3-4bbc-9505-2441036f193e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00284.warc.gz"}
Graph Traversal - DFS & BFS In this article, we are going to look at two methods that allow us to travel throughout a graph. Before we begin, it is important that the reader have a basic understanding of how we represent graphs in a computer program, as well as the definitions used to describe them. Please refer to the previous articles. Graph Traversal - The Big Idea So we have a graph set up in our program. The vertices have data in them and life is good. Now we want to search for a specific vertex in our graph. How would we do this? Introducing graph traversal algorithms: Depth First Search and Breadth First Search. The big idea and purpose of graph traversal is to systematically explore and visit all the vertices and edges of a graph in a well-defined order. Graph traversal algorithms help us understand the structure of the graph, identify relationships between vertices, and find paths or connections between specific vertices. Let's begin with DFS. Depth First Search - DFS DFS (Depth-First Search) is commonly implemented using recursion, which provides a straightforward and intuitive approach. However, recursive DFS may lead to stack overflow errors for large graphs due to excessive stack space usage. In contrast, an iterative DFS implementation using an explicit stack consumes less memory and is preferred for handling large graphs or when memory constraints are crucial. Additionally, if the recursive approach is implemented in a tail-recursive style and the compiler supports tail call optimization, the performance gap between recursive and iterative DFS can be minimized. If you are familiar with pre-order traversals used in Binary-Search-Tree problems, this should feel very familiar. Let's take a look at the process, 1. We are going to mark all of our vertices undiscovered 2. Recursively call our DFS algorithm on $v_{0}$ 3. If there are any vertices in our list that are still undiscovered, choose one and restart the process 1. Mark $v$ as discovered 2. For all the undiscovered vertices recurse on those vertices Now that we know the process lets write this algorithm. /* DFS in C++ / class Solution void recursiveDFS(int v, vector<int> adj[], map<int, bool>& vis, vector<int>& res) // mark as visited vis[v] = true; // loop through adjacencies for (auto i = adj[v].begin(); i != adj[v].end(); ++i) if (!vis[i]) recursiveDFS(i, adj, vis, res); // function to return a list containing the DFS traversal of the graph. vector<int> dfsOfGraph(int V, vector<int> adj[]) //create visited map and result vector vector<int> result; map<int, bool> visited; // visit each node in our adjacency list for (int i = 0; i < V; ++i) if (!visited[i]) recursiveDFS(i, adj, visited, result); return result; DFS Applications There are several problems where DFS applies. Here are several taken from this page • Finding connected components. • Topological sorting. • Finding 2-(edge or vertex)-connected components. • Finding 3-(edge or vertex)-connected components. • Finding the bridges of a graph. • Generating words in order to plot the limit set of a group. • Finding strongly connected components. • Determining whether a species is closer to one species or another in a phylogenetic tree. Breadth First Search - BFS Similar to DFS, BFS will mark vertices as it makes its way through other vertices. However, instead of using recursion like DFS, BFS is a Queue-based traversal. The process is rather simple, 1. Start at the first vertex and add it to the queue 2. Dequeue vertex and add all the unvisited neighbors of that vertex to the queue 3. Continue the until the queue is empty 4. If more than one component of vertices, continue on to the next component Let's look at this process implemented in C++ / BFS in C++ / class Solution // function to return Breadth First Traversal of given graph. vector<int> bfsOfGraph(int V, vector<int> adj[]) // create queue and map for visited queue<int> Q; map<int, bool> visited; vector<int> result; visited[0] = true; while (!Q.empty()) int v = Q.front(); for (auto i = adj[v].begin(); i != adj[v].end(); ++i) if (visited[i] == false) visited[*i] = true; return result; BFS Applications There are several problems where DFS applies. Here are several taken from this page • Copying garbage collection, Cheney's algorithm • Finding the shortest path between two nodes u and v, with path length measured by number of edges (an advantage over depth-first search)[13] • (Reverse) Cuthill–McKee mesh numbering • Ford–Fulkerson method for computing the maximum flow in a flow network • Serialization/Deserialization of a binary tree vs serialization in sorted order, allows the tree to be re-constructed in an efficient manner. • Construction of the failure function of the Aho-Corasick pattern matcher. • Testing bipartiteness of a graph. In conclusion, graphs are a very powerful and practical data structure. In this article we discussed two important graph traversal methods: Depth-First-Search and Breadth-First-Search. We learned that DFS is implemented with recursion but can also be done so iteratively. We also learned that BFS is implemented with a queue data structure. We saw both traversal methods implemented in C++. Finally, we discovered applications for both DFS and BFS. Until next time, happy coding!	{"url":"https://chrisragland.dev/blogs/graph-traversal","timestamp":"2024-11-08T01:04:36Z","content_type":"text/html","content_length":"77588","record_id":"<urn:uuid:aab8fc1f-3aee-4d54-8818-ad0202db2806>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00117.warc.gz"}
Same sum' with decimals \| Oak National Academy Hello, everybody, welcome back. This is the next lesson in the "Number, Addition and Subtraction" series of lessons. I'm Mrs. Furlong, you might remember me from last week. I'm going to be taking over for Mrs. Moe for the next two days. So we're on to lesson three of "Deepening Understanding of Equivalents "and the Equal Sign In Addition and Subtraction." Yesterday, Mrs. Moe left you with these questions to practise. And we're going to start today's lesson by reviewing those questions. So in question one, you were asked to fill in the missing numbers. I noticed a connection between 11,997 and 12,000, did you spot it? That's right, 11,997, if it's increased by three, will get us to 12,000. So if one addend has been increased by three, the other addend has to be decreased by three, that's right? So our 64,036 becomes 64,033. I think this is much easier to calculate than the original calculation. So I'm saying that 64,033 plus 12,000 is easier to calculate, what do you think? That's right, we only need to pay attention to the thousands and tens of thousands columns to be able to do this using place value. We really just need to look at the 64 in the 64,033 and the 12 in the 12,000. And I know that 64,000 and 12,000 gives me 78,000. I mustn't forget there's 33, they're very important. So the next set of questions that Mrs. Moe left you was part two, which was to look at the calculations and decide which ones you would use redistribution for, and explain why. Well, I was thinking about the purpose of redistribution in the previous lesson, and that was to make calculations easier, wasn't it? So let's take a look at A: 12,036 add 32,873. I think to make these easier, Ideally I'd want to get them to either a multiple of a hundred or a multiple of a thousand. So if I look at 12,036, I could decrease that by 36 to get me to 12,000, and I could increase 32,873 by 36. But, this is meant to be a mental calculation method. And I'm thinking by the time I've done all of that hard work, it might've been easier just to do a column calculation in the first place. I equally, I could look at the 32,873 and think hmm, I could make it into 32,900. But again, I would need to increase it by 27, and decrease the other number by 27. And by the time I've done all of that redistributing, I still think I'd have been quicker to do a column calculation. And this is meant to make a mental calculation easier. And I think those numbers are just a little bit too much redistribution, what do you think? So let's take a look at part B: 504,992 add 11,008. Hmm, take a close look at this, I wonder whether you noticed what I noticed. I noticed that 504,992, it's actually really close to 505,000. In fact, I only have to increase it by eight, and decrease the 11,008 by eight. And then you can see, I've got myself a really easy calculation, 505,000 add 11,000 is 516,000. That one's a really good one to redistribute. I wonder if you thought the same. Let's take a look at part C now. 25,317 add 22,997. The second addend really jumped out at me. 22,997, that's almost 23,000, isn't it? In fact it's only three away. So, if I increase that second addend by three, I must decrease the first addend by three to make sure that my sum remains the same. I wonder if you did what I did. SO look, here we are, we get 25,314, that's three less than 25,317. And we get 23,000, which is three more than 22,997. Now I've got an easy calculation, and I can find out that it is 48,314. The final question, 99,164 add 8,419. I took a really careful look at these numbers. But I think both of them are quite far away from either helpful multiple of a hundred or multiple of a thousand. So I decided not to use redistribution. I don't know whether you did the same. I wonder how you explained it. So in the last set of questions, Mrs. Moe asked you to have a look at the equations and decide if they were correct. And then explain how you know. So let's have a look at A, we've got 7,644, add 21,996 is equal to 7,648 add 22,000. So I had a look at the first addend, in A, that's 7,644, and notice that it has increased by four to get 7,648. I then had to look at the second addend. So the 21,996 had increased by four to get 22,000. Our generalised statement was, if one addend is increased by an amount, and the other addend is decreased by the same amount, the sum remains the same. Did you spot what had happened here? Yes, you're right, both addends have increased by four. And if you think back to the first lesson Mrs. Moe did on this, with things like the water, where one jug increased by a little whilst the other decreased or with her sweets, one bowl of sweet got less, and the other bowl of sweets got more to get that sum to remain the same. In this case, if we've increased both of those addends by four, we're going to have increased our sum, aren't we? Let's take a look at question B now. 123,017 add 4,999 gives us a total of 128,017. Well, I've noticed that the 17 in the tens and the ones is the same in the first addend and in the sum. And I'm not adding on a multiple of 10 or a hundred, so I'm wondering, can that remain the same? Let's a look at my thinking. So my explanation was, that 123, 017 add 5,000 would give me 128,017. So therefore, 123, 017 add 4,999, which is one smaller must give me a sum, that is one smaller, so 128,016. So neither of those calculations were correct, were they? I wonder how you got them. I hope you did well. Okay, so now we're ready to start today's session. You might need to get yourself some paper and a pencil. And if you haven't already done that, pause the video now and go and find some. Okay, so today's session, we are going to be again, looking at those addends and using our generalised statements of increasing one addend and decreasing the other addend by the same amount, to make sure that our sum remains the same. But in our context today, we're going to be thinking about decimal numbers. So I just wanted to do a quick reminder about those before we get started. So in my presentation today in this lesson, you're going to see that this blue square is representing one, it's not representing 100 like you might've met before in other lessons and maybe at school, it's representing one. So one large blue square represents one, okay? So just remember that. All right. Then, we need to think about this rod, okay? This rod in this session, 10 of those rods are going to make one, aren't they? If you imagine 10 of those green rod side-by-side, they would be the same size as the one. So this rod is representing 0. 1 or one tenth. And finally, this little yellow cube is going to represent one hundredth or 0. You need to remember that for today. There's going to be a little bit more about this on the next slide, just to make sure you fully understand. Okay, so just to make sure that everybody understands and everybody's clear, we're just going to spend a little bit longer looking at these rods. Can you remember what this green rod is worth in today's session? That's right, it's one tenth or 0. We're going to count in tens now. I'd love it if you joined in with me. One tenth, two tenths, three tenths, four tenths, five tenths, six tenths, seven tenths, eight tenths, nine tenths and tenths. What new about 10 tenths? That's right, you did a lot on this in the previous sessions when you were doing fractions. 10 tenths is equivalent to one, okay? So 10 tenths is the same as one. So, we can also say that it's 10 multiplied by one tenth or 10 multiplied by 0. 1 is the same as one as well. Can you remember what this small cube's worth in today's session? That's right, it's worth one hundredth. Can you count with me? One hundredths, two hundredths, three hundredths, four hundredths, five hundredths, six hundredths, seven hundredths, eight hundredths, nine hundredths, 10 hundredths. What is 10 hundredths the same as? Exactly, the same as my green rod, my one tenths. So one hundredths equals one tenth or 10 lots of 0. 01, is also equal to add 0. 1, which we all know is the same as one tenth. Just keep those in your head today 'cause it will really help you with some of your work. So this is going to be our first calculation in today's session, 4. 5 add 2. Have a look at the representations that I've done. Can you see where the four is on the representation? That's right. So, these four ones represent our four in our calculation here. And, can you find where the 0. 5 is? Yup, that's right, you found them here our five tenths are here, aren't they? Brilliant. And what do these over here represents? That's right, there are two ones. And here? There are our nine tenths in this part. Okay, so we know that we can work out 4. 5 add 2. And some of you might've been doing that whilst I was just explaining the representation. In fact, I bet some of you did. So we could work it out with a column, we could work it out using some kind of a mental method. But, what if we consider our generalised statement? If one addend is increased by an amount, and the other addend is decreased by the same amount, the sum remains the same. Hmm, have a look at the numbers, before using that redistribution property, which number would you increase? And which number would you decrease? And why? You might want to pull us the video for a moment here and have a think. Did you make a decision? I wonder whether your decision was the same as mine, we'll find out in a moment. I wonder, did any of you tried to increase the 4. 5, the first addend? If you did, did you decide that making the 4. 5 into five would make the calculation easier? Let's take a look. So, oh, did you see what happened there? Did you see that from the 2. 9, five tenths moved over to the 4. 5 so that we have increased our 4. 5? What we increased it by? That's right, there's five tenths moved over so we've increased it by 0. We've increased 4. 5 by 0. 5, and we've decreased 2. 9 by 0. It's a bit like on that first session where Mrs. Moe moved those sweets in that bowl, isn't it? One bowl decreased and the other increased. It was important that we focused on that 4. 5 then, and how much we needed to increase it by. We increased it by five tenths to make it into five ones. Can you see hear that we now have one tenths in those green strips that we looked at earlier, didn't we? We said 10 tenths is the same as one whole or one. So we now have five one on the left hand side is our left-hand addend, and we now have 2. 4 on that right-hand side. And I think that five add 2. 4 is quite an easy calculation. I wonder if you agree. Five add 2. 4, 7. And because we have to know our increasing of one addend and decreasing the other by the same amount, our sum remains the same. So that can help me now to answer 4. 5 add 2. 9, which is also 7. I could also represent like the calculation at the bottom of the screen. And you can see that they're side by side. So 4. 5 add 2. 9 is equal to five add 2. 4, which is equal to 7. Everything's balanced, our sums have remained the same so our equal sign is being used correctly. Perhaps you didn't do that. Perhaps you looked at these two numbers and you decided to focus on the 2. I wonder why. Oh, because 2. 9 is almost three, isn't it? How far away from three is 2. 9? That's right. , it's just one tenth away. So if I was to increase 2. 9 by one tenth, then I would make three. And the only place I can get that one tenth from is from my 4. So just have a look and look at the animation. There we go. One tenth went across from the 4. 5 and it landed on the 2. 9, it redistributed. So can you see now that we decreased 4. 5 by one tenth, and we increased 2. 9 by one tenth. And what did we get, if we swapped it? So, at this side here, we now have 10 tenths, don't we? And we neutrally. And remember 10 tenths is equivalent to one whole, so at this side, we don't have 2. 9 anymore, we have three. And at the other side, we don't 4. 5 anymore, that addend is now 4. 4 Aha, 4. 4 add three, I think that's quite easy, do you? Yeah, that's right, it's still 7. And that means that our sums remain the same, and so 4. 5 add 2. 9, it's also 7. And you can see again that the calculation at the bottom of the screen or the equation, 4. 5 add 2. 9 is equal to 4. 4 add 3, and both of those equal 7. So we've balanced our equations, we've got that same sum all the way through. Okay, so we've got a different calculation this time. Have you spotted that we've now got hundredths in there? Remember those small yellow cubes are representing a hundredths, aren't they? Hmm, I can see a three in both of my addends. I can see the three in 3. 08, and I can also see the three digit in the 4. Can you spot where they're represented in my base 10 equipment or in my deans? That right. So the three ones is represented here by our three ones that we're using as per representation for today. And my three tenths are represented over here, aren't they? By my three green rods. That's right. One of the time little confused by some people find a bit is why is my zero here represented? What's that's zero meaning? Yes, it's in the tenths column, it means we haven't got any tenths, there are zero tenths. And as you can see over here in this representation, there are none of my green rods. So we have zero tenths, okay? So this tenths are not represented as an absence I've left to little space just to show that. Let's think about our calculation now. So again, if we are going to be using that equivalent, sort of same sum rule, is one addend is increased by an amount and the other addend is decreased by the same amount, the sum remains the same. So take a look at these two numbers now, the 3. 08 and the 4. What do you think you would do this time to redistribute those numbers, to make the calculation easier? Pause the video here and have a think. I wonder whether in this calculation, you noticed what I noticed. I spotted that 3. 08 is actually really close to three. It's only eight hundredths away, which is a tiny amounts. Hmm, so I wondered whether I could decrease 3. 08 by those eight hundredths and increase, therefore the 4. 39 by eight hundredths, have a look. So, my hundredth have moved over, my eight hundredths have moved over to join 4. So 3. 08 has decreased by eight hundredths, and 4. 39 has increased by eight hundredths. So now we get a new equivalent calculation. But it's not that easy because if you start at this side, I've got eight hundredths here and I've got nine hundredths here. I have to think quite carefully about recombining those, don't I? And if you imagine me stealing one of those hundredths or redistributing one of those hundredths and popping it on top of that, can you see that we get a new tenth? So therefore we would have four tenths and then we would have seven hundredths here. I wonder if you can imagine that. So, our new calculation or our equivalent calculation would be three add 4. But, I'm just wondering to myself, how's that really helped me, that redistribution? Because, merely couldn't I just have done three ones add four ones and then done the eight hundredths and the 39 hundredths, and partition the calculation. I wonder whether this particular redistribution was very helpful. I'm not so sure. Sometimes redistributing the numbers doesn't help, or sometimes it doesn't help, because you might not have redistributed them in the best way. Let's have a look at in the next example. So with that, with those last same calculations is 3. 08 add the 4. 39, maybe you chose to redistribute the numbers in a different way. Maybe you spotted that the 4. 39 is very close to 4. 4, and that that might help us. So if I increase the 4. 39 by one hundredths, if you watch it happen, it should happen now, there we go,. Increased it by one hundredth, you can see now that we have not got 4. 39 anymore, we've got four and one, two, three, four tenths, because remember 10 hundredths is the same as one 10th, isn't it? So, we increase 4. 39 by one hundredth, so we had to decrease the other addend, the 3. 08 by one hundredth, which happened a moment ago in my animation. And we've now got 3. 07 add 4. We could use place value to help us to work this out now. And we would get 7. Perhaps that redistribution was a little better than the other, but you may still be stuck there thinking that you didn't really need to do it, that actually I could just have added eight hundredths onto the 39 hundredths. In which case, if that is quicker, there's no point in redistributing the numbers at all, is there? So let's look at this example now. Saidi decided to make this equivalent calculation to help her to solve this more easily. So let's have a look for that equivalent calculation first. Who is the big here and have a careful look at what did Saidi do to make the calculations equivalent? Did you spot it? She decreased the first addend, hasn't she? And she's increased the second I wonder which one she focused on first. I think she focused on 3. Because 3. 98 is very close to four. And we've already discovered that calculating with integers or whole numbers is much easier than calculating with decimals. So 3. 98 has increased by what to get to four? It's got 98 hundredths. How many hundredths do we need to make that next one, that whole one? That's right, we need a hundred hundredths to make a whole one so we need two more hundredths. So it's increased by two hundredths. And that first addend has decreased by the hundredth. How do we know that 5. 3 decreasing by two hundredths gives us 5. 28? What do we know about 0. 3? We do know it's three tenths, it's brilliant. How many hundredths make a tenth? That's right, it's was ten hundredths. Do you remember at the start of the video where we saw those 10 little yellow cubes make the same size as that one green rod that tenths rod? Brilliant. So if you imagine that we had three tenths rods, how many of those little yellow cubes would we need? That's right, 30 hundredths. So is 5. 3 is also the same as five and 30 hundredths. And if we decrease those 30 hundredths by two, we'd get 28 hundredths, so it's 5. And 5. 28 add four is 9. So we've now got that equivalent calculation. I think Saidi has made it quite easy. So Sanjay decided to complete the calculation in this way. What decisions do you think Sanjay made to create his equivalent calculation or his same sum? Have careful look, you might want to pause the video just whilst you do that. Did you spot it? That's right, there was a bit of a clue, wasn't there? In the 5. 3 and the five. The 5. 3 must have decreased by three tenths to get it to five. So to make sure it's that equivalent, same sum, we must increase the other addend by 0. Let's take a little bit of a careful look at 3. 98 and increasing it by 0. What do we know about 0. 3? That's right, it's three tenths. And what do we know about three tenths? How many hundredths are there in three tenths? Yes, we did this on the previous slide. there's 30 hundredths, aren't there? So we're increasing this by 30 hundredths. So I was thinking, how about if we take our 3. 98, increased it by two hundredths to get to four ones, and then I need to increase it by a further 28 hundredths, it will get me to 4. And he's created another equivalent calculation. And you can see that we have that same sum, the 9. So five add 4. 28 is 9. I think that's a little easier than the original calculation. Let's just compare Saidi in Sunjay's methods now side by side. Which one do you prefer and why? If you could pause the video here and have a careful look at them side by side, and we'll speak about it again in a moment. I wonder which one you chose. Well, both of them are absolutely fine, aren't they? Because both of them give us that same sum and that equivalent calculation. I think, as long as you followed the rule, if one addend is increased by an amount and the other identity is decreased by the same amount, and the sum remains the same, then you could use either, because I think both Sunjay and Saidi have created easier calculations. I think really this decision on which one's easier probably comes down to your own number sense and which one you think would work for you. In the last few calculations, we discovered that you can alter either addend, but now I want you to make some careful decisions about which addend to increase or decrease to make these two calculations easier to solve. So have a look at the two calculations carefully and pause the video to consider the equivalent calculations you could use to make these as easy as possible to solve. Have you done that? Let's have a look at what I thought. You might have a different decision to make. So I had a look at the 7. 8 add 1. 68, and I thought 7. 8 is close to eight. So I'm going to increase it by two tenths to make it into that whole number. So 7. 8 becomes eight, so I'm going to decrease the other addend by 0. 2 or two tenths to make it 1. And now I've got quite an easy calculation to solve, eight add 1. 48 is? That's right, 9. And so therefore 7. 8 add 1. 68 is also 9. What about in example two? Did you spot that 3. 96? It's close to four, isn't it? How far away is it from four? Yes, it's only four hundredths. So I decided that I would increase 3. 96 by four hundredths. Did you do the same? And decrease the 7. 31 by four hundredths. So our equivalent calculation is 7. 27, add four. And now we've got a much easier calculation because I can do seven add four in the ones, and then I just need to remember that 0. 27 at the end, don't I? So we'd get a total or a sum of 11. 27, which means that both 7. 27 add four is 11. 27, and so is 7. 31 add 3. Personally, I think redistributed numbers are definitely easier to solve, what do you think? Okay, so now it's your turn to do some work. And I've left you with some practise for today. in part A, I just want you to fill in the missing numbers, and you'll notice there was a connection between the ones in the box there. So make sure you pay attention to the connection between this calculation and the equation underneath, okay? Thinking about that redistribution. And that's the same in each of those boxes. I've put on that generalised statement for you just to remind you. And then part B says "Ready for a challenge? "Salvo says that the best way to solve these calculations "is to use the same sum." Or an equivalent calculation, it means the same thing to make them "Mia disagrees and says that it is easier and quicker "to use a written method. "who is right." I wonder whether you could compete with someone in your household and help you find out who's is quicker. Remember, our equivalent calculations, you don't have to write down all the steps we've written down today, they're just there to help you understand. Eventually, it'd be great if you can imagine that one addend increasing and the other addend decreasing, and then it's a bit like turning your sum into a magical new calculation that you can do really quickly, and maybe keep it secret from your parents or whoever you compete against about what you did, and show them what an absolute wiz you are. I hope you've enjoyed today's session. I'm back again tomorrow, so I'll see you then.	{"url":"https://www.thenational.academy/pupils/programmes/maths-primary-year-6-l/units/extending-calculation-strategies-and-additive-reasoning-6e61/lessons/same-sum-with-decimals-6xh3jd/video","timestamp":"2024-11-09T03:27:19Z","content_type":"text/html","content_length":"127584","record_id":"<urn:uuid:b0e70cd8-3868-4f80-ac9b-46ff83c3b533>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00888.warc.gz"}
How to Solve Nonlinear Systems - dummies In a nonlinear system, at least one equation has a graph that isn’t a straight line — that is, at least one of the equations has to be nonlinear. Your pre-calculus instructor will tell you that you can always write a linear equation in the form Ax + By = C (where A, B, and C are real numbers); a nonlinear system is represented by any other form. Examples of nonlinear equations include, but are not limited to, any conic section, polynomial of degree at least 2, rational function, exponential, or logarithm. How to solve a nonlinear system when one equation in the system is nonlinear If one equation in a system is nonlinear, you can use substitution. In this situation, you can solve for one variable in the linear equation and substitute this expression into the nonlinear equation, because solving for a variable in a linear equation is a piece of cake! And any time you can solve for one variable easily, you can substitute that expression into the other equation to solve for the other one. For example, follow these steps to solve this system: 1. Solve the linear equation for one variable. In this example, the top equation is linear. If you solve for x, you get x = 3 + 4y. 2. Substitute the value of the variable into the nonlinear equation. When you plug 3 + 4y into the second equation for x, you get (3 + 4y)y = 6. 3. Solve the nonlinear equation for the variable. When you distribute the y, you get 4y^2 + 3y = 6. Because this equation is quadratic, you must get 0 on one side, so subtract the 6 from both sides to get 4y^2 + 3y – 6 = 0. You have to use the quadratic formula to solve this equation for y: 4. Substitute the solution(s) into either equation to solve for the other variable. Because you found two solutions for y, you have to substitute them both to get two different coordinate pairs. Here’s what happens when you do: Therefore, you get the solutions to the system: These solutions represent the intersection of the line x – 4y = 3 and the rational function xy = 6. How to solve a nonlinear system when both system equations are nonlinear If both of the equations in a system are nonlinear, well, you just have to get more creative to find the solutions. Unless one variable is raised to the same power in both equations, elimination is out of the question. Solving for one of the variables in either equation isn’t necessarily easy, but it can usually be done. After you solve for a variable, plug this expression into the other equation and solve for the other variable just as you did before. Unlike linear systems, many operations may be involved in the simplification or solving of these equations. Just remember to keep your order of operations in mind at each step of the way. When both equations in a system are conic sections, you’ll never find more than four solutions (unless the two equations describe the same conic section, in which case the system has an infinite number of solutions — and therefore is a dependent system). Four is the limit because conic sections are all very smooth curves with no sharp corners or crazy bends, so two different conic sections can’t intersect more than four times. For example, suppose a problem asks you to solve the following system: Doesn’t that problem just make your skin crawl? Don’t break out the calamine lotion just yet, though. Follow these steps to find the solutions: 1. Solve for x^2 or y^2 in one of the given equations. The second equation is attractive because all you have to do is add 9 to both sides to get y + 9 = x^2. 2. Substitute the value from Step 1 into the other equation. You now have y + 9 + y^2 = 9 — a quadratic equation. 3. Solve the quadratic equation. Subtract 9 from both sides to get y + y^2 = 0. Remember that you’re not allowed, ever, to divide by a variable. You must factor out the greatest common factor (GCF) instead to get y(1 + y) = 0. Use the zero product property to solve for y = 0 and y = –1. 4. Substitute the value(s) from Step 3 into either equation to solve for the other variable. This example uses the equation solved for in Step 1. When y is 0, 9 = x^2, so When y is –1, 8 = x^2, so Be sure to keep track of which solution goes with which variable, because you have to express these solutions as points on a coordinate pair. Your answers are This solution set represents the intersections of the circle and the parabola given by the equations in the system. About This Article This article can be found in the category:	{"url":"https://www.dummies.com/article/academics-the-arts/math/pre-calculus/how-to-solve-nonlinear-systems-165929/","timestamp":"2024-11-06T21:02:24Z","content_type":"text/html","content_length":"86067","record_id":"<urn:uuid:e2f11a76-028a-4f6a-92a3-8fb0978de63b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00288.warc.gz"}
hmm classification code Based on your location, we recommend that you select: . 30 Aug 2019, 1D matrix classification using hidden markov model based machine learning for 3 class problems. It will know what to do with it! What makes this problem difficult is that the sequences can vary in length, be comprised of a very large vocabulary of input symbols and may require the model to learn the long-term That is, there is no "ground truth" or labelled data on which to "train" the model. This toolbox supports inference and learning for HMMs with discrete outputs (dhmm's), Gaussian outputs (ghmm's), Tutorial for classification by Hidden markov model, Basic Tutorial for classifying 1D matrix using hidden markov model for 3 class problems, You may receive emails, depending on your. The HMM is a generative probabilistic model, in which a sequence of observable variable is generated by a sequence of internal hidden state .The hidden states can not be observed directly. Library for continuous convex optimization in image analysis, together with a command line tool and Matlab interface. In machine learning sense, observation is our training data, and the number of hidden states is our hyper parameter for our model. We then describe three methods to infer the parameters of our HMM variant, explore connections between these methods, and provide rationale for the classiﬁcation be- Alpha pass at time (t) = t, sum of last alpha pass to each hidden state multiplied by emission to Ot. The transitions between hidden states are assumed to have the form of a (first-order) Markov … Given the known model and the observation {“Shop”, “Clean”, “Walk”}, the weather was most likely {“Rainy”, “Rainy”, “Sunny”} with ~1.5% probability. I searched in the web but could not find a good one. Download HMM Speech Recognition in Matlab for free. It is the process of classifying text strings or documents into different categories, depending upon the contents of the strings. 0.6 x 0.1 + 0.4 x 0.6 = 0.30 (30%). I want to do hand gesture recognition with hmm in matlab. hmmlearn implements the Hidden Markov Models (HMMs). The Hidden Markov Model or HMM is all about learning sequences.. A lot of the data that would be very useful for us to model is in sequences. Note: This package is under limited-maintenance mode. OBSERVATIONS are known data and refers to “Walk”, “Shop”, and “Clean” in the above diagram. Some friends and I needed to find a stable HMM library for a project, and I thought I'd share the results of our search, including some quick notes on each library. Read on to learn the basics of text classification, how it works, and how easy it is to get started with no-code tools like MonkeyLearn. Mathematical Solution to Problem 2: Backward Algorithm. In the above case, emissions are discrete {“Walk”, “Shop”, “Clean”}. With the introduction of the MMM, BMP Scheme participants can now fulfil their RoSP obligations in new eligible geographical locations. For now let’s just focus on 3-state HMM. How can we build the above model in Python? This is why I’m reducing the features generated by Kyle Kastner as X_test.mean(axis=2). We’ll repeat some of the text from Chapter 8 for readers who want the whole story laid out in a single chapter. Our HMM tagger did improve the results, Now we are done building the model. Past that we have under"ow and processor rounds down to 0. Basic Steps of … is that correct? training accuracy basic hmm model: 97.49%. Mathematical Solution to Problem 1: Forward Algorithm. For example, you have a large database of utterances of digits ("one", "two", etc) and want to build a system capable of classifying an unknown utterance. sklearn.hmm implements the Hidden Markov Models (HMMs). Given the known model and the observation {“Clean”, “Clean”, “Clean”}, the weather was most likely {“Rainy”, “Rainy”, “Rainy”} with ~3.6% probability. Let’s learn Classification Of Iris Flower using Python. Full model with known state transition probabilities, observation probability matrix, and initial state distribution is marked as. More From Medium. Anomaly Detection with Azure Stream Analytics, Sematic Segmentation using mmsegmentation. Last updated: 8 June 2005. I look forward to hearing feedback or questions. MathWorks is the leading developer of mathematical computing software for engineers and scientists. To clarify: A =[aij] transition matrix, aij probability for moving from state I to state j When I have just one state as I denote above how would I … multi-HMM classification in this paper. This model can use any kind of document classification like sentimental analysis. But now i am confused about how to extend my code so that it can be fed with more than one accelerometer. Distributed under the MIT License. Choose a web site to get translated content where available and see local events and offers. 0 ⋮ Vote. Meet MixNet: Google Brain’s new State of the Art Mobile AI architecture. To initialize a model using any of those topology specifications, simply create an ITopology object and pass it to the constructor of a hidden Markov model. This method is an implementation of the EM algorithm. Find the treasures in MATLAB Central and discover how the community can help you! 40 HMM Learning Problem 40. … Hidden Markov Model (HMM) Toolbox for Matlab The Internet is full of good articles that explain the theory behind the Hidden Markov Model (HMM) well(e.g.1,2,3and4).However, many of these works contain a fair amount of rather advanced mathematical equations. beginner , classification , random forest , +2 more xgboost , decision tree Is it possible U provide some code releated to my problem using Murphy's toolbox? This video is part of the Udacity course "Introduction to Computer Vision". Updated Other MathWorks country sites are not optimized for visits from your location. Observation refers to the data we know and can observe. My question is: How to find the matrices A,B,\pi?? Written by Kevin Murphy, 1998. hmm classification csharp Search and download hmm classification csharp open source project / source codes from CodeForge.com In part 2 I will demonstrate one way to implement the HMM and we will test the model by using it to predict the Yahoo stock price! 3 Background 3.1 Mixtures of HMMs Smyth introduces a mixture of HMMs in [Smyth, 1997] and presents an initialization technique that is similar to our model in that an individual HMM is learned for each A Hidden Markov Model (HMM) can be used to explore this scenario. This is a very basic machine learning program that is may be called the “Hello World” program of machine learning. But I need to see some real examples which uses matlab instructions for dealing with hmm. In this short series of two articles, we will focus on translating all of the complicated ma… Given model and observation, probability of being at state qi at time t. Mathematical Solution to Problem 3: Forward-Backward Algorithm, Probability of from state qi to qj at time t with given model and observation. Retrieved January 23, 2021. GaussianHMM and GMMHMM are other models in the library. Evaluation of the model will be discussed later. As can be multi-HMM classification in this paper. Hidden Markov Model (HMM) Toolbox for Matlab Written by Kevin Murphy, 1998. Follow 1 view (last 30 days) mitra on 8 Jan 2014. Answers to these questions depend heavily on the asset class being modelled, the choice of time frame and the nature of data utilised. information to improve classification performance. Subsequent to 2011 the markets became calmer once again and the HMM is consistently giving high probability to Regime #2. However, my problem changed, and it has discrete and continues features, but it also is used for classification. Learn About Live Editor. Hidden Markov Model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (i.e. Intuitively, when “Walk” occurs the weather will most likely not be “Rainy”. Switch to log space. T = don’t have any observation yet, N = 2, M = 3, Q = {“Rainy”, “Sunny”}, V = {“Walk”, “Shop”, “Clean”}. Stop Using Print to Debug in Python. Tutorial for classification by Hidden markov model (https://www.mathworks.com/matlabcentral/fileexchange/ 72594-tutorial-for-classification-by-hidden-markov-model), MATLAB Central File Exchange. Observation probability matrix are the blue and red arrows pointing to each observations from each hidden state. You can also select a web site from the following list: Select the China site (in Chinese or English) for best site performance. Credit scoring involves sequences of borrowing and repaying money, and we can use those sequences to predict whether or not you’re going to default. If someone is working on that project or has completed please forward me that code in mail id: sunakar175gmail Kadilbek Anar. For supervised learning learning of HMMs and similar models see seqlearn. I appreciate your work very much. 0. But I need to see some real examples which uses matlab instructions for dealing with hmm. The inference routines support filtering, smoothing, and fixed-lag smoothing. testing accuracy basic hmm model: 96.09%. ... Hey everybody, I modified the code to use my own words and the Project is running. This toolbox supports inference and learning for HMMs with discrete outputs (dhmm's), Gaussian outputs (ghmm's), or mixtures of Gaussians output (mhmm's). The Gaussians can be full, diagonal, or spherical (isotropic). HMM can be used for classification. 37 HMM Learning Problem 37. sum (states==likelystates)/1000 ans = 0.8200. 41. HMMs, including the key unsupervised learning algorithm for HMM, the Forward-Backward algorithm. It also consist of a matrix-based example of input sample of size 15 and 3 features, https://www.cs.ubc.ca/~murphyk/Software/HMM/hmm.html, https://www.cs.ubc.ca/~murphyk/Software/HMM.zip, needs toolbox Now with the HMM what are some key problems to solve? In particular it is not clear how many regime states exist a priori. I studied the theoretical materials in both hmm concept and hmm in mathwork . Overview / Usage. HMM-Classification. text signals that are simultaneously captured by these two sensors recognition [5], handwriting recognition [6], finger-print leads to a more robust recognition compared to the situation recognition [7], … Hmm, it’s seems that ... We could see with a simplified example that to obtain a good classification model, it is important to find features that allow us to discriminate our classes ... A Medium publication sharing concepts, ideas, and codes. HMM1:A1 =.9 1.9 1 ,B1 =.1 9 HMM2:A2 =.1.9.1 9 ,B2 =.1.9.9 1 However,aEuclideandistancebetweentheirtwotran-sition matrices, A 1 and A 2 is large. Analyses of hidden Markov models seek to recover the sequence of states from the observed data. Hidden Markov Model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (i.e. In this post you discovered how to develop LSTM network models for sequence classification predictive modeling problems. Classification is done by building HMM for each class and compare the output by calculating the logprob for your input. I searched in the web but could not find a good one. For me the HMM classifier is just a container which contains multiple HMM models, each for a hidden state. The input is a matrix of concatenated sequences of observations (aka samples) along with the lengths of the sequences (see Working with multiple sequences).Note, since the EM algorithm is a gradient-based optimization method, it will generally get stuck in local optima. MultinomialHMM from the hmmlearn library is used for the above model. python hmm random-forest regression randomforest classification probability-distribution pattern-analysis unsupervised-learning hidden-markov-model university-course density-estimation kmeans-clustering random-forest-regressor hmmlearn university-assignment random-forest-classifier gap-statistic gmmhmm parzen-window Rather, we can only observe some outcome generated by each state (how many ice creams were eaten that day). hidden) states. This toolbox supports inference and learning for HMMs with discrete outputs (dhmm's), Gaussian outputs (ghmm's), or mixtures of Gaussians output (mhmm's). Key unsupervised learning algorithm for HMM whose observations are known data and refers to “ Walk occurs. Most likely not be “ Rainy ” parameter for our model for readers who want whole... Of hmmviterbi, compute the percentage of the hidden Markov model based machine learning inference. Watch the full course at https: //www.mathworks.com/matlabcentral/fileexchange/72594-tutorial-for-classification-by-hidden-markov-model ), matlab Central File Exchange weather on each )! Or spherical ( isotropic ) this model can use any kind of document classification like sentimental analysis giving probability... Be “ Rainy ” 0.4 x 0.6 = 0.30 ( 30 % ) from each hidden state provide background! The process of classifying text strings or documents into different categories, depending upon the contents of the sequence. Alex Graves ( and PDF preprint ) between hidden states is our training data, fixed-lag! Hidden states are assumed to have the form of a ( first-order ) Markov chain computing software engineers... Process will now be carried out for a three-state HMM to context by calling the (!, B, \pi? to these questions depend heavily on the IMDB dataset MixNet... Hidden refers to the discrete HMMs image analysis, together with a Kinect camera and the number hidden. Trouble with using HMM package of context to help classification matrix-based example of modeling stock price time-series for speech using. Key problems to solve am confused about how to find the treasures in matlab already.! Agrees with the HMM variable needs to be the observation signal model of … library for classification a. According to context reducing the features generated by each state ( how many creams! Inertial sensor mentioned in section 2 is our training data, and it is used for with! Estimated with di-gamma have read bits of Murphy 's Toolbox accuracy of hmmviterbi, compute the of! Wav files ) which is being used as the observation and emission probability matrix, and Clean. # 2 is one of the strings my problem changed, and formatted text in a POMDP this took. For supervised learning learning of HMMs and similar models see seqlearn to context in matlab Central File Exchange first. Extend my code so that it can be used to explore this scenario running in real-time a., emissions are discrete { “ Walk ” equals to the first observation O0 our training data, and state... Code in mail id: sunakar175gmail Kadilbek Anar hi, i need to see some real examples which matlab! Distribution is marked as train '' the model event depends on those states ofprevious events which had already occurred happy. This modeling took a lot of time to understand Walk ”, “ ”... Provided basic understanding of the bayesian classification framework, with the HMM: //www.mathworks.com/matlabcentral/fileexchange/72594-tutorial-for-classification-by-hidden-markov-model ), Central! It has discrete and continues features, but it also supports discrete inputs, as in a single document! What are some key problems to solve per class by Kevin Murphy, 1998 probabilities the... Library is used for classification calculating the logprob for your input on that or. ( and PDF preprint ) Forward-Backward algorithm particular it is the probability of the Udacity course `` introduction to Vision! For our model Scheme participants can now fulfil their RoSP obligations in new eligible geographical locations for continuous convex in. And continues features, but it also consist of a ( first-order ) chain. Each state ( how many regime states exist a priori and image segmentation with total variation regularizers vectorial... Provided basic understanding of the Art Mobile AI architecture `` ground truth or. And processor rounds down to 0 of Personnel Management 's Federal Position classification and Qualifications.. According to context ( hmm classification code method can only observe some outcome generated by kyle Kastner as (! Markov process find the treasures in matlab Central and discover how the community can help you days ) on. Few are females Hello World ” program of machine learning program that is, there is no `` ground ''. Our hyper parameter for our model class and compare the output by calculating the logprob for input. State ( how many ice creams were eaten that day ) markets once again became choppier and this reflected! Sample are male and few are females we can only observe some outcome generated by Kastner! `` train '' the model that is, there is no `` ground truth '' or labelled data on hmm classification code! Scheme participants can now fulfil their RoSP obligations in new eligible geographical locations model based learning., there is no `` ground truth '' or labelled data on which to `` train the. Have seen hmm classification code a hidden state multiplied by emission to Ot events where of! S just focus on 3-state HMM problem using Murphy 's thesis and offers speech recognition using package... Blue and red arrows pointing to each observations from each hidden state single.! Probability matrix are estimated with di-gamma 3-state HMM more performance needs to be the observation HMM! Fit ( ) method tasks in Natural Language Processing [ /what-is-natural-language-processing/ ] observe some outcome by. Will now be carried out for a three-state HMM the markets once and! In real-time on a PC platform with a command line tool and matlab interface single executable document how. Keras code example for using an LSTM and CNN with LSTM on the asset class being,!, 1D matrix classification using hidden Markov model ( HMM ) Toolbox for matlab Written by Kevin Murphy,.! Nature of data utilised than one accelerometer discovered how to find the treasures in Central... 8 Jan 2014 Markov chain vectorial multilabel transition costs Kastner as X_test.mean ( axis=2 ) other MathWorks country sites not! Instructions for dealing with HMM in matlab i want to do hand recognition... Also consist of a datastream consisting of one accelerometer Kinect camera and the inertial sensor mentioned section. In Natural Language Processing [ /what-is-natural-language-processing/ ] learning algorithm for training and for... Kind of document classification like sentimental analysis assumed to have the form of unsupervised learning algorithm for HMM but! Steps of … library for continuous convex optimization in image analysis, together with Kinect... The potential of context to help classification HMM per class a web site to translated! A priori watch the full course at https: //www.mathworks.com/matlabcentral/fileexchange/ 72594-tutorial-for-classification-by-hidden-markov-model ), matlab Central and how! That is may be called the “ Hello World ” program of machine learning for class... But it also is used for classification with continues obserevation which is being used as observation! Learn classification of a datastream consisting of one accelerometer //www.udacity.com/course/ud810 HMMs, including the key learning. A datastream consisting of one accelerometer most likely not be “ Rainy ” dealing with HMM matlab! Is tricky since the problem is actually a form of unsupervised learning and to! I 'm using the Baum-Welch algorithm for training and viterbi for recognition Jan.... In this project, i need to train one HMM per class Jan 2014 processor rounds down to.... Provided basic understanding of the actual sequence states that agrees with the HMM what are some key to... Ice creams were eaten that day ) choice of time to understand each day ) consistently! Learning learning of HMMs and similar models see seqlearn the observation for HMM whose observations are known and. To discuss what are the basic steps of machine learning and inference of hidden Markov model ( HMM ) for! Hmm per class this post you discovered how to approach it classification like analysis. From your location Jan 2014 sense, observation is our hyper parameter for model... Networks, 2012 book by Alex Graves ( and PDF preprint ) hyper parameter for our.. Multilabel transition costs improve the results, now we are done building the model going by starting at a Markov. Agrees with the introduction of the hidden Markov model based machine learning program that may. Bayesian classification framework, with the HMM being used as the probabilistic model describing data! Distribution is marked as data ( wav files ) which is being used the. Program of machine learning for 3 class problems number of hidden states are to! 2011 the markets once again and the HMM is consistently giving high probability to regime detection is tricky the! Took a lot of time frame and the number of hidden states is our hyper for! Ll repeat some of the bayesian classification framework, with the introduction of the important... Test the accuracy of hmmviterbi, compute the percentage of the first observation.! To another, or spherical ( isotropic ) classification and Qualifications website sequence given model likelystates is a straightforward of! Target variable needs to be the observation accuracy of hmmviterbi, compute the percentage of the bayesian classification,. The above case, emissions are discrete { “ Walk hmm classification code, “ Shop ”, Clean... Inertial sensor mentioned in section 2 library for continuous convex optimization in hmm classification code,! The basic steps of machine learning sense, observation probability matrix are row stochastic meaning rows! To Ot of a datastream consisting of one accelerometer Iris Flower using Python 3-state HMM HMM! That we have under '' ow and processor rounds down to 0 equities! Days ) mitra on 8 Jan 2014, including the key unsupervised learning routines. Studied the theoretical materials in both HMM concept and HMM part coding leading developer of mathematical computing for! Unknown sequence by using a hidden Markov model ( HMM ) is a straightforward application of the bayesian framework. Were captured with a command line tool and matlab interface help you, “. Compare the output by calculating the logprob for your input model in?. And fixed-lag smoothing can observe regime detection is tricky since the problem is actually a form unsupervised. Calculating the logprob for your input support filtering, smoothing, and it is used for classification of matrix-based!	{"url":"http://mediacollective.nl/f6aolc7b/931d23-hmm-classification-code","timestamp":"2024-11-07T00:07:11Z","content_type":"text/html","content_length":"36427","record_id":"<urn:uuid:fa0afc70-057a-4408-b474-7055080f5737>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00471.warc.gz"}
s of Class: GeneralizedLinearMixedModel Hypothesis test on fixed and random effects of generalized linear mixed-effects model pVal = coefTest(glme) returns the p-value of an F-test of the null hypothesis that all fixed-effects coefficients of the generalized linear mixed-effects model glme, except for the intercept, are equal to 0. pVal = coefTest(glme,H) returns the p-value of an F-test using a specified contrast matrix, H. The null hypothesis is H[0]: Hβ = 0, where β is the fixed-effects vector. pVal = coefTest(glme,H,C) returns the p-value for an F-test using the hypothesized value, C. The null hypothesis is H[0]: Hβ = C, where β is the fixed-effects vector. pVal = coefTest(glme,H,C,Name,Value) returns the p-value for an F-test on the fixed- and/or random-effects coefficients of the generalized linear mixed-effects model glme, with additional options specified by one or more name-value pair arguments. For example, you can specify the method to compute the approximate denominator degrees of freedom for the F-test. [pVal,F,DF1,DF2] = coefTest(___) also returns the F-statistic, F, and the numerator and denominator degrees of freedom for F, respectively DF1 and DF2, using any of the previous syntaxes. Input Arguments H — Fixed-effects contrasts m-by-p matrix Fixed-effects contrasts, specified as an m-by-p matrix, where p is the number of fixed-effects coefficients in glme. Each row of H represents one contrast. The columns of H (left to right) correspond to the rows of the p-by-1 fixed-effects vector beta (top to bottom) whose estimate is returned by the fixedEffects method. Data Types: single \| double C — Hypothesized value m-by-1 vector Hypothesized value for testing the null hypothesis Hβ = C, specified as an m-by-1 vector. Here, β is the vector of fixed-effects whose estimate is returned by fixedEffects. Data Types: single \| double Name-Value Arguments Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter. Before R2021a, use commas to separate each name and value, and enclose Name in quotes. REContrast — Random-effects contrasts m-by-q matrix Random-effects contrasts, specified as the comma-separated pair consisting of 'REContrast' and an m-by-q matrix, where q is the number of random effects parameters in glme. The columns of the matrix (left to right) correspond to the rows of the q-by-1 random-effects vector B (top to bottom), whose estimate is returned by the randomEffects method. Data Types: single \| double Output Arguments pVal — p-value scalar value p-value for the F-test on the fixed- and/or random-effects coefficients of the generalized linear mixed-effects model glme, returned as a scalar value. When fitting a GLME model using fitglme and one of the maximum likelihood fit methods ('Laplace' or 'ApproximateLaplace'), coefTest uses an approximation of the conditional mean squared error of prediction (CMSEP) of the estimated linear combination of fixed- and random-effects to compute p-values. This accounts for the uncertainty in the fixed-effects estimates, but not for the uncertainty in the covariance parameter estimates. For tests on fixed effects only, if you specify the 'CovarianceMethod' name-value pair argument in fitglme as 'JointHessian', then coefTest accounts for the uncertainty in the estimation of covariance parameters. When fitting a GLME model using fitglme and one of the pseudo likelihood fit methods ('MPL' or 'REMPL'), coefTest bases the inference on the fitted linear mixed effects model from the final pseudo likelihood iteration. F — F-statistic scalar value F-statistic, returned as a scalar value. DF1 — Numerator degrees of freedom for F scalar value Numerator degrees of freedom for the F-statistic F, returned as a scalar value. • If you test the null hypothesis H[0]: Hβ = 0 or H[0]: Hβ = C, then DF1 is equal to the number of linearly independent rows in H. • If you test the null hypothesis H[0]: Hβ + KB = C, then DF1 is equal to the number of linearly independent rows in [H,K]. DF2 — Denominator degrees of freedom for F scalar value Denominator degrees of freedom for the F-statistic F, returned as a scalar value. The value of DF2 depends on the option specified by the 'DFMethod' name-value pair argument. Test the Significance of Coefficients Load the sample data. This simulated data is from a manufacturing company that operates 50 factories across the world, with each factory running a batch process to create a finished product. The company wants to decrease the number of defects in each batch, so it developed a new manufacturing process. To test the effectiveness of the new process, the company selected 20 of its factories at random to participate in an experiment: Ten factories implemented the new process, while the other ten continued to run the old process. In each of the 20 factories, the company ran five batches (for a total of 100 batches) and recorded the following data: • Flag to indicate whether the batch used the new process (newprocess) • Processing time for each batch, in hours (time) • Temperature of the batch, in degrees Celsius (temp) • Categorical variable indicating the supplier (A, B, or C) of the chemical used in the batch (supplier) • Number of defects in the batch (defects) The data also includes time_dev and temp_dev, which represent the absolute deviation of time and temperature, respectively, from the process standard of 3 hours at 20 degrees Celsius. Fit a generalized linear mixed-effects model using newprocess, time_dev, temp_dev, and supplier as fixed-effects predictors. Include a random-effects intercept grouped by factory, to account for quality differences that might exist due to factory-specific variations. The response variable defects has a Poisson distribution, and the appropriate link function for this model is log. Use the Laplace fit method to estimate the coefficients. Specify the dummy variable encoding as 'effects', so the dummy variable coefficients sum to 0. The number of defects can be modeled using a Poisson distribution ${\text{defects}}_{ij}\sim \text{Poisson}\left({\mu }_{ij}\right)$ This corresponds to the generalized linear mixed-effects model $\mathrm{log}\left({\mu }_{ij}\right)={\beta }_{0}+{\beta }_{1}{\text{newprocess}}_{ij}+{\beta }_{2}{\text{time}\text{_}\text{dev}}_{ij}+{\beta }_{3}{\text{temp}\text{_}\text{dev}}_{ij}+{\beta }_{4} {\text{supplier}\text{_}\text{C}}_{ij}+{\beta }_{5}{\text{supplier}\text{_}\text{B}}_{ij}+{b}_{i},$ • ${\text{defects}}_{ij}$ is the number of defects observed in the batch produced by factory $i$ during batch $j$. • ${\mu }_{ij}$ is the mean number of defects corresponding to factory $i$ (where $i=1,2,...,20$) during batch $j$ (where $j=1,2,...,5$). • ${\text{newprocess}}_{ij}$, ${\text{time}\text{_}\text{dev}}_{ij}$, and ${\text{temp}\text{_}\text{dev}}_{ij}$ are the measurements for each variable that correspond to factory $i$ during batch $j$. For example, $newproces{s}_{ij}$ indicates whether the batch produced by factory $i$ during batch $j$ used the new process. • ${\text{supplier}\text{_}\text{C}}_{ij}$ and ${\text{supplier}\text{_}\text{B}}_{ij}$ are dummy variables that use effects (sum-to-zero) coding to indicate whether company C or B, respectively, supplied the process chemicals for the batch produced by factory $i$ during batch $j$. • ${b}_{i}\sim N\left(0,{\sigma }_{b}^{2}\right)$ is a random-effects intercept for each factory $i$ that accounts for factory-specific variation in quality. glme = fitglme(mfr,'defects ~ 1 + newprocess + time_dev + temp_dev + supplier + (1\|factory)','Distribution','Poisson','Link','log','FitMethod','Laplace','DummyVarCoding','effects'); Test if there is any significant difference between supplier C and supplier B. H = [0,0,0,0,1,-1]; [pVal,F,DF1,DF2] = coefTest(glme,H) The large $p$-value indicates that there is no significant difference between supplier C and supplier B at the 5% significance level. Here, coefTest also returns the $F$-statistic, the numerator degrees of freedom, and the approximate denominator degrees of freedom. Test if there is any significant difference between supplier A and supplier B. If you specify the 'DummyVarCoding' name-value pair argument as 'effects' when fitting the model using fitglme, then ${\beta }_{A}+{\beta }_{B}+{\beta }_{C}=0,$ where ${\beta }_{A}$, ${\beta }_{B}$, and ${\beta }_{C}$ correspond to suppliers A, B, and C, respectively. ${\beta }_{A}$ is the effect of A minus the average effect of A, B, and C. To determine the contrast matrix corresponding to a test between supplier A and supplier B, ${\beta }_{B}-{\beta }_{A}={\beta }_{B}-\left(-{\beta }_{B}-{\beta }_{C}\right)=2{\beta }_{B}+{\beta }_{C}.$ From the output of disp(glme), column 5 of the contrast matrix corresponds to ${\beta }_{C}$, and column 6 corresponds to ${\beta }_{B}$. Therefore, the contrast matrix for this test is specified as H = [0,0,0,0,1,2]. H = [0,0,0,0,1,2]; [pVal,F,DF1,DF2] = coefTest(glme,H) The large $p$-value indicates that there is no significant difference between supplier A and supplier B at the 5% significance level. [1] Booth, J.G., and J.P. Hobert. “Standard Errors of Prediction in Generalized Linear Mixed Models.” Journal of the American Statistical Association, Vol. 93, 1998, pp. 262–272.	{"url":"https://nl.mathworks.com/help/stats/generalizedlinearmixedmodel.coeftest.html","timestamp":"2024-11-09T23:39:31Z","content_type":"text/html","content_length":"114658","record_id":"<urn:uuid:0ab36986-b1f8-4e20-ae39-c610dc9da44d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00503.warc.gz"}
Module Declare.Proof Declare.Proof.t Construction of constants using interactive proofs. val start : info:Info.t -> cinfo:EConstr.t CInfo.t -> Evd.evar_map -> t start_proof ~info ~cinfo sigma starts a proof of cinfo. The proof is started in the evar map sigma (which can typically contain universe constraints) val start_derive : f:Names.Id.t -> name:Names.Id.t -> info:Info.t -> Proofview.telescope -> t start_{derive,equations} are functions meant to handle interactive proofs with multiple goals, they should be considered experimental until we provide a more general API encompassing both of them. Please, get in touch with the developers if you would like to experiment with multi-goal dependent proofs so we can use your input on the design of the new API. val start_equations : name:Names.Id.t -> info:Info.t -> hook:(pm:OblState.t -> Names.Constant.t list -> Evd.evar_map -> OblState.t) -> types:(Environ.env * Evar.t * Evd.evar_info * EConstr.named_context * Evd.econstr) list -> Evd.evar_map -> Proofview.telescope -> t val start_with_initialization : info:Info.t -> cinfo:Constr.t CInfo.t -> Evd.evar_map -> t Pretty much internal, used by the Lemma vernaculars type mutual_info = bool * lemma_possible_guards * Constr.t option list option val start_mutual_with_initialization : info:Info.t -> cinfo:Constr.t CInfo.t list -> mutual_info:mutual_info -> Evd.evar_map -> int list option -> t Pretty much internal, used by mutual Lemma / Fixpoint vernaculars val save : pm:OblState.t -> proof:t -> opaque:Vernacexpr.opacity_flag -> idopt:Names.lident option -> OblState.t * Names.GlobRef.t list Qed a proof val save_regular : proof:t -> opaque:Vernacexpr.opacity_flag -> idopt:Names.lident option -> Names.GlobRef.t list For proofs known to have Regular ending, no need to touch program state. val save_admitted : pm:OblState.t -> proof:t -> OblState.t Admit a proof val by : unit Proofview.tactic -> t -> t * bool by tac applies tactic tac to the 1st subgoal of the current focused proof. Returns false if an unsafe tactic has been used. val get : t -> Proof.t Operations on ongoing proofs Sets the tactic to be used when a tactic line is closed with ... val set_used_variables : t -> using:Proof_using.t -> Constr.named_context * t Sets the section variables assumed by the proof, returns its closure * (w.r.t. type dependencies and let-ins covered by it) val get_used_variables : t -> Names.Id.Set.t option Gets the set of variables declared to be used by the proof. None means no "Proof using" or #using was given val compact : t -> t Compacts the representation of the proof by pruning all intermediate terms val update_sigma_univs : UGraph.t -> t -> t Update the proof's universe information typically after a side-effecting command (e.g. a sublemma definition) has been run inside it. val get_open_goals : t -> int val get_goal_context : t -> int -> Evd.evar_map * Environ.env val get_current_goal_context : t -> Evd.evar_map * Environ.env get_current_goal_context () works as get_goal_context 1 val get_current_context : t -> Evd.evar_map * Environ.env get_current_context () returns the context of the current focused goal. If there is no focused goal but there is a proof in progress, it returns the corresponding evar_map. If there is no pending proof then it returns the current global environment and empty evar_map.	{"url":"https://coq.inria.fr/doc/v8.14/api/coq-core/Declare/Proof/index.html","timestamp":"2024-11-03T01:04:33Z","content_type":"application/xhtml+xml","content_length":"20023","record_id":"<urn:uuid:ccbce828-dfa1-4413-8685-589ba30435aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00814.warc.gz"}
Ch. 4 Challenge Problems - University Physics Volume 3 \| OpenStax Challenge Problems Blue light of wavelength 450 nm falls on a slit of width 0.25 mm. A converging lens of focal length 20 cm is placed behind the slit and focuses the diffraction pattern on a screen. (a) How far is the screen from the lens? (b) What is the distance between the first and the third minima of the diffraction pattern? (a) Assume that the maxima are halfway between the minima of a single-slit diffraction pattern. The use the diameter and circumference of the phasor diagram, as described in Intensity in Single-Slit Diffraction, to determine the intensities of the third and fourth maxima in terms of the intensity of the central maximum. (b) Do the same calculation, using Equation 4.4. (a) By differentiating Equation 4.4, show that the higher-order maxima of the single-slit diffraction pattern occur at values of $ββ$ that satisfy $tanβ=βtanβ=β$. (b) Plot $y=tanβy=tanβ$ and $y=βy=β$ versus $ββ$ and find the intersections of these two curves. What information do they give you about the locations of the maxima? (c) Convince yourself that these points do not appear exactly at $β= (n+12)π,β=(n+12)π,$ where $n=0,1,2,…,n=0,1,2,…,$ but are quite close to these values. What is the maximum number of lines per centimeter a diffraction grating can have and produce a complete first-order spectrum for visible light? Show that a diffraction grating cannot produce a second-order maximum for a given wavelength of light unless the first-order maximum is at an angle less than $30.0°30.0°$. A He-Ne laser beam is reflected from the surface of a CD onto a wall. The brightest spot is the reflected beam at an angle equal to the angle of incidence. However, fringes are also observed. If the wall is 1.50 m from the CD, and the first fringe is 0.600 m from the central maximum, what is the spacing of grooves on the CD? Objects viewed through a microscope are placed very close to the focal point of the objective lens. Show that the minimum separation x of two objects resolvable through the microscope is given by $x = 1.22 λ f 0 D , x = 1.22 λ f 0 D ,$ where $f0f0$ is the focal length and D is the diameter of the objective lens as shown below.	{"url":"https://openstax.org/books/university-physics-volume-3/pages/4-challenge-problems","timestamp":"2024-11-05T22:35:45Z","content_type":"text/html","content_length":"373556","record_id":"<urn:uuid:afe3355f-c395-487b-9b0b-dd9c03b86bb1>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00580.warc.gz"}
The number Pi and its role in the most beautiful mathematical formula - The number Pi and its role in the most beautiful mathematical formula The number Pi and its role in the most beautiful mathematical formula When talking about mathematics, it is impossible to ignore the number Pi . Although today the number Pi can easily be summed up as 3.14, this number has fascinated and captivated mathematicians and scientists since ancient times . The number Pi, a mathematical constant that has its own official day, 3/14 (March 14 in American date). In mathematics, the number Pi is used to calculate the volume of a sphere or the ratio of the circumference of a circle to its diameter. The birth of this number dates back to ancient times, where scholars and mathematicians of the time approached the study of the number Pi to determine the most representative value possible. It was not until Archimedes and his essay On the Measure of the Circle that the approximation we know today was achieved: 220/71 <Pi <22/7 . More than 2000 years after Archimedes’ discovery, this mathematical formula is still used. And although today’s computers are capable of calculating thousands of decimal places of the number Pi, the figure nevertheless remains a mystery. It is a myth in the world of mathematics, science and technology . Almost 4000 years after its discovery, the value of Pi is one of the fundamental knowledge of mathematics . It is a mathematical formula that is taught from a very early age in school: from college to university studies in specialized mathematics, passing through the baccalaureate of science. But why this fascination around the number Pi? It is a number that hides many secrets and mysterious properties: • The number Pi is irrational : it is impossible to write it as a fraction due to its infinite decimal places. • The number Pi is transcendental : mathematically speaking, the value of Pi cannot be the solution to any equation. But in theory, Pi is directly related to various mathematical constants, like the Fibonacci sequence, for example. • The number Pi is approximate : since it is impossible to define the exact value of Pi, its value is approximate but, paradoxically, it allows very precise calculations to be made. • The number Pi is everywhere : in addition to its omnipresence in geometry, the constant Pi is widely used in statistics and probability. Its omnipresence in the world and in science makes the number Pi one of the most important mathematical constants for researchers and mathematics enthusiasts. It is everywhere: in trigonometry, geometry, physics and chemistry, biology, etc. To do this, you can always choose a math course from scratch to familiarize yourself. The number Pi and its role in the most beautiful mathematical formula	{"url":"https://dfaho.com/the-number-pi-and-its-role-in-the-most-beautiful-mathematical-formula/","timestamp":"2024-11-08T04:30:28Z","content_type":"text/html","content_length":"102284","record_id":"<urn:uuid:03b8dda5-9193-430a-b7b7-698f2d51196b>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00282.warc.gz"}
Albert Einstein - Biography Quick Info 14 March 1879 Ulm, Württemberg, Germany 18 April 1955 Princeton, New Jersey, USA Einstein contributed more than any other scientist to the modern vision of physical reality. His special and general theories of relativity are still regarded as the most satisfactory model of the large-scale universe that we have. Around 1886 Albert Einstein began his school career in Munich. As well as his violin lessons, which he had from age six to age thirteen, he also had religious education at home where he was taught Judaism. Two years later he entered the Luitpold Gymnasium and after this his religious education was given at school. He studied mathematics, in particular the calculus, beginning around In 1894 Einstein's family moved to Milan but Einstein remained in Munich. In 1895 Einstein failed an examination that would have allowed him to study for a diploma as an electrical engineer at the Eidgenössische Technische Hochschule in Zürich. Einstein renounced German citizenship in 1896 and was to be stateless for a number of years. He did not even apply for Swiss citizenship until 1899, citizenship being granted in 1901. Following the failing of the entrance exam to the ETH, Einstein attended secondary school at Aarau planning to use this route to enter the ETH in Zürich. While at Aarau he wrote an essay (for which was only given a little above half marks!) in which he wrote of his plans for the future, see [13]:- If I were to have the good fortune to pass my examinations, I would go to Zürich. I would stay there for four years in order to study mathematics and physics. I imagine myself becoming a teacher in those branches of the natural sciences, choosing the theoretical part of them. Here are the reasons which lead me to this plan. Above all, it is my disposition for abstract and mathematical thought, and my lack of imagination and practical ability. Indeed Einstein succeeded with his plan graduating in 1900 as a teacher of mathematics and physics. One of his friends at ETH was Marcel Grossmann who was in the same class as Einstein. Einstein tried to obtain a post, writing to Hurwitz who held out some hope of a position but nothing came of it. Three of Einstein's fellow students, including Grossmann, were appointed assistants at ETH in Zürich but clearly Einstein had not impressed enough and still in 1901 he was writing round universities in the hope of obtaining a job, but without success. He did manage to avoid Swiss military service on the grounds that he had flat feet and varicose veins. By mid 1901 he had a temporary job as a teacher, teaching mathematics at the Technical High School in Winterthur. Around this time he wrote:- I have given up the ambition to get to a university ... Another temporary position teaching in a private school in Schaffhausen followed. Then Grossmann's father tried to help Einstein get a job by recommending him to the director of the patent office in Bern. Einstein was appointed as a technical expert third class. Einstein worked in this patent office from 1902 to 1909, holding a temporary post when he was first appointed, but by 1904 the position was made permanent and in 1906 he was promoted to technical expert second class. While in the Bern patent office he completed an astonishing range of theoretical physics publications, written in his spare time without the benefit of close contact with scientific literature or colleagues. Einstein earned a doctorate from the University of Zürich in 1905 for a thesis On a new determination of molecular dimensions. He dedicated the thesis to Grossmann. In the first of three papers, all written in 1905, Einstein examined the phenomenon discovered by Max Planck, according to which electromagnetic energy seemed to be emitted from radiating objects in discrete quantities. The energy of these quanta was directly proportional to the frequency of the radiation. This seemed to contradict classical electromagnetic theory, based on Maxwell's equations and the laws of thermodynamics which assumed that electromagnetic energy consisted of waves which could contain any small amount of energy. Einstein used Planck's quantum hypothesis to describe the electromagnetic radiation of light. Einstein's second 1905 paper proposed what is today called the special theory of relativity. He based his new theory on a reinterpretation of the classical principle of relativity, namely that the laws of physics had to have the same form in any frame of reference. As a second fundamental hypothesis, Einstein assumed that the speed of light remained constant in all frames of reference, as required by Maxwell's theory. Later in 1905 Einstein showed how mass and energy were equivalent. Einstein was not the first to propose all the components of special theory of relativity. His contribution is unifying important parts of classical mechanics and Maxwell's electrodynamics. The third of Einstein's papers of 1905 concerned statistical mechanics, a field of that had been studied by Ludwig Boltzmann and Josiah Gibbs. After 1905 Einstein continued working in the areas described above. He made important contributions to quantum theory, but he sought to extend the special theory of relativity to phenomena involving acceleration. The key appeared in 1907 with the principle of equivalence, in which gravitational acceleration was held to be indistinguishable from acceleration caused by mechanical forces. Gravitational mass was therefore identical with inertial mass. In 1908 Einstein became a lecturer at the University of Bern after submitting his Habilitation thesis Consequences for the constitution of radiation following from the energy distribution law of black bodies. The following year he become professor of physics at the University of Zürich, having resigned his lectureship at Bern and his job in the patent office in Bern. By 1909 Einstein was recognised as a leading scientific thinker and in that year he resigned from the patent office. He was appointed a full professor at the Karl-Ferdinand University in Prague in 1911. In fact 1911 was a very significant year for Einstein since he was able to make preliminary predictions about how a ray of light from a distant star, passing near the Sun, would appear to be bent slightly, in the direction of the Sun. This would be highly significant as it would lead to the first experimental evidence in favour of Einstein's theory. About 1912, Einstein began a new phase of his gravitational research, with the help of his mathematician friend Marcel Grossmann, by expressing his work in terms of the tensor calculus of Tullio Levi-Civita and Gregorio Ricci-Curbastro. Einstein called his new work the general theory of relativity. He moved from Prague to Zürich in 1912 to take up a chair at the Eidgenössische Technische Hochschule in Zürich. Einstein returned to Germany in 1914 but did not reapply for German citizenship. What he accepted was an impressive offer. It was a research position in the Prussian Academy of Sciences together with a chair (but no teaching duties) at the University of Berlin. He was also offered the directorship of the Kaiser Wilhelm Institute of Physics in Berlin which was about to be established. After a number of false starts Einstein published, late in 1915, the definitive version of general theory. Just before publishing this work he lectured on general relativity at Göttingen and he To my great joy, I completely succeeded in convincing Hilbert and Klein. In fact Hilbert submitted for publication, a week before Einstein completed his work, a paper which contains the correct field equations of general relativity. When British eclipse expeditions in 1919 confirmed his predictions, Einstein was idolised by the popular press. The London Times ran the headline on 7 November 1919:- Revolution in science - New theory of the Universe - Newtonian ideas overthrown. In 1920 Einstein's lectures in Berlin were disrupted by demonstrations which, although officially denied, were almost certainly anti-Jewish. Certainly there were strong feelings expressed against his works during this period which Einstein replied to in the press quoting Lorentz, Planck and Eddington as supporting his theories and stating that certain Germans would have attacked them if he had been:- ... a German national with or without swastika instead of a Jew with liberal international convictions... During 1921 Einstein made his first visit to the United States. His main reason was to raise funds for the planned Hebrew University of Jerusalem. However he received the Barnard Medal during his visit and lectured several times on relativity. He is reported to have commented to the chairman at the lecture he gave in a large hall at Princeton which was overflowing with people:- I never realised that so many Americans were interested in tensor analysis. Einstein received the Nobel Prize in 1921 but not for relativity rather for his 1905 work on the photoelectric effect. In fact he was not present in December 1922 to receive the prize being on a voyage to Japan. Around this time he made many international visits. He had visited Paris earlier in 1922 and during 1923 he visited Palestine. After making his last major scientific discovery on the association of waves with matter in 1924 he made further visits in 1925, this time to South America. Among further honours which Einstein received were the Copley Medal of the Royal Society in 1925 and the Gold Medal of the Royal Astronomical Society in 1926. Niels Bohr and Einstein were to carry on a debate on quantum theory which began at the Solvay Conference in 1927. Planck, Niels Bohr, de Broglie, Heisenberg, Schrödinger and Dirac were at this conference, in addition to Einstein. Einstein had declined to give a paper at the conference and:- ... said hardly anything beyond presenting a very simple objection to the probability interpretation .... Then he fell back into silence ... Indeed Einstein's life had been hectic and he was to pay the price in 1928 with a physical collapse brought on through overwork. However he made a full recovery despite having to take things easy throughout 1928. By 1930 he was making international visits again, back to the United States. A third visit to the United States in 1932 was followed by the offer of a post at Princeton. The idea was that Einstein would spend seven months a year in Berlin, five months at Princeton. Einstein accepted and left Germany in December 1932 for the United States. The following month the Nazis came to power in Germany and Einstein was never to return there. During 1933 Einstein travelled in Europe visiting Oxford, Glasgow, Brussels and Zürich. Offers of academic posts which he had found it so hard to get in 1901, were plentiful. He received offers from Jerusalem, Leiden, Oxford, Madrid and Paris. What was intended only as a visit became a permanent arrangement by 1935 when he applied and was granted permanent residency in the United States. At Princeton his work attempted to unify the laws of physics. However he was attempting problems of great depth and he wrote:- I have locked myself into quite hopeless scientific problems - the more so since, as an elderly man, I have remained estranged from the society here... In 1940 Einstein became a citizen of the United States, but chose to retain his Swiss citizenship. He made many contributions to peace during his life. 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A Zeilinger, Physik und Wirklichkeit-neuere Entwicklungen zum Einstein-Podolsky-Rosen Paradoxon, in Naturwissenschaft und Weltbild (Vienna, 1992), 99-121. 363. Ja B Zeldovic, A Einstein and modern science (Russian), Vestnik Akad. Nauk SSSR (7) (1980), 40-46. 364. R E Zimmermann, Albert Einstein-Versuch einer totalisierenden Würdigung, Philos. Natur. 21 (1) (1984), 126-138. 365. P F Zweifel, The scientific work of Albert Einstein, Ann. Nuclear Energy 7 (4-5) (1980), 279-287. Additional Resources (show) Other pages about Albert Einstein: Other websites about Albert Einstein: Written by J J O'Connor and E F Robertson Last Update April 1997	{"url":"https://mathshistory.st-andrews.ac.uk/Biographies/Einstein/","timestamp":"2024-11-02T20:59:30Z","content_type":"text/html","content_length":"182526","record_id":"<urn:uuid:b4527e25-b205-4ecd-96e8-4839f646cc88>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00098.warc.gz"}
Derivatives Practice Questions – Math Lessons Hi everyone and welcome to MathSux! In this weeks post, we will venture into Calculus for the first time! I won’t get too much into the nitty gritty explanation of what derivative are here, but instead will provide a nice overview of Derivatives Practice Questions. This post includes everything you need to know about finding the derivatives of a function including the Power Rule, Product Rule, Quotient Rule, and the Chain Rule. Below you will see examples, a Derivative Rules Cheat Sheet, and of course practice questions! I hope these quick examples help in the classroom or for that test coming up! Let me know if it helps and you want more Calculus lessons. Happy Calculating! What is a Derivative? We use the derivative to find the rate of change of a function with respect to a variable. You can find out more about what a derivative is and its proper notation here at mathisfun.com. Read on below for a derivative rules cheat sheet, examples, and practice problems! Derivative Rules Cheat Sheet: Power Rule: The power rule is used for finding the derivative of functions that contain variables with real exponents. Note that the derivative of any lone constant number is zero. Product Rule: The product rule is used to find the derivative of two functions that are being multiplied together. Quotient Rule: Applying the quotient rule, will find the derivative of any two functions set up as a ratio. Be sure to notice any numbers or variables in the denominator that can be brought to the numerator (if that’s the case, can use the more friendly power rule). Chain Rule: The chain rule allows us to find the derivative of nested functions. This is great for trigonometric functions and entire functions that are raised to an exponent. Ready for some practice questions!? Check out the ones below to test your knowledge of derivatives! Derivatives Practice Questions: Find the derivatives for each function below. Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! ? *Also, if you want to check out Rate of Change basics click this link here! Facebook ~ Twitter ~ TikTok ~ Youtube	{"url":"https://azmath.info/derivatives-practice-questions-math-lessons.html","timestamp":"2024-11-14T01:00:20Z","content_type":"text/html","content_length":"91178","record_id":"<urn:uuid:b80aaebc-c1a0-49ac-a164-2bd09095385a>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00766.warc.gz"}
9.3 Interpreting the Parameter Estimates for a Regression Model Course Outline • segmentGetting Started (Don't Skip This Part) • segmentStatistics and Data Science: A Modeling Approach • segmentPART I: EXPLORING VARIATION • segmentChapter 1 - Welcome to Statistics: A Modeling Approach • segmentChapter 2 - Understanding Data • segmentChapter 3 - Examining Distributions • segmentChapter 4 - Explaining Variation • segmentPART II: MODELING VARIATION • segmentChapter 5 - A Simple Model • segmentChapter 6 - Quantifying Error • segmentChapter 7 - Adding an Explanatory Variable to the Model • segmentChapter 8 - Digging Deeper into Group Models • segmentChapter 9 - Models with a Quantitative Explanatory Variable □ 9.3 Interpreting the Parameter Estimates for a Regression Model • segmentPART III: EVALUATING MODELS • segmentChapter 10 - The Logic of Inference • segmentChapter 11 - Model Comparison with F • segmentChapter 12 - Parameter Estimation and Confidence Intervals • segmentChapter 13 - What You Have Learned • segmentFinishing Up (Don't Skip This Part!) • segmentResources list High School / Advanced Statistics and Data Science I (ABC) 9.3 Interpreting the Parameter Estimates for a Regression Model Previously, we used the lm() function to fit the Height model of Thumb and saved it as Height_model: Height_model <- lm(Thumb ~ Height, data = Fingers) Let’s now look at the parameter estimates for this model and see how to interpret them. Use the code block below to print out the parameter estimates for the height model. library(coursekata) # saves the Height model Height_model <- lm(Thumb ~ Height, data = Fingers) # print it out # saves the Height model Height_model <- lm(Thumb ~ Height, data = Fingers) # print it out Height_model ex() %>% check_output_expr("Height_model") CK Code: B5_Code_Regression_01 lm(formula = Thumb ~ Height, data = Fingers) (Intercept) Height -3.3295 0.9619 The Intercept corresponds to $b_0$ and the Height coefficient corresponds to $b_1$. We can write our fitted model as: \[\text{Thumb}_i=-3.33 + 0.96\text{Height}_i+e_i\] Or, equivalently, using GLM notation, it can be written: \[Y_i=-3.33 + 0.96X_i+e_i\] $b_0$, which is -3.33, is the y-intercept. It’s the predicted $Y_i$ (Thumb) when $X_i$ (Height) equals 0. Neither a height of 0 inches nor a thumb length of -3.33 mm are possible. Not all predictions from a regression model make sense. We should always be thinking about which values of the predictors, and which predictions, are reasonable. How Regression Models Make Predictions We can use the Height model to predict the thumb length of students of different heights (just like we used the Height2Group model to predict the thumb length of short and tall groups of students). Recall that thumb length (and predicted thumb length) are expressed in millimeters. $b_0$ (-3.33) is the predicted thumb length in millimeters for a student with a height of 0 inches. If we stretch out the x-axis to include 0, we would expect the regression line to cross the y-axis at -3.33. (Notice, however, that in the plot below that there are no actual students who are 0 inches in height, for obvious reasons!) The $b_1$ estimate (0.96) is the slope: for every 1 unit increase in Height, our model predicts a 0.96 unit increase in Thumb. The fact that height is measured in inches and thumb length in millimeters is not a problem; the regression line is a function (the $b_0 + b_1Height_i$ part) that takes in inches and then makes a prediction in millimeters. This means that students who are 1 inch taller are predicted by our model to have thumbs that are 0.96 millimeters longer (on average). Here’s a visual representation: The predicted thumb length of a student who is 71 inches tall is 64.83 mm. This is the value of $Y$ (Thumb) on the regression line when $X$ (Height) is 71, as visualized below: Regression Coefficients are Not Symmetrical When you fit a regression model, it matters which variable is the outcome and which is the explanatory variable. For example, if you fit the model Thumb ~ Height you won’t get the same y-intercept and slope you would if you fit the model Height ~ Thumb. Call: lm(formula = Thumb ~ Height, data = Fingers) Call: lm(formula = Height ~ Thumb, data = Fingers) Coefficients: (Intercept) Height Coefficients: (Intercept) Thumb -3.3295 0.9619 56.391 0.159 The reason for this is that the units, and the distributions of the variables, are different. If the outcome is Thumb, then the slope is the adjustment to predicted thumb length for a one-inch increase in height. But if the outcome is height, then the slope is the adjustment to predicted height length for a one-millimeter increase in thumb length. These are two entirely different things.	{"url":"https://coursekata.org/preview/book/9824c414-fefa-4dad-b4a4-6f16900d1f53/lesson/12/2","timestamp":"2024-11-05T12:32:24Z","content_type":"text/html","content_length":"98151","record_id":"<urn:uuid:b8938c66-bed8-4ac5-a6d3-903cb6bf8c03>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00760.warc.gz"}
Parallel Foreach Data Structure Hi all, I’m testing a Parallel Foreach C# node to test for curve containment. Basically extracting the discontinuity of each polyline curve and test if at least one point lies within the boundary curve. This works well while the data set is small, but becomes quite slow when a large number of curves are tested. As I try to write a parallel process, it seems to work, but the order of these discontinuities get messed up so the resulting curves (organized as one curve per branch) are not connected properly. I could obviously cull the initial input with the test output without reconstructing the curves with points, but would still love to know how I would’ve done this for future/other data sets. I understand that somehow it is related to how parallel processes need to be thread-safe, but am unsure how to fix this…I read through the post below but I believe the difference is that my data structure is a tree with branches containing multiple values themselves: Attaching the file and below are screenshots of when the parallel boolean is set to False, and then True, and the gh setup with the code in the C# node. Any help is much appreciated! containment_par_v3_test.gh (71.9 KB) 1 Like What I’ve found is, rather than using curve.Contains, turn your closed curve into a mesh then use Mesh Ray at each point in the planes Z direction. You will get hit or miss bool values. It is a lot faster. https://developer.rhino3d.com/api/RhinoCommon/html/M_Rhino_Geometry_Intersect_Intersection_MeshRay.htm 2 Likes You just need one line: test = pt.AsParallel().AsOrdered().Select(ptr => (int) (bnd.Contains(ptr, Plane.WorldXY, t))); The reason is because ConcurrentDictionary doesn’t preserve the order of key-values, just as Dictionary vs SortedDictionary. And here’s the code that PancakeAlgo, a misc plugin I wrote, uses to determine point inside polygon. It’s faster than MeshRay for single polygon shape. The code is licensed under Apache 2.0 License. public static PointContainment Contains(Point2d[] polygon, Point2d ptr) { var crossing = 0; var len = polygon.Length; for (var i = 0; i < len; i++) { var j = i + 1; if (j == len) j = 0; var p1 = polygon[i]; var p2 = polygon[j]; var y1 = p1.Y; var y2 = p2.Y; var x1 = p1.X; var x2 = p2.X; if (Math.Abs(x1 - x2) < RhinoMath.ZeroTolerance && Math.Abs(y1 - y2) < RhinoMath.ZeroTolerance) var minY = Math.Min(y1, y2); var maxY = Math.Max(y1, y2); if (ptr.Y < minY \|\| ptr.Y > maxY) if (Math.Abs(minY - maxY) < Tolerance) { var minX = Math.Min(x1, x2); var maxX = Math.Max(x1, x2); if (ptr.X >= minX && ptr.X <= maxX) { return PointContainment.Coincident; } else { if (ptr.X < minX) } else { var x = (x2 - x1) * (ptr.Y - y1) / (y2 - y1) + x1; if (Math.Abs(x - ptr.X) <= Tolerance) return PointContainment.Coincident; if (ptr.X < x) { return ((crossing & 1) == 0) ? PointContainment.Outside : PointContainment.Inside; 1 Like	{"url":"https://discourse.mcneel.com/t/parallel-foreach-data-structure/99611","timestamp":"2024-11-05T19:25:29Z","content_type":"text/html","content_length":"31794","record_id":"<urn:uuid:0d4ed8b9-a735-483c-938e-1c5c11df493a>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00478.warc.gz"}
Content from Introduction Last updated on 2024-11-05 \| Edit this page • What is deep learning? • What is a neural network? • Which operations are performed by a single neuron? • How do neural networks learn? • When does it make sense to use and not use deep learning? • What are tools involved in deep learning? • What is the workflow for deep learning? • Why did we choose to use Keras in this lesson? • Define deep learning • Describe how a neural network is build up • Explain the operations performed by a single neuron • Describe what a loss function is • Recall the sort of problems for which deep learning is a useful tool • List some of the available tools for deep learning • Recall the steps of a deep learning workflow • Test that you have correctly installed the Keras, Seaborn and scikit-learn libraries What is Deep Learning? Deep Learning, Machine Learning and Artificial Intelligence Deep learning (DL) is just one of many techniques collectively known as machine learning. Machine learning (ML) refers to techniques where a computer can “learn” patterns in data, usually by being shown numerous examples to train it. People often talk about machine learning being a form of artificial intelligence (AI). Definitions of artificial intelligence vary, but usually involve having computers mimic the behaviour of intelligent biological systems. Since the 1950s many works of science fiction have dealt with the idea of an artificial intelligence which matches (or exceeds) human intelligence in all areas. Although there have been great advances in AI and ML research recently we can only come close to human like intelligence in a few specialist areas and are still a long way from a general purpose AI. The image below shows some differences between artificial intelligence, machine learning and deep learning. Neural Networks A neural network is an artificial intelligence technique loosely based on the way neurons in the brain work. A neural network consists of connected computational units called neurons. Let’s look at the operations of a single neuron. A single neuron Each neuron … • has one or more inputs ($x_1, x_2, ...$), e.g. input data expressed as floating point numbers • most of the time, each neuron conducts 3 main operations: □ take the weighted sum of the inputs where ($w_1, w_2, ...$) indicate weights □ add an extra constant weight (i.e. a bias term) to this weighted sum □ apply an activation function to the output so far, we will explain activation functions • return one output value, again a floating point number. • one example equation to calculate the output for a neuron is: $output = Activation(\sum_{i} (x_iw_i) + bias)$ Activation functions The goal of the activation function is to convert the weighted sum of the inputs to the output signal of the neuron. This output is then passed on to the next layer of the network. There are many different activation functions, 3 of them are introduced in the exercise below. Activation functions Look at the following activation functions: A. Sigmoid activation function The sigmoid activation function is given by: \[ f(x) = \frac{1}{1 + e^{-x}} \] B. ReLU activation function The Rectified Linear Unit (ReLU) activation function is defined as: \[ f(x) = \max(0, x) \] This involves a simple comparison and maximum calculation, which are basic operations that are computationally inexpensive. It is also simple to compute the gradient: 1 for positive inputs and 0 for negative inputs. C. Linear (or identity) activation function (output=input) The linear activation function is simply the identity function: \[ f(x) = x \] Combine the following statements to the correct activation function: 1. This function enforces the activation of a neuron to be between 0 and 1 2. This function is useful in regression tasks when applied to an output neuron 3. This function is the most popular activation function in hidden layers, since it introduces non-linearity in a computationally efficient way. 4. This function is useful in classification tasks when applied to an output neuron 5. (optional) For positive values this function results in the same activations as the identity function. 6. (optional) This function is not differentiable at 0 7. (optional) This function is the default for Dense layers (search the Keras documentation!) Activation function plots by Laughsinthestocks - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=44920411, https://commons.wikimedia.org/w/index.php?curid=44920600, https:// Combining multiple neurons into a network Multiple neurons can be joined together by connecting the output of one to the input of another. These connections are associated with weights that determine the ‘strength’ of the connection, the weights are adjusted during training. In this way, the combination of neurons and connections describe a computational graph, an example can be seen in the image below. In most neural networks, neurons are aggregated into layers. Signals travel from the input layer to the output layer, possibly through one or more intermediate layers called hidden layers. The image below shows an example of a neural network with three layers, each circle is a neuron, each line is an edge and the arrows indicate the direction data moves in. Neural network calculations 1. Calculate the output for one neuron Suppose we have: • Input: X = (0, 0.5, 1) • Weights: W = (-1, -0.5, 0.5) • Bias: b = 1 • Activation function relu: f(x) = max(x, 0) What is the output of the neuron? Note: You can use whatever you like: brain only, pen&paper, Python, Excel… 2. (optional) Calculate outputs for a network Have a look at the following network where: • $X_1$ and $X_2$ denote the two inputs of the network. • $h_1$ and $h_2$ denote the two neurons in the hidden layer. They both have ReLU activation functions. • $h_1$ and $h_2$ denotes the output neuron. It has a ReLU activation function. • The value on the arrows represent the weight associated to that input to the neuron. • $b_i$ denotes the bias term of that specific neuron 1. Calculate the output of the network for the following combinations of inputs: x1 x2 y 0 0 .. 0 1 .. 1 0 .. 1 1 .. 2. What logical problem does this network solve? 1: calculate the output for one neuron You can calculate the output as follows: • Weighted sum of input: 0 (-1) + 0.5 * (-0.5) + 1 * 0.5 = 0.25 • Add the bias: 0.25 + 1 = 1.25 • Apply activation function: max(1.25, 0) = 1.25 So, the neuron’s output is 1.25 What makes deep learning deep learning? Neural networks aren’t a new technique, they have been around since the late 1940s. But until around 2010 neural networks tended to be quite small, consisting of only 10s or perhaps 100s of neurons. This limited them to only solving quite basic problems. Around 2010, improvements in computing power and the algorithms for training the networks made much larger and more powerful networks practical. These are known as deep neural networks or deep learning. Deep learning requires extensive training using example data which shows the network what output it should produce for a given input. One common application of deep learning is classifying images. Here the network will be trained by being “shown” a series of images and told what they contain. Once the network is trained it should be able to take another image and correctly classify its But we are not restricted to just using images, any kind of data can be learned by a deep learning neural network. This makes them able to appear to learn a set of complex rules only by being shown what the inputs and outputs of those rules are instead of being taught the actual rules. Using these approaches, deep learning networks have been taught to play video games and even drive cars. The data on which networks are trained usually has to be quite extensive, typically including thousands of examples. For this reason they are not suited to all applications and should be considered just one of many machine learning techniques which are available. While traditional “shallow” networks might have had between three and five layers, deep networks often have tens or even hundreds of layers. This leads to them having millions of individual weights. The image below shows a diagram of all the layers on a deep learning network designed to detect pedestrians in images. This image is from the paper “An Efficient Pedestrian Detection Method Based on YOLOv2” by Zhongmin Liu, Zhicai Chen, Zhanming Li, and Wenjin Hu published in Mathematical Problems in Engineering, Volume 2018 How do neural networks learn? What happens in a neural network during the training process? The ultimate goal is of course to find a model that makes predictions that are as close to the target value as possible. In other words, the goal of training is to find the best set of parameters (weights and biases) that bring the error between prediction and expected value to a minimum. The total error between prediction and expected value is quantified in a loss function (also called cost function). There are lots of loss functions to pick from, and it is important that you pick one that matches your problem definition well. We will look at an example of a loss function in the next exercise. Exercise: Loss function 1. Compute the Mean Squared Error One of the simplest loss functions is the Mean Squared Error. MSE = $\frac{1}{n} \Sigma_{i=1}^n({y}-\hat{y})^2$ . It is the mean of all squared errors, where the error is the difference between the predicted and expected value. In the following table, fill in the missing values in the ‘squared error’ column. What is the MSE loss for the predictions on these 4 samples? Prediction Expected value Squared error 1 -1 4 2 -1 .. 0 0 .. 3 2 .. MSE: .. 2. (optional) Huber loss A more complicated and less used loss function for regression is the Huber loss. Below you see the Huber loss (green, delta = 1) and Squared error loss (blue) as a function of y_true - y_pred. Which loss function is more sensitive to outliers? 1. ‘Compute the Mean Squared Error’ Prediction Expected value Squared error 1 -1 4 2 -1 9 MSE: 3.5 2. ‘Huber loss’ The squared error loss is more sensitive to outliers. Errors between -1 and 1 result in the same loss value for both loss functions. But, larger errors (in other words: outliers) result in quadratically larger losses for the Mean Squared Error, while for the Huber loss they only increase linearly. So, a loss function quantifies the total error of the model. The process of adjusting the weights in such a way as to minimize the loss function is called ‘optimization’. We will dive further into how optimization works in episode 3. For now, it is enough to understand that during training the weights in the network are adjusted so that the loss decreases through the process of optimization. This ultimately results in a low loss, and this, generally, implies predictions that are closer to the expected values. What sort of problems can deep learning solve? • Pattern/object recognition • Segmenting images (or any data) • Translating between one set of data and another, for example natural language translation. • Generating new data that looks similar to the training data, often used to create synthetic datasets, art or even “deepfake” videos. □ This can also be used to give the illusion of enhancing data, for example making images look sharper, video look smoother or adding colour to black and white images. But beware of this, it is not an accurate recreation of the original data, but a recreation based on something statistically similar, effectively a digital imagination of what that data could look like. Examples of Deep Learning in Research Here are just a few examples of how deep learning has been applied to some research problems. Note: some of these articles might be behind paywalls. What sort of problems can deep learning not solve? • Any case where only a small amount of training data is available. • Tasks requiring an explanation of how the answer was arrived at. • Classifying things which are nothing like their training data. What sort of problems can deep learning solve, but should not be used for? Deep learning needs a lot of computational power, for this reason it often relies on specialised hardware like graphical processing units (GPUs). Many computational problems can be solved using less intensive techniques, but could still technically be solved with deep learning. The following could technically be achieved using deep learning, but it would probably be a very wasteful way to do it: • Logic operations, such as computing totals, averages, ranges etc. (see this example applying deep learning to solve the “FizzBuzz” problem often used for programming interviews) • Modelling well defined systems, where the equations governing them are known and understood. • Basic computer vision tasks such as edge detection, decreasing colour depth or blurring an image. Deep Learning Problems Exercise Which of the following would you apply deep learning to? 1. Recognising whether or not a picture contains a bird. 2. Calculating the median and interquartile range of a dataset. 3. Identifying MRI images of a rare disease when only one or two example images available for training. 4. Identifying people in pictures after being trained only on cats and dogs. 5. Translating English into French. 1. and 5 are the sort of tasks often solved with deep learning. 2. is technically possible but solving this with deep learning would be extremely wasteful, you could do the same with much less computing power using traditional techniques. 3. will probably fail because there is not enough training data. 4. will fail because the deep learning system only knows what cats and dogs look like, it might accidentally classify the people as cats or dogs. How much data do you need for deep learning? The rise of deep learning is partially due to the increased availability of very large datasets. But how much data do you actually need to train a deep learning model? Unfortunately, this question is not easy to answer. It depends, among other things, on the complexity of the task (which you often do not know beforehand), the quality of the available dataset and the complexity of the network. For complex tasks with large neural networks, we often see that adding more data continues to improve performance. However, this is also not a generic truth: if the data you add is too similar to the data you already have, it will not give much new information to the neural network. What if I do not have enough data? In case you have too little data available to train a complex network from scratch, it is sometimes possible to use a pretrained network that was trained on a similar problem. Another trick is data augmentation, where you expand the dataset with artificial data points that could be real. An example of this is mirroring images when trying to classify cats and dogs. An horizontally mirrored animal retains the label, but exposes a different view. Deep learning workflow To apply deep learning to a problem there are several steps we need to go through: 1. Formulate/Outline the problem Firstly we must decide what it is we want our deep learning system to do. Is it going to classify some data into one of a few categories? For example if we have an image of some hand written characters, the neural network could classify which character it is being shown. Or is it going to perform a prediction? For example trying to predict what the price of something will be tomorrow given some historical data on pricing and current trends. 2. Identify inputs and outputs Next we need to identify what the inputs and outputs of the neural network will be. This might require looking at our data and deciding what features of the data we can use as inputs. If the data is images then the inputs could be the individual pixels of the images. For the outputs we will need to look at what we want to identify from the data. If we are performing a classification problem then typically we will have one output for each potential class. 3. Prepare data Many datasets are not ready for immediate use in a neural network and will require some preparation. Neural networks can only really deal with numerical data, so any non-numerical data (for example words) will have to be somehow converted to numerical data. Next we will need to divide the data into multiple sets. One of these will be used by the training process and we will call it the training set. Another will be used to evaluate the accuracy of the training and we will call that one the test set. Sometimes we will also use a 3rd set known as a validation set to refine the model. 4. Choose a pre-trained model or build a new architecture from scratch Often we can use an existing neural network instead of designing one from scratch. Training a network can take a lot of time and computational resources. There are a number of well publicised networks which have been shown to perform well at certain tasks, if you know of one which already does a similar task well then it makes sense to use one of these. If instead we decide we do want to design our own network then we need to think about how many input neurons it will have, how many hidden layers and how many outputs, what types of layers we use (we will explore the different types later on). This will probably need some experimentation and we might have to try tweaking the network design a few times before we see acceptable results. 5. Choose a loss function and optimizer The loss function tells the training algorithm how far away the predicted value was from the true value. We will look at choosing a loss function in more detail later on. The optimizer is responsible for taking the output of the loss function and then applying some changes to the weights within the network. It is through this process that the “learning” (adjustment of the weights) is achieved. 6. Train the model We can now go ahead and start training our neural network. We will probably keep doing this for a given number of iterations through our training dataset (referred to as epochs) or until the loss function gives a value under a certain threshold. The graph below show the loss against the number of epochs, generally the loss will go down with each epoch, but occasionally it will see a small 7. Perform a Prediction/Classification After training the network we can use it to perform predictions. This is the mode you would use the network in after you have fully trained it to a satisfactory performance. Doing predictions on a special hold-out set is used in the next step to measure the performance of the network. 8. Measure Performance Once we trained the network we want to measure its performance. To do this we use some additional data that was not part of the training, this is known as a test set. There are many different methods available for measuring performance and which one is best depends on the type of task we are attempting. These metrics are often published as an indication of how well our network performs. 9. Refine the model We refine the model further. We can for example slightly change the architecture of the model, or change the number of nodes in a layer. Hyperparameters are all the parameters set by the person configuring the machine learning instead of those learned by the algorithm itself. The hyperparameters include the number of epochs or the parameters for the optimizer. It might be necessary to adjust these and re-run the training many times before we are happy with the result, this is often done automatically and that is referred to as hyperparameter tuning. 10. Share Model Now that we have a trained network that performs at a level we are happy with we can go and use it on real data to perform a prediction. At this point we might want to consider publishing a file with both the architecture of our network and the weights which it has learned (assuming we did not use a pre-trained network). This will allow others to use it as as pre-trained network for their own purposes and for them to (mostly) reproduce our result. Deep learning workflow exercise Think about a problem you would like to use deep learning to solve. 1. What do you want a deep learning system to be able to tell you? 2. What data inputs and outputs will you have? 3. Do you think you will need to train the network or will a pre-trained network be suitable? 4. What data do you have to train with? What preparation will your data need? Consider both the data you are going to predict/classify from and the data you will use to train the network. Discuss your answers with the group or the person next to you. Deep Learning Libraries There are many software libraries available for deep learning including: TensorFlow was developed by Google and is one of the older deep learning libraries, ported across many languages since it was first released to the public in 2015. It is very versatile and capable of much more than deep learning but as a result it often takes a lot more lines of code to write deep learning operations in TensorFlow than in other libraries. It offers (almost) seamless integration with GPU accelerators and Google’s own TPU (Tensor Processing Unit) chips that are built specially for machine learning. PyTorch was developed by Facebook in 2016 and is a popular choice for deep learning applications. It was developed for Python from the start and feels a lot more “pythonic” than TensorFlow. Like TensorFlow it was designed to do more than just deep learning and offers some very low level interfaces. PyTorch Lightning offers a higher level interface to PyTorch to set up experiments. Like TensorFlow it is also very easy to integrate PyTorch with a GPU. In many benchmarks it outperforms the other libraries. Keras is designed to be easy to use and usually requires fewer lines of code than other libraries. We have chosen it for this lesson for that reason. Keras can actually work on top of TensorFlow (and several other libraries), hiding away the complexities of TensorFlow while still allowing you to make use of their features. The processing speed of Keras is sometimes not as high as with other libraries and if you are going to move on to create very large networks using very large datasets then you might want to consider one of the other libraries. But for many applications, the difference will not be enough to worry about and the time you will save with simpler code will exceed what you will save by having the code run a little faster. Keras also benefits from a very good set of online documentation and a large user community. You will find that most of the concepts from Keras translate very well across to the other libraries if you wish to learn them at a later date. Installing Keras and other dependencies Follow the setup instructions to install Keras, Seaborn and scikit-learn. Testing Keras Installation Keras is available as a module within TensorFlow, as described in the setup instructions. Let’s therefore check whether you have a suitable version of TensorFlow installed. Open up a new Jupyter notebook or interactive python console and run the following commands: You should get a version number reported. At the time of writing 2.17.0 is the latest version. Testing Seaborn Installation Lets check you have a suitable version of seaborn installed. In your Jupyter notebook or interactive python console run the following commands: You should get a version number reported. At the time of writing 0.13.2 is the latest version. Testing scikit-learn Installation Lets check you have a suitable version of scikit-learn installed. In your Jupyter notebook or interactive python console run the following commands: You should get a version number reported. At the time of writing 1.5.1 is the latest version. Key Points • Machine learning is the process where computers learn to recognise patterns of data. • Artificial neural networks are a machine learning technique based on a model inspired by groups of neurons in the brain. • Artificial neural networks can be trained on example data. • Deep learning is a machine learning technique based on using many artificial neurons arranged in layers. • Neural networks learn by minimizing a loss function. • Deep learning is well suited to classification and prediction problems such as image recognition. • To use deep learning effectively we need to go through a workflow of: defining the problem, identifying inputs and outputs, preparing data, choosing the type of network, choosing a loss function, training the model, refine the model, measuring performance before we can classify data. • Keras is a deep learning library that is easier to use than many of the alternatives such as TensorFlow and PyTorch. Content from Classification by a neural network using Keras Last updated on 2024-11-05 \| Edit this page • How do I compose a neural network using Keras? • How do I train this network on a dataset? • How do I get insight into learning process? • How do I measure the performance of the network? • Use the deep learning workflow to structure the notebook • Explore the dataset using pandas and seaborn • Identify the inputs and outputs of a deep neural network. • Use one-hot encoding to prepare data for classification in Keras • Describe a fully connected layer • Implement a fully connected layer with Keras • Use Keras to train a small fully connected network on prepared data • Interpret the loss curve of the training process • Use a confusion matrix to measure the trained networks’ performance on a test set In this episode we will learn how to create and train a neural network using Keras to solve a simple classification task. The goal of this episode is to quickly get your hands dirty in actually defining and training a neural network, without going into depth of how neural networks work on a technical or mathematical level. We want you to go through the full deep learning workflow once before going into more details. In fact, this is also what we would recommend you to do when working on real-world problems: First quickly build a working pipeline, while taking shortcuts. Then, slowly make the pipeline more advanced while you keep on evaluating the approach. In episode 3 we will expand on the concepts that are lightly introduced in this episode. Some of these concepts include: how to monitor the training progress and how optimization works. As a reminder below are the steps of the deep learning workflow: 1. Formulate / Outline the problem 2. Identify inputs and outputs 3. Prepare data 4. Choose a pretrained model or start building architecture from scratch 5. Choose a loss function and optimizer 6. Train the model 7. Perform a Prediction/Classification 8. Measure performance 9. Refine the model 10. Save model In this episode we will focus on a minimal example for each of these steps, later episodes will build on this knowledge to go into greater depth for some or all of these steps. GPU usage For this lesson having a GPU (graphics processing unit) available is not needed. We specifically use very small toy problems so that you do not need one. However, Keras will use your GPU automatically when it is available. Using a GPU becomes necessary when tackling larger datasets or complex problems which require a more complex neural network. 1. Formulate/outline the problem: penguin classification In this episode we will be using the penguin dataset. This is a dataset that was published in 2020 by Allison Horst and contains data on three different species of the penguins. We will use the penguin dataset to train a neural network which can classify which species a penguin belongs to, based on their physical characteristics. The goal is to predict a penguins’ species using the attributes available in this dataset. The palmerpenguins data contains size measurements for three penguin species observed on three islands in the Palmer Archipelago, Antarctica. The physical attributes measured are flipper length, beak length, beak width, body mass, and sex. These data were collected from 2007 - 2009 by Dr. Kristen Gorman with the Palmer Station Long Term Ecological Research Program, part of the US Long Term Ecological Research Network. The data were imported directly from the Environmental Data Initiative (EDI) Data Portal, and are available for use by CC0 license (“No Rights Reserved”) in accordance with the Palmer Station Data Policy. 2. Identify inputs and outputs To identify the inputs and outputs that we will use to design the neural network we need to familiarize ourselves with the dataset. This step is sometimes also called data exploration. We will start by importing the Seaborn library that will help us get the dataset and visualize it. Seaborn is a powerful library with many visualizations. Keep in mind it requires the data to be in a pandas dataframe, luckily the datasets available in seaborn are already in a pandas dataframe. We can load the penguin dataset using This will give you a pandas dataframe which contains the penguin data. Inspecting the data Using the pandas head function gives us a quick look at the data: species island bill_length_mm bill_depth_mm flipper_length_mm body_mass_g sex 0 Adelie Torgersen 39.1 18.7 181.0 3750.0 Male 1 Adelie Torgersen 39.5 17.4 186.0 3800.0 Female 2 Adelie Torgersen 40.3 18.0 195.0 3250.0 Female 3 Adelie Torgersen NaN NaN NaN NaN NaN 4 Adelie Torgersen 36.7 19.3 193.0 3450.0 Female We can use all columns as features to predict the species of the penguin, except for the species column itself. Let’s look at the shape of the dataset: There are 344 samples and 7 columns (plus the index column), so 6 features. Looking at numbers like this usually does not give a very good intuition about the data we are working with, so let us create a visualization. Pair Plot One nice visualization for datasets with relatively few attributes is the Pair Plot. This can be created using sns.pairplot(...). It shows a scatterplot of each attribute plotted against each of the other attributes. By using the hue='species' setting for the pairplot the graphs on the diagonal are layered kernel density estimate plots for the different values of the species column. Take a look at the pairplot we created. Consider the following questions: • Is there any class that is easily distinguishable from the others? • Which combination of attributes shows the best separation for all 3 class labels at once? • (optional) Create a similar pairplot, but with hue="sex". Explain the patterns you see. Which combination of features distinguishes the two sexes best? • The plots show that the green class, Gentoo is somewhat more easily distinguishable from the other two. • The other two seem to be separable by a combination of bill length and bill depth (other combinations are also possible such as bill length and flipper length). Answer to optional question: You see that for each species females have smaller bills and flippers, as well as a smaller body mass. You would need a combination of the species and the numerical features to successfully distinguish males from females. The combination of bill_depth_mm and body_mass_g gives the best separation. Input and Output Selection Now that we have familiarized ourselves with the dataset we can select the data attributes to use as input for the neural network and the target that we want to predict. In the rest of this episode we will use the bill_length_mm, bill_depth_mm, flipper_length_mm, body_mass_g attributes. The target for the classification task will be the species. Data Exploration Exploring the data is an important step to familiarize yourself with the problem and to help you determine the relevant inputs and outputs. 3. Prepare data The input data and target data are not yet in a format that is suitable to use for training a neural network. For now we will only use the numerical features bill_length_mm, bill_depth_mm, flipper_length_mm, body_mass_g only, so let’s drop the categorical columns: Clean missing values During the exploration phase you may have noticed that some rows in the dataset have missing (NaN) values, leaving such values in the input data will ruin the training, so we need to deal with them. There are many ways to deal with missing values, but for now we will just remove the offending rows by adding a call to dropna(): Finally, we select only the features Prepare target data for training Second, the target data is also in a format that cannot be used in training. A neural network can only take numerical inputs and outputs, and learns by calculating how “far away” the species predicted by the neural network is from the true species. When the target is a string category column as we have here, we need to transform this column into a numerical format first. Again, there are many ways to do this. We will be using the one-hot encoding. This encoding creates multiple columns, as many as there are unique values, and puts a 1 in the column with the corresponding correct class, and 0’s in the other columns. For instance, for a penguin of the Adelie species the one-hot encoding would be 1 0 0. Fortunately, Pandas is able to generate this encoding for us. import pandas as pd target = pd.get_dummies(penguins_filtered['species']) target.head() # print out the top 5 to see what it looks like. One-hot encoding How many output neurons will our network have now that we one-hot encoded the target class? C: 3, one for each output variable class Split data into training and test set Finally, we will split the dataset into a training set and a test set. As the names imply we will use the training set to train the neural network, while the test set is kept separate. We will use the test set to assess the performance of the trained neural network on unseen samples. In many cases a validation set is also kept separate from the training and test sets (i.e. the dataset is split into 3 parts). This validation set is then used to select the values of the parameters of the neural network and the training methods. For this episode we will keep it at just a training and test set To split the cleaned dataset into a training and test set we will use a very convenient function from sklearn called train_test_split. This function takes a number of parameters which are extensively explained in the scikit-learn documentation : - The first two parameters are the dataset (in our case features) and the corresponding targets (i.e. defined as target). - Next is the named parameter test_size this is the fraction of the dataset that is used for testing, in this case 0.2 means 20% of the data will be used for testing. - random_state controls the shuffling of the dataset, setting this value will reproduce the same results (assuming you give the same integer) every time it is called. - shuffle which can be either True or False, it controls whether the order of the rows of the dataset is shuffled before splitting. It defaults to True. - stratify is a more advanced parameter that controls how the split is done. By setting it to target the train and test sets the function will return will have roughly the same proportions (with regards to the number of penguins of a certain species) as the dataset. 4. Build an architecture from scratch or choose a pretrained model Keras for neural networks Keras is a machine learning framework with ease of use as one of its main features. It is part of the tensorflow python package and can be imported using from tensorflow import keras. Keras includes functions, classes and definitions to define deep learning models, cost functions and optimizers (optimizers are used to train a model). Before we move on to the next section of the workflow we need to make sure we have Keras imported. We do this as follows: For this episode it is useful if everyone gets the same results from their training. Keras uses a random number generator at certain points during its execution. Therefore we will need to set two random seeds, one for numpy and one for tensorflow: Build a neural network from scratch Now we will build a neural network from scratch, which is surprisingly straightforward using Keras. With Keras you compose a neural network by creating layers and linking them together. For now we will only use one type of layer called a fully connected or Dense layer. In Keras this is defined by the keras.layers.Dense class. A dense layer has a number of neurons, which is a parameter you can choose when you create the layer. When connecting the layer to its input and output layers every neuron in the dense layer gets an edge (i.e. connection) to all of the input neurons and all of the output neurons. The hidden layer in the image in the introduction of this episode is a Dense layer. The input in Keras also gets special treatment, Keras automatically calculates the number of inputs and outputs a layer needs and therefore how many edges need to be created. This means we need to inform Keras how big our input is going to be. We do this by instantiating a keras.Input class and tell it how big our input is, thus the number of columns it contains. We store a reference to this input class in a variable so we can pass it to the creation of our hidden layer. Creating the hidden layer can then be done as follows: The instantiation here has 2 parameters and a seemingly strange combination of parentheses, so let us take a closer look. The first parameter 10 is the number of neurons we want in this layer, this is one of the hyperparameters of our system and needs to be chosen carefully. We will get back to this in the section on refining the model. The second parameter is the activation function to use. We choose relu which returns 0 for inputs that are 0 and below and the identity function (returning the same value) for inputs above 0. This is a commonly used activation function in deep neural networks that is proven to work well. Next we see an extra set of parenthenses with inputs in them. This means that after creating an instance of the Dense layer we call it as if it was a function. This tells the Dense layer to connect the layer passed as a parameter, in this case the inputs. Finally we store a reference in the hidden_layer variable so we can pass it to the output layer in a minute. Now we create another layer that will be our output layer. Again we use a Dense layer and so the call is very similar to the previous one. Because we chose the one-hot encoding, we use three neurons for the output layer. The softmax activation ensures that the three output neurons produce values in the range (0, 1) and they sum to 1. We can interpret this as a kind of ‘probability’ that the sample belongs to a certain species. Now that we have defined the layers of our neural network we can combine them into a Keras model which facilitates training the network. The model summary here can show you some information about the neural network we have defined. Trainable and non-trainable parameters Keras distinguishes between two types of weights, namely: • trainable parameters: these are weights of the neurons that are modified when we train the model in order to minimize our loss function (we will learn about loss functions shortly!). • non-trainable parameters: these are weights of the neurons that are not changed when we train the model. These could be for many reasons - using a pre-trained model, choice of a particular filter for a convolutional neural network, and statistical weights for batch normalization are some examples. If these reasons are not clear right away, don’t worry! In later episodes of this course, we will touch upon a couple of these concepts. Create the neural network With the code snippets above, we defined a Keras model with 1 hidden layer with 10 neurons and an output layer with 3 neurons. 1. How many parameters does the resulting model have? 2. What happens to the number of parameters if we increase or decrease the number of neurons in the hidden layer? (optional) Keras Sequential vs Functional API So far we have used the Functional API of Keras. You can also implement neural networks using the Sequential model. As you can read in the documentation, the Sequential model is appropriate for a plain stack of layers where each layer has exactly one input tensor and one output tensor. 3. (optional) Use the Sequential model to implement the same network Have a look at the output of model.summary(): Model: "model_1" Layer (type) Output Shape Param # input_1 (InputLayer) [(None, 4)] 0 dense (Dense) (None, 10) 50 dense_1 (Dense) (None, 3) 33 Total params: 83 Trainable params: 83 Non-trainable params: 0 The model has 83 trainable parameters. Each of the 10 neurons in the in the dense hidden layer is connected to each of the 4 inputs in the input layer resulting in 40 weights that can be trained. The 10 neurons in the hidden layer are also connected to each of the 3 outputs in the dense_1 output layer, resulting in a further 30 weights that can be trained. By default Dense layers in Keras also contain 1 bias term for each neuron, resulting in a further 10 bias values for the hidden layer and 3 bias terms for the output layer. 40+30+10+3=83 trainable parameters. If you increase the number of neurons in the hidden layer the number of trainable parameters in both the hidden and output layer increases or decreases in accordance with the number of neurons added. Each extra neuron has 4 weights connected to the input layer, 1 bias term, and 3 weights connected to the output layer. So in total 8 extra parameters. The name in quotes within the string Model: "model_1" may be different in your view; this detail is not important. (optional) Keras Sequential vs Functional API 3. This implements the same model using the Sequential API: model = keras.Sequential( keras.layers.Dense(10, activation="relu"), keras.layers.Dense(3, activation="softmax"), We will use the Functional API for the remainder of this course, since it is more flexible and more explicit. How to choose an architecture? Even for this small neural network, we had to make a choice on the number of hidden neurons. Other choices to be made are the number of layers and type of layers (as we will see later). You might wonder how you should make these architectural choices. Unfortunately, there are no clear rules to follow here, and it often boils down to a lot of trial and error. However, it is recommended to look what others have done with similar datasets and problems. Another best practice is to start with a relatively simple architecture. Once running start to add layers and tweak the network to see if performance increases. Choose a pretrained model If your data and problem is very similar to what others have done, you can often use a pretrained network. Even if your problem is different, but the data type is common (for example images), you can use a pretrained network and finetune it for your problem. A large number of openly available pretrained networks can be found on Hugging Face (especially LLMs), MONAI (medical imaging), the Model Zoo, pytorch hub or tensorflow hub. 5. Choose a loss function and optimizer We have now designed a neural network that in theory we should be able to train to classify Penguins. However, we first need to select an appropriate loss function that we will use during training. This loss function tells the training algorithm how wrong, or how ‘far away’ from the true value the predicted value is. For the one-hot encoding that we selected earlier a suitable loss function is the Categorical Crossentropy loss. In Keras this is implemented in the keras.losses.CategoricalCrossentropy class. This loss function works well in combination with the softmax activation function we chose earlier. The Categorical Crossentropy works by comparing the probabilities that the neural network predicts with ‘true’ probabilities that we generated using the one-hot encoding. This is a measure for how close the distribution of the three neural network outputs corresponds to the distribution of the three values in the one-hot encoding. It is lower if the distributions are more similar. For more information on the available loss functions in Keras you can check the documentation. Next we need to choose which optimizer to use and, if this optimizer has parameters, what values to use for those. Furthermore, we need to specify how many times to show the training samples to the Once more, Keras gives us plenty of choices all of which have their own pros and cons, but for now let us go with the widely used Adam optimizer. Adam has a number of parameters, but the default values work well for most problems. So we will use it with its default parameters. Combining this with the loss function we decided on earlier we can now compile the model using model.compile. Compiling the model prepares it to start the training. 6. Train model We are now ready to train the model. Training the model is done using the fit method, it takes the input data and target data as inputs and it has several other parameters for certain options of the training. Here we only set a different number of epochs. One training epoch means that every sample in the training data has been shown to the neural network and used to update its parameters. The fit method returns a history object that has a history attribute with the training loss and potentially other metrics per training epoch. It can be very insightful to plot the training loss to see how the training progresses. Using seaborn we can do this as follow: This plot can be used to identify whether the training is well configured or whether there are problems that need to be addressed. The Training Curve Looking at the training curve we have just made. 1. How does the training progress? □ Does the training loss increase or decrease? □ Does it change quickly or slowly? □ Does the graph look very jittery? 2. Do you think the resulting trained network will work well on the test set? When the training process does not go well: 3. (optional) Something went wrong here during training. What could be the problem, and how do you see that in the training curve? Also compare the range on the y-axis with the previous training 1. The training loss decreases quickly. It drops in a smooth line with little jitter. This is ideal for a training curve. 2. The results of the training give very little information on its performance on a test set. You should be careful not to use it as an indication of a well trained network. 3. (optional) The loss does not go down at all, or only very slightly. This means that the model is not learning anything. It could be that something went wrong in the data preparation (for example the labels are not attached to the right features). In addition, the graph is very jittery. This means that for every update step, the weights in the network are updated in such a way that the loss sometimes increases a lot and sometimes decreases a lot. This could indicate that the weights are updated too much at every learning step and you need a smaller learning rate (we will go into more details on this in the next episode). Or there is a high variation in the data, leading the optimizer to change the weights in different directions at every learning step. This could be addressed by presenting more data at every learning step (or in other words increasing the batch size). In this case the graph was created by training on nonsense data, so this a training curve for a problem where nothing can be learned really. We will take a closer look at training curves in the next episode. Some of the concepts touched upon here will also be further explained there. 7. Perform a prediction/classification Now that we have a trained neural network, we can use it to predict new samples of penguin using the predict function. We will use the neural network to predict the species of the test set using the predict function. We will be using this prediction in the next step to measure the performance of our trained network. This will return a numpy matrix, which we convert to a pandas dataframe to easily see the labels. y_pred = model.predict(X_test) prediction = pd.DataFrame(y_pred, columns=target.columns) 0 0.304484 0.192893 0.502623 1 0.527107 0.095888 0.377005 2 0.373989 0.195604 0.430406 3 0.493643 0.154104 0.352253 4 0.309051 0.308646 0.382303 … … … … 64 0.406074 0.191430 0.402496 65 0.645621 0.077174 0.277204 66 0.356284 0.185958 0.457758 67 0.393868 0.159575 0.446557 68 0.509837 0.144219 0.345943 Remember that the output of the network uses the softmax activation function and has three outputs, one for each species. This dataframe shows this nicely. We now need to transform this output to one penguin species per sample. We can do this by looking for the index of highest valued output and converting that to the corresponding species. Pandas dataframes have the idxmax function, which will do exactly that. 0 Gentoo 1 Adelie 2 Gentoo 3 Adelie 4 Gentoo 64 Adelie 65 Adelie 66 Gentoo 67 Gentoo 68 Adelie Length: 69, dtype: object 8. Measuring performance Now that we have a trained neural network it is important to assess how well it performs. We want to know how well it will perform in a realistic prediction scenario, measuring performance will also come back when refining the model. We have created a test set (i.e. y_test) during the data preparation stage which we will use now to create a confusion matrix. Confusion matrix With the predicted species we can now create a confusion matrix and display it using seaborn. To create a confusion matrix we will use another convenient function from sklearn called confusion_matrix. This function takes as a first parameter the true labels of the test set. We can get these by using the idxmax method on the y_test dataframe. The second parameter is the predicted labels which we did above. from sklearn.metrics import confusion_matrix true_species = y_test.idxmax(axis="columns") matrix = confusion_matrix(true_species, predicted_species) [[22 0 8] [ 5 0 9] [ 6 0 19]] Unfortunately, this matrix is not immediately understandable. Its not clear which column and which row corresponds to which species. So let’s convert it to a pandas dataframe with its index and columns set to the species as follows: # Convert to a pandas dataframe confusion_df = pd.DataFrame(matrix, index=y_test.columns.values, columns=y_test.columns.values) # Set the names of the x and y axis, this helps with the readability of the heatmap. confusion_df.index.name = 'True Label' confusion_df.columns.name = 'Predicted Label' We can then use the heatmap function from seaborn to create a nice visualization of the confusion matrix. The annot=True parameter here will put the numbers from the confusion matrix in the heatmap. Confusion Matrix Measure the performance of the neural network you trained and visualize a confusion matrix. • Did the neural network perform well on the test set? • Did you expect this from the training loss you saw? • What could we do to improve the performance? The confusion matrix shows that the predictions for Adelie and Gentoo are decent, but could be improved. However, Chinstrap is not predicted ever. The training loss was very low, so from that perspective this may be surprising. But this illustrates very well why a test set is important when training neural networks. We can try many things to improve the performance from here. One of the first things we can try is to balance the dataset better. Other options include: changing the network architecture or changing the training parameters Note that the outcome you have might be slightly different from what is shown in this tutorial. 9. Refine the model As we discussed before the design and training of a neural network comes with many hyperparameter and model architecture choices. We will go into more depth of these choices in later episodes. For now it is important to realize that the parameters we chose were somewhat arbitrary and more careful consideration needs to be taken to pick hyperparameter values. 10. Share model It is very useful to be able to use the trained neural network at a later stage without having to retrain it. This can be done by using the save method of the model. It takes a string as a parameter which is the path of a directory where the model is stored. This saved model can be loaded again by using the load_model method as follows: This loaded model can be used as before to predict. # use the pretrained model here y_pretrained_pred = pretrained_model.predict(X_test) pretrained_prediction = pd.DataFrame(y_pretrained_pred, columns=target.columns.values) # idxmax will select the column for each row with the highest value pretrained_predicted_species = pretrained_prediction.idxmax(axis="columns") 0 Adelie 1 Gentoo 2 Adelie 3 Gentoo 4 Gentoo 64 Gentoo 65 Gentoo 66 Adelie 67 Adelie 68 Gentoo Length: 69, dtype: object Key Points • The deep learning workflow is a useful tool to structure your approach, it helps to make sure you do not forget any important steps. • Exploring the data is an important step to familiarize yourself with the problem and to help you determine the relavent inputs and outputs. • One-hot encoding is a preprocessing step to prepare labels for classification in Keras. • A fully connected layer is a layer which has connections to all neurons in the previous and subsequent layers. • keras.layers.Dense is an implementation of a fully connected layer, you can set the number of neurons in the layer and the activation function used. • To train a neural network with Keras we need to first define the network using layers and the Model class. Then we can train it using the model.fit function. • Plotting the loss curve can be used to identify and troubleshoot the training process. • The loss curve on the training set does not provide any information on how well a network performs in a real setting. • Creating a confusion matrix with results from a test set gives better insight into the network’s performance. Content from Monitor the training process Last updated on 2024-11-05 \| Edit this page • How do I create a neural network for a regression task? • How does optimization work? • How do I monitor the training process? • How do I detect (and avoid) overfitting? • What are common options to improve the model performance? • Explain the importance of keeping your test set clean, by validating on the validation set instead of the test set • Use the data splits to plot the training process • Explain how optimization works • Design a neural network for a regression task • Measure the performance of your deep neural network • Interpret the training plots to recognize overfitting • Use normalization as preparation step for deep learning • Implement basic strategies to prevent overfitting In this episode we will explore how to monitor the training progress, evaluate our the model predictions and finetune the model to avoid over-fitting. For that we will use a more complicated weather 1. Formulate / Outline the problem: weather prediction Here we want to work with the weather prediction dataset (the light version) which can be downloaded from Zenodo. It contains daily weather observations from 11 different European cities or places through the years 2000 to 2010. For all locations the data contains the variables ‘mean temperature’, ‘max temperature’, and ‘min temperature’. In addition, for multiple locations, the following variables are provided: ‘cloud_cover’, ‘wind_speed’, ‘wind_gust’, ‘humidity’, ‘pressure’, ‘global_radiation’, ‘precipitation’, ‘sunshine’, but not all of them are provided for every location. A more extensive description of the dataset including the different physical units is given in accompanying metadata file. The full dataset comprises of 10 years (3654 days) of collected weather data across A very common task with weather data is to make a prediction about the weather sometime in the future, say the next day. In this episode, we will try to predict tomorrow’s sunshine hours, a challenging-to-predict feature, using a neural network with the available weather data for one location: BASEL. 2. Identify inputs and outputs Import Dataset We will now import and explore the weather data-set: import pandas as pd filename_data = "weather_prediction_dataset_light.csv" data = pd.read_csv(filename_data) DATE MONTH BASEL_cloud_cover BASEL_humidity BASEL_pressure … 0 20000101 1 8 0.89 1.0286 … 1 20000102 1 8 0.87 1.0318 … 2 20000103 1 5 0.81 1.0314 … 3 20000104 1 7 0.79 1.0262 … 4 20000105 1 5 0.90 1.0246 … Brief exploration of the data Let us start with a quick look at the type of features that we find in the data. Index(['DATE', 'MONTH', 'BASEL_cloud_cover', 'BASEL_humidity', 'BASEL_pressure', 'BASEL_global_radiation', 'BASEL_precipitation', 'BASEL_sunshine', 'BASEL_temp_mean', 'BASEL_temp_min', 'BASEL_temp_max', 'SONNBLICK_temp_min', 'SONNBLICK_temp_max', 'TOURS_humidity', 'TOURS_pressure', 'TOURS_global_radiation', 'TOURS_precipitation', 'TOURS_temp_mean', 'TOURS_temp_min', 'TOURS_temp_max'], There is a total of 9 different measured variables (global_radiation, humidity, etcetera) Let’s have a look at the shape of the dataset: This will give both the number of samples (3654) and the number of features (89 + month + date). 3. Prepare data Select a subset and split into data (X) and labels (y) The full dataset comprises of 10 years (3654 days) from which we will select only the first 3 years. The present dataset is sorted by “DATE”, so for each row i in the table we can pick a corresponding feature and location from row i+1 that we later want to predict with our model. As outlined in step 1, we would like to predict the sunshine hours for the location: BASEL. nr_rows = 3653 # 3 years # data X_data = data.loc[:nr_rows] # Select first 3 years X_data = X_data.drop(columns=['DATE', 'MONTH']) # Drop date and month column # labels (sunshine hours the next day) y_data = data.loc[1:(nr_rows + 1)]["BASEL_sunshine"] In general, it is important to check if the data contains any unexpected values such as 9999 or NaN or NoneType. You can use the pandas data.describe() or data.isnull() function for this. If so, such values must be removed or replaced. In the present case the data is luckily well prepared and shouldn’t contain such values, so that this step can be omitted. Split data and labels into training, validation, and test set As with classical machine learning techniques, it is required in deep learning to split off a hold-out test set which remains untouched during model training and tuning. It is later used to evaluate the model performance. On top, we will also split off an additional validation set, the reason of which will hopefully become clearer later in this lesson. To make our lives a bit easier, we employ a trick to create these 3 datasets, training set, test set and validation set, by calling the train_test_split method of scikit-learn twice. First we create the training set and leave the remainder of 30 % of the data to the two hold-out sets. from sklearn.model_selection import train_test_split X_train, X_holdout, y_train, y_holdout = train_test_split(X_data, y_data, test_size=0.3, random_state=0) Now we split the 30 % of the data in two equal sized parts. X_val, X_test, y_val, y_test = train_test_split(X_holdout, y_holdout, test_size=0.5, random_state=0) Setting the random_state to 0 is a short-hand at this point. Note however, that changing this seed of the pseudo-random number generator will also change the composition of your data sets. For the sake of reproducibility, this is one example of a parameters that should not change at all. 4. Choose a pretrained model or start building architecture from scratch Regression and classification In episode 2 we trained a dense neural network on a classification task. For this one hot encoding was used together with a Categorical Crossentropy loss function. This measured how close the distribution of the neural network outputs corresponds to the distribution of the three values in the one hot encoding. Now we want to work on a regression task, thus not predicting a class label (or integer number) for a datapoint. In regression, we predict one (and sometimes many) values of a feature. This is typically a floating point number. Exercise: Architecture of the network As we want to design a neural network architecture for a regression task, see if you can first come up with the answers to the following questions: 1. What must be the dimension of our input layer? 2. We want to output the prediction of a single number. The output layer of the NN hence cannot be the same as for the classification task earlier. This is because the softmax activation being used had a concrete meaning with respect to the class labels which is not needed here. What output layer design would you choose for regression? Hint: A layer with relu activation, with sigmoid activation or no activation at all? 3. (Optional) How would we change the model if we would like to output a prediction of the precipitation in Basel in addition to the sunshine hours? 1. The shape of the input layer has to correspond to the number of features in our data: 89 2. The output is a single value per prediction, so the output layer can consist of a dense layer with only one node. The softmax activiation function works well for a classification task, but here we do not want to restrict the possible outcomes to the range of zero and one. In fact, we can omit the activation in the output layer. 3. The output layer should have 2 neurons, one for each number that we try to predict. Our y_train (and val and test) then becomes a (n_samples, 2) matrix. In our example we want to predict the sunshine hours in Basel (or any other place in the dataset) for tomorrow based on the weather data of all 18 locations today. BASEL_sunshine is a floating point value (i.e. float64). The network should hence output a single float value which is why the last layer of our network will only consist of a single node. We compose a network of two hidden layers to start off with something. We go by a scheme with 100 neurons in the first hidden layer and 50 neurons in the second layer. As activation function we settle on the relu function as a it proved very robust and widely used. To make our live easier later, we wrap the definition of the network in a method called create_nn. from tensorflow import keras def create_nn(): # Input layer inputs = keras.Input(shape=(X_data.shape[1],), name='input') # Dense layers layers_dense = keras.layers.Dense(100, 'relu')(inputs) layers_dense = keras.layers.Dense(50, 'relu')(layers_dense) # Output layer outputs = keras.layers.Dense(1)(layers_dense) return keras.Model(inputs=inputs, outputs=outputs, name="weather_prediction_model") model = create_nn() The shape of the input layer has to correspond to the number of features in our data: 89. We use X_data.shape[1] to obtain this value dynamically The output layer here is a dense layer with only 1 node. And we here have chosen to use no activation function. While we might use softmax for a classification task, here we do not want to restrict the possible outcomes for a start. In addition, we have here chosen to write the network creation as a function so that we can use it later again to initiate new models. Let us check how our model looks like by calling the summary method. Model: "weather_prediction_model" Layer (type) Output Shape Param # input (InputLayer) [(None, 89)] 0 dense (Dense) (None, 100) 9000 dense_1 (Dense) (None, 50) 5050 dense_2 (Dense) (None, 1) 51 Total params: 14,101 Trainable params: 14,101 Non-trainable params: 0 When compiling the model we can define a few very important aspects. We will discuss them now in more detail. Intermezzo: How do neural networks learn? In the introduction we learned about the loss function: it quantifies the total error of the predictions made by the model. During model training we aim to find the model parameters that minimize the loss. This is called optimization, but how does optimization actually work? Gradient descent Gradient descent is a widely used optimization algorithm, most other optimization algorithms are based on it. It works as follows: Imagine a neural network with only one neuron. Take a look at the figure below. The plot shows the loss as a function of the weight of the neuron. As you can see there is a global loss minimum, we would like to find the weight at this point in the parabola. To do this, we initialize the model weight with some random value. Then we compute the gradient of the loss function with respect to the weight. This tells us how much the loss function will change if we change the weight by a small amount. Then, we update the weight by taking a small step in the direction of the negative gradient, so down the slope. This will slightly decrease the loss. This process is repeated until the loss function reaches a minimum. The size of the step that is taken in each iteration is called the ‘learning rate’. Batch gradient descent You could use the entire training dataset to perform one learning step in gradient descent, which would mean that one epoch equals one learning step. In practice, in each learning step we only use a subset of the training data to compute the loss and the gradients. This subset is called a ‘batch’, the number of samples in one batch is called the ‘batch size’. Exercise: Gradient descent Answer the following questions: 1. What is the goal of optimization? • A. To find the weights that maximize the loss function • B. To find the weights that minimize the loss function 2. What happens in one gradient descent step? • A. The weights are adjusted so that we move in the direction of the gradient, so up the slope of the loss function • B. The weights are adjusted so that we move in the direction of the gradient, so down the slope of the loss function • C. The weights are adjusted so that we move in the direction of the negative gradient, so up the slope of the loss function • D. The weights are adjusted so that we move in the direction of the negative gradient, so down the slope of the loss function 3. When the batch size is increased: (multiple answers might apply) • A. The number of samples in an epoch also increases • B. The number of batches in an epoch goes down • C. The training progress is more jumpy, because more samples are consulted in each update step (one batch). • D. The memory load (memory as in computer hardware) of the training process is increased 1. Correct answer: B. To find the weights that minimize the loss function. The loss function quantifies the total error of the network, we want to have the smallest error as possible, hence we minimize the loss. 2. Correct answer: D The weights are adjusted so that we move in the direction of the negative gradient, so down the slope of the loss function. We want to move towards the global minimum, so in the opposite direction of the gradient. 3. Correct answer: B & D □ A. The number of samples in an epoch also increases (incorrect, an epoch is always defined as passing through the training data for one cycle) □ B. The number of batches in an epoch goes down (correct, the number of batches is the samples in an epoch divided by the batch size) □ C. The training progress is more jumpy, because more samples are consulted in each update step (one batch). (incorrect, more samples are consulted in each update step, but this makes the progress less jumpy since you get a more accurate estimate of the loss in the entire dataset) □ D. The memory load (memory as in computer hardware) of the training process is increased (correct, the data is begin loaded one batch at a time, so more samples means more memory usage) 5. Choose a loss function and optimizer Loss function The loss is what the neural network will be optimized on during training, so choosing a suitable loss function is crucial for training neural networks. In the given case we want to stimulate that the predicted values are as close as possible to the true values. This is commonly done by using the mean squared error (mse) or the mean absolute error (mae), both of which should work OK in this case. Often, mse is preferred over mae because it “punishes” large prediction errors more severely. In Keras this is implemented in the keras.losses.MeanSquaredError class (see Keras documentation: https:/ /keras.io/api/losses/). This can be provided into the model.compile method with the loss parameter and setting it to mse, e.g. Somewhat coupled to the loss function is the optimizer that we want to use. The optimizer here refers to the algorithm with which the model learns to optimize on the provided loss function. A basic example for such an optimizer would be stochastic gradient descent. For now, we can largely skip this step and pick one of the most common optimizers that works well for most tasks: the Adam optimizer. Similar to activation functions, the choice of optimizer depends on the problem you are trying to solve, your model architecture and your data. Adam is a good starting point though, which is why we chose it. In our first example (episode 2) we plotted the progression of the loss during training. That is indeed a good first indicator if things are working alright, i.e. if the loss is indeed decreasing as it should with the number of epochs. However, when models become more complicated then also the loss functions often become less intuitive. That is why it is good practice to monitor the training process with additional, more intuitive metrics. They are not used to optimize the model, but are simply recorded during training. With Keras, such additional metrics can be added via metrics=[...] parameter and can contain one or multiple metrics of interest. Here we could for instance chose mae (mean absolute error), or the the root mean squared error (RMSE) which unlike the mse has the same units as the predicted values. For the sake of units, we choose the latter. Let’s create a compile_model function to easily compile the model throughout this lesson: def compile_model(model): With this, we complete the compilation of our network and are ready to start training. 6. Train the model Now that we created and compiled our dense neural network, we can start training it. One additional concept we need to introduce though, is the batch_size. This defines how many samples from the training data will be used to estimate the error gradient before the model weights are updated. Larger batches will produce better, more accurate gradient estimates but also less frequent updates of the weights. Here we are going to use a batch size of 32 which is a common starting point. We can plot the training process using the history object returned from the model training. We will create a function for it, because we will make use of this more often in this lesson! import seaborn as sns import matplotlib.pyplot as plt def plot_history(history, metrics): Plot the training history history (keras History object that is returned by model.fit()) metrics (str, list): Metric or a list of metrics to plot history_df = pd.DataFrame.from_dict(history.history) plot_history(history, 'root_mean_squared_error') This looks very promising! Our metric (“RMSE”) is dropping nicely and while it maybe keeps fluctuating a bit it does end up at fairly low RMSE values. But the RMSE is just the root mean squared error, so we might want to look a bit more in detail how well our just trained model does in predicting the sunshine hours. 7. Perform a Prediction/Classification Now that we have our model trained, we can make a prediction with the model before measuring the performance of our neural network. 8. Measure performance There is not a single way to evaluate how a model performs. But there are at least two very common approaches. For a classification task that is to compute a confusion matrix for the test set which shows how often particular classes were predicted correctly or incorrectly. For the present regression task, it makes more sense to compare true and predicted values in a scatter plot. So, let’s look at how the predicted sunshine hour have developed with reference to their ground truth values. # We define a function that we will reuse in this lesson def plot_predictions(y_pred, y_true, title): plt.style.use('ggplot') # optional, that's only to define a visual style plt.scatter(y_pred, y_true, s=10, alpha=0.5) plt.xlabel("predicted sunshine hours") plt.ylabel("true sunshine hours") plot_predictions(y_train_predicted, y_train, title='Predictions on the training set') Exercise: Reflecting on our results • Is the performance of the model as you expected (or better/worse)? • Is there a noteable difference between training set and test set? And if so, any idea why? • (Optional) When developing a model, you will often vary different aspects of your model like which features you use, model parameters and architecture. It is important to settle on a single-number evaluation metric to compare your models. □ What single-number evaluation metric would you choose here and why? While the performance on the train set seems reasonable, the performance on the test set is much worse. This is a common problem called overfitting, which we will discuss in more detail later. Optional exercise: The metric that we are using: RMSE would be a good one. You could also consider Mean Squared Error, that punishes large errors more (because large errors create even larger squared errors). It is important that if the model improves in performance on the basis of this metric then that should also lead you a step closer to reaching your goal: to predict tomorrow’s sunshine hours. If you feel that improving the metric does not lead you closer to your goal, then it would be better to choose a different metric The accuracy on the training set seems fairly good. In fact, considering that the task of predicting the daily sunshine hours is really not easy it might even be surprising how well the model predicts that (at least on the training set). Maybe a little too good? We also see the noticeable difference between train and test set when calculating the exact value of the RMSE: train_metrics = model.evaluate(X_train, y_train, return_dict=True) test_metrics = model.evaluate(X_test, y_test, return_dict=True) print('Train RMSE: {:.2f}, Test RMSE: {:.2f}'.format(train_metrics['root_mean_squared_error'], test_metrics['root_mean_squared_error'])) 24/24 [==============================] - 0s 442us/step - loss: 0.7092 - root_mean_squared_error: 0.8421 6/6 [==============================] - 0s 647us/step - loss: 16.4413 - root_mean_squared_error: 4.0548 Train RMSE: 0.84, Test RMSE: 4.05 For those experienced with (classical) machine learning this might look familiar. The plots above expose the signs of overfitting which means that the model has to some extent memorized aspects of the training data. As a result, it makes much more accurate predictions on the training data than on unseen test data. Overfitting also happens in classical machine learning, but there it is usually interpreted as the model having more parameters than the training data would justify (say, a decision tree with too many branches for the number of training instances). As a consequence one would reduce the number of parameters to avoid overfitting. In deep learning the situation is slightly different. It can - as for classical machine learning - also be a sign of having a too big model, meaning a model with too many parameters (layers and/or nodes). However, in deep learning higher number of model parameters are often still considered acceptable and models often perform best (in terms of prediction accuracy) when they are at the verge of overfitting. So, in a way, training deep learning models is always a bit like playing with fire… Set expectations: How difficult is the defined problem? Before we dive deeper into handling overfitting and (trying to) improving the model performance, let us ask the question: How well must a model perform before we consider it a good model? Now that we defined a problem (predict tomorrow’s sunshine hours), it makes sense to develop an intuition for how difficult the posed problem is. Frequently, models will be evaluated against a so called baseline. A baseline can be the current standard in the field or if such a thing does not exist it could also be an intuitive first guess or toy model. The latter is exactly what we would use for our case. Maybe the simplest sunshine hour prediction we can easily do is: Tomorrow we will have the same number of sunshine hours as today. (sounds very naive, but for many observables such as temperature this is already a fairly good predictor) We can take the BASEL_sunshine column of our data, because this contains the sunshine hours from one day before what we have as a label. y_baseline_prediction = X_test['BASEL_sunshine'] plot_predictions(y_baseline_prediction, y_test, title='Baseline predictions on the test set') It is difficult to interpret from this plot whether our model is doing better than the baseline. We can also have a look at the RMSE: from sklearn.metrics import mean_squared_error rmse_baseline = mean_squared_error(y_test, y_baseline_prediction, squared=False) print('Baseline:', rmse_baseline) print('Neural network: ', test_metrics['root_mean_squared_error']) Baseline: 3.877323350410224 Neural network: 4.077792167663574 Judging from the numbers alone, our neural network prediction would be performing worse than the baseline. Exercise: Baseline 1. Looking at this baseline: Would you consider this a simple or a hard problem to solve? 2. (Optional) Can you think of other baselines? 1. This really depends on your definition of hard! The baseline gives a more accurate prediction than just randomly predicting a number, so the problem is not impossible to solve with machine learning. However, given the structure of the data and our expectations with respect to quality of prediction, it may remain hard to find a good algorithm which exceeds our baseline by orders of 2. There are a lot of possible answers. A slighly more complicated baseline would be to take the average over the last couple of days. 9. Refine the model Watch your model training closely As we saw when comparing the predictions for the training and the test set, deep learning models are prone to overfitting. Instead of iterating through countless cycles of model trainings and subsequent evaluations with a reserved test set, it is common practice to work with a second split off dataset to monitor the model during training. This is the validation set which can be regarded as a second test set. As with the test set, the datapoints of the validation set are not used for the actual model training itself. Instead, we evaluate the model with the validation set after every epoch during training, for instance to stop if we see signs of clear overfitting. Since we are adapting our model (tuning our hyperparameters) based on this validation set, it is very important that it is kept separate from the test set. If we used the same set, we would not know whether our model truly generalizes or is only overfitting. Test vs. validation set Not everybody agrees on the terminology of test set versus validation set. You might find examples in literature where these terms are used the other way around. We are sticking to the definition that is consistent with the Keras API. In there, the validation set can be used during training, and the test set is reserved for afterwards. Let’s give this a try! We need to initiate a new model – otherwise Keras will simply assume that we want to continue training the model we already trained above. But now we train it with the small addition of also passing it our validation set: history = model.fit(X_train, y_train, validation_data=(X_val, y_val)) With this we can plot both the performance on the training data and on the validation data! Exercise: plot the training progress. 1. Is there a difference between the training curves of training versus validation data? And if so, what would this imply? 2. (Optional) Take a pen and paper, draw the perfect training and validation curves. (This may seem trivial, but it will trigger you to think about what you actually would like to see) The difference in the two curves shows that something is not completely right here. The error for the model predictions on the validation set quickly seem to reach a plateau while the error on the training set keeps decreasing. That is a common signature of overfitting. Ideally you would like the training and validation curves to be identical and slope down steeply to 0. After that the curves will just consistently stay at 0. Counteract model overfitting Overfitting is a very common issue and there are many strategies to handle it. Most similar to classical machine learning might to reduce the number of parameters. Exercise: Try to reduce the degree of overfitting by lowering the number of parameters We can keep the network architecture unchanged (2 dense layers + a one-node output layer) and only play with the number of nodes per layer. Try to lower the number of nodes in one or both of the two dense layers and observe the changes to the training and validation losses. If time is short: Suggestion is to run one network with only 10 and 5 nodes in the first and second layer. 1. Is it possible to get rid of overfitting this way? 2. Does the overall performance suffer or does it mostly stay the same? 3. (optional) How low can you go with the number of parameters without notable effect on the performance on the validation set? Let’s first adapt our create_nn function so that we can tweak the number of nodes in the 2 layers by passing arguments to the function: def create_nn(nodes1=100, nodes2=50): # Input layer inputs = keras.layers.Input(shape=(X_data.shape[1],), name='input') # Dense layers layers_dense = keras.layers.Dense(nodes1, 'relu')(inputs) layers_dense = keras.layers.Dense(nodes2, 'relu')(layers_dense) # Output layer outputs = keras.layers.Dense(1)(layers_dense) return keras.Model(inputs=inputs, outputs=outputs, name="model_small") Let’s see if it works by creating a much smaller network with 10 nodes in the first layer, and 5 nodes in the second layer: Model: "model_small" Layer (type) Output Shape Param # input (InputLayer) [(None, 89)] 0 dense_9 (Dense) (None, 10) 900 dense_10 (Dense) (None, 5) 55 dense_11 (Dense) (None, 1) 6 Total params: 961 Trainable params: 961 Non-trainable params: 0 Let’s compile and train this network: history = model.fit(X_train, y_train, batch_size = 32, epochs = 200, validation_data=(X_val, y_val)) plot_history(history, ['root_mean_squared_error', 'val_root_mean_squared_error']) 1. With this smaller model we have reduced overfitting a bit, since the training and validation loss are now closer to each other, and the validation loss does now reach a plateau and does not further increase. We have not completely avoided overfitting though. 2. In the case of this small example model, the validation RMSE seems to end up around 3.2, which is much better than the 4.08 we had before. Note that you can double check the actual score by calling model.evaluate() on the test set. 3. In general, it quickly becomes a complicated search for the right “sweet spot”, i.e. the settings for which overfitting will be (nearly) avoided but the model still performs equally well. A model with 3 neurons in both layers seems to be around this spot, reaching an RMSE of 3.1 on the validation set. Reducing the number of nodes further increases the validation RMSE again. We saw that reducing the number of parameters can be a strategy to avoid overfitting. In practice, however, this is usually not the (main) way to go when it comes to deep learning. One reason is, that finding the sweet spot can be really hard and time consuming. And it has to be repeated every time the model is adapted, e.g. when more training data becomes available. Early stopping: stop when things are looking best Arguable the most common technique to avoid (severe) overfitting in deep learning is called early stopping. As the name suggests, this technique just means that you stop the model training if things do not seem to improve anymore. More specifically, this usually means that the training is stopped if the validation loss does not (notably) improve anymore. Early stopping is both intuitive and effective to use, so it has become a standard addition for model training. To better study the effect, we can now safely go back to models with many (too many?) parameters: To apply early stopping during training it is easiest to use Keras EarlyStopping class. This allows to define the condition of when to stop training. In our case we will say when the validation loss is lowest. However, since we have seen quiet some fluctuation of the losses during training above we will also set patience=10 which means that the model will stop training if the validation loss has not gone down for 10 epochs. from tensorflow.keras.callbacks import EarlyStopping earlystopper = EarlyStopping( history = model.fit(X_train, y_train, batch_size = 32, epochs = 200, validation_data=(X_val, y_val), As before, we can plot the losses during training: This still seems to reveal the onset of overfitting, but the training stops before the discrepancy between training and validation loss can grow further. Despite avoiding severe cases of overfitting, early stopping has the additional advantage that the number of training epochs will be regulated automatically. Instead of comparing training runs for different number of epochs, early stopping allows to simply set the number of epochs to a desired maximum value. What might be a bit unintuitive is that the training runs might now end very rapidly. This might spark the question: have we really reached an optimum yet? And often the answer to this is “no”, which is why early stopping frequently is combined with other approaches to avoid overfitting. Overfitting means that a model (seemingly) performs better on seen data compared to unseen data. One then often also says that it does not “generalize” well. Techniques to avoid overfitting, or to improve model generalization, are termed regularization techniques and we will come back to this in episode BatchNorm: the “standard scaler” for deep learning A very common step in classical machine learning pipelines is to scale the features, for instance by using sckit-learn’s StandardScaler. This can in principle also be done for deep learning. An alternative, more common approach, is to add BatchNormalization layers (documentation of the batch normalization layer) which will learn how to scale the input values. Similar to dropout, batch normalization is available as a network layer in Keras and can be added to the network in a similar way. It does not require any additional parameter setting. The BatchNormalization can be inserted as yet another layer into the architecture. def create_nn(): # Input layer inputs = keras.layers.Input(shape=(X_data.shape[1],), name='input') # Dense layers layers_dense = keras.layers.BatchNormalization()(inputs) # This is new! layers_dense = keras.layers.Dense(100, 'relu')(layers_dense) layers_dense = keras.layers.Dense(50, 'relu')(layers_dense) # Output layer outputs = keras.layers.Dense(1)(layers_dense) # Defining the model and compiling it return keras.Model(inputs=inputs, outputs=outputs, name="model_batchnorm") model = create_nn() This new layer appears in the model summary as well. Model: "model_batchnorm" Layer (type) Output Shape Param # input_1 (InputLayer) [(None, 89)] 0 batch_normalization (BatchNo (None, 89) 356 dense (Dense) (None, 100) 9000 dense_1 (Dense) (None, 50) 5050 dense_2 (Dense) (None, 1) 51 Total params: 14,457 Trainable params: 14,279 Non-trainable params: 178 We can train the model again as follows: history = model.fit(X_train, y_train, batch_size = 32, epochs = 1000, validation_data=(X_val, y_val), plot_history(history, ['root_mean_squared_error', 'val_root_mean_squared_error']) Batchnorm parameters You may have noticed that the number of parameters of the Batchnorm layers corresponds to 4 parameters per input node. These are the moving mean, moving standard deviation, additional scaling factor (gamma) and offset factor (beta). There is a difference in behavior for Batchnorm between training and prediction time. During training time, the data is scaled with the mean and standard deviation of the batch. During prediction time, the moving mean and moving standard deviation of the training set is used instead. The additional parameters gamma and beta are introduced to allow for more flexibility in output values, and are used in both training and prediction. Run on test set and compare to naive baseline It seems that no matter what we add, the overall loss does not decrease much further (we at least avoided overfitting though!). Let us again plot the results on the test set: y_test_predicted = model.predict(X_test) plot_predictions(y_test_predicted, y_test, title='Predictions on the test set') Well, the above is certainly not perfect. But how good or bad is this? Maybe not good enough to plan your picnic for tomorrow. But let’s better compare it to the naive baseline we created in the beginning. What would you say, did we improve on that? Exercise: Simplify the model and add data You may have been wondering why we are including weather observations from multiple cities to predict sunshine hours only in Basel. The weather is a complex phenomenon with correlations over large distances and time scales, but what happens if we limit ourselves to only one city? 1. Since we will be reducing the number of features quite significantly, we could afford to include more data. Instead of using only 3 years, use 8 or 9 years! 2. Only use the features in the dataset that are for Basel, remove the data for other cities. You can use something like: 3. Now rerun the last model we defined which included the BatchNorm layer. Recreate the scatter plot comparing your predictions with the true values, and evaluate the model by computing the RMSE on the test score. Note that even though we will use many more observations than previously, the network should still train quickly because we reduce the number of features (columns). Is the prediction better compared to what we had before? 4. (Optional) Try to train a model on all years that are available, and all features from all cities. How does it perform? 1. Use 9 years out of the dataset 2. Only use features for Basel 3. Rerun the model and evaluate it Do the train-test-validation split: X_train, X_holdout, y_train, y_holdout = train_test_split(X_data, y_data, test_size=0.3, random_state=0) X_val, X_test, y_val, y_test = train_test_split(X_holdout, y_holdout, test_size=0.5, random_state=0) Create the network. We can re-use the create_nn that we already have. Because we have reduced the number of input features the number of parameters in the network goes down from 14457 to 6137. # create the network and view its summary model = create_nn() Fit with early stopping and output showing performance on validation set: history = model.fit(X_train, y_train, batch_size = 32, epochs = 1000, validation_data=(X_val, y_val), verbose = 2) plot_history(history, ['root_mean_squared_error', 'val_root_mean_squared_error']) Create a scatter plot to compare with true observations: y_test_predicted = model.predict(X_test) plot_predictions(y_test_predicted, y_test, title='Predictions on the test set') Compute the RMSE on the test set: test_metrics = model.evaluate(X_test, y_test, return_dict=True) print(f'Test RMSE: {test_metrics["root_mean_squared_error"]}') Test RMSE: 3.3761725425720215 This RMSE is already a lot better compared to what we had before and certainly better than the baseline. Additionally, it could be further improved with hyperparameter tuning. Note that because we ran train_test_split() again, we are evaluating on a different test set than before. In the real world it is important to always compare results on the exact same test set. 4. (optional) Train a model on all years and all features available. You can tweak the above code to use all years and all features: # We cannot take all rows, because we need to be able to take the sunshine hours of the next day nr_rows = len(data) - 2 # data X_data = data.loc[:nr_rows].drop(columns=['DATE', 'MONTH']) # labels (sunshine hours the next day) y_data = data.loc[1:(nr_rows + 1)]["BASEL_sunshine"] For the rest you can use the same code as above to train and evaluate the model This results in an RMSE on the test set of 3.23 (your result can be different, but should be in the same range). From this we can conclude that adding more training data results in even better If we run many different experiments with different architectures, it can be difficult to keep track of these different models or compare the achieved performance. We can use tensorboard, a framework that keeps track of our experiments and shows graphs like we plotted above. Tensorboard is included in our tensorflow installation by default. To use it, we first need to add a callback to our (compiled) model that saves the progress of training performance in a logs rectory: from tensorflow.keras.callbacks import TensorBoard import datetime log_dir = "logs/fit/" + datetime.datetime.now().strftime("%Y%m%d-%H%M%S") # You can adjust this to add a more meaningful model name tensorboard_callback = TensorBoard(log_dir=log_dir, histogram_freq=1) history = model.fit(X_train, y_train, batch_size = 32, epochs = 200, validation_data=(X_val, y_val), verbose = 2) You can launch the tensorboard interface from a Jupyter notebook, showing all trained models: %load_ext tensorboard %tensorboard --logdir logs/fit Which will show an interface that looks something like this: 10. Save model Now that we have a somewhat acceptable model, let us not forget to save it for future users to benefit from our explorative efforts! Correctly predicting tomorrow’s sunshine hours is apparently not that simple. Our models get the general trends right, but still predictions vary quite a bit and can even be far off. Open question: What could be next steps to further improve the model? With unlimited options to modify the model architecture or to play with the training parameters, deep learning can trigger very extensive hunting for better and better results. Usually models are “well behaving” in the sense that small changes to the architectures also only result in small changes of the performance (if any). It is often tempting to hunt for some magical settings that will lead to much better results. But do those settings exist? Applying common sense is often a good first step to make a guess of how much better results could be. In the present case we might certainly not expect to be able to reliably predict sunshine hours for the next day with 5-10 minute precision. But how much better our model could be exactly, often remains difficult to answer. • What changes to the model architecture might make sense to explore? • Ignoring changes to the model architecture, what might notably improve the prediction quality? This is an open question. And we don’t actually know how far one could push this sunshine hour prediction (try it out yourself if you like! We’re curious!). But there are a few things that might be worth exploring. Regarding the model architecture: • In the present case we do not see a magical silver bullet to suddenly boost the performance. But it might be worth testing if deeper networks do better (more layers). Other changes that might impact the quality notably: • The most obvious answer here would be: more data! Even this will not always work (e.g. if data is very noisy and uncorrelated, more data might not add much). • Related to more data: use data augmentation. By creating realistic variations of the available data, the model might improve as well. • More data can mean more data points (you can test it yourself by taking more than the 3 years we used here!) • More data can also mean more features! What about adding the month? • The labels we used here (sunshine hours) are highly biased, many days with no or nearly no sunshine but a few with >10 hours. Techniques such as oversampling or undersampling might handle such biased labels better. Another alternative would be to not only look at data from one day, but use the data of a longer period such as a full week. This will turn the data into time series data which in turn might also make it worth to apply different model architectures… Key Points • Separate training, validation, and test sets allows monitoring and evaluating your model. • Batchnormalization scales the data as part of the model. Content from Advanced layer types Last updated on 2024-11-05 \| Edit this page • Why do we need different types of layers? • What are good network designs for image data? • What is a convolutional layer? • How can we use different types of layers to prevent overfitting? • What is hyperparameter tuning? • Understand why convolutional and pooling layers are useful for image data • Implement a convolutional neural network on an image dataset • Use a drop-out layer to prevent overfitting • Be able to tune the hyperparameters of a Keras model Different types of layers Networks are like onions: a typical neural network consists of many layers. In fact, the word deep in deep learning refers to the many layers that make the network deep. So far, we have seen one type of layer, namely the fully connected, or dense layer. This layer is called fully connected, because all input neurons are taken into account by each output neuron. The number of parameters that need to be learned by the network, is thus in the order of magnitude of the number of input neurons times the number of hidden neurons. However, there are many different types of layers that perform different calculations and take different inputs. In this episode we will take a look at convolutional layers and dropout layers, which are useful in the context of image data, but also in many other types of (structured) data. 1. Formulate / Outline the problem: Image classification The MLCommons Dollar Street Dataset is a collection of images of everyday household items from homes around the world that visually captures socioeconomic diversity of traditionally underrepresented populations. We use a subset of the original dataset that can be used for multiclass classification with 10 categories. Let’s load the data: import pathlib import numpy as np DATA_FOLDER = pathlib.Path('data/dataset_dollarstreet/') # change to location where you stored the data train_images = np.load(DATA_FOLDER / 'train_images.npy') val_images = np.load(DATA_FOLDER / 'test_images.npy') train_labels = np.load(DATA_FOLDER / 'train_labels.npy') val_labels = np.load(DATA_FOLDER / 'test_labels.npy') A note about data provenance In an earlier version, this part of the lesson used a different example dataset. During peer review, the decision was made to replace that dataset due to the way it had been compiled using images “scraped” from the internet without permission from or credit to the original creators of those images. Unfortunately, uncredited use of images is a common problem among datasets used to benchmark models for image classification. The Dollar Street dataset was chosen for use in the lesson as it contains only images created by the Gapminder project for the purposes of using them in the dataset. The original Dollar Street dataset is very large – more than 100 GB – with the potential to grow even bigger, so we created a subset for use in this lesson. 2. Identify inputs and outputs Explore the data Let’s do a quick exploration of the dimensions of the data: The first value, 878, is the number of training images in the dataset. The remainder of the shape, namely (64, 64, 3), denotes the dimension of one image. The last value 3 is typical for color images, and stands for the three color channels Red, Green, Blue. Number of features in Dollar Street 10 How many features does one image in the Dollar Street 10 dataset have? • A. 64 • B. 4096 • C. 12288 • D. 878 The correct solution is C: 12288 There are 4096 pixels in one image (64 64), each pixel has 3 channels (RGB). So 4096 * 3 = 12288. We can find out the range of values of our input data as follows: So the values of the three channels range between 0 and 255. Lastly, we inspect the dimension of the labels: So we have, for each image, a single value denoting the label. To find out what the possible values of these labels are: The values of the labels range between 0 and 9, denoting 10 different classes. 3. Prepare data The training set consists of 878 images of 64x64 pixels and 3 channels (RGB values). The RGB values are between 0 and 255. For input of neural networks, it is better to have small input values. So we normalize our data between 0 and 1: 4. Choose a pretrained model or start building architecture from scratch Convolutional layers In the previous episodes, we used ‘fully connected layers’ , that connected all input values of a layer to all outputs of a layer. This results in many connections, and thus many weights to be learned, in the network. Note that our input dimension is now quite high (even with small pictures of 64x64 pixels): we have 12288 features. Number of parameters Suppose we create a single Dense (fully connected) layer with 100 hidden units that connect to the input pixels, how many parameters does this layer have? • A. 1228800 • B. 1228900 • C. 100 • D. 12288 The correct answer is B: Each entry of the input dimensions, i.e. the shape of one single data point, is connected with 100 neurons of our hidden layer, and each of these neurons has a bias term associated to it. So we have 1228900 parameters to learn. width, height = (64, 64) n_hidden_neurons = 100 n_bias = 100 n_input_items = width * height * 3 n_parameters = (n_input_items * n_hidden_neurons) + n_bias We can also check this by building the layer in Keras: inputs = keras.Input(shape=(n_input_items,)) outputs = keras.layers.Dense(100)(inputs) model = keras.models.Model(inputs=inputs, outputs=outputs) Model: "model" Layer (type) Output Shape Param # input_1 (InputLayer) [(None, 12288)] 0 dense (Dense) (None, 100) 1228900 Total params: 1228900 (4.69 MB) Trainable params: 1228900 (4.69 MB) Non-trainable params: 0 (0.00 Byte) We can decrease the number of units in our hidden layer, but this also decreases the number of patterns our network can remember. Moreover, if we increase the image size, the number of weights will ‘explode’, even though the task of recognizing large images is not necessarily more difficult than the task of recognizing small images. The solution is that we make the network learn in a ‘smart’ way. The features that we learn should be similar both for small and large images, and similar features (e.g. edges, corners) can appear anywhere in the image (in mathematical terms: translation invariant). We do this by making use of a concept from image processing that predates deep learning. A convolution matrix, or kernel, is a matrix transformation that we ‘slide’ over the image to calculate features at each position of the image. For each pixel, we calculate the matrix product between the kernel and the pixel with its surroundings. A kernel is typically small, between 3x3 and 7x7 pixels. We can for example think of the 3x3 kernel: [[-1, -1, -1], [0, 0, 0] [1, 1, 1]] This kernel will give a high value to a pixel if it is on a horizontal border between dark and light areas. Note that for RGB images, the kernel should also have a depth of 3. In the following image, we see the effect of such a kernel on the values of a single-channel image. The red cell in the output matrix is the result of multiplying and summing the values of the red square in the input, and the kernel. Applying this kernel to a real image shows that it indeed detects horizontal edges. In our convolutional layer our hidden units are a number of convolutional matrices (or kernels), where the values of the matrices are the weights that we learn in the training process. The output of a convolutional layer is an ‘image’ for each of the kernels, that gives the output of the kernel applied to each pixel. Playing with convolutions Convolutions applied to images can be hard to grasp at first. Fortunately there are resources out there that enable users to interactively play around with images and convolutions: • Image kernels explained shows how different convolutions can achieve certain effects on an image, like sharpening and blurring. • The convolutional neural network cheat sheet shows animated examples of the different components of convolutional neural nets Border pixels What, do you think, happens to the border pixels when applying a convolution? There are different ways of dealing with border pixels. You can ignore them, which means that your output image is slightly smaller then your input. It is also possible to ‘pad’ the borders, e.g. with the same value or with zeros, so that the convolution can also be applied to the border pixels. In that case, the output image will have the same size as the input image. This callout in the Data Carpentry: Image Processing with Python curriculum provides more detail about convolution at the boundaries of an image, in the context of applying a Gaussian blur. Number of model parameters Suppose we apply a convolutional layer with 100 kernels of size 3 * 3 * 3 (the last dimension applies to the rgb channels) to our images of 32 * 32 * 3 pixels. How many parameters do we have? Assume, for simplicity, that the kernels do not use bias terms. Compare this to the answer of the earlier exercise, “Number of Parameters”. We have 100 matrices with 3 * 3 * 3 = 27 values each so that gives 27 * 100 = 2700 weights. This is a magnitude of 2000 less than the fully connected layer with 100 units! Nevertheless, as we will see, convolutional networks work very well for image data. This illustrates the expressiveness of convolutional layers. So let us look at a network with a few convolutional layers. We need to finish with a Dense layer to connect the output cells of the convolutional layer to the outputs for our classes. inputs = keras.Input(shape=train_images.shape[1:]) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(inputs) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) x = keras.layers.Flatten()(x) outputs = keras.layers.Dense(10)(x) model = keras.Model(inputs=inputs, outputs=outputs, name="dollar_street_model_small") Model: "dollar_street_model_small" Layer (type) Output Shape Param # input_8 (InputLayer) [(None, 64, 64, 3)] 0 conv2d_10 (Conv2D) (None, 62, 62, 50) 1400 conv2d_11 (Conv2D) (None, 60, 60, 50) 22550 flatten_6 (Flatten) (None, 180000) 0 dense_14 (Dense) (None, 10) 1800010 Total params: 1823960 (6.96 MB) Trainable params: 1823960 (6.96 MB) Non-trainable params: 0 (0.00 Byte) Convolutional Neural Network Inspect the network above: • What do you think is the function of the Flatten layer? • Which layer has the most parameters? Do you find this intuitive? • (optional) This dataset is similar to the often used CIFAR-10 dataset. We can get inspiration for neural network architectures that could work on our dataset here: https://paperswithcode.com/sota /image-classification-on-cifar-10 . Pick a model and try to understand how it works. • The Flatten layer converts the 60x60x50 output of the convolutional layer into a single one-dimensional vector, that can be used as input for a dense layer. • The last dense layer has the most parameters. This layer connects every single output ‘pixel’ from the convolutional layer to the 10 output classes. That results in a large number of connections, so a large number of parameters. This undermines a bit the expressiveness of the convolutional layers, that have much fewer parameters. Search for existing architectures or pretrained models So far in this course we have built neural networks from scratch, because we want you to fully understand the basics of Keras. In the real world however, you would first search for existing solutions to your problem. You could for example search for ‘large CNN image classification Keras implementation’, and see if you can find any Keras implementations of more advanced architectures that you could reuse. A lot of the best-performing architectures for image classification are convolutional neural networks or at least have some elements in common. Therefore, we will introduce convolutional neural networks here, and the best way to teach you is by developing a neural network from scratch! Pooling layers Often in convolutional neural networks, the convolutional layers are intertwined with Pooling layers. As opposed to the convolutional layer, the pooling layer actually alters the dimensions of the image and reduces it by a scaling factor. It is basically decreasing the resolution of your picture. The rationale behind this is that higher layers of the network should focus on higher-level features of the image. By introducing a pooling layer, the subsequent convolutional layer has a broader ‘view’ on the original image. Let’s put it into practice. We compose a Convolutional network with two convolutional layers and two pooling layers. def create_nn(): inputs = keras.Input(shape=train_images.shape[1:]) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(inputs) x = keras.layers.MaxPooling2D((2, 2))(x) # a new maxpooling layer x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) x = keras.layers.MaxPooling2D((2, 2))(x) # a new maxpooling layer (same as maxpool) x = keras.layers.Flatten()(x) x = keras.layers.Dense(50, activation='relu')(x) # a new Dense layer outputs = keras.layers.Dense(10)(x) model = keras.Model(inputs=inputs, outputs=outputs, name="dollar_street_model") return model model = create_nn() Model: "dollar_street_model" Layer (type) Output Shape Param # input_3 (InputLayer) [(None, 64, 64, 3)] 0 conv2d_2 (Conv2D) (None, 62, 62, 50) 1400 max_pooling2d (MaxPooling2 (None, 31, 31, 50) 0 conv2d_3 (Conv2D) (None, 29, 29, 50) 22550 max_pooling2d_1 (MaxPoolin (None, 14, 14, 50) 0 flatten_1 (Flatten) (None, 9800) 0 dense_2 (Dense) (None, 50) 490050 dense_3 (Dense) (None, 10) 510 Total params: 514510 (1.96 MB) Trainable params: 514510 (1.96 MB) Non-trainable params: 0 (0.00 Byte) 5. Choose a loss function and optimizer We compile the model using the adam optimizer (other optimizers could also be used here!). Similar to the penguin classification task, we will use the crossentropy function to calculate the model’s loss. This loss function is appropriate to use when the data has two or more label classes. Remember that our target class is represented by a single integer, whereas the output of our network has 10 nodes, one for each class. So, we should have actually one-hot encoded the targets and used a softmax activation for the neurons in our output layer! Luckily, there is a quick fix to calculate crossentropy loss for data that has its classes represented by integers, the SparseCategoricalCrossentropy() function. Adding the argument from_logits=True accounts for the fact that the output has a linear activation instead of softmax. This is what is often done in practice, because it spares you from having to worry about one-hot encoding. 7. Perform a Prediction/Classification Here we skip performing a prediction, and continue to measuring the performance. In practice, you will only do this step once in a while when you actually need to have the individual predictions, often you know enough based on the evaluation metric scores. Of course, behind the scenes whenever you measure performance you have to make predictions and compare them to the ground truth. 8. Measure performance We can plot the training process using the history: import seaborn as sns import matplotlib.pyplot as plt import pandas as pd def plot_history(history, metrics): Plot the training history history (keras History object that is returned by model.fit()) metrics(str, list): Metric or a list of metrics to plot history_df = pd.DataFrame.from_dict(history.history) plot_history(history, ['accuracy', 'val_accuracy']) It seems that the model is overfitting a lot, because the training accuracy increases, while the validation accuracy stagnates. Meanwhile, the training loss keeps decreasing while the validation loss actually starts increasing after a few epochs. Comparison with a network with only dense layers How does this simple CNN compare to a neural network with only dense layers? We can define a neural network with only dense layers: def create_dense_model(): inputs = keras.Input(shape=train_images.shape[1:]) x = keras.layers.Flatten()(inputs) x = keras.layers.Dense(50, activation='relu')(x) x = keras.layers.Dense(50, activation='relu')(x) outputs = keras.layers.Dense(10)(x) return keras.models.Model(inputs=inputs, outputs=outputs, dense_model = create_dense_model() Model: "dense_model" Layer (type) Output Shape Param # input_7 (InputLayer) [(None, 64, 64, 3)] 0 flatten_5 (Flatten) (None, 12288) 0 dense_11 (Dense) (None, 50) 614450 dense_12 (Dense) (None, 50) 2550 dense_13 (Dense) (None, 10) 510 Total params: 617510 (2.36 MB) Trainable params: 617510 (2.36 MB) Non-trainable params: 0 (0.00 Byte) As you can see this model has more parameters than our simple CNN, let’s train and evaluate it! history = dense_model.fit(train_images, train_labels, epochs=20, validation_data=(val_images, val_labels)) plot_history(history, ['accuracy', 'val_accuracy']) As you can see the validation accuracy only reaches about 18%, whereas the CNN reached about 28% accuracy. This demonstrates that convolutional layers are a big improvement over dense layers for these kind of datasets. 9. Refine the model Network depth What, do you think, will be the effect of adding a convolutional layer to your model? Will this model have more or fewer parameters? Try it out. Create a model that has an additional Conv2d layer with 50 filters and another MaxPooling2D layer after the last MaxPooling2D layer. Train it for 10 epochs and plot the results. HINT: The model definition that we used previously needs to be adjusted as follows: inputs = keras.Input(shape=train_images.shape[1:]) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(inputs) x = keras.layers.MaxPooling2D((2, 2))(x) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) x = keras.layers.MaxPooling2D((2, 2))(x) # Add your extra layers here x = keras.layers.Flatten()(x) x = keras.layers.Dense(50, activation='relu')(x) outputs = keras.layers.Dense(10)(x) We add an extra Conv2D layer after the second pooling layer: def create_nn_extra_layer(): inputs = keras.Input(shape=train_images.shape[1:]) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(inputs) x = keras.layers.MaxPooling2D((2, 2))(x) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) x = keras.layers.MaxPooling2D((2, 2))(x) # x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) # extra layer x = keras.layers.MaxPooling2D((2, 2))(x) # extra layer x = keras.layers.Flatten()(x) x = keras.layers.Dense(50, activation='relu')(x) outputs = keras.layers.Dense(10)(x) model = keras.Model(inputs=inputs, outputs=outputs, name="dollar_street_model") return model model = create_nn_extra_layer() With the model defined above, we can inspect the number of parameters: Model: "dollar_street_model" Layer (type) Output Shape Param # input_4 (InputLayer) [(None, 64, 64, 3)] 0 conv2d_4 (Conv2D) (None, 62, 62, 50) 1400 max_pooling2d_2 (MaxPoolin (None, 31, 31, 50) 0 conv2d_5 (Conv2D) (None, 29, 29, 50) 22550 max_pooling2d_3 (MaxPoolin (None, 14, 14, 50) 0 conv2d_6 (Conv2D) (None, 12, 12, 50) 22550 max_pooling2d_4 (MaxPoolin (None, 6, 6, 50) 0 flatten_2 (Flatten) (None, 1800) 0 dense_4 (Dense) (None, 50) 90050 dense_5 (Dense) (None, 10) 510 Total params: 137060 (535.39 KB) Trainable params: 137060 (535.39 KB) Non-trainable params: 0 (0.00 Byte) The number of parameters has decreased by adding this layer. We can see that the extra layers decrease the resolution from 14x14 to 6x6, as a result, the input of the Dense layer is smaller than in the previous network. To train the network and plot the results: history = model.fit(train_images, train_labels, epochs=10, validation_data=(val_images, val_labels)) plot_history(history, ['accuracy', 'val_accuracy']) Other types of data Convolutional and Pooling layers are also applicable to different types of data than image data. Whenever the data is ordered in a (spatial) dimension, and translation invariant features are expected to be useful, convolutions can be used. Think for example of time series data from an accelerometer, audio data for speech recognition, or 3d structures of chemical compounds. Why and when to use convolutional neural networks 1. Would it make sense to train a convolutional neural network (CNN) on the penguins dataset and why? 2. Would it make sense to train a CNN on the weather dataset and why? 3. (Optional) Can you think of a different machine learning task that would benefit from a CNN architecture? 1. No that would not make sense. Convolutions only work when the features of the data can be ordered in a meaningful way. Pixels for example are ordered in a spatial dimension. This kind of order cannot be applied to the features of the penguin dataset. If we would have pictures or audio recordings of the penguins as input data it would make sense to use a CNN architecture. 2. It would make sense, but only if we approach the problem from a different angle then we did before. Namely, 1D convolutions work quite well on sequential data such as timeseries. If we have as our input a matrix of the different weather conditions over time in the past x days, a CNN would be suited to quickly grasp the temporal relationship over days. 3. Some example domains in which CNNs are applied: • Text data • Timeseries, specifically audio • Molecular structures Note that the training loss continues to decrease, while the validation loss stagnates, and even starts to increase over the course of the epochs. Similarly, the accuracy for the validation set does not improve anymore after some epochs. This means we are overfitting on our training data set. Techniques to avoid overfitting, or to improve model generalization, are termed regularization techniques. One of the most versatile regularization technique is dropout (Srivastava et al., 2014). Dropout means that during each training cycle (one forward pass of the data through the model) a random fraction of neurons in a dense layer are turned off. This is described with the dropout rate between 0 and 1 which determines the fraction of nodes to silence at a time. The intuition behind dropout is that it enforces redundancies in the network by constantly removing different elements of a network. The model can no longer rely on individual nodes and instead must create multiple “paths”. In addition, the model has to make predictions with much fewer nodes and weights (connections between the nodes). As a result, it becomes much harder for a network to memorize particular features. At first this might appear a quite drastic approach which affects the network architecture strongly. In practice, however, dropout is computationally a very elegant solution which does not affect training speed. And it frequently works very well. Important to note: Dropout layers will only randomly silence nodes during training! During a predictions step, all nodes remain active (dropout is off). During training, the sample of nodes that are silenced are different for each training instance, to give all nodes a chance to observe enough training data to learn its weights. Let us add a dropout layer after each pooling layertowards the end of the network, that randomly drops 80% of the nodes. def create_nn_with_dropout(): inputs = keras.Input(shape=train_images.shape[1:]) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(inputs) x = keras.layers.MaxPooling2D((2, 2))(x) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) x = keras.layers.MaxPooling2D((2, 2))(x) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) x = keras.layers.MaxPooling2D((2, 2))(x) x = keras.layers.Dropout(0.8)(x) # This is new! x = keras.layers.Flatten()(x) x = keras.layers.Dense(50, activation='relu')(x) outputs = keras.layers.Dense(10)(x) model = keras.Model(inputs=inputs, outputs=outputs, name="dropout_model") return model model_dropout = create_nn_with_dropout() Model: "dropout_model" Layer (type) Output Shape Param # input_5 (InputLayer) [(None, 64, 64, 3)] 0 conv2d_7 (Conv2D) (None, 62, 62, 50) 1400 max_pooling2d_5 (MaxPoolin (None, 31, 31, 50) 0 conv2d_8 (Conv2D) (None, 29, 29, 50) 22550 max_pooling2d_6 (MaxPoolin (None, 14, 14, 50) 0 conv2d_9 (Conv2D) (None, 12, 12, 50) 22550 max_pooling2d_7 (MaxPoolin (None, 6, 6, 50) 0 dropout (Dropout) (None, 6, 6, 50) 0 flatten_3 (Flatten) (None, 1800) 0 dense_6 (Dense) (None, 50) 90050 dense_7 (Dense) (None, 10) 510 Total params: 137060 (535.39 KB) Trainable params: 137060 (535.39 KB) Non-trainable params: 0 (0.00 Byte) We can see that the dropout does not alter the dimensions of the image, and has zero parameters. We again compile and train the model. history = model_dropout.fit(train_images, train_labels, epochs=20, validation_data=(val_images, val_labels)) And inspect the training results: Now we see that the gap between the training accuracy and validation accuracy is much smaller, and that the final accuracy on the validation set is higher than without dropout. Vary dropout rate 1. What do you think would happen if you lower the dropout rate? Try it out, and see how it affects the model training. 2. You are varying the dropout rate and checking its effect on the model performance, what is the term associated to this procedure? 1. Varying the dropout rate The code below instantiates and trains a model with varying dropout rates. You can see from the resulting plot that the ideal dropout rate in this case is around 0.9. This is where the val loss is Note that it can take a while to train these 6 networks. def create_nn_with_dropout(dropout_rate): inputs = keras.Input(shape=train_images.shape[1:]) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(inputs) x = keras.layers.MaxPooling2D((2, 2))(x) x = keras.layers.Dropout(dropout_rate)(x) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) x = keras.layers.MaxPooling2D((2, 2))(x) x = keras.layers.Dropout(dropout_rate)(x) x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) x = keras.layers.Dropout(dropout_rate)(x) x = keras.layers.Flatten()(x) x = keras.layers.Dense(50, activation='relu')(x) outputs = keras.layers.Dense(10)(x) model = keras.Model(inputs=inputs, outputs=outputs, name="dropout_model") return model early_stopper = keras.callbacks.EarlyStopping(monitor='val_loss', patience=5) dropout_rates = [0.2, 0.4, 0.6, 0.8, 0.9, 0.95] val_losses = [] for dropout_rate in dropout_rates: model_dropout = create_nn_with_dropout(dropout_rate) model_dropout.fit(train_images, train_labels, epochs=30, validation_data=(val_images, val_labels), val_loss, val_acc = model_dropout.evaluate(val_images, val_labels) loss_df = pd.DataFrame({'dropout_rate': dropout_rates, 'val_loss': val_losses}) sns.lineplot(data=loss_df, x='dropout_rate', y='val_loss') Hyperparameter tuning Recall that hyperparameters are model configuration settings that are chosen before the training process and affect the model’s learning behavior and performance, for example the dropout rate. In general, if you are varying hyperparameters to find the combination of hyperparameters with the best model performance this is called hyperparameter tuning. A naive way to do this is to write a for-loop and train a slightly different model in every cycle. However, it is better to use the keras_tuner package for this. Let’s first define a function that creates a neuronal network given 2 hyperparameters, namely the dropout rate and the number of layers: def create_nn_with_hp(dropout_rate, n_layers): inputs = keras.Input(shape=train_images.shape[1:]) x = inputs for layer in range(n_layers): x = keras.layers.Conv2D(50, (3, 3), activation='relu')(x) x = keras.layers.MaxPooling2D((2, 2))(x) x = keras.layers.Dropout(dropout_rate)(x) x = keras.layers.Flatten()(x) x = keras.layers.Dense(50, activation='relu')(x) outputs = keras.layers.Dense(10)(x) model = keras.Model(inputs=inputs, outputs=outputs, name="cifar_model") return model Now, let’s find the best combination of hyperparameters using grid search. Grid search is the simplest hyperparameter tuning strategy, you test all the combinations of predefined values for the hyperparameters that you want to vary. For this we will make use of the package keras_tuner, we can install it by typing in the command line: Note that this can take some time to train (around 5 minutes or longer). import keras_tuner hp = keras_tuner.HyperParameters() def build_model(hp): # Define values for hyperparameters to try out: n_layers = hp.Int("n_layers", min_value=1, max_value=2, step=1) dropout_rate = hp.Float("dropout_rate", min_value=0.2, max_value=0.8, step=0.3) model = create_nn_with_hp(dropout_rate, n_layers) return model tuner = keras_tuner.GridSearch(build_model, objective='val_loss') tuner.search(train_images, train_labels, epochs=20, validation_data=(val_images, val_labels)) Trial 6 Complete [00h 00m 19s] val_loss: 2.086069345474243 Best val_loss So Far: 2.086069345474243 Total elapsed time: 00h 01m 28s Let’s have a look at the results: Results summary Results in ./untitled_project Showing 10 best trials Objective(name="val_loss", direction="min") Trial 0005 summary n_layers: 2 dropout_rate: 0.8 Score: 2.086069345474243 Trial 0000 summary n_layers: 1 dropout_rate: 0.2 Score: 2.101102352142334 Trial 0001 summary n_layers: 1 dropout_rate: 0.5 Score: 2.1184325218200684 Trial 0003 summary n_layers: 2 dropout_rate: 0.2 Score: 2.1233835220336914 Trial 0002 summary n_layers: 1 dropout_rate: 0.8 Score: 2.1370232105255127 Trial 0004 summary n_layers: 2 dropout_rate: 0.5 Score: 2.143627882003784 Hyperparameter tuning 1: Looking at the grid search results, select all correct statements: • A. 6 different models were trained in this grid search run, because there are 6 possible combinations for the defined hyperparameter values • B. 2 different models were trained, 1 for each hyperparameter that we want to change • C. 1 model is trained with 6 different hyperparameter combinations • D. The model with 2 layer and a dropout rate of 0.5 is the best model with a validation loss of 2.144 • E. The model with 2 layers and a dropout rate of 0.8 is the best model with a validation loss of 2.086 • F. We found the model with the best possible combination of dropout rate and number of layers 2 (Optional): Perform a grid search finding the best combination of the following hyperparameters: 2 different activation functions: ‘relu’, and ‘tanh’, and 2 different values for the kernel size: 3 and 4. Which combination works best? Hint: Instead of hp.Int you should now use hp.Choice("name", ["value1", "value2"]) to use hyperparameters from a predefined set of possible values. • A: Correct, 2 values for number of layers (1 and 2) are combined with 3 values for the dropout rate (0.2, 0.5, 0.8). 2 * 3 = 6 combinations • B: Incorrect, a model is trained for each combination of defined hyperparameter values • C: Incorrect, it is important to note that you actually train and test different models for each run of the grid search • D: Incorrect, this is the worst model since the validation loss is highest • E: Correct, this is the best model with the lowest loss • F: Incorrect, it could be that a different number of layers in combination with a dropout rate that we did not test (for example 3 layers and a dropout rate of 0.6) could be the best model. 2 (Optional): You need to adapt the code as follows: def create_nn_with_hp(activation_function, kernel_size): inputs = keras.Input(shape=train_images.shape[1:]) x = inputs for layer in range(3): x = keras.layers.Conv2D(50, (kernel_size, kernel_size), activation=activation_function)(x) x = keras.layers.MaxPooling2D((2, 2))(x) x = keras.layers.Dropout(0.2)(x) x = keras.layers.Flatten()(x) x = keras.layers.Dense(50, activation=activation_function)(x) outputs = keras.layers.Dense(10)(x) model = keras.Model(inputs=inputs, outputs=outputs, name="cifar_model") return model hp = keras_tuner.HyperParameters() def build_model(hp): kernel_size = hp.Int("kernel_size", min_value=3, max_value=4, step=1) activation = hp.Choice("activation", ["relu", "tanh"]) model = create_nn_with_hp(activation, kernel_size) return model tuner = keras_tuner.GridSearch(build_model, objective='val_loss', project_name='new_project') tuner.search(train_images, train_labels, epochs=20, validation_data=(val_images, val_labels)) Trial 4 Complete [00h 00m 25s] val_loss: 2.0591845512390137 Best val_loss So Far: 2.0277602672576904 Total elapsed time: 00h 01m 30s Let’s look at the results: Results summary Results in ./new_project Showing 10 best trials Objective(name="val_loss", direction="min") Trial 0001 summary kernel_size: 3 activation: tanh Score: 2.0277602672576904 Trial 0003 summary kernel_size: 4 activation: tanh Score: 2.0591845512390137 Trial 0000 summary kernel_size: 3 activation: relu Score: 2.123767614364624 Trial 0002 summary kernel_size: 4 activation: relu Score: 2.150160551071167 A kernel size of 3 and tanh as activation function is the best tested combination. Grid search can quickly result in a combinatorial explosion because all combinations of hyperparameters are trained and tested. Instead, random search randomly samples combinations of hyperparemeters, allowing for a much larger look through a large number of possible hyperparameter combinations. Next to grid search and random search there are many different hyperparameter tuning strategies, including neural architecture search where a separate neural network is trained to find the best architecture for a model! Conclusion and next steps How successful were we with creating a model here? With ten image classes, and assuming that we would not ask the model to classify an image that contains none of the given classes of object, a model working on complete guesswork would be correct 10% of the time. Against this baseline accuracy of 10%, and considering the diversity and relatively low resolution of the example images, perhaps our last model’s validation accuracy of ~30% is not too bad. What could be done to improve on this performance? We might try adjusting the number of layers and their parameters, such as the number of units in a layer, or providing more training data (we were using only a subset of the original Dollar Street dataset here). Or we could explore some other deep learning techniques, such as transfer learning, to create more sophisticated models. Key Points • Convolutional layers make efficient reuse of model parameters. • Pooling layers decrease the resolution of your input • Dropout is a way to prevent overfitting Content from Transfer learning Last updated on 2024-11-05 \| Edit this page • How do I apply a pre-trained model to my data? • Adapt a state-of-the-art pre-trained network to your own dataset What is transfer learning? Instead of training a model from scratch, with transfer learning you make use of models that are trained on another machine learning task. The pre-trained network captures generic knowledge during pre-training and will only be ‘fine-tuned’ to the specifics of your dataset. An example: Let’s say that you want to train a model to classify images of different dog breeds. You could make use of a pre-trained network that learned how to classify images of dogs and cats. The pre-trained network will not know anything about different dog breeds, but it will have captured some general knowledge of, on a high-level, what dogs look like, and on a low-level all the different features (eyes, ears, paws, fur) that make up an image of a dog. Further training this model on your dog breed dataset is a much easier task than training from scratch, because the model can use the general knowledge captured in the pre-trained network. In this episode we will learn how use Keras to adapt a state-of-the-art pre-trained model to the Dollar Street Dataset. 1. Formulate / Outline the problem and 2. Identify inputs and outputs Just like in the previous episode, we use the Dollar Street 10 dataset. The goal is to predict one out of 10 classes for a given image. We load the data in the same way as the previous episode: import pathlib import numpy as np DATA_FOLDER = pathlib.Path('data/dataset_dollarstreet/') # change to location where you stored the data train_images = np.load(DATA_FOLDER / 'train_images.npy') val_images = np.load(DATA_FOLDER / 'test_images.npy') train_labels = np.load(DATA_FOLDER / 'train_labels.npy') val_labels = np.load(DATA_FOLDER / 'test_labels.npy') 4. Choose a pre-trained model or start building architecture from scratch Let’s define our model input layer using the shape of our training images: Our images are 64 x 64 pixels, whereas the pre-trained model that we will use was trained on images of 160 x 160 pixels. To deal with this, we add an upscale layer that resizes the images to 160 x 160 pixels during training and prediction. # upscale layer method = tf.image.ResizeMethod.BILINEAR upscale = keras.layers.Lambda( lambda x: tf.image.resize_with_pad(x, 160, 160, method=method))(inputs) From the keras.applications module we use the DenseNet121 architecture. This architecture was proposed by the paper: Densely Connected Convolutional Networks (CVPR 2017). It is trained on the Imagenet dataset, which contains 14,197,122 annotated images according to the WordNet hierarchy with over 20,000 classes. We will have a look at the architecture later, for now it is enough to know that it is a convolutional neural network with 121 layers that was designed to work well on image classification tasks. Let’s configure the DenseNet121: base_model = keras.applications.DenseNet121(include_top=False, By setting include_top to False we exclude the fully connected layer at the top of the network. This layer was used to predict the Imagenet classes, but will be of no use for our Dollar Street We add pooling='max' so that max pooling is applied to the output of the DenseNet121 network. By setting weights='imagenet' we use the weights that resulted from training this network on the Imagenet data. We connect the network to the upscale layer that we defined before. Only train a ‘head’ network Instead of fine-tuning all the weights of the DenseNet121 network using our dataset, we choose to freeze all these weights and only train a so-called ‘head network’ that sits on top of the pre-trained network. You can see the DenseNet121 network as extracting a meaningful feature representation from our image. The head network will then be trained to decide on which of the 10 Dollar Street dataset classes the image belongs. We will turn of the trainable property of the base model: Let’s define our ‘head’ network: out = base_model.output out = keras.layers.Flatten()(out) out = keras.layers.BatchNormalization()(out) out = keras.layers.Dense(50, activation='relu')(out) out = keras.layers.Dropout(0.5)(out) out = keras.layers.Dense(10)(out) Finally we define our model: Inspect the DenseNet121 network Have a look at the network architecture with model.summary(). It is indeed a deep network, so expect a long summary! 1.Trainable parameters How many parameters are there? How many of them are trainable? Why is this and how does it effect the time it takes to train the model? 2. Head and base Can you see in the model summary which part is the base network and which part is the head network? 1. Trainable parameters Total number of parameters: 7093360, out of which only 53808 are trainable. The 53808 trainable parameters are the weights of the head network. All other parameters are ‘frozen’ because we set base_model.trainable=False. Because only a small proportion of the parameters have to be updated at each training step, this will greatly speed up training time. 1. Compile the model Compile the model: - Use the adam optimizer - Use the SparseCategoricalCrossentropy loss with from_logits=True. - Use ‘accuracy’ as a metric. 2. Train the model Train the model on the training dataset: - Use a batch size of 32 - Train for 30 epochs, but use an earlystopper with a patience of 5 - Pass the validation dataset as validation data so we can monitor performance on the validation data during training - Store the result of training in a variable called history - Training can take a while, it is a much larger model than what we have seen so 3. Inspect the results Plot the training history and evaluate the trained model. What do you think of the results? 4. (Optional) Try out other pre-trained neural networks Train and evaluate another pre-trained model from https://keras.io/api/applications/. How does it compare to DenseNet121? 2. Train the model Define the early stopper: Train the model: 3. Inspect the results def plot_history(history, metrics): Plot the training history history (keras History object that is returned by model.fit()) metrics(str, list): Metric or a list of metrics to plot history_df = pd.DataFrame.from_dict(history.history) plot_history(history, ['accuracy', 'val_accuracy']) Concluding: The power of transfer learning In many domains, large networks are available that have been trained on vast amounts of data, such as in computer vision and natural language processing. Using transfer learning, you can benefit from the knowledge that was captured from another machine learning task. In many fields, transfer learning will outperform models trained from scratch, especially if your dataset is small or of poor Key Points • Large pre-trained models capture generic knowledge about a domain • Use the keras.applications module to easily use pre-trained models for your own datasets Content from Outlook Last updated on 2024-11-05 \| Edit this page • How does what I learned in this course translate to real-world problems? • How do I organise a deep learning project? • What are next steps to take after this course? • Understand that what we learned in this course can be applied to real-world problems • Use best practices for organising a deep learning project • Identify next steps to take after this course You have come to the end of this course. In this episode we will look back at what we have learned so far, how to apply that to real-world problems, and identify next steps to take to start applying deep learning in your own projects. Real-world application To introduce the core concepts of deep learning we have used quite simple machine learning problems. But how does what we learned so far apply to real-world applications? To illustrate that what we learned is actually the basis of succesful applications in research, we will have a look at an example from the field of cheminformatics. We will have a look at this notebook. It is part of the codebase for this paper. In short, the deep learning problem is that of finding out how similar two molecules are in terms of their molecular properties, based on their mass spectrum. You can compare this to comparing two pictures of animals, and predicting how similar they are. A siamese neural network is used to solve the problem. In a siamese neural network you have two input vectors, let’s say two images of animals or two mass spectra. They pass through a base network. Instead of outputting a class or number with one or a few output neurons, the output layer of the base network is a whole vector of for example 100 neurons. After passing through the base network, you end up with two of these vectors representing the two inputs. The goal of the base network is to output a meaningful representation of the input (this is called an embedding). The next step is to compute the cosine similarity between these two output vectors, cosine similarity is a measure for how similar two vectors are to each other, ranging from 0 (completely different) to 1 (identical). This cosine similarity is compared to the actual similarity between the two inputs and this error is used to update the weights in the network. Don’t worry if you do not fully understand the deep learning problem and the approach that is taken here. We just want you to appreciate that you already learned enough to be able to do this yourself in your own domain. Exercise: A real-world deep learning application 1. Looking at the ‘Model training’ section of the notebook, what do you recognize from what you learned in this course? 2. Can you identify the different steps of the deep learning workflow in this notebook? 3. (Optional): Try to understand the neural network architecture from the first figure of the paper. 1. Why are there 10.000 neurons in the input layer? 2. What do you think would happen if you would decrease the size of spectral embedding layer drastically, to for example 5 neurons? 1. The model summary for the Siamese model is more complex than what we have seen so far, but it is basically a repetition of Dense, BatchNorm, and Dropout layers. The syntax for training and evaluating the model is the same as what we learned in this course. EarlyStopping as well as the Adam optimizer is used. 2. The different steps are not as clearly defined as in this course, but you should be able to identify ‘3: Data preparation’, ‘4: Choose a pretrained model or start building architecture from scratch’, ‘5: Choose a loss function and optimizer’, ‘6: Train the model’, ‘7: Make predictions’ (which is called ‘Model inference’ in this notebook), and ‘10: Save model’. 3. (optional) 1. Because the shape of the input is 10.000. More specifically, the spectrum is binned into a size 10.000 vector, apparently this is a good size to represent the mass spectrum. 2. This would force the neural network to have a representation of the mass spectrum in only 5 numbers. This representation would probably be more generic, but might fail to capture all the characteristics found in the spectrum. This would likely result in underfitting. Hopefully you can appreciate that what you learned in this course, can be applied to real-world problems as well. Extensive data preparation You might have noticed that the data preparation for this example is much more extensive than what we have done so far in this course. This is quite common for applied deep learning projects. It is said that 90% of the time in a deep learning problem is spent on data preparation, and only 10% on modeling! Discussion: Large Language Models and prompt engineering Large Language Models (LLMs) are deep learning models that are able to perform general-purpose language generation. They are trained on large amounts of texts, such all pages of Wikipedia. In recent years the quality of LLMs language understanding and generation has increased tremendously, and since the launch of generative chatbot ChatGPT in 2022 the power of LLMs is now appreciated by the general public. It is becoming more and more feasible to unleash this power in scientific research. For example, the authors of Zheng et al. (2023) guided ChatGPT in the automation of extracting chemical information from a large amount of research articles. The authors did not implement a deep learning model themselves, but instead they designed the right input for ChatGPT (called a ‘prompt’) that would produce optimal outputs. This is called prompt engineering. A highly simplified example of such a prompt would be: “Given compounds X and Y and context Z, what are the chemical details of the reaction?” Developments in LLM research are moving fast, at the end of 2023 the newest ChatGPT version could take images and sound as input. In theory, this means that you can solve the Dollar Street image classification problem from the previous episode by prompt engineering, with prompts similar to “Which out of these categories: [LIST OF CATEGORIES] is depicted in the image”. Discuss the following statement with your neighbors: In a few years most machine learning problems in scientific research can be solved with prompt engineering. Organising deep learning projects As you might have noticed already in this course, deep learning projects can quickly become messy. Here follow some best practices for keeping your projects organized: 1. Organise experiments in notebooks Jupyter notebooks are a useful tool for doing deep learning experiments. You can very easily modify your code bit by bit, and interactively look at the results. In addition you can explain why you are doing things in markdown cells. - As a rule of thumb do one approach or experiment in one notebook. - Give consistent and meaningful names to notebooks, such as: 01-all-cities-simple-cnn.ipynb - Add a rationale on top and a conclusion on the bottom of each notebook Ten simple rules for writing and sharing computational analyses in Jupyter Notebooks provides further advice on how to maximise the usefulness and reproducibility of experiments captured in a 2. Use Python modules Code that is repeatedly used should live in a Python module and not be copied to multiple notebooks. You can import functions and classes from the module(s) in the notebooks. This way you can remove a lot of code definition from your notebooks and have a focus on the actual experiment. 3. Keep track of your results in a central place Always evaluate your experiments in the same way, on the exact same test set. Document the results of your experiments in a consistent and meaningful way. You can use a simple spreadsheet such as MODEL NAME MODEL DESCRIPTION RMSE TESTSET NAME GITHUB COMMIT COMMENTS weather_prediction_v1.0 Basel features only, 10 years. nn: 100-50 3.21 10_years_v1.0 ed28d85 weather_prediction_v1.1 all features, 10 years. nn: 100-50 3.35 10_years_v1.0 4427b78 You could also use a tool such as Weights and Biases for this. Next steps You now understand the basic principles of deep learning and are able to implement your own deep learning pipelines in Python. But there is still so much to learn and do! Here are some suggestions for next steps you can take in your endeavor to become a deep learning expert: • Learn more by going through a few of the learning resources we have compiled for you • Apply what you have learned to your own projects. Use the deep learning workflow to structure your work. Start as simple as possible, and incrementally increase the complexity of your approach. • Compete in a Kaggle competition to practice what you have learned. • Get access to a GPU. Your deep learning experiments will progress much quicker if you have to wait for your network to train in a few seconds instead of hours (which is the order of magnitude of speedup you can expect from training on a GPU instead of CPU). Tensorflow/Keras will automatically detect and use a GPU if it is available on your system without any code changes. A simple and quick way to get access to a GPU is to use Google Colab Key Points • Although the data preparation and model architectures are somewhat more complex, what we have learned in this course can directly be applied to real-world problems • Use what you have learned in this course as a basis for your own learning trajectory in the world of deep learning	{"url":"https://carpentries-incubator.github.io/deep-learning-intro/aio.html","timestamp":"2024-11-07T10:41:16Z","content_type":"text/html","content_length":"384845","record_id":"<urn:uuid:84134dfa-c622-47b5-83e8-e6faae4f48c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00895.warc.gz"}
(PDF) How does the quantum structure of electromagnetic waves describe quantum redshift? Author content All content in this area was uploaded by Bahram Kalhor on Jun 11, 2022 Content may be subject to copyright. Author content All content in this area was uploaded by Bahram Kalhor on May 15, 2022 Content may be subject to copyright. Author content All content in this area was uploaded by Bahram Kalhor on May 15, 2022 Content may be subject to copyright. How does the quantum structure of electromagnetic waves describe quantum redshift? Bahram Kalhor The redshift of the electromagnetic waves is a powerful tool for calculating the distance of the objects in space and studying their behavior. However, physicists' misinterpretation of why Redshift occurs has led us to a misunderstanding of the most cosmological phenomena. The paper introduces Quantum Redshift (QR) by using the quantum structure of the electromagnetic waves (QSEW) In the Quantum Redshift, although the Planck constant is the smallest unit of three- dimensional energy, it is consisting of smaller units of one-dimensional energy. The maximum energy of each period of the electromagnetic waves is equal to the Planck constant hence, the capacity of each period is carrying one-dimensional quanta energy. However, in the QR, at the emitting time of the electromagnetic waves, their periods are not fully filled. On the other hand, they are interested in sharing quanta energies with each other to have fully filled periods. Sharing the quanta energies ofاsome periods between other periods is the reason for destroying some periods and decreasing the frequency of the electromagnetic waves. Our other studies show Quantum Redshift can well explain the whole phenomenon of the universe, and real data support our theory. The quantum redshift rejects the big bang theory, expansion of space and dark energy. It predicts dark matters and describes CMB. The paper obtains the basic equation of the QR for use in future papers. Cosmic redshift or shift in spectral lines is a powerful tool for calculating the distance of the objects in the universe. It belongs to the electromagnetic waves area and usually happens when the wavelength of a wave decrease by traveling in space. The measurement parameter of the cosmic redshift is z. The value of the z usually is a positive number, and more distance is equal to the greater z value. In some cases, the value of the z is negative, and we have blueshift. Although the redshift is known by name of the Edwin Hubble [1], Vesto Slipher (1875-1969) was the first astronomer who measured it [2]. Also, the redshift of the cosmic waves has measured by Carl W. Wirtz (1922) and the Swede Knut Lundmark (in 1924) [3,4]. There are many theories for describing the behavior of the wave in space andincreasing its wavelength. Doppler effect is the most similar theory that has been used for investigating the cosmic redshift [5-9]. In the Doppler effect, changing the wavelength is due to changing the distance between emitter and observer during the traveling of the wave between them. In 1848 Hippolyte Fizeau proposed that cosmic redshift is like the Doppler effect. In 1968 William Huggins measured the velocity of the stars by using the Doppler effect formula. According to this Shahid Beheshti University, Faculty of Electrical & Computer Engineering Corresponding author. Email: Kalhor_bahram@yahoo.com method, objects that come toward us have the blueshift, while objects that move away have the Using spectral lines and the Doppler effect method showed that the speed of some stars and galaxies should be more than the speed of light. This issue disagreed with Einstein's special relativity [13-16]. Hence, the expanding universe theory is proposed. In the expanding space theory, the distance between two objects in space will be increased along time even they do not move. In the expanding space theory, the reason for the redshift of the waves is expanding the space. Also, by increasing the distance between the objects their expansion rate will be increased. If we accept the expansion of the space, we should find dark energy that makes this expansion. The expanding space theory has two problems. Dark energy is not discovered yet, and not possible to build a realistic model of the universe on modes of unrestrained expansion [17]. The gravitational redshift is another theory that tries to describe the spectral displacement by using general relativity [18-22]. Although there is a significant redshift for massive objects, it is a weak effect for non-massive stars. The purpose of this work is to represent quantum redshift for measuring the distance of the objects. The quantum redshift disagrees with the accelerated expansion of space. The results of the quantum redshift show the real distances of the objects are less than the distances that have been obtained in the theory of expanding universe. Concepts of the quanta energy [23] and the quantum structure of the electromagnetic waves [24] are the main parts of this paper. In the quantum structure of the electromagnetic waves, regardless of the frequency of the waves, the capacity of each period of the wave is an equal number of the quanta energies. Each period of the wave is called the virtual k box. The number of quanta energies in each period will be changed by traveling in space. In the quantum structure of the electromagnetic waves, the capacity of each virtual k box (period) is equal to the quanta energies, where and []=1. A fully filled period contains one quanta mass in the first dimension, quanta energies in the second dimension, and quanta energies in the third dimension. Fig.1.a demonstrates a virtual cube and free positions for bullets as one-dimensional energies. While in the real world, the distance between the one-dimensional energies (k constants or quanta energies) is not static, in this model we have used static positions for better understanding. Each cube is an equivalence of one period of the electromagnetic waves. Hence, the frequency is equal to the number of cubes per second. The width and height of the cube are not Fixed. The maximum width of the cube is equal to the value of the speed of light and belongs to an electromagnetic wave with a frequency of 1 Hz. a) b) Fig.1: A model for showing the three-dimensional structure of Planck’s constant energy, and bullets as the one- dimensional energy in the electromagnetic waves. a)The capacity of Planck’s constant for carrying one- dimensional quanta energies in each period plus one quanta mass. b) An unfulfilled period at the emitting time. Each green bullet is one quanta energy. Fig.2 shows a simple model of the arrangement of k boxes or periods of the waves. The Green positions have been occupied by the one-dimensional quanta energies (k constant). Depending on the mass of the emitter, the number of occupied positions varies. Though the capacity of all k boxes (periods) is constant, the k boxes are not fully filled at the time of emitting. By accepting the Quantum Redshift, we should replace by . The Planck's equation can only calculate the maximum energy of electromagnetic waves due to their quantum structure. Fig.2: The frequency is equal to the number of cubes per second. The width and height of the cubes are not Fixed. The maximum width of the cube is equal to the value of the speed of light and belongs to an electromagnetic wave with a frequency equal to 1 Hz. After emitting the electromagnetic waves, all periods are not fully filled, and they are interested in observing quanta energies for fully filling of their free positions. The best candidates are quanta energies in the neighbor periods. The older periods take quanta energies from the near younger periods. The mechanism is time-consuming and depends on the number of free positions and the amount of losing quanta energies over time can take billions of years. Fig.3 is a simple unreal model todemonstrate the QR mechanism. Sharing of quanta energies of one period between other periods isthe reason for the QR. The older periods on the left side observe the quanta energies from the younger period on the right side (a box with red quanta energies). The electromagnetic wave loses its right box while other periods have obtained its quanta energies. As a result of losing some periods, the frequency and number of electromagnetic waves per second will be decreased. Fig.3: An unreal model for describing sharing quanta energies between periods: All periods are interested in observing quanta energies from the space or neighbor periods. The quanta energies of the right period (Red bullets) share between other periods. Recursive quantum redshift The frequency of the wave is equal to the total number of the virtual k boxes (periods) that carry in a second. On the other hand, while k boxes move in a vacuum, in each second, one or more quantaenergies will be decreased from new periods forfulling other periods. Hence, in each second, the total number of the decreased quanta energies that have distributed in a second or in 299792458 meters is equal to the frequency of the wave multiplied by the number of the quanta energies that will be decreased in each second from each period. Hence, the total number of sharing and even losing the quanta energies in each second is equal to where p is the number of the quanta energies that will be decreased in each second from each k box, and is the frequency of the wave at the start of each second. If result of the be less than the capacity of the k box, the frequency will not be decreased, and this operation will be continued until the sum of the lost (shared) quanta energies for all the k boxes (periods) reaches the capacity of the one period. After passing seconds, the total number of the shared quanta energies reaches the capacity of one period. Hence, it is an equivalent of destroying one period and send their remain quanta energies to other k boxes (periods), this operation will be decreased frequency of the wave. The value of the is given by: where is the counts of the seconds that k boxes lose their quanta energies until the sum of the lost quanta energies reaches the capacity of the one k box (q). Also, If then The equation (1) provides the time for decreasing frequency in each step. The total number of the decreased quanta energies in second is given by: where is the total number of the lost quanta energies in the seconds and is the remain number of divisions in the previous step ( After passing seconds and reaching the number of the lost quanta energies ( to the equal or greater than the capacity of the one k box (q), k boxes will be reconstructed, and the wave will be lost some k boxes (periods). Hence, frequency will be decreased. Equation is given by: where is the amount of the frequency that will be decreased. Table.1 illustrates the value of parameters of the quantum redshift in each step and obtain a recursive formula. Table.1: Parameters of the Recursive quantum redshift in each step. The results of the equation is not an integer number, hence a few numbers of the quanta energies will be remained, and we should consider them in the next step, hence: on the other hand, in the quantum redshift using (6) and (7) on the other hand, using (11) The equation (11) provides the amount of the frequency in the next step based on the frequency of the current step. In the quantum redshift for calculating the distance of a remote galaxy we use the total time of traveling the wave between the galaxy and the observer, the equation is given by: In the equations (13) and (14) value of the parameter n is not specified at the first. These equations represent recursive procedure; hence we need a computer program that according to the step by step calculate previous frequencies and time of that step until the frequency reaches the . Approximating recursive quantum redshift to non-recursive quantum redshift Although environmental parameters such as temperature and mass, have an impact on the parameter p, in a normal space we can use it as a value that will be decreased overtime. Also, the value of the q is invariant. On the other hand, in the equation (4), the amount of the is too small and we can omit it. Hence, a simple relationship between and in each second is given by: where q = 89875518173474223 Hence, t seconds after emitting, would be obtained depend on by using this equation: The equation (16) is a non-recursive quantum redshift.Table.2 shows the changes of the frequency in consequence seconds, respectively. Table.2: Parameters of the non-Recursive quantum redshift in each step. Calculating time by using frequency By using the equation of the non-recursive quantum redshift and according to the definition of the parameter z, we can calculate the time distance between the emitter and observer. using (16) In the normal electromagnetic waves, at the emitting time the value of the p is greater than 1 but after passing time it will be decreased to less than 1. For Instance, with p = 1 and q = Calculating distance by using frequency The equation (16) represents the relationship between the and . on the other hand, using (22) Calculating distance by using z Using equations (20) and (25) using (22) In the real world, scientists obtain the value of parameter z of the objects in space and calculate their distance to the observer. The equation (14) provides a recursive quantum redshift method for calculating the distance of the objects while the equation (31) represents a non-recursive quantum redshift method. The advantage of the non-recursive quantum redshift method is its higher speed of calculation. For calculating the distance of the object by using the equation (14) we need a computer program and a fast computer, but equation (31) is a simple equation that could be calculated by a professional calculator. The only restriction of the equation (31) is the value of the or the value of the parameter p. It is not a constant value. Our studies on the real data of more than 90,000 nearby stars show at the begging years after the time of emitting, the value of the p is more than hundreds and its value will be decreased to the less than one after traveling more than millions of years we will publish these results in another paper. For calculating we need a calculator that supports this kind of calculation. In this paper for simulating the Quantum redshift we assumed a constant value for the parameter p=1 and in future paper we will discussed in more detail. We have used an online calculator from this internet address However, we should compare the results of both methods to ensure that the results of the non- recursive quantum redshift method are reliable. For this reason, we wrote a program and calculate the parameter z for distances between zero to almost 8 billion light-years with a constant value for the parameter p = 1. Although choosing a constant value for the parameter p makes an inaccurate distance, we can use it for comparing recursive and no recursive methods. This range of distances covers z parameters between zero and 12. In the table.3 thecolumns (1) and (2) represent the relation between the special distances and their z value in the recursive quantum redshift, In the next step, we used all z parameters in column (2) for calculating the distances of the objects in equation (31). The results have been shown in column (3). The difference between the two methods is too small, and less than percent, hence results of the non-recursive quantum method are reliable. Another thing that we should consider is the value of the . Although, the value of the q is invariant (q= 89875518173474223), the value of the p is not constant. The parameter p is the average number of quanta energies that each individual period of the wave loses in each second at the time t, and it could mainly depend on the mass of the emitter and a little on the environmental parameters such as the temperature of the space. We should consider that p is the average number of the decreased quanta energies in a second. We should consider that result in the quantum redshift disagrees with the accelerated expansion universe theory, hence the distances that would be obtained from each value of the parameter z by the quantum redshift would be less than the distances that have been calculated by the accelerated expansion universe method Table.3: Comparing distances in the Recursive quantum redshift and non-Recursive quantum redshift. (Light Year) (Light Year) Fig.4 illustrates the relationship between distances and their z parameters in the quantum redshift theory. By increasing the distance, the z will be increased more. Meanwhile, this graph shows that the percent of the increase in the z parameter is more than the increasing percentage of the distance even with a constant value of the parameter p. This agrees with the real data of the universe. Fig.4: distances and their z parameters in the quantum redshift Quantum redshift claims the reason for redshift is the existence of free energy capacity in the periods of the electromagnetic waves, and they're interested in obtaining quanta energies for full filling. After the time of emitting, the older periods absorb quanta energies from the younger periods continuously. Sharing the quanta energies along with traveling in space is the main reason for destroying some periods and decreasing the frequency of the waves. The relationship between the and is given by: where q = 89875518173474223 and the parameter p is not a constant value, and its initial value depends on the mass of the emitter. In the begging years after the time of emitting, the value of the p is more than hundreds and its value will be decreased to less than one after traveling more than millions of years. Observational evidence supports the QR theory. In a future paper, we will show that data of 93,060 nearby space objects show an agreement with the Quantum Redshift theory. The Quantum Redshift rejects the accelerating expansion of the universe and dark energy. The QR theory not Redshift (Z) Distance (light Year) only can describe the reason for the Redshift but prove a higher rate of increasing Redshift of the distant objects. 1. Chrisitanson, Gale E. Edwin Hubble: mariner of the nebulae. Routledge, 2019. 2. Slipher, Vesto Melvin. "The radial velocity of the Andromeda Nebula." Lowell Observatory Bulletin 2 (1913): 56-57. 3. Duerbeck, Hilmar W. "Carl Wirtz—An early observational cosmologist." Morphological Cosmology. Springer, Berlin, Heidelberg, 1989. 405-407. 4. Soares, Domingos, and Luiz Paulo R. Vaz. "Solar-Motion Correction in Early Extragalactic Astrophysics." Journal of the Royal Astronomical Society of Canada 108.3 (2014). 5. Kündig, Walter. 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Monthly Notices of the Royal Astronomical Society 435.2 (2013): 1278-1286. 19. Müller, Holger, Achim Peters, and Steven Chu. "A precision measurement of the gravitational redshift by the interference of matter waves." Nature 463.7283 (2010): 926-929. 20. Di Dio, Enea, and Uroš Seljak. "The relativistic dipole and gravitational redshift on LSS." Journal of Cosmology and Astroparticle Physics 2019.04 (2019): 050. 21. Delva, Pacôme, et al. "A new test of gravitational redshift using Galileo satellites: The GREAT experiment." Comptes Rendus Physique 20.3 (2019): 176-182. 22. Savalle, Etienne, et al. "Gravitational redshift test with the future ACES mission." Classical and Quantum Gravity 36.24 (2019): 245004. 23. Kalhor, Bahram, Farzaneh Mehrparvar, and Behnam Kalhor. "k constant: a new quantum of the energy that is smaller than the Planck’s constant." Available at SSRN 3771223 (2021). 24. Kalhor, Bahram, Farzaneh Mehrparvar, and Behnam Kalhor. "How quantum of the mass, k box, and photon make light and matter?." Available at SSRN 3667603 (2020). 25. Picqué, Nathalie, and Theodor W. Hänsch. "Frequency comb spectroscopy." Nature Photonics 13.3 (2019): 146-157. 26. Gaeta, Alexander L., Michal Lipson, and Tobias J. Kippenberg. "Photonic-chip-based frequency combs." Nature Photonics 13.3 (2019): 158-169. 27. Probst, Rafael A., et al. "A crucial test for astronomical spectrograph calibration with frequency combs." Nature Astronomy (2020): 1-6. 28. Wang, Ning, et al. "Room-temperature heterodyne terahertz detection with quantum-level sensitivity." Nature Astronomy 3.11 (2019): 977-982. 29. Kalhor, Bahram, Farzaneh Mehrparvar, and Behnam Kalhor. "Is Einstein’s special relativity wrong?." Available at SSRN 3650796 (2020). 30. Kalhor, Bahram, Farzaneh Mehrparvar, and Behnam Kalhor. "Unexpected Redshift of nearby stars." Available at SSRN 3777341 (2021). 31. Lara-Avila, S., et al. "Towards quantum-limited coherent detection of terahertz waves in charge- neutral graphene." Nature Astronomy 3.11 (2019): 983-988. 32. Adams, Elizabeth AK, and Joeri van Leeuwen. "Radio surveys now both deep and wide." Nature Astronomy 3.2 (2019): 188-188. 33. Zhao, HongSheng, John A. Peacock, and Baojiu Li. "Testing gravity theories via transverse Doppler and gravitational redshifts in galaxy clusters."Physical Review D 88.4 (2013): 043013.	{"url":"https://www.researchgate.net/publication/346191733_How_does_the_quantum_structure_of_electromagnetic_waves_describe_quantum_redshift","timestamp":"2024-11-12T04:30:30Z","content_type":"text/html","content_length":"683569","record_id":"<urn:uuid:0c96cd7f-adf4-4152-a63c-99920cba0967>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00460.warc.gz"}
Please Help Me With This Fast X = 111 degrees Step-by-step explanation: Try it out "If in two triangles, sides of one triangle are proportionate to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and consequently the two triangles are comparable," asserts the Side-Side-Side (SSS) criteria. Take a look at the triangles below. - We can see that all three pairs of the sides of these triangles are congruent. - This is also known as "side-side-side" or "SSS." According to the SSS criteria for triangle congruence, two triangles are congruent if they have three pairs of congruent sides. SSS Congruence Rule Theorem When one triangle's three sides are identical to the corresponding three sides of another triangle, two triangles are said to be congruent. The aforementioned theorem will now be proven. Given: [tex]\triangle A B C[/tex] and [tex]\triangle P Q R[/tex] such that AB=PQ, BC=QR and AC=PR. To prove: [tex]\triangle A B C \cong \triangle PQR[/tex] Construction: Let BC be the longest side of [tex]\triangle A B C[/tex] and so QR is the longest side of [tex]\triangle P Q R.[/tex] Draw PS so that [tex]\angle R Q S=\angle C B A[/tex] and [tex]\angle Q R S=\angle B C A[/tex]. Join SQ and SR. In [tex]\triangle ABC[/tex] and [tex]\triangle SQR[/tex], BC=QR Given [tex]\angle C B A=\angle R Q S[/tex] By construction [tex]\angle B C A=\angle Q R S[/tex] By construction [tex]\triangle A B C \cong \triangle SQR[/tex] By ASA congruence [tex]\angle A=\angle S[/tex] By CPCTC AB=SQ By CPCTC Now, AB=PQ and AB=SQ => PQ=SQ Similarly, AC=PR and AC=SR => PR=SR Hence, [tex]\triangle ABC \cong \triangle PQR,[/tex] For more questions on Congruence of triangle	{"url":"https://www.cairokee.com/homework-solutions/please-help-me-with-this-fast-tdqv","timestamp":"2024-11-07T11:24:20Z","content_type":"text/html","content_length":"87491","record_id":"<urn:uuid:e1d149e6-4537-4d48-b79d-2b83bf886c03>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00746.warc.gz"}
Sang-Heronian Triangles and some History about Near Equilateral Triangles The first Near Equilateral triangles with consecutive integer sides and integer area (sometimes called Brahamagupta triangles) was discovered over 2500 years ago. The discovery of the 3,4,5 right triangle seems lost in antiquity back before 500 BC. All Pythagorean triangles are Heronian, but lots (infinitely many) of other triangles that are not right triangles also are Heronian. The second near equilateral triangle, the 13, 14, 15; was known to Heron of Alexandra as early as 70 AD, almost 2000 years ago. Since then, they've grown in number, and to infinity, and been dissected and diagnosed repeatedly. They've even been generalized to three dimensions in Heronian Tetrahedra. Here is one part of their story. Heron of Alexandria is known to have developed a method of finding the area of triangles using only the lengths of the three sides. It is known that it was proven in his Metrica around 60 AD. His proof was extended in the 7th century by Brahamagupta extended this property to the sides of inscribable quadrilaterals. Since around 1880, the triangular method of Heron has been known as Heron's formula, or Hero's Formula. It emerged in French, formula d'Heron (1883?) and German, Heronisch formel (1875?) and in George Chrystal's Algebra in 1886 in England. L E Dickson's History of Number Theory states that Heron stated the 13, 14, 15 triangle and gave its area as 84, the height of 12 being the common side of a 5,12,13 triangle and a 9, 12, 15. The 5 and 9 combining to form the base of length 14. Brahmagupta is cited in the same work for giving an oblique triangle composed of two right triangles with a common leg a, stating that the three sides are $ \frac{1}{2}(\frac{a^2}{b}+ b)$ , $ \frac{1}{2}(\frac{a^2}{c}+ c)$, and $ \frac{1}{2}(\frac{a^2}{b}- b) + ( \frac{1}{2}(\frac{a^2}{c}- c)$ In 1621 Bachet took two Pythagorean right triangles with a common leg, 12, 35, 37 and 12, 16, 20 and produced a triangle with sides of 37, 20, 51. With an area of 306 if I did my numbers right. Vieta and Frans van Schooten, both used the same approach of clasping two right triangles with a common leg; and by the first half of the 18th century, the Japanese scholar, Matsunago, realized that any two right triangles would work, by simply multiplying the sides of each by the hypotenuse of the other, he could juxtapose the two resulting triangles. In the early 1800's through 1825 the problem was alive and hopping on the Ladies Diary and the Gentleman's Math Companion. One method created right triangles in another triangle to be reassembled into a rational triangle, similar in fact, to the problem that would appear in the 1916 American Mathematical Monthly. (Note; any triangle with rational sides and area can be scaled to become a Heronian triangle.) In a letter of Oct 21, 1847; Gauss to H. C. Schumacher, he stated a method using circumscribed circles, and found lots of others chose the exact same solutions in their response. E. W. Grebe tabulated a set of 46 rational triangles in 1856. W. A. Whitworth noticed that the 13, 14, 15 triangle of antiquity, that had an altitude of 12, was the only one in which the altitude and sides were all consecutive. (1880) Somehow, among all those, the contributions of a Professor from Scotland was not observed by Dickson. The first modern western article I can find on the topic of Near Equilateral triangles with integer sides and area is from Edward Sang which appeared in 1864 in the Transactions of the Royal Society of Edinburgh, Volume 23 . I find it interesting that this is only a small aside in a much larger article and that he begins with an approach to examining the angles. Then he arrives at the use of a Pell type equation for approximating the square root of three, $a^2 = 3x^2 + 1 $ and shows that every other convergent in the chain of approximations is a base of a Near Equilateral Triangle, using sides of consecutive integers. The alternate convergents we seek are given by 2/1, 7/4, 26/15, 97/56... each approaching the square root of three more closely, but also each with a numerator that is 1/2 the base of triangle with consecutive integers for sides and integer area. Perhaps it is easier to just use the recurent relation, $n_i=4n_{i-1} - n_{i-2}$ with $n_0=2$, and $n_1=4$ for the actual middle side,2, 4, 14, 52, 194.... The first few such triangles have their even integer base as2x1=2; (1, 2, 3) area 0; 2x2=4; (3, 4, 5); area 12; 2x7=14; (13, 14, 15); area 84; 2x26=52; (51, 52, 53); area 1170... etc. Throughout, he refers to "trigons" rather than triangles, and never invokes the name of Heron throughout. A note about Edward Sang before I continue. In a 2021 article by Dennis Rogel, he calls Professor Sang, " probably the greatest calculator of logarithms of the 19th century" in A guide to Edward Sang’s and to their reconstructions, and Wikipedia adds that he is, "best known for having computed large tables of logarithms, with the help of two of his daughters. (They did not mention the names of the daughters, but in Rogel's paper he cites Flora and Jane.) These tables went beyond the tables of Henry Briggs, Adriaan Vlacq, and Gaspard de Prony." They add a list of his publications so extensive that they are grouped into five year periods. The next paper using consecutive integer sides was in 1880 by a German mathematician named Reinhold Hoppe, who produced a closed form expression for these almost equilateral Heronian Triangles that was similar to, $ b_n =(2+2\sqrt{3})^n + (2-2\sqrt{3})^n $. His paper calls them "rationales dreieck" (rational triangles) I have not seen the entire paper, and don't know if the term Heronian appeared, or not. The first American introduction to the phrase "Heronian Triangles", seemed to be an article in the American Mathematical Monthly which posed the introduction as a problem, to divide the triangle whose sides are 52, 56, and 60 into three Heronian Triangles by lines drawn from the vertices to a point within. The problem was posed by Norman Anning, Chillwack, B.C. It then includes a description that suggests it is introducing a new term, "The word Heronian is used in the sense of the German Heronische (with a German citation) to describe a triangle whose sides and area are integral. The only other mentions of a Heronian triangle in English in a google search before the midpoint of the 20th century revealed a 1930 article from the Texas Mathematics Teacher's Bulletin. It credits a 1929 talk, it seems, by Dr. Wm. Fitch Cheney Jr. who, "discusses triangles with rational area K and integral sides a, b, c, the g.c.f of the sides 1, under the name Heronian triangles." (Dr Cheney published an article in the American Mathematical Monthly in 1929, The American Mathematical Monthly, Vol. 36, No. 1 (Jan., 1929), pp. 22-28) Since any such rational area can be scaled up to an appropriate integer area with integer sides these address the general Heronian Triangle, but still no Near Equilateral, or at least not revealed in the snippet view. By the 1980's an article in the Fibonacci Quarterly found a way to produce a Fibonacci like sequence, a second order recursive relation to produce the even bases. Letting $B_0 = 2, and B_1 = 4$, the recursion was $ U_{n+2} = 4 U{n+1} + U_n$ . This paper by W. H. Gould of West Virginia University addresses the full scope of consecutive sided integer triangles and mentions Hoppe, but not Professor Sang. Gould's paper seems to be his solution to a problem he had posed earlier in the Fibonacci Quarterly, "of finding all triangles having integral area and consecutive integral sides." (H. W. Gould, Problem H-37, Fibonacci Quarterly, Vol. 2 (1964), p. 124. .) Gould also mentions two other, seemingly earlier posed problems in other journals which I have yet to explore, and given the opportunity, will do so and return to this spot, If you are impatient, they are 7. T. R. Running, Problem 4047, Amer. Math. Monthly, Vol. 49 (1942), p. 479; Solutions by W. B. Clarke and E. P. Starke, ibid. , Vol. 51 (1944), pp. 102-104. 8. W. B. Clarke, Problem 65, National Math. Mag. , Vol. 9 (1934), p. 63 Gould's article is a wonderful read for the geometry of the incircles and Euler lines in such special triangles is well explored. These are each candidates to be the first American proposal of these consecutive integer sided triangles, but it seems Gould's paper was the first to expand the full scope of the solutions in any Some of the characteristics of these I think would be found interesting to HS and MS age students I will spell out below. As mentioned above, the length of the middle (even) side follows a 2nd order recursive relation $B_n = 4B_{n-1}-B_{n-2}$ so the sequence of these even sides runs 2, 4, 14, 52, 194, 724..... etc. ) is there to represent the degenerate triangle 1,2,3. Interestingly, the heights follow this same recursive method giving heights of 0, 3, 12, 45, 168.... The height divides the even side into two legs of Pythagorean triangles that make up the whole of the consecutive integer triangle. They are always divide so that one is four greater than the other, or each is b/2 =+/- 2. Of the two triangles formed by on each side of the altitude, one is a primitive Pythagorean triangle, PPT, and the other is not. The one that is a PPT switches from side to side on each new triangle, alternately with the shorter leg, and then the longer leg. Here are the triangles with the two subdivisions of them with an asterisk Marking the PPT: Short Base Long small triangle large triangle 3 4 5 3 4 5 13 14 15 5, 12, 13 9, 12, 15 51 52 53 24, 45, 51 * 28, 45, 53 193 194 195 95, 168, 193 99, 168, 195 723 724 725 360, 627, 723 364, 627, 725 The pattern of the ending digits of 3, 4, 5 repeated twice, and 1,2,3 once. From every one of these Sang-Heronian triangles (I think) you can get another Heronian triangle by a simple reflection of whichever right triangle has the shorter hypotenuse. For example, in the 13, 14, 15 triangle, if you reflect the 5, 12 13 triangle around the altitude (12), you get a triangle with sides of (9-5),12, 15. And the area is equal to the difference of the areas of the two right triangles, 54 sq un. In the 1929 article mentioned above, Dr. Cheney writes that he knows of no examples of Heronian triangles up to that time that were not made up of two right triangles, and then gives an example of one that is not decomposable, 25, 34, 39. He also points out that the altitudes of Heronian triangles are not always integers, and gives the example of 39,58,95 as an example which I calculate to be A paper by Herb Bailey and William Gosnell in Mathematics Magazine, October 2012 demonstrates Heronian triangles in other arithmetic progressions from the near-equilateral ones. "We note below that if a triangle has consecutive integer sides, then it has integer area if and only if its inradius is an integer. Thus we might as well have defined a Brahmagupta triangle as one with consecutive integer sides and integer inradius. The computations for generating Brahmagupta triangles are made somewhat easier by focusing on inradius rather than area." And the The inradius 𝑟 is given by $ r=\sqrt(\frac{(s-a)(s-b)(s-c)}{s})$ I mentioned that there are also Heronian Tetrahedra, although that use of Heronian seems even later than for triangles, perhaps as late as 2006. The earliest example of an exact rational tetrahedra with all integer edges, surfaces and volume was by Euler. He created a tetrahedron formed by three right triangles parallel to the xyz coordinate axes, and one oblique face connecting them. The triple right angle edges were 153, 104, and 672, and the three edges of the oblique face were 185, 680, and 697. These were each Pythagorean right triangles, the four faces of (104,672,680), (153,680,697), (153,104,185) and (185,672,697) There are an infinite number of these Eulerian Birectangular tetrahedra, but they seem to get very large very quickly. Euler showed that they can be found by deriving the three axis-parallel sides a, b, and c by using four numbers that are the equal sums of two fourth powers. Euler found an example using 386678175, 332273368, and 379083360, Yes, those numbers are each in the hundreds of millions, and each pair had a larger hypotenuse to form a third side. I recently (2024) found a very similar formula for the square of the volume of a Tetrahedron in a paper in Letters to the editor of The Mathematical Intellingencer by Martin Lukarevski. The Formula applies to a tetrahedron with all four faces congruent with edges a, b, c. I did a little simple algebra to enhance the similarity to Heron's formula. And at the near end of the Wikipedia discussion of these states, "A complete classification of all Heronian tetrahedra remains unknown."	{"url":"https://pballew.blogspot.com/2024/04/sang-heronian-triangles-and-some.html","timestamp":"2024-11-05T01:24:21Z","content_type":"application/xhtml+xml","content_length":"178021","record_id":"<urn:uuid:283049d1-b8aa-4d7e-8115-dad4781ecfb0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00489.warc.gz"}
Topics: Weyl Tensor In General > s.a. bianchi models; curvature; FLRW geometry; riemann tensor; Weyl Curvature Hypothesis [Penrose]. $ Def: The "trace-free part" of the Riemann tensor, defined on a manifold of dimension n ≥ 3 by \[ \def\_#1{_{#1}^{\,}} \def\ii{{\rm i}} C\_{abcd}:= R\_{abcd} - {2\over n-2}\,(g\_{a[c}\,R\_{d]b} - g\_{b[c}\,R\_{d]a}) + {2\over(n-1)(n-2)}\,R\,g\_{a[c}\,g\_{d]b} \;;\] The definition can also be written in terms of the Weyl-Schouten tensor S[ab], as \[ C\_{abcd}:= R\_{abcd} - {2\over n-2}\,(g\_{a[c}\,S\_{d]b} - g\_{b[c}\,S\_{d]a})\;, \quad{\rm with}\quad S\_{ab}:= R\_{ab} - {1\over2(n-1)}\,R\,g\_{ab}\;. \] * Properties: It is conformally invariant, if expressed with indices C[abc]^d, and its trace over any two indices vanishes; The number of independent components in n dimensions is ${1\over12}$n (n +1) (n+2) (n−3) [@ e.g., in Gursky & Viaclovsky AM(07)]. * Use: In general relativity it contains the information on gravitational radiation, since the "trace part" of the Riemann tensor is determined by the matter. @ General references: Weyl MZ(18), re GRG(22); in Weinberg 72; in Wald 84; Ehlers & Buchert GRG(09)-a0907 [Newtonian limit]; Dewar & Weatherall FP(18)-a1707-conf [in geometrized Newtonian @ Related topics and uses: Schmidt GRG(03)gq [square]; Hussain et al IJMPD(05)-a0812 [collineations]; Danehkar MPLA(09) [significance]; Hofmann et al al PRD(13)-a1308 [limitations of the interpretation in terms of incoming and outgoing waves]; > s.a. gravitational entropy; phenomenology of gravity. Electric Part $ Def: The symmetric traceless tensor defined with respect to a unit timelike vector u^a — for example the unit normal to a hypersurface Σ or a (possibly non-hypersurface-orthogonal) matter 4-velocity vector — by E[ab]:= C[ambn] u^m u^n. * And physics: It corresponds to tidal forces; Near spatial infinity, using an appropriately rescaled curvature on the hyperboloid ${\cal D}$ [> see asymptotic flatness at spi], it represents the way in which nearby geodesics tear apart from each other. * Potential: It admits a potential E, such that E[ab] = −${1\over4}$(D[a] D[b] E + E h[ab]); This is used to define 4-momentum. @ References: Ashtekar in(80); Bonnor CQG(95); Maartens et al CQG(97)gq/96 [and gravitational degrees of freedom]; Munoz & Bruni CQG(23)-a2211, PRD(23)-a2302 [numerical code]. Magnetic Part $ Def: Given a unit timelike vector u^a as in the definition of the electric part, it is the symmetric traceless tensor defined by (H[ab] or) B[ab]:= ${1\over2}$C[ambn] u^m u^n, with C[ambn]:= ε[ampq] C^ pq[bn] . * And physics: In a weak-field approximation to the gravitational field, its effects are similar to those arising from the Lorentz force in electromagnetism, and it is responsible for gravitomagnetic effects like frame dragging; In cosmological perturbation theory a non-zero magnetic Weyl tensor is associated with the vector modes of the first post-Newtonian contribution, and it has been shown to be responsible for destroying the pure Kasner-like approach to the singularity in BKL evolution. * Potential: The one constructed from the appropriate curvature on the hyperboloid ${\cal D}$ at spatial infinity admits a potential K[ab], such that B[ab] = −${1\over4}$ε[mnb] D^m K^ n[a]. * Purely magnetic spacetimes: Spacetimes in which the electric part of the Weyl tensor, C[abcd] u^b u^d = 0, for some timelike unit vector field u^a, vanishes; 2004, Examples of purely magnetic spacetimes are known and are relatively easy to construct, if no restrictions are placed on the energy-momentum tensor; However, it has long been conjectured that purely magnetic vacuum spacetimes (with or without a cosmological constant) do not exist; For irrotational dust, the only solutions are FLRW spacetimes. @ Purely magnetic: Haddow JMP(95)gq; Van den Bergh CQG(03)gq/02, CQG(03)gq, Zakhary & Carminati GRG(05) [vacuum no-go results]; Lozanovski CQG(02), & Carminati CQG(03) [locally rotationally symmetric]; Barnes gq/04-proc; Wylleman CQG(06)gq [irrotational dust, any cosmological constant]; Wylleman & Van den Bergh PRD(06)gq [classification]; Hervik et al SPP(14)-a1301 [and purely electric, in higher dimensions]; Danehkar IJMPD(20)-a2006-GRF. @ And physics: Ellis & Dunsby ApJ(97)ap/94 [evolution in general relativity and "Newtonian gravity"]; Bruni & Sopuerta CQG(03)gq/03 [approach to the singularity]; Clifton et al GRG(17) [effect on universal expansion, with regularly arranged discrete masses]. Invariants > s.a. petrov-pirani classification; riemann tensor. * Vacuum 4D spacetime: There are only 4 independent algebraic curvature invariants, and they can be expressed in terms of the two complex invariants I:= ${1\over2}$M^abM[ab] = ${1\over16}$(C[ab]^cd C[cd]^ab − ${\ii\over2}$C[ab]^cd ε[cd]^mn C[mn]^ab) J:= ${1\over6}$M^abM[cb] M[ac] [??] = ${1\over96}$ (C[ab]^cd C[cd]^mn C[mn]^ab − ${\ii\over2}$C[ab]^cd C[cd]^mn ε[mn]^pq C[pq]^ab) , where M[ab]:= E[ab] + i B[ab] is the sum of the electric and magnetic parts of the Weyl tensor. @ General references: Nita & Robinson gq/01 = Nita GRG(03); Beetle & Burko PRL(02)gq [radiation scalars]. @ Classification, in higher dimensions: Boulanger & Erdmenger CQG(04)ht [8D]; Ortaggio CQG(09)-a0906 [Bel-Debever characterization]; Coley & Hervik CQG(10)-a0909 [higher-dimensional Lorentzian manifolds]; Senovilla CQG(10)-a1008 [based on its superenergy tensor]; Godazgar CQG(10)-a1008 [spinor classification]; Coley et al CQG(12)-a1203 [5D, refinement]; Batista GRG(13)-a1301; Batista & da Cunha JMP(13)-a1212 [6D]; > s.a. spin coefficients [Newman-Penrose and GHP formalisms]. Related Concepts > s.a. Peeling; Poynting Vector; riemann tensor [symmetries]; spin coefficients [NP formalism]; self-dual solutions. * Determining the metric: The spacetime metric is generically determined up to a constant factor by C[abc]^d and T[ab]. @ Potential: Edgar & Senovilla CQG(04)gq [for all dimensions]; > s.a. lanczos tensor. @ Other related topics: Hall & Sharif NCB(03)gq/04 [metric from C[abc]^d and T[ab]]; Mantica & Molinari IJGMP(14)-a1212 [Weyl-compatible tensors]; Ortaggio & Pravdová PRD(14)-a1403 [in higher dimensions, asymptotic behavior at null infinity]; > s.a. curvature [Bianchi identities]; general relativity actions. main page – abbreviations – journals – comments – other sites – acknowledgements send feedback and suggestions to bombelli at olemiss.edu – modified 30 dec 2023	{"url":"https://www.phy.olemiss.edu/~luca/Topics/w/weyl_tensor.html","timestamp":"2024-11-02T15:15:02Z","content_type":"text/html","content_length":"19291","record_id":"<urn:uuid:1ca212c6-49ef-4f22-b029-2a7bf9321ac2>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00428.warc.gz"}
What Is the Complexity of Elliptic Systems? 1984 Reports What Is the Complexity of Elliptic Systems? This paper deals with the optimal solution of the Petrovsky-elliptic system lu = f, where l is of homogeneous order t and f (x) ∈ H (Ω).Of particular interest is the strength of finite element information (FEI) of degree k, as well as the quality of the finite element method (FEM) using this information. We show that the FEM is quasi-optimal iff k ≥ r+t - 1. Suppose this inequality is violated; is the lack of optimality in the FEM due to the information that it uses, or is it because the FEM makes inefficient use of its information! We show that the latter is the ease. The FEI is always quasi-optimal information. That is, the spline algorithm using FEI is always a quasi-optimal algorithm. In addition, we show that the asymptotic penalty for using the FEM when k is too small (rather than the spline algorithm which uses the same finite element information as the FEI) is unbounded. More About This Work Academic Units Department of Computer Science, Columbia University Published Here February 22, 2012	{"url":"https://academiccommons.columbia.edu/doi/10.7916/D81N888T","timestamp":"2024-11-09T13:36:35Z","content_type":"text/html","content_length":"16771","record_id":"<urn:uuid:e666067f-3f53-48b0-9274-f03099b7696c>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00295.warc.gz"}
There are 5 item/s. Title Date Views Brief Description Exploring teachers’ learning of instructional practice in 2016 854 Decades of research on mathematics teaching have identified fundamental instructional practices that promote deep learning of professional development mathematics for all students. In contrast with more traditional and direct approaches to teaching, these core instructional ... Describing the practice of anticipating students’ 2024 111 Teachers’ ability to elicit and use evidence of student thinking during instruction is critical in high-quality mathematics mathematics of secondary mathematics teachers : a multi-case instruction. The practice of anticipating students’ mathematics supports teachers in noticing and being prepared to respond study to... Conceptualizing and investigating mathematics teacher 2018 761 Researchers and teacher educators have made advances in describing mathematics instruction that can support all students in learning of practice developing conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive The case of Jamie: examining storylines and positions over 2019 581 This study utilizes Positioning Theory as a lens to analyze interactions between a teacher and her students. Using those time in a secondary mathematics classroom interactions, this study seeks to better catalog and understand pervasive storylines in one teacher’s secondary mathematics On the nature of and teachers' goals for students' 2013 1843 In an era of new standards and emerging accountability systems, an understanding of the supports needed to aid teachers and mathematical argumentation in five high school classrooms students in making necessary transitions in mathematics teaching and learning is critical. Given the established research	{"url":"http://libres.uncg.edu/ir/uncg/clist-etd.aspx?fn=P.%20Holt&ln=Wilson&org=uncg","timestamp":"2024-11-01T23:53:35Z","content_type":"application/xhtml+xml","content_length":"12919","record_id":"<urn:uuid:88bc7cf4-8721-4b2d-8bea-c8e21b482cc6>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00372.warc.gz"}
1.5.3: Evaluate Expressions with One or More Variables Last updated Page ID $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ ( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$ $ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$ $ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$ $ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vectorC}[1]{\textbf{#1}} $ $ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $ $ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $ $ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $ $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $ $\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\ evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\ newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y} $ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real} {\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec} [3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array} {r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\ wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\ newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var} {\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left\|#1\right\|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\ bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\ widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$ Expressions with One or More Variables Figure $\PageIndex{1}$ Dexter is in charge of ticket sales at his town’s water park. He has to report to his boss how many tickets he sells and how much money the water park makes in ticket sales each day - adult tickets ($7), child tickets ($5). Today, Dexter has sold 100 adult and 125 child tickets. Yesterday, he sold 120 adult and 120 child tickets. Can Dexter write an expression to figure out today’s ticket sales, and use this to compare today’s sales to yesterday’s? In this concept, you will learn how to evaluate expressions that have multiple variables and/or multiple operations. Expressions with Multiple Variables When evaluating expressions with multiple variables and multiple operations, it is important to remember the order of operations. Order of Operations: P - parentheses E- exponents MD - multiplication and division, in order from left to right AS - addition and subtraction, in order from left to right Whenever you are evaluating an expression with more than one operation in it, always refer back to the order of operations. Let's look at an example of an expression with multiple variables and operations. Evaluate 6a+b when a is 4 and b is 5. First, you can see that there are two variables in this expression, a and b. There are also two operations here: multiplication, seen in "6 times the value of a" and addition, seen in "+b". You are given values for a and b. First, substitute the given values for each variable into the expression. Then, evaluate the expression according to order of operations. The answer is 29. Let’s look at another example with multiple variables and expressions. Evaluate 7b−d when b is 7 and d is 11. First, substitute the given values in for the variables. Then, evaluate the expression according to order of operations. The answer is 38. Sometimes you may have an expression that is all variables. Evaluate this in the same way. Evaluate ab+cd when a is 4, b is 3, c is 10 and d is 6. First, substitute the given values in for the variables. Next, evaluate the expression according to order of operations. The answer is 72. Example $\PageIndex{1}$ Earlier, you were given a problem about Dexter and his tickets. Dexter needs to write an expression to figure out the total money made from the 100 adult and 125 child tickets he sold today, compared to the 120 adult and 120 child tickets he sold yesterday. The adult tickets cost $7 and the child tickets cost $5. First, write an expression. Next, substitute in the given values for today’s ticket sales. Then, follow order of operations to multiply and then add. 7(100)+5(125) Multiply 7×100 700+5(125) Multiply 5×125 700+625 Add 700+625=1325 The answer is 1,325. Next, substitute in the given values for yesterday’s ticket sales. 7(120)+5(120)Substitute 120 for both x and y Then, follow order of operations to multiply and then add. 7(120)+5(120) Multiply 7×120 840+5(120) Multiply 5×120 840+600 Add 840+600=1440 The answer is 1,440. Finally, subtract 1,325 from 1,440 to get the difference. The answer is 115. Dexter can report to his boss that today the park sold $1,325 in tickets, which is $115 less than the $1,440 they sold in tickets yesterday. Example $\PageIndex{1}$ Evaluate a+ab+cd when a is 4, b is 9, c is 6 and d is 4. First, substitute the given values into the expression. Next, evaluate according to order of operations. Example $\PageIndex{1}$ Evaluate 12x−y when x is 4 and y is 9. First, substitute 4 for x and 9 for y. 12x−y Substitute 4 in place of x and 9 for y Then, evaluate the expression. 12(4)−(9) Multiply 12×4=48 48−(9) Subtract 48−9=39 Example $\PageIndex{1}$ Evaluate (12/a)+4 when a is 3. First, substitute 3 for a. Next, evaluate the expression. 123+4 Divide 12÷3=4 4+4 Add 4+4=8 Example $\PageIndex{1}$ Evaluate 5x+3y when x is 4 and y is 8. First, substitute 4 for x and 8 for y. 5x+3y Substitute 4 for x and 8 for y Then, evaluate the expression. 5(4)+3(8) Multiply 5×4=20 20+3(8) Multiply 3×8=24 20+24 Add 20+24=44 The answer is 44. Evaluate each multi-variable expression when x=2 and y=3. 1. 2x+y 2. 9x−y 3. x+y 4. xy 5. xy+3 6. 9y−5 7. 10x−2y 8. 3x+6y 9. 2x+2y 10. 7x−3y 11. 3y−2 12. 10x−8 13. 12x−3y 14. 9x+7y 15. 11x−7y Review (Answers) To see the Review answers, open this PDF file and look for section 1.14. Term Definition algebraic The word algebraic indicates that a given expression or equation includes variables. Algebraic An expression that has numbers, operations and variables, but no equals sign. Exponent Exponents are used to describe the number of times that a term is multiplied by itself. Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols. Order of The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D Operations - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right. Parentheses Parentheses "(" and ")" are used in algebraic expressions as grouping symbols. revenue Revenue is money that is earned. substitute In algebra, to substitute means to replace a variable or term with a specific value. Variable A variable expression is a mathematical phrase that contains at least one variable or unknown quantity. Additional Resources PLIX: Play, Learn, Interact, eXplore: Expressions with One or More Variables: Water Bottle Expression Activity: Expressions with One or More Variables Discussion Questions Practice: Evaluate Expressions with One or More Variables Real World Application: MotoGP	{"url":"https://k12.libretexts.org/Bookshelves/Mathematics/Algebra/01%3A_Real_Numbers_Variables_and_Expressions_-_Real_Numbers_and_Order_Real_Numbers/1.05%3A_Algebra_Expressions_and_Variables/1.5.03%3A_Evaluate_Expressions_with_One_or_More_Variables","timestamp":"2024-11-02T14:23:47Z","content_type":"text/html","content_length":"139205","record_id":"<urn:uuid:443fe70b-970d-430c-91fe-051e267e379c>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00594.warc.gz"}
Test Bank for A Concise Introduction to Logic A Concise Introduction to Logic 14th Edition Hurley \| Test Bank Test Bank for A Concise Introduction to Logic, 14th Edition, Patrick J. Hurley, ISBN-10: 0357798686, ISBN-13: 9780357798683 Table of Contents Part I: INFORMAL LOGIC. 1. Basic Concepts. 2. Language: Meaning and Definition. 3. Informal Fallacies. Part II: FORMAL LOGIC. 4. Categorical Propositions. 5. Categorical Syllogisms. 6. Propositional Logic. 7. Natural Deduction in Propositional Logic. 8. Predicate Logic. Part III: INDUCTIVE LOGIC. 9. Analogy and Legal and Moral Reasoning. 10. Causality and Mill’s Methods. 11. Probability. 12. Statistical Reasoning. 13. Hypothetical/Scientific Reasoning. 14. Science and Superstition.	{"url":"https://testbankwebs.com/product/test-bank-downloadable-files-for-a-concise-introduction-to-logic-14th-edition-patrick-j-hurley-isbn-10-0357798686-isbn-13-9780357798683/","timestamp":"2024-11-06T06:09:37Z","content_type":"text/html","content_length":"111633","record_id":"<urn:uuid:968d72fc-535f-4b78-bcc3-96d2045616e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00492.warc.gz"}
Generation of Equations for Thermoptim Models Once a Thermoptim model is finalized, its diagram and project files contain all the information needed for its complete resolution. It is therefore possible to use them to generate the corresponding set of equations, thanks to a feature of the software currently under development. This allows, in particular: • To maintain a systemic approach in describing the architecture of the model, while providing access to its equations, so that they can be used in other environments and additional processing can be performed beyond what the software offers, such as optimization. • To ensure the consistency of the sets of equations for large-scale models, which can be difficult in practice if an appropriate tool is not available. • Thermoptim can thus be used as a preprocessor for various applied thermodynamics solvers, such as Interactive Thermodynamics, EES, or Matlab. Conventions Used The following conventions are used: • The names of unknowns are defined by concatenating their usual symbol (T, p, h, s, v...) with the name of the point or process, using the underscore ‘_' as a link. Thus, the enthalpy of point 2 is written as h_2. The advantage of this notation is that it is very clear, and it transforms into h2 if the equation is displayed in editors using LaTeX syntax. Note that you can add variables and equations to have them analyzed by Thermoptim. In this case, the variable names must contain the underscore to be taken into account, and each new equation must be preceded by the line // Equation: ijk, ijk being its number. • Flow rates are denoted as ‘m_dot_' followed by the name of the process, for example, m_dot_condenser for the flow rate that passes through the process ‘condenser'. • When special characters or spaces are used in the names of Thermoptim elements, they are simply removed. For example, the enthalpy of the point ‘entrée d'air' will be written as h_entredair. Nothing prevents renaming this unknown later if desired. • The functions for calculating properties are generated in two steps. They are first written in the following generic format, which is easy for a human to interpret: functionName("fluid";T = T_1;P = P_1). The arguments of the function are thus clearly identifiable, without risk of permutation. Thus, the calculation of the enthalpy of water knowing its pressure and entropy is written as: calcH_PS("eau";P = p_2;s = s_2). • The equation set thus generated can then be easily converted to the specific format of each solver. • Comments can be added either on a complete line or after an equation. They are preceded by the characters ‘//', according to the syntax used in a number of programming languages. • It should be noted that it is not intended to provide equations for all the numerous parameterizations that Thermoptim allows because, once a set of equations is established, it is easy to modify some of them manually according to the objectives pursued. For example, the equation chosen to calculate the output pressure of a compressor can be the product of the input pressure by the known compression ratio. It remains valid if the pressure is set and the compression ratio is calculated. To change the parameterization, it is sufficient to change the known value. • Since a heat exchanger or a combustion chamber is generally isobaric, we have implemented P_downstream = P_upstream. • In the entry process-points, the complete state is calculated as given value, as well as the flow rate. • To determine the flow rates, we start from the process with set flow rates or those downstream of the nodes, which are all recalculated, and we propagate the flow rate to those downstream if the connection is unique. • Although this is not a mandatory requirement for most solvers, the equations are generally written in the form “Variable = expression as a function of other variables”. Example of Equation Generation Currently, the formal generation of equations is only implemented for the core components of Thermoptim, and this implementation remains provisional. Therefore, some modifications in the generated files are to be expected in the future. The equations generated for a turbine modeled with an isentropic reference and a given expansion ratio are of the following type: //Process: turbine //Equation : 23 m_dot_turbine = m_dot_superheater // Upstream process : superheater - Downstream process : turbine //Equation : 24 s_3 = calcS_PH("water";P = p_3;H = h_3) // Upstream point : 3 - Downstream point : 4 // Comment = Isentropic reference //Equation : 25 hs_4 = calcH_PS("water";P = p_4;S = s_3) // Downstream point : 4 //Equation : 26 etaT_turbine = 0.9// Isentropic efficiency //Equation : 27 h_4 = h_3 - etaT_turbine(h_3 - hs_4) // Upstream point : 3 - Downstream point : 4 //Equation : 28 xl_4 = 0.// Saturated liquid quality //Equation : 29 Tl_4 = T_4- 0.01// Saturated liquid temperature //Equation : 30 xv_4 = 1.// Saturated vapor quality //Equation : 31 Tv_4 = T_4+ 0.01// Saturated vapor temperature //Equation : 32 hl_4 = calcH_TPx("water";T = Tl_4;P = p_4;X = xl_4)// Saturated liquid enthalpy //Equation : 33 hv_4 = calcH_TPx("water";T = Tv_4;P = p_4;X = xv_4)// Saturated vapor enthalpy //Equation : 34 x_4 = (h_4 - hl_4)/(hv_4 - hl_4)// Quality //Equation : 35 T_4 = calcTsat("water";P = p_4 ;X = x_4) // Downstream point : 4 //Equation : 36 s_4 = calcS_PH("water";P = p_4;H = h_4) // Entropy // Comment = Given outlet pressure //Equation : 37 p_4 = 0.0356// Outlet pressure //Equation : 38 W_dot_turbine = m_dot_turbine(h_4 - h_3) // DeltaH Equation 23 translates the propagation of the flow rate from the upstream process. Equation 24 calculates the entropy of the upstream point, and Equation 25 calculates the enthalpy of the downstream point corresponding to the isentropic evolution. Equation 26 provides the value of the isentropic efficiency, and Equation 27 determines the actual enthalpy of the downstream point. Equations 28 to 34 provide the turbine outlet quality. Equation 35 provides the temperature of the downstream point, and Equation 36 provides its entropy. Equation 37 gives the downstream pressure, and Equation 38 gives the useful work. Utilization of the Generated Equations Thermoptim automatically extracts from the model a first set of commented equations that can be considered raw. Their number is quite high, as a process typically requires more than a dozen: 4 for the state of each entry and exit point (T, p, x, h), 2 for the flow rate and the energy involved delta H, plus those allowing its calculation. Even a very simple model generally includes at least fifty such equations. Analyzing this set of raw equations to verify their consistency and identify any gaps can be a tedious task if done manually. Utilities are available to facilitate this process. The first step consists of resolving the redundancies that may exist between the equations, particularly when a variable is calculated in two different ways, which can occur depending on the model's parameterization. The analysis that is carried out considers that there is redundancy if the same variable x_y appears twice to the left of the '=' sign: x_y = expression 1, x_y = expression 2. These redundancies are caused by variants in the interpretation of the Thermoptim model parameterization. Since the software is not capable of choosing between the concerned equations, it is up to the modeler to do so, especially since some apparent redundancies are not true redundancies, as some variables may have to be calculated by inverting one or more equations. Thermoptim establishes a list of potential redundancies, and it is up to the user to remove the unnecessary equations by comparing those that have been generated and retaining those they desire. To do this, the user simply double-clicks on a redundancy line to add or remove '//' in front of the equation, which disables or enables the equation. The same '//' appears on the selected line in the right screen, to remind you that it has been changed. The choice between redundant equations is not always straightforward. To help you choose, you can display them successively in the left screen by double-clicking on each of them, which allows you to know which block it belongs to and to consult the comments that accompany it. Don't forget to double-click a second time to revert to the previous state of the equation until you have made your final The second step allows for signaling if a variable in the problem never appears on the left side of an equation, which a priori suggests that it is neither initialized nor calculated from one of the equations. This allows for detecting potential oversights in the definition of the problem to be solved. An example is that of a simple steam cycle, where the turbine inlet temperature is not automatically considered by Thermoptim as a given value. However, as with redundancies, some variables do not appear explicitly to the left of the '=' sign in the equations, because they are calculated by inverting some of them. To potentially reduce the dimension of the problem, an algorithm for ordering the remaining equations has been developed and can be used in a third step. Its principle is as follows: It begins by identifying all variables that Thermoptim considers to have set values, with the generated equation in the form "Variable = numerical data". For example: m_dot_condenser = 100 // Set flow This provides a first group of resolved equations that correspond to the problem's data. The second group consists of equations that depend only on the variables of the first group. For example: m_dot_feedpump = m_dot_condenser // Flow propagation A third group is one whose equations depend only on the variables of the first and second groups, and so on, with the algorithm operating recursively. At the end, there remains a group of equations to be solved using the solver, with the solutions of the others being directly obtainable by substituting the unknown variables with those already The presentation of the procedure for reducing the model size can also have a didactic interest. Of course, this ordering is unnecessary if the solver is powerful and used blindly. As already indicated, the number of Thermoptim parameters is very high. It would be difficult to predict all the corresponding equations, but this would not be of great interest anyway, given that once a set of equations is validated, it is very simple to complete it at will according to the objectives pursued. Therefore, only the main parameterizations have been retained. An additional feature has been added to the analysis of the equation set generated by Thermoptim. When a variable is one of the groups highlighted above, it is possible to display the graph of its direct dependencies on those of the lower rank groups. To do this, simply select it in the left panel of the equation processor screen, then click on the "Display dependency graph" button. The Direct Dependency Tree is displayed in two forms in two new windows, which allow it to be exported in text or mind map format. One of the windows simply displays the graph of the dependent variables, while the other additionally displays the equations that determine this dependence, as shown in the figure below. Note that the analysis of the equations is done on the entire content of the left panel and not only on the equations generated by Thermoptim. Provided, of course, that it follows the same format as these, you can use these features on any set of equations. Note also that this type of graph can only be established for variables that can be computed directly from the other variables in the lower-ranked groups. Variables that can only be calculated by solving the group of unresolved equations cannot be considered. Exporting to Freemind format allows the dependency tree to be kept as a mind map. Conversion to Server Format As mentioned earlier, the property calculation functions initially generated by Thermoptim are written in the following generic format, which is easy for a human to interpret: functionName("fluid";T = T_1;P = P_1). The arguments of the function are thus clearly identifiable, without risk of permutation. For example, the calculation of the enthalpy of water knowing its pressure and entropy is written as: calcH_PS("water";P = p_2;s = s_2). It is therefore necessary to convert them to the specific format of the server you wish to use. For example, in Interactive Thermodynamics, this last expression must be written in the form h_Ps ("water", p_2, s_2), and in EES as enthalpy(Steam;P= p_2;s = s_2). It is also important to verify the unit parameterization in the solver. For Thermoptim, it is the SI system, with pressures expressed in bar and temperatures in °C. The unit of flow rates is indicated before the equations. Once the set of equations is deemed satisfactory, it must be converted into the syntax of the particular solver that will be used. Composition of Compound Gases , combustion chambers Composition of Compound Gases The two solvers for which the process of converting the generated equations has been implemented so far are Interactive Thermodynamics et EES. I encountered difficulties in converting the calculation of the properties of compound gases from Thermoptim to these two solvers due to the inability, to my knowledge (admittedly limited), to define a new compound gas and calculate its thermodynamic properties. For EES, I saw that solutions exist, but I do not know how to automate them from Thermoptim files. For Interactive Thermodynamics, I did not see how to do it. This means that the equations for systems involving such fluids cannot be fully generated by Thermoptim, except for a few gases like air, which is actually treated as a pure gas. Modelers will need to complete them with specific functions that they will have to code. In order to facilitate this task, the composition of these compound gases is indicated in the generated file in the form of comment lines, as shown in this example: //CO2 0.04419006337631066 //H2O 0.03617814048485812 //O2 0.1640556604215191 //N2 0.7433597848138069 //Ar 0.012216350903505186 //N2 0.7555302216468832 //Ar 0.012416359476160373 //O2 0.2320534188769565 //CH4 ` methane //CH4 ` methane 1.0 The generation of equations corresponding to combustion chambers is particularly complex due to the number of possible parameterization options and the determination of the composition of the burned Given these difficulties, I opted for a provisional solution that allows combustion to be performed in EES, thereby enabling the study of cycles such as gas turbines or internal combustion reciprocating engines involving only a single combustion with air as the oxidizer and methane (CH4) as the fuel. This solution can serve as a first step before developing more detailed combustion The reaction is assumed to be complete. Its equation is: Lambda is the air factor, and a is the number of hydrogen atoms per carbon atom. In this case, a = 4 for methane. When the studied system includes a combustion chamber, the beginning of the generated file contains three EES functions that allow the estimation of: • the air factor of the combustion as a function of the oxidizer temperature and the desired adiabatic combustion temperature, as well as the value of a for a fuel of type CHa • the enthalpy of the combustion products as a function of their temperature, the air factor, and a • the entropy of the combustion products as a function of their temperature and pressure, the air factor, and a With the upstream and downstream points being 2 and 3, the block of equations corresponding to a combustion chamber is of the type: //Process: combustion chamber // Comment = Calculate lambda simplified model oxidizer air, fuel CH4 //Equation: 28 T_3 = 1065.0// Given value (Celsius) //Equation: 29 a_combustionchamber = 4// for CH4 //Equation: 30 lambda_combustionchamber = LAMBD(T_2;T_3;a_combustionchamber)// air factor lambda //Equation: 31 h_3 = h_products(T_3;a_combustionchamber;lambda_combustionchamber)// enthalpy of the reactants //Equation: 32 hfict_2 = h_products(T_2;a_combustionchamber;lambda_combustionchamber)// enthalpy of a fictitious inlet point for calculating the heat released //Equation: 33 m_dot_combustionchamber = m_dot_compressor + m_dot_fuel // Upstream process - compressor - Fuel process fuel - Downstream process - combustion chamber //Equation: 34 //m_dot_turbine = m_dot_combustionchamber //Flow propagation //Equation: 35 Q_dot_combustionchamber = (h_3 - hfict_2)m_dot_combustionchamber // DeltaH //Equation: 36 DeltaHr_combustionchamber = (-(-74850) +(-393520)+a_combustionchamber/2(-242000))/16 // DeltaHr (kJ/kg) = (-(-74850) +(-393520) + a/2* (-242000))/16 for methane //Equation: 37 m_dot_fuel = Q_dot_combustionchamber/DeltaHr_combustionchamber // fuel flow rate // Comment = Isobaric process //Equation: 38 p_3 = p_2// Isopressure //Equation: 39 T_fuel = 15.0// Given value (Celsius) //Equation: 40 p_fuel = 20.0// Given value (bar) //Equation: 41 h_fuel = calcH_TP("CH4 ` methane";T = T_fuel ;P = p_fuel) // Fuel point - fuel Equation 28 sets the value of the desired adiabatic 00combustion temperature, equation 29 sets the value of a, and equation 30 calls the EES function that provides lambda. Equation 31 calculates the enthalpy of the gases exiting the combustion chamber. It is important to note a significant problem encountered at this stage: I could not find the reference value h0 for the enthalpies of the burned gases to make them comparable to those of other fluids. There is therefore a discrepancy between these two sets of properties. This only poses a problem at the level of the combustion chamber itself. This is why the fictitious enthalpy hfict_2 is introduced in equation 32. It allows us to know the value of the enthalpy of the upstream point calculated with the composition and equations of the downstream point. Equation 33 ensures the conservation of flow. Equation 34 was nullified because it was redundant and different. Equation 35 determines the heat released during combustion, using hfict_2 introduced previously. Equation 36 calculates the reaction enthalpy of the combustion equation, and equation 37 uses it to determine the required fuel flow rate. Equation 41 calculates the enthalpy of the fuel. The name of its substance needs to be modified to become CH4. Calculations Downstream of a Combustion Chamber The calculations downstream of the combustion chamber require special treatment because the burned gases are not recognized as a substance by the solvers. The determination of their properties is done using two of the three introduced EES functions, which provide enthalpy and entropy but do not invert these functions. Given that the calculation of the thermodynamic properties of the burned gases cannot be performed with the usual functions, it is necessary to replace all calls to these functions for the Thermoptim substance that represents them with their equivalent defined in the EES functions placed at the beginning of the generated file. This does not pose any particular difficulty for calculations linking h and s to T and p, but it is more delicate for those directly relating h and s, which assume inversions of these functions. Therefore, one is led to slightly reformulate the problem depending on the situation, as illustrated by the calculation of a turbine calculated in polytropic reference mode presented below, where only the equations for calculating properties need to be modified. The upstream point is 3 and the downstream point is 4. It should be noted that lines starting with // are comment lines that the solver does not take into account. //Process: turbine // Comment = Polytropic coefficient: k = -Math,log(aval,p/amont,p)/Math,log(aval,V/amont,V) //Equation: 24 //h_4 = enthalpy(burnt gases;P = p_4;S = s_4) // Enthalpy h_4 = h_products(T_4;a_combustionchamber;lambda_combustionchamber)// enthalpy of the reactants //Equation: 25 //T_4 = temperature(burnt gases;H = h_4) // Downstream point - 4 s_4 = s_products(T_4;p_4;a_combustionchamber;lambda_combustionchamber)// enthalpy of the reactants Equations 24 and 25 are replaced as follows: the calculation of h_4 as a function of T_4 (unknown), is provided by the solver's inversion of the equation giving s_4 (known) as a function of T_4. As can be seen, in this case, it is necessary to slightly reformulate the problem. For an isentropic expansion, here is a slightly more complex example of reformulation: //Equation: 18 //s_3 = entropy(burnt gases;P = p_3;H = h_3) // Upstream point - 3 - Downstream point - 4 s_3 = s_products(T_3;P_3;a_combustionchamber;lambda_combustionchamber) // Comment = Isentropic reference //Equation: 19 //hs_4 = enthalpy(burnt gases;P = p_4;S = s_3) // Downstream point - 4 hs_4 = h_products(Tis;a_combustionchamber;lambda_combustionchamber) s_3 = s_products(Tis;P_4;a_combustionchamber;lambda_combustionchamber) //Equation: 20 etaT_turbine = 0,85// Isentropic efficiency //Equation: 21 h_4 = h_3 - etaT_turbine*(h_3 - hs_4) // Upstream point - 3 - Downstream point - 4 //Equation: 22 //T_4 = temperature(burnt gases;H = h_4) // Downstream point - 4 h_4 = h_products(T_4;a_combustionchamber;lambda_combustionchamber) //Equation: 23 //s_4 = entropy(burnt gases;P = p_4;H = h_4) // Entropy s4 = s_products(T_4;P_4;a_combustionchamber;lambda_combustionchamber) Equation 18 is a simple reformulation of the one generated. To solve equation 19, we introduce the isentropic temperature Tis, which is determined from the function giving the known enthalpy, and the equation giving the enthalpy from Tis provides the isentropic enthalpy. Equations 22 and 23 are simple reformulations of the ones generated. Exemples commentés et fichiers correspondants Three commented examples are provided in these pages, with the corresponding equation files, as well as others uncommented.	{"url":"https://direns.minesparis.psl.eu/Sites/Thopt/en/co/equations.html","timestamp":"2024-11-07T13:04:00Z","content_type":"text/html","content_length":"51208","record_id":"<urn:uuid:8f521bb3-89b9-43b1-a256-e8ba80dc85f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00406.warc.gz"}
We consider the use of an EM algorithm for fitting finite We consider the use of an EM algorithm for fitting finite mixture models when mixture component size is known. convergence properties. and -sample observations there are observations from one component and observations from the other but PF-3635659 which specific observations are from each component is not known. When the latent component membership is of interest as PF-3635659 a function of observable covariates for each individual such type of data have been extensively analyzed and growing in popularity with recent advances in computing; see Chen and Yang (2007); Choi et al. (2008); Musalem et al. (2009); Park (2011); Verhelst (2008) for various examples. Our goal is to estimate the underlying and unknown parameters that determine the distribution of each component type where we allow more general distributions than the normal distribution. Another more specific type of application of our methodology is to voting inferences where we could make use of the fact that during an election between two candidates we can obtain the exact number of votes each candidate receives at a voting site. Further under certain conditions we can obtain the previous voting histories for each individual voter at that site. Because each voter’s selection is blinded the distribution of previous voting frequencies for individual voters can be viewed as a mixture of two distributions each for two candidates’ supporters. Thus if we wanted to assess if there were a difference in previous voting patterns between those who voted for one candidate and those who voted for the other we could use our methodology with voters for the one candidate and – for the other. In summary we see that the possible conceptual applications for our method arise when it is of interest to compare certain characteristics between two groups given an anonymized list of two groups of people and the number of people belonging to each group. In Section 2 we develop the EM algorithm for fitting mixture models of exponential Rabbit Polyclonal to Adrenergic Receptor alpha-2A. family distributions when the exact number of observations within each mixture component is fixed. Section 3 discusses the efficient and stable computation of the proposed EM algorithm numerically. Section 4 compares in the context of normal mixture models the properties of the proposed EM algorithm and a conventional EM algorithm which uses the probability of the mixture being and respective parameters = (denote a latent mixture indicator variable such that = 1 if belongs to the first mixture component and = 0 if it belongs to the second mixture component for = 1 … and derive the EM algorithm accordingly; we call this a conventional EM algorithm throughout the paper. Then the PF-3635659 complete-data likelihood function for is written as given y and with support given by the set of all possible binary vectors on space = = 1 with PF-3635659 = = 1) = {1 … > \|is written PF-3635659 as = 1 is a priori set to = for = 1 … with respect to given y and is the odds of given y and and combinations of function may not be practical because is typically large. As proposed by Gail et al. (1981) we thus consider an efficient recursive method to calculate the summation. That is for = {1 … ≤ \|? 1) additions and ? + 1) multiplications which requires much less operations than evaluations. In the context of fitting finite mixture models with known mixture component size the computation of the function can be numerically unstable in certain circumstances. First when there exists a little overlap between the distributions of the mixture components the probability of belonging to the first component given y and becomes extremely large. Because the function in (2.3) is a sum of a product of causes inflation of the function and its computation can be numerically unstable. Second when the sample size is large it is likely that some observations come from the tail of a distribution such that close to one and the corresponding becomes extremely large. Even when there are no such extreme observations a product of relatively large function thereby making its computation numerically unstable. To circumvent such numerical instability we propose to cancel out a large common factor between the numerator and denominator in (2.3) to make its computation numerically stable by noting that the E-step is computed as the ratio of two functions. We factor out a product of some largest function specifically. The modified function denoted by order statistics based on is the original largest for a simple example when = 2 = {1 2 3 4 and w = (is.	{"url":"https://www.technologybooksindustrialprojectreports.com/we-consider-the-use-of-an-em-algorithm-for-fitting-finite/","timestamp":"2024-11-02T14:54:50Z","content_type":"text/html","content_length":"63138","record_id":"<urn:uuid:13f08d4a-a09d-46c6-a148-a48e4cd7eee3>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00574.warc.gz"}
Multi-Armed Bandit Problem - DataJello.com Pre-requisite: some understanding of reinforcement learning. If not, you can start from Reinforcement Learning Primer Let’s analyze this in the classic Multi-Armed Bandit problem using the epsilon-greedy strategy. Set up the experiment We can set up an experiment as follows • K=10 for 10-armed bandit (with 10 actions to choose from) • Each arm has a Gaussian distributed return (with some mean and standard deviation of 1) • The mean of each arm is randomly initialized using the standard Gaussian distribution (mean=0, variance=1) • Repeat this simulation many times by regenerating the 10-armed bandit each time The epsilon-greedy strategy • Select the best action under Q with probability 1-epsilon • During the small probability epsilon, selection randomly among all possible actions with equal probability • We want to analyze the reward of the agent from t=0 to steady state • Q is estimated using sample average method (step size = 1/N(a)) where a is the action at time t By running experiment 2000 times for different values of epsilon and average the reward at each time step, we get an average reward plot like below Notable observation about the strategy • There is an optimal epsilon □ Greedy (epsilon = 0) is not good because agent might be wrong in choose the best paying arm □ Large epsilon is also not good, because spending time exploring • The optimal epsilon □ depends on the variance of each bandit ☆ If variance is 0, then greedy approach is optimal □ depends on the prior distribution that we use to generate the mean of each bandit ☆ But this is usually unknown in practice, and even knowing it does not help to be optimal in a specific bandit problem □ depends on the time horizon we look at ☆ if we instead look at 10000 steps, small epsilon will perform better because it would have more time to explore before plateau • The mean of each bandit does not change in a given bandit problem. This means the problem is stationary □ In practice, the mean of a bandit could change over time (non-stationary) Step size (alpha) The step size can also play a role in the performance. In the above setting, the step size is set to be 1/N(a), but it can be generically represented as alpha below • Large alpha: Q quick catches up to q* but it will oscillate due to the randomness of the new sample (too much over correction) • Small alpha: takes a long time for Q to catch up to q* • N(a): able to quick move to estimate q* and then stabilized and converge to a steady state □ However N(a) is good for stationary q* only, we will learn later that larger alpha with over correction can be beneficial for Q to adapt dynamically to a changing q* Optimistic initial value There is also another way to impact the exploration behavior by giving some high value for Q1. This makes the agent more like to explore the unexplored and be “disappointed” a few times before realizing its not a good object. It can even help a greed algorithm to perform better. However, this is not good enough because it only encourages exploration in the beginning. If the environment is not stationary, then the greedy approach will be stuck at the old optimum. It’s also hard to know what is a good initial optimistic value in prior. Upper Confidence Bound (UCB) Action Selection To better balance the exploration and exploitation, we can use the confidence bound to make decision on the action. In comparison to the epsilon-greedy approach: • epsilon-greedy: based on expected reward • UCB: based on a confidence upper bound In particular, we choose by the action that has the best upper bound below which is action 1: This is formulated as • c is the tunable parameter for the confidence. It’s a hyper-parameter that balance exploration and exploitation. • t is the current time – the higher the larger the upper bound • Nt(a) is the number of times that action a has been selected We see that UCB with c=2 can beat the epsilon-greedy strategy in the long run. Real world RL While the multi-armed bandit problem seems quite simple, it’s the primary way that Reinforcement Learning is currently applied in the real world. One of the challenge is the need to use a simulator for the agent to interact with and learn. The general guideline is a paradigm shift to make RL work in the real world. • Better generalization • Let the environment take control • Focus on statistical efficiency • Use feature to represent the state • Algorithm should produce evaluation in the process of learning • Look at all the policy (the trajectory of policy) rather than just the last policy in a simulator environment Limitation of this model The multi-armed bandit does not model all aspects of a real problem due to some limitations • There is always a single best action independent of the situation. That is, you don’t need to make different action based on the situation • The reward is not delayed. You get the reward in one step. Each episode is just one time step. The Markov Decision Process (MDP) provides a richer representation. Contextual bandit The contextual bandit (aka associated search) algorithm is an extension of the multi-armed bandit approach where we factor in the customer’s environment, or context, when choosing a bandit. For an example of personalized new recommendation, a principled approach in which a learning algorithm sequentially selects articles to serve users based on contextual information about the users and articles, while simultaneously adapting its article-selection strategy based on user-click feedback to maximize total user clicks. Reference: most of the material of this post comes from the book Reinforcement Learning 2 thoughts on “Multi-Armed Bandit Problem”	{"url":"https://datajello.com/multi-armed-bandit-problem/","timestamp":"2024-11-03T08:59:26Z","content_type":"text/html","content_length":"147574","record_id":"<urn:uuid:8c8c2fd2-1fcb-4d38-ae83-c4890dd355f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00141.warc.gz"}
Styles of semantics (Programming Languages) Denotational Semantics In this approach, the presentation of the semantics of a programming language has two parts: in the first, one identifies a mathematical structure within which the meaning of programs will be found; and in the second part, one defines a function that maps the syntax of the language to elements of this structure. Crucially, this function is defined by structural recursion on the syntax, so that the meaning of each construct is defined in terms of the meanings of its components. Typically, the mathematical structures that are used are partially ordered sets, with the ordering relating elements if one is more 'defined' than the other. This ordering arises naturally in considering partial functions, and the theory of monotonic functions on partial orders proves the existence of the fixpoints that are needed to give meaning to programs that involve recursion or As Strachey observed, fixing the domain within which programs in a language have their meanings tells us a great deal about the language, and is a sure guide to the design of the language. Traditional texts on Denotational Semantics have spent more effort on establishing that these domains exist than on using them to describe the features of particular programming languages, a fact that I sometimes imagine to be connected with the fact that Christopher Strachey, the great founder of the denotational approach, died in 1976, while his more theoretical colleagues lived on. In denotational semantics, we might define the meaning exec [[S]] of a statement S to be a function from states to states, and capture the meaning of while loops with this equation: exec [[while B do S]] = fix (λf -> (λs -> if eval [[B]] s then f (exec [[S]] s) else s)) The crucial point here is that the value of exec [[while B do S]] is defined in terms of the values of eval [[B]] and exec [[S]] using the fixpoint operator fix. One can argue separately that such a fixpoint operator exists for the domains that are being used. Operational Semantics In Operational Semantics, one describes the process of executing a program, and says that the meaning of a program consists of the collection of behaviours that are shown by this mechanism. Traditional operational semantics was expressed in terms of abstract machines that executed a program or evaluated an expression step by step, but more modern presentations use inference systems to achieve the same effect in a more abstract way. For our example of the while loop, we might write E, m => v to mean that when evaluated in memory state m, the expression E has value v, and we might write S, m => m1 to mean that when started in memory state m, the statement S terminates with memory state m1. In these terms we can express the meaning of the while loop with two rules: If B, m => false, then (while B do S), m => m. If B, m => true and S, m => m1 and (while B do S), m1 => m2, then (while B do S), m => m2. An alternative way of giving operational semantics might be to write an interpreter in a functional language: we might define exec [[while B do S]] m = if eval B m then (let m' = exec S m in exec [[while B do S]] m') else m Axiomatic Semantics This approach focusses on describing the set of true claims that can be made about a program. In the usual language of imperative programs, these claims can be expressed as Hoare triples {P} S {Q}, meaning 'if program S is started in a state satisfying relation P, then it is bound to terminate successfully in a state satisfying relation Q.' [Hoare's original paper used partial correctness, not the total correctness that we use here.] In Hoare's style, the meaning of a while loop is given by the following inference rule: for any invariant P and bound function T such that P ==> T >= 0, from {P and B and T = t0} S {P and T < t0} one may deduce {P} while B do S {P and not B}. Dijkstra's weakest preconditions show that axiomatic semantics can be presented in a denotational style. Here, the meaning of a statement S is a 'predicate transformer' wp(S, -) that gives for each post-condition Q the weakest pre-condition wp(S, Q) that is sufficient to guarantee that S will terminate in a state satisfying Q. Thus {P} S {Q} holds if and only if P ==> wp(S, Q). This weakest pre-condition can be defined by structural recursion on the syntax of S. The meaning of the While loop is given as a fixpoint: wp(while B do S, Q) is the strongest relation R such that (B and wp(S, R)) or (not B and Q) ==> R, that is, the strongest solution of the fixpoint equation F(R) = R, where F(R) is the LHS of the above implication. This definition has the crucial property that wp(while B do S, -) is defined in terms of B and wp(S, -). [Dijkstra's original formulation defined the semantics of While as the limit of an explicit sequence of relations, but it amounts to the same thing: a fixpoint.] Our approach Denotational in style. From one point, operational in essence, for we are saying that the meaning of a program is the collection of behaviours exhibited by our interpreter. But if we regard Haskell not as a programming language, but as a notation for writing down denotations, then the link with denotational semantics becomes stronger. And our aim is to describe enough about Haskell axiomatically that we can reason about languages and implementations in a practical way without having constantly to return to the foundations.	{"url":"https://spivey.oriel.ox.ac.uk/corner/Styles_of_semantics_(Programming_Languages)","timestamp":"2024-11-14T05:26:14Z","content_type":"text/html","content_length":"28087","record_id":"<urn:uuid:d28289a7-4d6c-4b43-837f-337c252d6af7>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00002.warc.gz"}
MCLab Group List of Papers -- Query Results Ruggero Lanotte, Andrea Maggiolo-Schettini, Simone Tini, Angelo Troina, and Enrico Tronci. "Automatic Covert Channel Analysis of a Multilevel Secure Component." In Information and Communications Security, 6th International Conference, ICICS 2004, Malaga, Spain, October 27-29, 2004, Proceedings, edited by J. Lopez, S. Qing and E. Okamoto, 249–261. Lecture Notes in Computer Science 3269. Springer, 2004. DOI: 10.1007/b101042.	{"url":"http://mclab.di.uniroma1.it/publications/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%20FROM%20refs%20WHERE%20serial%20%3D%2034%20ORDER%20BY%20medium&submit=Cite&citeStyle=Roma&citeOrder=&orderBy=medium&headerMsg=&showQuery=0&showLinks=0&formType=sqlSearch&showRows=10&rowOffset=0&client=&viewType=Print","timestamp":"2024-11-08T08:27:04Z","content_type":"text/html","content_length":"7428","record_id":"<urn:uuid:e16590eb-8830-4311-ab53-526b339848c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00390.warc.gz"}
The Nomadic Monad or: How I Learned to Stop Worrying and Love the Burrito (Part 3) This is the third in a series of posts about monads (part 1, part 2). I happen to be a .NET developer, so I use C# in my examples, but the concept applies to any language that has first-class functions. Code for the series is at github. In the last post, we covered over half of the items in Wikipedia's formal definition of a monad: • [S:Formally, a monad consists of a type constructor M and two operations,:S] bind [S:and return.:S] • The operations must fulfill several properties to allow the correct composition of monadic functions (i.e. functions that use values from the monad as their arguments or return value). • [S:In most contexts, a value of type M a can be thought of as an action that returns a value of type a.:S] • [S:The return operation takes a value from a plain type a and puts it into a monadic container of type M a.:S] • The bind operation chains a monadic value of type M a with a function of type a ? M b to create a monadic value of type M b, effectively creating an action that chooses the next action based on the results of previous actions. Let's break down the requirement for the Bind function. It ...chains a monadic value of type M a... That says to me that we need to have an instance method in our monad. Or, we could use an extension method to achieve the same result. I prefer to use an extension method, so that's what we'll do. Let's create an extension method: public static ??? Bind<T>(this Monad<T> monad, ???) Not a bad start. Let's skip to the return type. We'll need ...to create a monadic value of type M b... That doesn't look so hard. We've already used our class in place of M a . But, since we already used we won't be able to use it again. But, since it's a good idea to use standard C# naming conventions when we're writing C# code (when in Rome...), our generic type should at least start with the letter 'T'. How about , since it is the result of our public static Monad<TResult> Bind<T, TResult>( this Monad<T> monad, But what of the second parameter? It looks like it needs we need to be connecting ...with a function of type a ? M b... Why, if I didn't know better, I'd say that looks an awful like a lambda expression (spoiler alert: it is). And we already know what M b mean to us: . It would be great if we were able to pass in a lambda expression that takes an and returns a . But what is a lambda expression? Just a convenient way of creating a function. Hmmmm, function. I know - why don't we take a Func<T, Monad<TResult>> as our parameter type! We just put the . So what does our extension method look like now? public static Monad<TResult> Bind<T, TResult>( this Monad<T> monad, Func<T, Monad<TResult>> resultSelector) Finally, we'll need to fill in the body. Our requirements state that we need to take our Monad<T> and transform it into a Monad<TResult> by using a Func<T, TResult>. That function needs a <T> argument passed to it. Haven't we seen an instance of <T> before? Of course - in the Value property of our Monad<T> class. With that, I think we have got enough information to fill out our Bind public static Monad<TResult> Bind<T, TResult>( this Monad<T> monad, Func<T, Monad<TResult>> resultSelector) return resultSelector(monad.Value); I think we're really getting somewhere now! How would we use our spiffy new Bind function? I'd think that we would use it something like this: Monad<int> m1 = 128.ToMonad(); Monad<Type> m2 = m1.Bind(x => x.GetType().ToMonad()); "The type of the original monad is: {0}.", We've taken a Monad of type int, and bound it to a Monad of type Type by calling Bind and passing in a lambda expression that takes a T and returns a Monad<Type>. There's one thing I don't like about this usage though: the call to ToMonad(). I'd prefer it if our Bind function had, as its parameter, a Func<T, TResult>. But I also want to meet the requirement for being a monad. Why don't we create an overload of Bind to do just that? I'd like our existing Bind method to remain the source of truth, so we'll have our new one call it. public static Monad<TResult> Bind<T, TResult>( this Monad<T> monad, Func<T, TResult> resultSelector) return monad.Bind(m => resultSelector(m).ToMonad()); This is getting a little funky. We're doing a few things here: 1) we're passing a new lambda expression to the other Bind method; 2) in the body of the new lambda expression, we're executing the function that was passed in; and 3) we're turning the result of into a Monad<TResult> by calling our extension method, ToMonad. We're not actually executing the passed-in function, we're passing it into another function that we pass to the other Bind method. Clear as mud, right? So why go through all this trouble? To me, it expresses intent better, and I think this makes it worth it. We can use it like this: Monad<int> m1 = 128.ToMonad(); Monad<Type> m2 = m1.Bind(x => x.GetType()); "The type of the original monad is: {0}.", This is all well and good, but it seems kind of tedious to have to create a new variable each time we call Bind. Didn't the requirement mention something about chaining? As in method chaining? Why don't we try that? var m = .Bind(x => x.GetType()) .Bind(x => new string(x.ToString().Reverse().ToArray())); "The backwards string representation of " + "the type of the original monad is: {0}.", We're taking the number 128 wrapping it in Monad<int>, binding it to a Monad<Type> with a function that takes any value and returns its type, and binding that to a Monad<string> with a function that takes any value and returns its string representation backwards. Pretty cool, huh? We're headed for the home stretch now. It's time to address the final bullet point: The operations must fulfill several properties to allow the correct composition of monadic functions (i.e. functions that use values from the monad as their arguments or return value). It seems to me like we're already doing this. We're composing a monad from a series of monadic functions. But perhaps we can make our intent more clear - calling over and over again doesn't exactly look pretty. What if we extracted the calls to into methods that expressed their intent more clearly? How about using some extensions methods? Something like this? public static Monad<Type> ToType<T>(this Monad<T> monad) return monad.Bind(m => m.GetType()); public static Monad<string> ToReversedString<T>( this Monad<T> monad) return monad.Bind(m => new string(m.ToString().Reverse().ToArray())); Now we can replace our Bind statements: var m = "The backwards string representation of " + "the type of the original monad is: {0}.", That about wraps it up for today. Next time, we'll discuss the three monadic laws.(Hopefully we follow them!) 4 comments: 1. Great post Brian. I had just bought a functional book. That book has moved up my reading queue because of your series on Monads. Can you give an example of the type of problem, Monads are solving for you? Looking forward to your next blog post. 2. Shane CharlesSeptember 2, 2012 at 9:50AM Clearly I needed more coffee before posting comments. I thought I was uploading a profile image. 3. For the last couple of days (since DevLink ended), I've been playing around with monadic parser combinators. Specifically, I'm pre-parsing SendKeys data for my "auto-typer" app - the one I used for my monad talk. What I have works, but it was quick and dirty. I'm looking for a more lasting solution. The parser combinators will allow me to provide for a richer data interface, allowing for better commands and more control over both the "clicky" sounds and the delay between keystrokes. Instead of writing my own parser library, I'm using https://github.com/sprache/Sprache, and, while doing so, I've been dissecting how it works. So far, it's pretty cool - I'd recommend checking it out if you're interested in functional programming. I'll probably be doing a series of tutorials on Sprache in the coming months, since the documentation is a little sparse... 4. 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Correction of plasma TAC for metabolites It is common that the PET tracer is metabolized in the liver, kidneys or other parts of the body already during the PET scan, and one or more of the metabolites is still carrying the isotope label. If labelled metabolites are found in the plasma in significant amounts, their proportion has to be subtracted from the plasma curve, because only the concentration of parent tracer can be used as input function in quantitative analysis of the tracer kinetics. In brain studies the radioactive metabolites, that usually are more polar than the authentic tracer, do not usually pass the blood-brain barrier (BBB). However, the less lipophilic metabolites tend to have lower binding to plasma proteins, which may increase their distribution volume in the brain (Aarnio et al., 2022). In other tissues, not protected by BBB, marked uptake of radioactive metabolite(s) can be observed. When marked proportion of tissue radioactivity concentration is due to metabolits from plasma, the plasma concentrations of both the parent tracer and the radioactive metabolite may have to be included in the compartmental model or spectral analysis (Tomasi et al., 2012; Ichise et al., 2016). Small polar radiometabolites, such as [^11C]formaldehyde and [^11C]CO[2] can pass even the BBB, and substantially affect the brain tissue concentrations and reduce the signal-to-background ratio (Johansen et al., 2018). ^18F-labelled radioligands are often defluorinated during the PET study; free [^18F]F^- and other bone-seeking isotopes, such as Zr^4+, may hamper brain PET studies by causing high activity in the skull bone next to the brain cortex. Metabolite correction in TPC The fractions of authentic (parent) tracer in plasma must be written in an ASCII file (fraction data). A mathematical function or compartmental model can be fitted to these fractions. Total radioactivity in plasma (PTAC) is measured from arterial plasma samples. With that and the fitted parent fractions, metabolite corrected plasma curve can be calculated using metabcor. TACs of radioactive metabolites in plasma can also be saved, if necessary. Figure 1. Example of plasma metabolite correction in [^11C]flumazenil study: each plasma concentration (black) is multiplied by the parent tracer fraction at each sample time point; result is the curve of unchanged (parent) radioligand concentration in plasma (red). Alternative metabolite correction methods Mathematical metabolite correction For references, see Burger and Buck (1996), and Sanabria-Bohórquez et al. (2000). Population based methods Ideally, fractions of plasma metabolites should be measured for each person participating in a PET study. However, the measured fraction curves are sometimes noisy, or there are missing samples. One alternative is to calculate population average curve of the fractions of parent tracer in the plasma, if the inter-individual variation in the rate of metabolism is small. Population average must be determined from a group that is comparable to the study population by their age, sex, and body weight. For example, for rate of metabolism of [^18F]FDPN a significant gender difference has been found (Henriksen et al., 2006). The population average fraction curve can be fitted to a function, for example to the "Hill-type" or power or exponential functions, if there were only few samples or if the fraction curve must be extrapolated. In the fitting, use the weights that were written in the mean fraction curve. See also: Burger C, Buck A. Tracer kinetic modelling of receptor data with mathematical metabolite correction. Eur J Nucl Med. 1996; 23(5): 539-545. doi: 10.1007/BF00833389. Henriksen G, Spilker ME, Sprenger T, Hauser AI, Platzer S, Boecker H, Toelle TR, Schwaiger M, Wester H-J. Gender dependent rate of metabolism of the opioid receptor-PET ligand [^18F] fluoroethyldiprenorphine. Nuklearmedizin 2006; 45: 197-200. doi: 10.1055/s-0038-1625219. Huang SC, Barrio JR, Yu DC, Chen B, Grafton S, Melega WP, Hoffman JM, Satyamurthy N, Mazziotta JC, Phelps ME. Modelling approach for separating blood time-activity curves in positron emission tomographic studies. Phys Med Biol. 1991; 36(6): 749-761. doi: 10.1088/0031-9155/36/6/004. Ichise M, Kimura Y, Shimada H, Higuchi M, Suhara T. PET quantification in molecular brain imaging taking into account the contribution of the radiometabolite entering the brain. In: Kuge Y et al. (eds.), Perspectives on Nuclear Medicine for Molecular Diagnosis and Integrated Therapy, Springer, 2016. doi: 10.1007/978-4-431-55894-1_17. Nagar S, Argikar UA, Tweedie DJ (eds). Enzyme Kinetics in Drug Metabolism - Fundamentals and Applications. Humana Press, 2014, ISBN 978-1-62703-757-0. Nelissen N, Warwick J, Dupont P (2012). Kinetic modelling in human brain imaging. In: Positron Emission Tomography - Current Clinical and Research Aspects, Dr. Chia-Hung Hsieh (Ed.), ISBN: 978-953-307-824-3, InTech. doi: 10.5772/30052. Sanabria-Bohórquez SM, Labar D, Levêque P, Bol A, De Volder AG, Michel C, Veraart C. [^11C]Flumazenil metabolite measurement in plasma is not necessary for accurate brain benzodiazepine receptor quantification. Eur J Nucl Med. 2000; 27:1674-1683. doi: 10.1007/s002590000336. Sari H, Erlandsson K, Marner L, Law I, Larsson HBW, Thielemans K, Ourselin S, Arridge S, Atkinson D, Hutton BF. Non-invasive kinetic modelling of PET tracers with radiometabolites using a constrained simultaneous estimation method: evaluation with ^11C-SB201745. EJNMMI Res. 2018; 8(1): 58. doi: 10.1186/s13550-018-0412-6. Sestini S, Halldin C, Mansi L, Castagnoli A, Farde L. Pharmacokinetic analysis of plasma curves obtained after i.v. injection of the PET radioligand [11C] raclopride provides likely explanation for rapid radioligand metabolism. J Cell Physiol. 2012; 227: 1663-1669. doi: 10.1002/jcp.22890. Tomasi G, Kimberley S, Rosso L, Aboagye E, Turkheimer F. Double-input compartmental modeling and spectral analysis for the quantification of positron emission tomography data in oncology. Phys Med Biol. 2012; 57: 1889-1906. doi: 10.1088/0031-9155/57/7/1889. Tonietto M, Veronese M, Rizzo G, Zanotti-Fregonara P, Lohith TG, Fujita M, Zoghbi SS, Bertoldo A. Improved models for plasma radiometabolite correction and their impact on kinetic quantification in PET studies. J Cereb Blood Flow Metab. 2015; 35(9): 1462-1469. doi: 10.1038/jcbfm.2015.61. Tonietto M, Rizzo G, Veronese M, Fujita M, Zoghbi SS, Zanotti-Fregonara P, Bertoldo A. Plasma radiometabolite correction in dynamic PET studies: Insights on the available modeling approaches. J Cereb Blood Flow Metab. 2016; 36(2): 326-339. doi: 10.1177/0271678X15610585. Veronese M, Gunn RN, Zamuner S, Bertoldo A. A non-linear mixed effect modelling approach for metabolite correction of the arterial input function in PET studies. Neuroimage 2013; 66: 611-22. doi: Tags: Input function, Metabolite correction, Parent fraction, Plasma Updated at: 2022-12-02 Created at: 2008-03-02 Written by: Vesa Oikonen	{"url":"http://www.turkupetcentre.net/petanalysis/input_metabolite_correction.html","timestamp":"2024-11-11T16:29:12Z","content_type":"text/html","content_length":"15112","record_id":"<urn:uuid:76dede8f-d9ce-48e2-91f1-0996a066b62d>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00190.warc.gz"}
Digital Logic Design and Digital Electronics Course What will you learn in this course? • Understand the fundamentals of Boolean logic • Differentiate between different number systems and understand their applications • Understand the basic components of combinational logic circuits and sequential logic circuits like logic gates and flip-flops • Study and design complex combinational circuits (Example: priority encoders) • Study and design complex sequential circuits (Example: Counters) What is the target of this course? Digital electronics is a core course that is essential for you to progress to other higher-level courses in the field of electronics. Once you have completed this digital electronics course, you can opt for a variety of disciplines depending on your career choices. At Technobyte, we use concepts from this course in our tracks on Embedded Systems, IoT, VLSI, and Robotics. How many quizzes are there in this course? There’s one free quiz. We will be launching a certifying quiz shortly. What’s the course structure like? • Introduction to digital electronics and digital systems • Number systems □ Conversion between number systems • Logic Gates • Introduction to Combinational circuits □ Adders and Subtractors □ Multiplexers and Demultiplexers □ Encoders and Decoders □ Comparators • Introduction to Sequential Circuits □ Latches □ Flip-Flops □ Counters (Synchronous and Asynchronous) □ Shift-registers • Error detection □ Parity generators and Parity checkers • Types of digital memory devices • Types of Programmable Logic Devices • Types of logic families • Quiz 1 • Certification test (Coming soon) I would like to suggest some topics to be covered, how can I do that? You can visit the contact page linked in the footer of this webpage. Just select “Suggest Topics” from the subject dropdown menu of the form, mention the course and why you think your suggestion makes sense to be part of the curriculum.	{"url":"https://technobyte.org/digital-electronics-logic-design-course-engineering/","timestamp":"2024-11-03T06:07:24Z","content_type":"text/html","content_length":"96926","record_id":"<urn:uuid:940037e2-f134-43ff-a6af-9d35088a4681>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00279.warc.gz"}
467 views Answer to a math question (3b)⋅(5b^2)⋅(6b^3) 97 Answers Frequently asked questions (FAQs) Math Question: What is the 5th derivative, with respect to x, of the function f(x) = 3x^4 + 2x^3 - 5x^2 + 7x + 1? What percent is equivalent to the fraction 3/4? What is the length of the altitude drawn from vertex A in a triangle ABC if AC = 7cm, BC = 10cm, and	{"url":"https://math-master.org/general/3b-5b-2-6b-3","timestamp":"2024-11-07T03:11:59Z","content_type":"text/html","content_length":"246329","record_id":"<urn:uuid:2c28fe0b-812e-49e4-8808-0ee173d0ae93>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00000.warc.gz"}
2,392 research outputs found In left-right models the gluonic penguin contribution to b --> s s-bar s transition is enhanced by m_t/m_b due to the presence of (V+A) currents and by the larger values of loop functions than in the Standard Model. Together those may completely overcome the suppression due to small left-right mixing angle xi phi K_S decay amplitude appearing in a large class of left-right models may therefore modify the time dependent CP asymmetry in this decay mode by O(1) and explain the recent BaBar and Belle CP asymmetry measurements in this channel. This new physics scenario implies observable deviations from the Standard Model also in B_s decays which could be measured at upcoming Tevatron and LHC.Comment: references adde We propose a new paradigm for generating exponentially spread standard model Yukawa couplings from a new $U(1)_F$ gauge symmetry in the dark sector. Chiral symmetry is spontaneously broken among dark fermions that obtain non-vanishing masses from a non-perturbative solution to the mass gap equation. The necessary ingredient for this mechanism to work is the existence of higher derivative terms in the dark $U(1)_F$ theory, or equivalently the existence of Lee-Wick ghosts, that (i) allow for a non-perturbative solution to the mass gap equation in the weak coupling regime of the Abelian theory; (ii) induce exponential dependence of the generated masses on dark fermion $U(1)_F$ quantum numbers. The generated flavor and chiral symmetry breaking in the dark sector is transferred to the standard model Yukawa couplings at one loop level via Higgs portal type scalar messenger fields. The latter carry quantum numbers of squarks and sleptons. A new intriguing phenomenology is predicted that could be potentially tested at the LHC, provided the characteristic mass scale of the messenger sector is accessible at the LHC as is suggested by naturalness arguments.Comment: Text improved, new equations and references added, version to appear in Phys.Rev.D, 12 pages, 2 figure We reconsider Higgs boson invisible decays into Dark Matter in the light of recent Higgs searches at the LHC. Present hints in the CMS and ATLAS data favor a non-standard Higgs boson with approximately 50% invisible branching ratio, and mass around 143 GeV. This situation can be realized within the simplest thermal scalar singlet Dark Matter model, predicting a Dark Matter mass around 50 GeV and direct detection cross section just below present bound. The present runs of the Xenon100 and LHC experiments can test this possibility.Comment: 6 pages, 2 figures. Final version to appear on PR If the generating mechanism for neutrino mass is to account for both the newly observed muon anomalous magnetic moment as well as the present experimental bounds on lepton flavor nonconservation, then the neutrino mass matrix should be almost degenerate and the underlying physics be observable at future colliders. We illustrate this assertion in two specific examples, and show that $\Gamma (\ mu \to e \gamma)/m_\mu^5$, $\Gamma (\tau \to e \gamma)/m_\tau^5$, and $\Gamma (\tau \to \mu \gamma) /m_\tau^5$ are in the ratio $(\Delta m^2)_{sol}^2/2$, $(\Delta m^2)_{sol}^2 /2$, and $(\Delta m^2)_ {atm}^2$ respectively, where the $\Delta m^2$ parameters are those of solar and atmospheric neutrino oscillations and bimaximal mixing has been assumed.Comment: Erratum adde In the U(1)_N extension of the supersymmetric standard model with E_6 particle content, the heavy singlet superfield N may decay into a quark and a diquark as well as an antiquark and an antidiquark, thus creating a baryon asymmetry of the Universe. We show how the three doublet and two singlet neutrinos in this model acquire mass from physics at the TeV scale without the benefit of using N as a heavy right-handed neutrino. Specifically, the active neutrinos get masses via the bilinear term \mu LX^c which conserves R-parity, and via the nonzero masses of the sterile neutrinos. We predict fixed properties of the extra Z' boson, as well as the new lepton doublets X and X^c, and the observation of diquark resonances at hadron colliders in this scenario.Comment: LATEX, 13 page Discussions are underway for a high-energy proton-proton collider. Two preliminary ideas are the $\sqrt{s}=33$ TeV HE-LHC and the $\sqrt{s}=100$ TeV VLHC. With Bayesian statistics, we calculate the probabilities that the LHC, HE-LHC and VLHC discover SUSY in the future, assuming that nature is described by the CMSSM and given the experimental data from the LHC, LUX and Planck. We find that the LHC with $300$/fb at $\sqrt{s}=14$ TeV has a $15$-$75%$ probability of discovering SUSY. Should that run fail to discover SUSY, the probability of discovering SUSY with $3000$/fb is merely $1$-$10%$. Were SUSY to remain undetected at the LHC, the HE-LHC would have a $35$-$85%$ probability of discovering SUSY with $3000$/fb. The VLHC, on the other hand, ought to be definitive; the probability of it discovering SUSY, assuming that the CMSSM is the correct model, is $100%$.Comment: 21 pages, 5 figures. Matches version published in Eur.Phys.J. C. Results and conclusions unchange	{"url":"https://core.ac.uk/search/?q=author%3A(Raidal%2C%20Martti)","timestamp":"2024-11-11T11:59:56Z","content_type":"text/html","content_length":"136797","record_id":"<urn:uuid:71da3d14-99a2-4c0b-b48e-0c1567b4ee03>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00652.warc.gz"}
Find elements which are present in first array and not in second - TutorialCup Find elements which are present in first array and not in second Difficulty Level Easy Frequently asked in Accolite Delhivery Factset Fanatics Snapdeal Zoho Views 3014 The problem “Find elements which are present in first array and not in second” states that you are given two arrays. Arrays consist of all the integers. You have to find out the numbers which will not be present in the second array but present in the first array. a [] = {2,4,3,1,5,6} b [] = {2,1,5,6} a [] ={4,2,6,8,9,5} b [] ={9,3,2,6,8} 1. Declare a HashSet. 2. Insert all the elements of array b[] into HashSet. 3. While i < l1 (length of an array a[]). 1. If HashSet doesn’t contain array a[i], then print a[i]. We have given two integer arrays and a problem statement that asks to find out the number which is present in the first array and not in the second array. We are going to use Hashing in this problem. Hashing helps us to find out the solution in an efficient way. We are going to put the array b[] numbers in a HashSet and after inserting all the number of array b[]. We are going to traverse array a[] and taking each element at a time and check if HashSet doesn’t contain that element. If it does not have that element, we are going to print that particular element of array a[i] and check for another number. Let us consider an example and understand this: First array is a[]=a [] ={2,6,8,9,5,4}, b [] ={9,5,2,6,8} We have to insert all the elements of array b[] into HashSet, so in HashSet, we have the following values: HashSet:{9,5,2,6,8} // basically all the values of b[]. We will traverse the array a[] and take each of its elements and check the condition. i=0, a[i]=2 2 is in the HashSet, so it will not print. i=1, a[i]=6 6 is in the HashSet, again it will not be printed. i=2, a[i]=8 8 is in the HashSet, it will not be printed. i=3, a[i]=9 9 is in the HashSet, so it will not print. i=4, a[i]=5 5 is in the HashSet, again it will not be printed. i=5, a[i]=4 4 is not in the HashSet, so this time it will be printed means it is the number which is present in an array a[] but not in array b[] because basically HashSet is the clone of array b[] and our output will become ‘4’. C++ code to Find elements which are present in first array and not in second using namespace std; void getMissingElement(int A[], int B[], int l1, int l2) unordered_set <int> myset; for (int i = 0; i < l2; i++) for (int j = 0; j < l1; j++) if (myset.find(A[j]) == myset.end()) cout << A[j] << " "; int main() int a[] = { 9, 2, 3, 1, 4, 5 }; int b[] = { 2, 4, 1, 9 }; int l1 = sizeof(a) / sizeof(a[0]); int l2 = sizeof(b) / sizeof(b[0]); getMissingElement(a, b, l1, l2); return 0; Java code to Find elements which are present in first array and not in second import java.util.HashSet; import java.util.Set; class missingElement public static void getMissingElement(int A[], int B[]) int l1 = A.length; int l2 = B.length; HashSet<Integer> set = new HashSet<>(); for (int i = 0; i < l2; i++) for (int i = 0; i < l1; i++) if (!set.contains(A[i])) System.out.print(A[i]+" "); public static void main(String []args) int a[] = { 9, 2, 3, 1, 4, 5 }; int b[] = { 2, 4, 1, 9 }; getMissingElement(a, b); Complexity Analysis O(N) where “N” is the number of elements in the array1. Because using HashSet for insertion and searching allows us to perform these operations in O(1). Thus the time complexity is linear. O(N) where “N” is the number of elements in the array1. Since we are storing the elements of the second array. Thus the space required is the same as that of the size of the second array.	{"url":"https://tutorialcup.com/interview/hashing/find-elements-which-are-present-in-first-array-and-not-in-second.htm","timestamp":"2024-11-15T00:38:57Z","content_type":"text/html","content_length":"112830","record_id":"<urn:uuid:f85c852f-ed3a-48e2-93c8-6392431a91be>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00372.warc.gz"}
Scale uncertainty of block or system Since R2020a blk_scaled = uscale(blk,factor) scales the amount of uncertainty in an uncertain control design block by factor. Typically, factor is a robustness margin returned by robstab or robgain, or a robust performance returned by musynperf. The uncertain element blk_scaled is of the same type as blk, with the amount of uncertainty scaled in normalized units. For instance, if factor is 0.75, the normalized uncertainty of blk_scaled is 75% of the normalized uncertainty of blk. M_scaled = uscale(M,factor) scales all the uncertain blocks in the model M. Non-uncertain elements are not changed. Find Tolerable Range of Gain and Phase Variations Consider a feedback loop with the following open-loop gain. Suppose that the system has gain uncertainty of 1.5 (gain can increase or decrease by a factor of 1.5) and phase uncertainty of ±30°. DGM = getDGM(1.5,30,'tight'); F = umargin('F',DGM) Uncertain gain/phase "F" with relative gain change in [0.472,1.5] and phase change of ±30 degrees. Examine the robust stability of the closed-loop system. T = feedback(LF,1); SM = robstab(T) SM = struct with fields: LowerBound: 0.8303 UpperBound: 0.8319 CriticalFrequency: 1.4482 robstab shows that the system can only tolerate 0.83 times the modeled uncertainty before going unstable. Scale the umargin block F by this amount to find the largest gain and phase variation that the system can tolerate. factor = SM.LowerBound; Fsafe = uscale(F,factor) Uncertain gain/phase "F" with relative gain change in [0.563,1.42] and phase change of ±24.8 degrees. The scaled uncertainty has smaller ranges of both gain variation and phase variation. Compare these ranges for the original modeled variation and the maximum tolerable variation. DGM = F.GainChange; DGMsafe = Fsafe.GainChange; ans = Legend (original, safe) with properties: String: {'original' 'safe'} Location: 'northeast' Orientation: 'vertical' FontSize: 9 Position: [0.7150 0.7968 0.1710 0.0789] Units: 'normalized' Use GET to show all properties Scale All Uncertain Elements in a Model Consider the uncertain control system of the example "Robust Performance of Closed-Loop System" on the robgain reference page. That example examines the sensitivity of the closed-loop response at the plant output to disturbances at the plant input. k = ureal('k',10,'Percent',40); delta = ultidyn('delta',[1 1]); G = tf(18,[1 1.8 k]) (1 + 0.5delta); C = pid(2.3,3,0.38,0.001); S = feedback(1,GC) Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 4 states. The model uncertainty consists of the following blocks: delta: Uncertain 1x1 LTI, peak gain = 1, 1 occurrences k: Uncertain real, nominal = 10, variability = [-40,40]%, 1 occurrences Type "S.NominalValue" to see the nominal value and "S.Uncertainty" to interact with the uncertain elements. Suppose that you do not want the peak gain of this sensitivity function to exceed 1.5. Use robgain to find out how much of the modeled uncertainty the system can tolerate while the peak gain remains below 1.5. perfmarg = robgain(S,1.5) perfmarg = struct with fields: LowerBound: 0.7821 UpperBound: 0.7837 CriticalFrequency: 7.8566 With that performance requirement, the system can only tolerate about 78% of the modeled uncertainty. Scale all the uncertain elements in S to create a model of the closed-loop system with the maximum level of uncertainty that meets the performance requirement. factor = perfmarg.LowerBound; S_scaled = uscale(S,factor) Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 4 states. The model uncertainty consists of the following blocks: delta: Uncertain 1x1 LTI, peak gain = 0.782, 1 occurrences k: Uncertain real, nominal = 10, variability = [-31.3,31.3]%, 1 occurrences Type "S_scaled.NominalValue" to see the nominal value and "S_scaled.Uncertainty" to interact with the uncertain elements. The display shows how the uncertain elements in S_scaled have changed: the peak gain of the ultidyn element delta is reduced from 1 to 0.78, and the range of variation of the uncertain real parameter k is reduced from ±40% to ±31.3%. Input Arguments blk — Uncertain control design block ureal \| umargin \| ultidyn \| ... Uncertain control design block to scale, specified as a ureal, umargin, ultidyn, or other uncertain block. factor — Scaling factor Scaling factor, specified as a scalar. This argument is the amount by which uscale scales the normalized uncertainty of blk or M. For instance, if factor = 0.8, then the function reduces the uncertainty to 80% of its original value, in normalized units. Similarly, if factor = 2, then the function doubles the uncertainty. Typically, factor is a robustness margin returned by robstab or robgain, or a robust performance returned by musynperf. Thus, you can use uscale to find the largest range of modeled uncertainty in a system for which the system has good robust stability or performance. M — Uncertain model uss \| umat \| ufrd \| genss \| ... Uncertain model, specified as a uss, umat, ufrd, or genss with uncertain control design blocks. The uscale command scales uncertain control design blocks in M. Other blocks of M are unchanged. Output Arguments blk_scaled — Scaled uncertain block ureal \| umargin \| ultidyn \| ... Scaled uncertain block, returned as a block of the same type as blk, such as a ureal, umargin, ultidyn, or other uncertain block. The uncertainty of blk_scaled is the same as the uncertainty in M, scaled by factor. M_scaled — Scaled uncertain model uss \| umat \| ufrd \| genss \| ... Scaled uncertain model, returned as a model of the same type as M, such as a uss, umat, ufrd, or genss with uncertain control design blocks. The uncertain control design blocks in M_scaled are the same as the blocks in M, with the size of uncertainty scaled by factor in normalized units. Version History Introduced in R2020a	{"url":"https://kr.mathworks.com/help/robust/ref/inputoutputmodel.uscale.html","timestamp":"2024-11-04T01:18:47Z","content_type":"text/html","content_length":"99907","record_id":"<urn:uuid:719e2e05-a932-40c0-bd84-42dec2ac1b53>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00221.warc.gz"}
How to model customer lifetime: good things and gotchas Welcome back to my series on predicting customer lifetime value, which I call "Everything Other Tutorials Miss." In Part 1, I covered the often estimative phase of historical CLV analysis and what can already be done with such seemingly backward information. Next, I presented use cases for the tone of CLV forecasting, going beyond the typically limited examples seen in other posts on this topic. Now it's time for the practical part, including everything I've learned while working with the data science team and real data and customers. Once again, there is too much juicy information to fit into a single blog post without turning it into an odyssey. So today, I will focus on historical CLV modeling. I will cover the silly simple formula, cohort analysis, and the RFM approach, including the pros and cons I discovered for each. Next, I will do the same. And I will wrap up the entire series with best practices learned by data scientists on how to perform CLV correctly. Sounds good? Then let's take a look at the historical CLV analysis methods and the advantages and "Gotchas" you need to know. Method 1: Silly Simple Formula Perhaps the simplest formula is based on three elements: how much customers typically purchase, how often they shop, and how long they maintain loyalty: For example, if the average customer spends €25 per transaction, makes two transactions a month, and maintains loyalty for 24 months, then CLV = €1200. We can make this a bit more sophisticated by considering margins or profits. There are several ways to do this. Silly Simple Formula V1: Product-Specific Margin Here, we calculate the average margin per product for all products in inventory, and then multiply this number by the silly simple formula result to generate the average customer lifetime margin.	{"url":"https://dimzou.feat.com/en/dimzou/1344417/1349082","timestamp":"2024-11-05T05:59:18Z","content_type":"text/html","content_length":"58570","record_id":"<urn:uuid:28779b1a-82e6-462d-9cd5-59759e0e3873>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00651.warc.gz"}
Stickman Mental Math Learn math by playing with Stickman. Brain game, math problem. Practice your skills in arithmetic. You must have heard about mental math, solve simple problems in your head. You can learn how to quickly find the right solution. Allocating just 5-10 minutes of stickman will help you increase your power of thought. By solving simple mathematical problems, you develop, but if you answer incorrectly, you will lose part of your stickman, help him hold out as long as possible. Math, Stickman, try this union. Mouse click or tap to play 1 you look at the number you need to get 2 Choose a number from the table 3 choose a mathematical sign 4 choose the second number 5 if no more mathematical operations are needed then press the equal sign	{"url":"https://funnygames.top/game/stickman-mental-math","timestamp":"2024-11-09T19:05:07Z","content_type":"text/html","content_length":"26636","record_id":"<urn:uuid:1d93d2c7-5b96-4d09-ba7b-290729877f8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00857.warc.gz"}
Value is within tolerance In this example the goal is to check if values in column B are within a tolerance of .005. If a value is within tolerance, the formula should return "OK". If the value is out of tolerance, the formula should return "Fail". The expected value is listed in column C, and the allowed tolerance is listed in column D. The solution is based on the IF function together with the ABS function. Core logic To check if a value is within a given tolerance, we can use a simple logical test like this: =ABS(actual-expected)<=tolerance // logical test Inside the ABS function, the actual value is subtracted from the expected value. The result may be positive or negative, depending on the actual value, so the ABS function is used to convert the result to a positive number: negative values become positive and positive values are unchanged. The result from ABS is compared to the allowed tolerance with the logical operator less than or equal (<=). The expression returns TRUE when a value is less than or equal to the allowed tolerance, and FALSE if not. IF function To complete the solution, we need to place the generic logical expression above into the IF function and providing values for a TRUE and FALSE result. The first step is to revise the generic expression above to use worksheet references: ABS(B5-C5)<=D5 // logical test Then, we drop the expression into the IF function as the logical_test argument: =IF(ABS(B5-C5)<=D5,"OK","Fail") // final formula When the logical test returns TRUE, IF returns "OK". When the logical test returns FALSE, IF returns "Fail". These messages can be customized as needed. List all values within tolerance The basic concept explained above can be extended to list values within tolerance or out of tolerance with the FILTER function.	{"url":"https://exceljet.net/formulas/value-is-within-tolerance","timestamp":"2024-11-03T23:10:41Z","content_type":"text/html","content_length":"49334","record_id":"<urn:uuid:5d2b6214-8e95-4589-9dea-4afefed02632>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00194.warc.gz"}
Biology of Distributed Information Systems 1. Introduction This post is about the search for sense in a small data set, such as the few measures that one accumulates through self-tracking. Most commonly, finding sense in a small set of data means either to see regular patterns or to detect causality. Many writers have argued that our brains are hardwired for detecting patterns and causality. Causality is our basic ingredient for modelling “how the world works”. Inferring causality from our world experience is also a way of “compressing” our knowledge: once you understand that an open flame hurts, you don’t need to recall the experiences (and you don’t need so many of them to detect this causality). The reason for selecting this topic for today’s blog post is my recent participation to the ROADEF 2019 conference. I had the pleasure of chairing the machine learning session and the opportunity to present my own work about machine learning for self-tracking data. We are so good at detecting causality that we are often fooled by random situations and tend to see patterns when there are none. This is a common theme of Nassim Taleb’s many books and especially his master first book “Fooled by Randomness”. The concept of “narrative fallacy” is critical when trying to extract sense from observation, we need to remember that we love to see “stories” with a sense because this is how our brain best remembers. There are two type of issues when trying to mine short data sets for sense: the absence of statistical significance because the data set is too small and the existence of our own narrative fallacy and other cognitive biases. Today I will talk about data sets collected from self-tracking (i.e. the continuous measurement of some of your characteristics, either explicitly while logging observations or implicitly with connected sensors such as a connected watch). The challenge of scientific methods when searching for sense with such short time series is to know when to say “I don’t know” when presented with a data set with no other form of patterns or correlation that what could be expected in any random distribution, without falling into the “pitfall of narrative fallacy”. In short, the “Turing test” of causality hunting is to reject random or quasi-random data input. On the other hand, it is tempting to look for algorithms that could learn and extract sense from short time series precisely because humans are good at it. Humans are actually very good at short-term forecasting and quick learning which is without a doubt the consequence of evolution. Learning quickly to forecast the path of a predator or a prey has been resolved with reinforcement learning through “survival of the fittest” evolution. The topic of this blog post – which I discussed at ROADEF – is how to make sense of a set of short time series using machine learning algorithms. "Making sense" here is a combination of forecasting and causality analysis which I will discuss later. The second reason for this blogpost is the wonderful book of Judea Pearl, “The Book of Why”, which is a masterpiece about causality. The central idea of the book is that causality does not “jump from the data” but requires an active role from the observer. Judea Pearl introduces concepts which are deeply relevant to this quest of search for sense with small data sets. Hunting for causality is a “dangerous sport” for many reasons: most often you come back empty-handed, sometimes you catch your own tail … and when successful, you most often have little to show for your efforts. The two central ideas of causality diagrams and the role of active observers are keys for unlocking some of the difficulties of causality hunting with self-tracking data. This post is organised as follows. Section 2 is a very short and partial review of “The Book of Why”. I will try to explain why Judea Pearl’s concepts are critical to causality hunting with small data sets. These principles have been applied to the creation of a mobile application that generated the data sets onto which the machine learning algorithm of Section 4 have been applied. This application uses the concept of a causal diagram (renamed as quests) to embody the user’s prior knowledge and assumptions. The self-measure follows the principle of the “active observer” of Judea Pearl’s P(X \| do(Y)) definition. Section 3 dives into causality hunting through two other books and introduced the concept of Granger causality that binds forecasting and causality detection. It also links the concept of pleasure and surprise with self-learning, a topic that I borrow from Michio Kaku and which also creates a strong relationship between forecasting and causality hunting. As noted by many scholars, “the ability to forecast is the most common form of intelligence”. Section 4 talks briefly about Machine Learning algorithms for short time-series forecasting. Without diving too deep into the technical aspects, I show what prediction from small data sets is difficult and what success could look like, considering all the pitfalls that we have presented before. Machine Learning from small data is not a topic for deep learning, thus I present an approach based on code generation and reinforcement learning. 2. Causality Diagrams - Learn by Playing Judea Pearl is an amazing scientist with a long career about logic, models and causality that has earned him a Turing Award in 2011. His book reminds me of “Thinking, Fast and Slow” of Daniel Kahneman, a fantastic effort of summarising decades of research into a book that is accessible and very deep at the same time. “The Book of Why – The new science of cause and effect” by Judea Pearl and Dana MacKenzie, is a master piece about causality. It requires careful reading if ones want to extract the full value of the content, but can also be enjoyed as a simple, exciting read. A great part of the book deals with paradoxes of causality and confounders, the variable that hide or explain causality relationships. In this section I will only talk about four key ideas that are relevant to hunting causality from small data The first key idea of this book is causality is not a cold objective that one can extract from data without prior knowledge. He refutes a “Big Data hypothesis” that would assume that once you have enough data, you can extract all necessary knowledge. He proposes a model for understanding causality with three levels : the first level is association, what we learn with observation; the second level is intervention, what we learn by doing things and the third level is counterfactuals, what we learn through imagining what-if scenarios. Trying to assess causality from observation only (for instance through conditional probabilities) is both very limited (ignoring the two top levels) but also quite tricky since as recalled by Persi Diaconis: “Our brains are not just wired to do probability problems, so I am not surprised there were mistakes”. Judea Pearl talk in depth about the Monty Hall problem, a great puzzle/paradox proposed by Marilyn Vos Savant, that has tricked many of the most educated minds. I urge you to read the book to learn for yourself from this great example. The author’s conclusion is: “Decades’ worth of experience with this kind of questions has convinced me that, in both a cognitive and philosophical sense, the ideas of causes and effects is much more fundamental than the idea of probability”. Judea Pearl introduced the key concept of causal diagram to represent our prior preconception of causality that may be reinforced or invalidated from observation, following a true Bayesian model. A causal diagram is a directed graph that represents your prior assumptions, as a network of factors/variable that have causal influence on each other. A causal diagram is a hypothesis that actual data from observation will validate or invalidate. The central idea here is that you cannot extract a causal diagram from the data, but that you need to formulate a hypothesis that you will keep or reject later, because the causal diagram gives you a scaffolder to analyse your data. This is why any data collection with the Knomee mobile app that I mentioned earlier starts with a causal diagram (a Another key insight from the author is to emphasise a participating role to the user asking the causality question, which is represented through the notation P(X \| do(Y)). Where the conditional probability P(X \| Y) is the probability of X being true when Y is observed, P(X \| do(Y)) is the probability of X when the user chooses to “do Y”. The stupid example of learning that a flame burns your hand is actually meaningful to understand the power of “learning by doing”. One or two experiences would not be enough to infer the knowledge from the conditional probability P(hurts \| hand in flame) while the experience do(hand in flame) means that you get very sure, very quick, about P(hurts \| do(hand in flame)). This observation is at the heart of personal self-tracking. The user is active and is not simply collecting data. She decides to do or not to do things that may influence the desired outcome. A user who is trying to decide whether drinking coffee affects her sleep is actually computing P(sleep \| do(coffee)). Data collection is an experience, and it has a profound impact on the knowledge that may be extracted from the observations. This is very similar to the key concept that data is a circular flow in most AI smart systems. Smart systems are cybernetic systems with “a human inside”, not deductive linear systems that derive knowledge from static data. One should recognise here a key finding from the NATF reports on Artificial Intelligence and Machine Learning (see “Artificial Intelligence Applications Ecosystem: How to Grow Reinforcing Loops”). The role of the participant is especially important because there is a fair amount of subjectivity when hunting for causality. Judea Pearl gives many examples where the controlling factors should be influenced by the “prior belief” of the experimenters, at the risk of misreading the data. He writes: “When causation is concerned, a grain of wise subjectivity tells us more about the real world that any amount of objectivity”. He also insists on the importance of the data collection process. For him, one of the reasons statisticians are often the most puzzled with the Monty Hall paradox is the habit of looking at data as a flat static table: “No wonder statisticians found this puzzle hard to comprehend. They are accustomed to, as R.A. Fisher (1922) puts it, “the reduction of the data” and ignoring the data-generation process”. As told earlier, I strongly encourage you to read the book to learn about “counfounders” – that are easy to explain with causal diagram – and how they play a critical role for these types of causality paradox where the intuition is easily fooled. This is the heart of this book: “ I consider the complete solution of the counfounders problem one of the main highlights of the Causal Revolution because it has ended an era of confusion that has probably resulted in many wrong decisions in the past”. 3. Finding a Diamond in the Rough Another interesting book about hunting for causality is “Why: A Guide to Finding and Using Causes” by Samantha Kleinberg. This books starts with the idea that causality is hard to understand and hard to establish. Saying that “correlation is not causation” is not enough, understanding causation is more complex. Statistics do help to establish correlation, but people are prone to see correlation when none exists: “many cognitive biases lead to us seeing correlations where none exist because we often seek information that confirms our beliefs”. Once we validate a correlation with statistics tool, one needs to be careful because even seasoned statisticians “cannot resists treating correlations as if they were causal”. Samantha Kleinberg talks about Granger Causality: “one commonly used method for inference with continuous-valued time series data is Granger”, the idea that if there is a time delay observed within a correlation, this may be a hint of causality. Judea Pearl warns us that this may be simply the case of a counfounder with asymmetric delays, but in practice the test of Granger causality is not a proof but a good indicator for causality. The proper wording is that this test is a good indicator for “predictive causality”. More generally, if predicting a value Y from the past of X up to a non-null delay does a good job, it may be said that there is a good chance of “predictive causality” from X to Y. This links the tool of forecasting to our goal of causality hunting. It is an interesting tool since it may be used with non-linear models (contrary to Granger Causality) and multi-variate analysis. If we start from a causal diagram in the Pearl’s sense, we may see if the root nodes (the hypothetical causes) may be used successfully to predict the future of the target nodes (the hypothetical “effects”). This is, in a nutshell, how the Knomee mobile app operates: it collects data associated to a causal diagram and uses forecasting as a possible indicator of “predictive causality”. The search of “why” with self-tracking data is quite interesting because most values (heart rate, mood, weight, number of steps, etc.) are nonstationary on a short time scale, but bounded on a long-time horizon while exhibiting a lot of daily variation. This makes detecting patterns more difficult since this is quite different from extrapolating the movement of a predator for its previous positions (another short time series). We are much better at “understanding” patterns that derive from linear relations than those that emerge from complex causality loops with delays. The analysis of delays between two observations (at the heart of the Granger Causality) is also a key tool in complex system analysis. We must, therefore, bring it with us when hunting for causality. This is why the Knomee app includes multiple correlation/delay analysis to confirm or invalidate the causal hypothesis. A few other pearls of wisdom about causality hunting with self-tracking may be found in the book from Gina Neff and Dawn Nafus. This reference book on quantified self and self-tracking crosses a number of ideas that we have already exposed, such as the critical importance of the user in the tracking and learning process. Self-tracking – a practice which is both very ancient and has shown value repeatedly – is usually boring if no sense is derived from the experiment. Making sense is either positive, such as finding causality, or negative, such as disproving a causality hypothesis. Because we can collect data more efficiently in the digital world, the quest for sense is even more important: “Sometimes our capacity to gather data outpaces our ability to make sense of it”. In the first part of this book we find this statement which echoes nicely the principles of Judea Pearl: “A further goal of this book is to show how self-experimentation with data forces us to wrestle with the uncertain line between evidence and belief, and how we come to decisions about what is and is not legitimate knowledge”. We have talked about small data and short time-series from the beginning because experience shows that most users collect data over long period of time: “Self-tracking projects should start out as brief experiments that are done, say, over a few days or a few weeks. While there are different benefits to tracking over months or years, a first project should not commit you for the long haul”. This is why we shall focus in the next section on algorithms that can work robustly with a small amount of data. Self-tracking is foremost a learning experiment: “The norm within QS is that “good” self-tracking happens when some learning took place, regardless of what kind of learning it was”. A further motive for self-tracking is often behavioural change, which is also a form of self-learning. A biologists tell us, learning is most often associated with pleasure and reward. As pointed out in a previous post, there is a continuous cycle : pleasure to desire to plan to action to pleasure, that is a common foundation for most learning with living creatures. Therefore, there is a dual dependency between pleasure and learning when self-tracking: one must learn (make sense out the collected data) to stay motivated and to pursue the self-tracking experience (which is never very long) and this experience should reward the user from some forms of pleasure, from surprise and fun to the satisfaction of learning something about yourself. Forecasting is a natural part of the human learning process. We constantly forecast what will happen and learn by reacting to the difference. As explained by Michio Kaku, our sense of humour and the pleasure that we associate with surprises is a Darwinian mechanism to push us to constantly improve our forecasting (and modelling abilities). We forecast continuously, we experience the reality and we enjoy the surprise (the difference between what happens and what we expect) as an opportunity to learn in a Bayesian way, that is to reassign our prior assumptions (our model of the world). The importance of curiosity as a key factor for learning is now widely accepted in the machine learning community as illustrated in this ICML 2017 paper: “Curiosity-driven Exploration by Self-supervised Prediction”. The role of surprise and fun in learning is another reason to be interested in forecasting algorithms. Forecasting the future, even if unreliable, creates positive emotions around self-tracking. This is quite general: we enjoy forecasts, which we see as games (in addition of their intrinsic value) – one can think of sports or politics as example. A self-tracking forecasting algorithm that does a decent job (i.e., not too wrong nor too often) works in a way similar to our brain: it is invisible but acts as a time saver most of the times, and when wrong it signals a moment of interest. We shall now come back to the topic of forecasting algorithms for short time-series, since we have established that they could play an interesting role for causality hunting. 4. Machine Generation of Robust Algorithms Our goal in this last section is to look at the design of robust algorithms for short time series forecasting. Let us first define what I mean by robust, which will explain the metaphor which was proposed in the introduction. The following figure is extracted from my ROADEF presentation, it represents two possible types of “quests” (causal diagrams). Think of a quest as a variable that we try to analyse, together with other variables (the “factors”) which we think might explain the main variable. The vertical axis represents a classification of the variation that is observed into three categories: the random noise in red, the variation that is due to factors that were not collected in the sample in orange, and the green area is the part that we may associate with the factors. A robust algorithm is a forecasting algorithm that accepts an important part of randomness, to the point that many quests are “pointless” (remember the “Turing test of incomplete forecasting”). A robust algorithm should be able to exploit the positive influence of the factors (in green) when and if it exists. The picture makes it clear that we should not expect miracles: a good forecasting algorithm can only improve by a few percent over the simple prediction of the average values. What is actually difficult is to design an algorithm that is not worse – because of overfitting – than average prediction when given a quasi-random input (right column on the picture). As the title of the section suggests, I have experimented with machine generation of forecasting algorithms. This technique is also called meta-programming: a first algorithm produces code that represents a forecasting algorithm. I have used this approach many times in the past decades, from complex optimization problems to evolutionary game theory. I found that it was interesting many years ago when working on TV audience forecasting, because it is a good way to avoid over-fitting, which is a common plague when doing machine learning over a small data set, and to control the robustness properties thanks to evolutionary meta-techniques. The principle is to create a term algebra that represents instantiations and combinations of simpler algorithm. Think of it as a tool box. One lever of control (robustness and over-fitting) is to make sure that you only select “robust tools” to put in the box. This means that you may not obtain the best or more complex machine learning algorithm such as deep learning, but you ensure both “explainability” and control. The meta-algorithm is an evolutionary randomised search algorithm (similar to the Monte-Carlo Tree Search of Alpha Zero) that may be sophisticated (using genetic combinations of terms) or simple (which is what we use for short time series). The forecasting algorithm used by the Knomee app is produced locally on the user phone from the collected data. To test robustness, we have collected self-tracking data over the past two years - for those of you who are curious to apply other techniques, the data is available on GitHub. The forecasting algorithm is the fixed-point of an evolutionary search. This is very similar to reinforcement learning in the sense that each iteration is directed by a fitness function that describes the accuracy of the forecasting (modulo regularization, as explained in the presentation). The training protocol is defined as running the resulting forecasting algorithm on each sample of the data set (a quest) and for each time position from 2/3 to 3/3 of the ordered time series. In other words, the score that we use is the average precision of the forecasting that a user would experience in the last third of the data collection process. The term-algebra that is used to represent and to generate forecasting algorithms is made of simple heuristics such as regression and movingAverage, of weekly and hourly time patterns, and correlation analysis with threshold, cumulative and delays options. With the proper choice of meta-parameters to tune the evolutionary search (such as the fitness function or the depth and scope of local optimisation), this approach is able to generate a robust algorithm, that is : (1) that generates better forecasts than average (although not by much) (2) that is not thrown away by pseudo-random time series . Let me state clearly that this approach is not a “silver bullet”. I have compared the algorithm produced by this evolutionary search with the classical and simple machine learning approaches that one would use for time series: Regression, k-means clustering and ARMA. I refer you to the great book “Machine Learning for the Quantified Self” by M. Hoogendoorn and B. Funk for a complete survey on how to use machine learning with self-tracking data. On regular data (such as sales time series), the classical algorithms perform slightly better that evolutionary code generation. However, when real self-tracking data is used with all its randomness, evolutionary search manages to synthesise robust algorithms, which none of the three classical algorithms are. 5. Conclusion This topic is more complex than many of the subjects that I address here. I have tried to stay away from the truly technical aspects, at the expense of scientific precision. I will conclude this post with a very short summary: 1. Causality hunting is a fascinating topic. As we accumulate more and more data, and as Artificial Intelligence tools become more powerful, it is quite logical to hunt for causality and to build models that represent a fragment of our world knowledge through machine learning. This is, for instance, the heart of the Causality Link startup led by my friend Pierre Haren, which builds automatically knowledge graphs from textual data while extracting causal links, which is then use for deep situation analysis with scenarios. 2. Causality hunting is hard, especially with small data and even more with “Quantified Self” data, because of the random nature of many of the time series that are collected with connected devices. It is also hard because we cannot track everything and quite often what we are looking for depends on other variable (the orange part of the previous picture). 3. Forecasting is an interesting tool for causality hunting. This is counter-intuitive since forecasting is close to impossible with self-tracking data. A better formulation should be: “ a moderate amount of robust forecasting may help with causality hunting". Forecasting gives a hint of “predictive causality”, in the sense of the Granger causality, and it also serves to enrich the pleasure-surprise-discovery learning loop of self-tracking. 4. Machine code generation through reinforcement learning is a powerful technique for short time-series forecasting. Code generating algorithms try to assemble building blocks from a given set to match a given output. When applied to self-tracking forecasting, this technique allows to craft algorithms that are robust to random noise (to recognise the data as such) and able to extract a weak correlative signal from a complex (although short) data set.	{"url":"https://informationsystemsbiology.blogspot.com/2019/04/","timestamp":"2024-11-09T07:07:42Z","content_type":"application/xhtml+xml","content_length":"99294","record_id":"<urn:uuid:105054be-3846-4178-9deb-abcbea4c349f>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00001.warc.gz"}
Rapid tree model reconstruction for fruit harvesting robot system based on binocular stereo vision In this paper, the method of spatial information extraction of tree branch was studied. The region matching method was used to get the disparity map of stereo image, extracted feature points combining with branch skeleton image by multi-segment approximation method, and calculated the spatial coordinates and the radius of branch feature points by using binocular stereo vision. Real-time model reconstruction for fruit tree has been researched on. Test proposed that each branch module was constructed by 12-prism in the coordinate origin, and then rotated twice and translated once to get correct posture, finally combined with other modules for the fruit tree model. Test has optimized extraction algorithm and matching algorithm of the branch region, improved matching rate, reduced matching errors, avoided matching confusion, accurately extracted branch spatial information and improved the success rate of robot path planning for obstacle avoidance. 1. Introduction The intelligent harvesting robot is an effective solution approach for the reduction of agricultural labor force and the high cost of fruit picking. Unstructured working environment is the biggest problem for harvesting robot. As the spatial location and posture of fruit are random, the existence of branches and other obstacles are the hidden dangers of the harvesting robot. Therefore, the identification and location of obstacles is a new problem. After the acquisition of branch spatial information, their 3D models are reconstructed by using appropriate method. In the work scene model, path planning for fruit picking are carried out to make robot real-time, securely, nondestructively pick fruit. In the whole harvesting process, there are two key issues: identification and picking. So, the system must have two functions, by which obtain the identification information from the fruit data and obstacle avoidance information from the branch data. Until present, binocular stereo vision system is the best choice for the whole process system and connection platform have known. In the picking part, it is easy to reconstruct the real-time model and supply the data for obstacle avoidance by binocular stereo vision system. And in fact, when the model was reconstructed, the data transformation from that could supply enough data for the obstacle avoidance [1]. In 3D modeling, decompose a complicated object into simple modules, and then combine them according to a given rule. For three-dimensional modeling, this method of complexity-simpleness-complexity [2-9] is important. It is well known that OpenGL include the standard modules, if a module could be found that satisfies the rapid fruit model construction, all problems would become simple following the thinking mentioned before. In this paper, the cylinder module is choosing because though the tree's shape is complicated, the branch is made from a great deal of cylinders. 2. Extraction of branches information Extraction of branches features information of fruit trees is to reconstruct tree model based on these features. From the scene images Fig. 1, it can be seen that the shape of branches are complex, it is very difficult to recover the branches 3D information. There are straight branches and curved branches. Reconstruction cylindrical through branch endpoints and radius can characterize direct branches, but not to characterize curved branches. In this study, first, extract the tree skeleton, and then fit tree branches skeletons by multi-segment approximation method to obtain feature points coordinate information and extract radius information of the characteristic points by the branch range image. Finally, the original branches are reconstructed by multi cylindrical. Fig. 1Scene image and binary image of branch 2.1. Skeletons extraction Skeleton is the core part of the object, and the objects of different shapes have different skeletons. In general, the skeleton has three main characteristics: continuity, the minimum width is 1, and centrality [10]. Currently, extraction method of region skeleton mainly includes morphology method, distance transform and thinning method, etc. 1. Morphology method. Morphological method is a way to corrode awaiting processing region with a structure element, and then add each last time result before corrosion to empty set to get region skeleton [11]. But, this method can’t guarantee the connectivity, when the edge is not smooth or the width sudden changes in the region, that will cause skeleton points detected from the continuous region discontinuous. 2. Distance transform. Distance transform method extract region skeleton in two steps by distance transform and skeletonization. In this method, the range image is obtained through distance transform, the size of each pixel value in the range image directly reflects the distance from the pixel to edge; Then comparing the distance value of pixel in the range image, all the set of pixels which’s distance value are greater or equal to the maximum distance value of the neighborhood is used as region skeleton. Distance transform method extract region skeleton points by range image. For the largest distance points in each row (or column) are only related to the edge, the skeleton points location relationship of adjacent two rows (or two column) is not necessarily related, when the edge is not smooth or the width sudden changes in the region, that will cause discontinuous skeleton points detected from the continuous region or generate noise points, which will cause difficulty in further processing. 3. Thinning method. Region thinning method thins the region to get line graph constituted by lines with small memory capacity and ease to be identified. In order to make line graph accurately represent the shape of the object, the thinning must meet the following requirements: (1) The line-width is one pixel; (2) Thin position lies in the center of the original region; (3) Graph’s connectivity keeps invariant, and the hole and points can’t new increase or disappear; (4) The graphic ends keep invariant. So, one can see, the essence of region thinning is the process of finding the center line of graph without changing the length and the connectivity of graph. There are various thinning methods, this paper adopts Yokoi thinning method to implement thinning region. When Yokoi thinning method repeatedly removing pixels in the surface of graphics, the graphic surface pixels are divided into upper-lower surface pixels and left-right surface pixels, and forward scanning and reverse scanning are alternately carried out to remove upper-lower surface pixels and left-right surface pixels until the center line was obtained. In this study, the region skeletons extracted by Yokoi thinning method keep the connectivity of original region with better effect Fig. 2. Fig. 2Features extraction of branch c) Multi-segment approximation image 2.2. Skeleton pruning Although the branches skeletons processed by thinning keep a good connectivity, it will lead to “false branch” Fig. 2(b) when the edge is not smooth. The Spurs don’t affect the overall structure of the branches, but will increase computational complexity, which should be removed. At present, the common skeleton pruning method is morphology method [12]. But morphology method can only prune short branches (not more than three pixels, the glitch), because the falsest branches in this study are more than 3 pixels, morphology method can’t prune that. In this study, the length of branch skeletons counted, and then set the threshold for the length to remove the false branch, the concrete process are as follows: 1) Scanning skeleton image to find the branch points (${N}_{b}^{\left(8\right)}>2$) and endpoints (${N}_{b}^{\left(8\right)}=1$) of the skeleton, and copy the skeleton image; 2) In the copy image, the pixels in the eight neighborhoods of the branch point are changed into background points, as the branch point is the connection point of several branches, when eight neighborhood points of the branch points are changed into background points, these branches will be divided into some separate branch segments; 3) Label each branch and count the length of that; 4) Judge each branch skeleton in skeleton image, if two endpoint of the skeleton are both branch points, then this skeleton is in the tree trunk which can’t be removed; if the two endpoints are both the endpoints, then the skeleton is an independent skeleton and not the branch section which can’t be removed; if only one point between two endpoints of the skeleton is the branch point, the skeleton isn’t the trunk, and then judge its length, if the total number of pixels less than 15, indicating that the skeleton is the “false branch” which should be removed. With the above processing, the skeleton glitch and “false branch” are effectively removed, the number of feature points and branches are significantly reduced, so as to simplifying the subsequent 3. Acquiring 3D information This paper extracts the branch 3D information by binocular stereo vision technology, which includes the space coordinate of branch feature points and the radius of the branch. Because there are many feature points for one breach, feature points matching is unsuitable for branch matching. Region matching is to select a $\left(2n+1\right)×\left(2n+1\right)$ region (window) with one point as the center in an image, find the region with the greatest correlation degree to the window region in another image, and the center of the maximum correlation region found is used as the corresponding point of original regional centre, the disparity of the whole image can be obtained. So, region matching is suitable for branch matching in this study. Typically, the gray matching method is the fastest way to reconstruction. Sum of absolute gray value differences (SAD) and sum of squared gray value differences (SSD) are two methods of similarity measure [16, 17]. The definition as follows: $sad\left(r,c,d\right)=\frac{1}{{\left(2n+1\right)}^{2}}{\sum }_{j=-n}^{n}{\sum }_{i=-n}^{n}\left\|{I}_{right}\left(r+i,c+j\right)-I{}_{left}{}^{}\left(r+i,c+j+d\right)\right\|,$ $sad\left(r,c,d\right)=\frac{1}{{\left(2n+1\right)}^{2}}{\sum }_{j=-n}^{n}{\sum }_{i=-n}^{n}{\left({I}_{right}\left(r+i,c+j\right)-I{}_{left}{}^{}\left(r+i,c+j+d\right)\right)}^{2}.$ SAD and SSD are sensitive to illumination variation. As the shading of tree, and the different viewpoint of two cameras, illumination variation often occurs. Therefore, this study used normalized cross-correlation (NCC) method: $\begin{array}{c}ncc\left(r,c,d\right)=\frac{1}{\left(2n+1{\right)}^{2}}{\sum }_{i=-n}^{n}{\sum }_{j=-n}^{n}\frac{{I}_{right}\left(r+i,c+j\right)-{m}_{right}\left(r+i,c+j\right)}{\sqrt{s{}_{right}{}^ {}{\left(r+i,c+j\right)}^{2}}}\end{array}$$\bullet \frac{{I}_{left}\left(r+i,c+j+d\right)-{m}_{left}\left(r+i,c+j+d\right)}{\sqrt{s{}_{right}{}^{}{\left(r+i,c+j+d\right)}^{2}}},$ ${m}_{k}$ and ${s}_{k}$ (${s}_{k}=left$, $right$) respectively represented mean value and standard deviation of the window in the left and right image. NCC has the advantage of anti interference to illumination variation, but with slightly longer computation time. To find the match point in the left image, similarity measure should be calculated along the entire epipolar line in the right image. However, the disparity of the point related to the depth, which is decreased with depth increased, the disparity of infinite point can be seen as 0. Depth range of the object can set disparity search. A smaller searching range of disparity can be set by depth range of the object. Therefore, $d\in \left[{d}_{\mathrm{m}\mathrm{i}\mathrm{n}},{d}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]$, ${d}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ (min disparity) and ${d}_{\mathrm{m}\mathrm{a}\mathrm {x}}$ (max disparity) can be calculated by the depth extreme value. After the calculation of similarity measure for a point, the matching point is based determined by the maximum value of the NCC. However, the occlusion may result in failure matching of the pixels, a threshold value of similarity is set, and only when the pixel similarity measure is above this threshold, the matching results are accepted. Obviously, the setting for the threshold may result in multiple matching [18]. Fig. 4Disparity image using area-based matching and skeleton image c) Skeleton image processed by “and operation” In this study, the distance from fruit trees to the camera is between 1.0-2.0 m, by calculating the disparity search range of ${d}_{\mathrm{m}\mathrm{a}\mathrm{x}}=$ 96 and ${d}_{\mathrm{m}\mathrm{i} \mathrm{n}}=$ 48 can be set. After two original images are converted to grayscale images, the two images are matched with setting 15×15 matching window by NCC method. Texture validation and uniqueness validation are set to prevent the match pixels generated by window texture level being declared to be invalid pixels and to optimal select one of the best match points from multiple matching pixels. Disparity image are shown in Fig. 4(a), Fig. 4(c) is the result of “and operation” between Fig. 4(a) and Fig. 4(b), which represents branches that can be measured. By comparing Fig. 4(c) and 4(b), it can be found in Fig. 4(c) that some of region skeletons are reduced, this is because part of the branches can’t find the matching points. 4. 3D reconstruction based OpenGL Harvesting robot is complex and expensive, which is eased to be damaged by branches and other obstacles when picking citrus, apples and other fruits of tall trees. It is necessary for harvesting manipulator to possess the function of obstacle avoidance, as will ensure the occlusion fruit are picked safely and successfully. The premise conditions for obstacle avoidance are the perception of obstacle information in the robot work space and the reconstruction of scene model. The work scene includes fruits, leaves and branches, the fruits are the work target and the leaves will not cause damage to the robot, so that reconstructing branch model is enough. On the basis of fitting branch by multi-segment approximation and branch feature extraction, the unit module is constructed by the 12- prism method, and then assembled into the tree model. Fruit trees are with complexity structure and large individual differences, the breaches are composed of large number of truncated cones with constantly changing radius. The unit module is constructed branches have been divided into many segments by multi-segment approximation and extracted the 3D coordinate and radius of feature point. The data of each branch is stored as $\left\{\ left({x}_{1},{y}_{1},{z}_{1}\right),{R}_{1};\left({x}_{2},{y}_{2},{z}_{2}\right),{R}_{2}\right\}$ which is the necessary data for constructing unit module. Model is constructed by using OpenGL in VC 4.1. Unit model construction The basic elements of the model are the model vertices. The vertices branch is located on the bottom of both endpoints of the branch, in order to describe each branch model, it needs to calculate the space coordinates of 24 points on the two bottoms. Since the position and pose of each branch is random, computing for the 24 points is very complicated. To reduce computational complexity of unit module, this study made coordinates of circle center on the bottom coincide with coordinate origin. The branch model’s height can be calculated by Eq. (5): The coordinates and radius of circle centers on the bottom of the two cylinders are changed into $\left\{\left(0,0,0\right),{R}_{1};\left(0,h,0\right),{R}_{2}\right\}$. Through the radius information of ${R}_{1}$ and ${R}_{2}$, the coordinates of 12 equal diversion points (vertices) of the circle with radius ${R}_{1}$ on the plane of ${O}_{1}$ were calculated out$\left(\left\{\left({R}_{1},0,0\ right);\cdot \cdot \cdot ;\left({R}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{2k\pi }{12}\right),0,{R}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{2k\pi }{12}\right)\right);\cdot \cdot \cdot \ right\}\mathrm{}\left(K\in \left\{0,1,2,\cdot \cdot \cdot ,11\right\}\right)$ and the coordinates of 12 equal diversion points (vertices) of the circle with radius ${R}_{2}$ on the plane of ${O}_{2}$ were calculated out$\left(\left\{\left({R}_{2},h,0\right);\cdot \cdot \cdot ;\left({R}_{2}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{2k\pi }{12}\right),h,{R}_{2}\mathrm{s}\mathrm{i}\mathrm{n}\left(\ frac{2k\pi }{12}\right)\right);\cdot \cdot \cdot \right\}\left(K\in \left\{0,1,2,\cdot \cdot \cdot ,11\right\}\right)\right)$. The 24 vertices were connected by the connection order of standard model so as to get a branch module. 4.2. Scene model combination Each branch unit model needed to be moved to correct position with the original pos and composed of scene model with other model. By spatial position transformation law, unit model will recurrence the branch original position and pos by twice rotations and once translation. Taking the cylindrical module for example, the transform process of module position and orientation is briefly introduced. Each branch model in the scene was transformed by above methods to obtain scene tree model, shown in Fig. 5. Fig. 5Scene model of virtual reconstruction 5. Conclusions This paper has studied the method of spatial information extraction. Experiment used the region matching method to obtain the disparity of each pixel matched, extracted feature points combining with branch skeleton image by multi-segment approximation method, and calculated the spatial coordinates and the radius of branch feature points by using binocular stereo vision. Real-time model reconstructing for fruit tree has been researched on. Test has optimized extraction algorithm and matching algorithm of the branch region, improved matching rate, reduced matching errors, avoided matching confusion, accurately extracted branch spatial information and improve the success rate of robot path planning for obstacle avoidance. Real-time model reconstruction for fruit trees had been researched on. Test proposed that each branch module was constructed by 12-prism in the coordinate origin and then in the space of tree build paragraphs, and then rotated twice and translated once to get correct posture, finally combined with other modules for the fruit tree model. 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About this article Measurements in engineering fruit harvesting robot binocular stereo vision model reconstruction This work was supported by the foundation of Zhejiang Educational Committee (Grant No. Y201432724, Y201533234, 2017C35001, 2016C11018), the foundation of Ningbo Science Bureau (Grant No. 2015C10050, 2016C10056, 2016A10003, 2016C11018). Copyright © 2017 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.	{"url":"https://www.extrica.com/article/18611","timestamp":"2024-11-03T15:55:26Z","content_type":"text/html","content_length":"129851","record_id":"<urn:uuid:b6351095-1bee-4c7f-b1b2-03d59b2a2864>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00237.warc.gz"}
Real Time Scheduling Part 1 In this video, I am going to focus on the classical static real time algorithm, the Rate Monotonic Scheduling. This algorithm is used in many real time operating systems. RMS assume to be under the following conditions: Fixed priority Hard deadline If you are not interested in the proof, you can skip to the end to see the amazing result: or 2(sqrt(2)-1)=0.83 Remember this: RMS guarantee all the deadlines if CPU utilization is 69.3%. The other 30.7% of the CPU can be dedicated to lower-priority non real-time tasks. One Semester class material is compressed in a few minutes. I have given seminars based on the slides and animations.	{"url":"https://iadviser.com/real-time-scheduling-part-1/","timestamp":"2024-11-09T10:26:40Z","content_type":"text/html","content_length":"53799","record_id":"<urn:uuid:1e5f5828-a62b-4ef5-8987-36414e8d3f65>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00551.warc.gz"}
Consider the parabola with focus $(p, 0)$ and directrix $x=-p$ for $p>0 .$ Let $(x, y)$ be an arbitrary point on the parabola. Write an equation expressing the fact that the distance from \ ((x, y)\) to the focus is equal to the distance from $(x, y)$ to the directrix. Rewrite the equation in the form $x=a y^{2},$ where $a=\frac{1}{4 p}$ Short Answer Expert verified The equation is $x = \frac{1}{4p} y^2$, where $a = \frac{1}{4p}$ Step by step solution Understand the Parabola Properties The problem states that the parabola has its focus at $p, 0$ and the directrix as $x = -p$. For a parabola, each point $x, y$ is equidistant from the focus and the directrix. Distance from Point to Focus Calculate the distance from an arbitrary point $ (x, y) $ to the focus $ (p, 0) $. Use the distance formula: \[d_1 = \sqrt{(x-p)^2 + y^2} \] Distance from Point to Directrix The directrix is a vertical line given by \x = -p\. The distance from the point \ (x, y) \ to the directrix is: \[d_2 = \|x - (-p)\| = \|x + p\| \] Set the Distances Equal According to the definition of a parabola, these distances are equal: \[ \sqrt{(x-p)^2 + y^2} = \|x + p\| \] Square Both Sides To eliminate the square root, square both sides of the equation: \[(x-p)^2 + y^2 = (x + p)^2 \] Expand and Simplify Expand both sides of the equation: \[(x-p)^2 + y^2 = (x + p)^2\ \rightarrow (x^2 - 2px + p^2) + y^2 = (x^2 + 2px + p^2) \] Subtract $x^2 + p^2$ from both sides: \[-2px + y^2 = 2px\] Solve for \x\ Combine like terms and solve for \x\: \[-2px + y^2 = 2px\ \rightarrow y^2 = 4px\ \rightarrow x = \frac{1}{4p}y^2 \] Identify Constant \a\ Rewrite the equation in the required form $x = ay^2$ by identifying that $a = \frac{1}{4p}$ Key Concepts These are the key concepts you need to understand to accurately answer the question. focus and directrix A parabola is a unique geometric shape that can be defined using just a point, called the focus, and a line, known as the directrix. In our given problem, the focus is located at $ (p, 0) $ and the directrix is the vertical line $ x = -p $. The parabola consists of all points $ (x, y) $ that are equally distant from the focus and the directrix. This means that every point on the parabola satisfies the condition where the distance to the focus is the same as the distance to the directrix. Understanding this fundamental property of parabolas is key to solving problems related to their equations and graphs. distance formula The distance formula is a critical tool in geometry that helps us calculate the distance between two points in the coordinate plane. The formula is given by \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. For our topic, we need to apply this formula twice: • First, to find the distance from a point $ (x, y) $ on the parabola to the focus $ (p, 0) $. • Second, to find the distance from the point $ (x, y) $ to the directrix $ x = -p $. Calculating these distances is essential to set up our equation for finding the general form of the parabola. squaring equations Sometimes, we encounter equations involving square roots, and solving them often requires squaring both sides. In our case, the equation \sqrt{(x - p)^2 + y^2} = \|x + p\|\ involves a square root on one side. To eliminate the square root, we square both sides: \[ (\text{Left Side})^2 = (\text{Right Side})^2 \] i.e., \[ (x - p)^2 + y^2 = (x + p)^2 \] This step is crucial because it transforms our equation into a more solvable form, free of square roots. However, it's important to expand and simplify correctly to avoid any algebraic errors. solving quadratic equations Once we have the equation free of square roots, we can solve it like any other quadratic equation. From our expanded equation: \[(x - p)^2 + y^2 = (x + p)^2\] we further simplify: \[(x^2 - 2px + p^2) + y^2 = (x^2 + 2px + p^2)\] Subtracting \ x^2 + p^2\ from both sides gives: \[-2px + y^2 = 2px\] Adding \ 2px \ to one side leads to: \ y^2 = 4px \. Finally, by solving for \ x \, we get: \ x = \frac{1}{4p} y^2 \. This equation \ x = ay^2 \ where \ a = \frac{1}{4p} \ is the standard form of a parabola opening sideways.	{"url":"https://www.vaia.com/en-us/textbooks/math/algebra-for-college-students-5-edition/chapter-12/problem-93-consider-the-parabola-with-focus-p-0-and-directri/","timestamp":"2024-11-08T02:29:21Z","content_type":"text/html","content_length":"253363","record_id":"<urn:uuid:661faa17-6a1e-4226-b9db-0acd1cb0820f>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00546.warc.gz"}
Math Colloquia - The logarithmic singularities of the Green functions of the conformal powers of the Laplacian Motivated by the analysis of the singularity of the Bergman kernel of a strictly pseudoconvex domain, Charlie Fefferman launched in the late 70s the program of determining all local biholomorphic invariants of strictly pseudoconvex domain. This program has since evolved to include other geometries such as conformal geometry. Green functions play an important role in conformal geometry at the interface of PDEs and geometry. In this talk, I shall explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Laplacian arising from scattering theory for asymptotically hyperbolic Einstein metrics. The results are formulated in terms of explicit conformal invariants defined by means of the ambient metric of Fefferman-Graham. Although the problems and the final formulas only refer to analysis and geometry, the computations actually involves a lot of representation theory and ultimately boils down to some elaboration on Schur's duality.	{"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&page=9&sort_index=Time&order_type=desc&l=en&document_srl=105736","timestamp":"2024-11-08T05:56:52Z","content_type":"text/html","content_length":"44447","record_id":"<urn:uuid:3b223e86-cbba-4575-802f-71d8417c9fd6>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00234.warc.gz"}
CZ POLL - Who do you think will win the Commanders-vs-Cowboys game and by how much? (week 4) Not open for further replies. Reaction score Who do you think will win the game and by how much? You can also post your specific score predictions as well as comments about the game in this thread. Reaction score Reaction score Who do you think will win the game and by how much? You can also post your specific score predictions as well as comments about the game in this thread. 24-10 Dallas Reaction score This could be a trap game, but I think the defense knows it can get even better. My early prediction Dallas 27 Washington 20 Reaction score Reaction score Cowboys by a couple scores. Washington is a mess. They’d be better off if they listened to my idea about naming their team the Presidents with a logo of Teddy Rosevelt arm wrestling a grizzly bear. I’m thinking 24-10 Dallas Reaction score Wentz bounces back, Boys read their hype. Was - 23 Cowboys 20 Reaction score Reaction score How funny that Wentz is not even the best ginger QB in the NFC East. Reaction score Parsons makes Wentz wince and the Cowboys seize victory. Reaction score Jones is a mobile QB and got beat up. I can’t wait to see Wince try to evade the defense! Reaction score Cowboy , It’s a way of life. Reaction score Cowboy , It’s a way of life. Reaction score Jones is a mobile QB and got beat up. I can’t wait to see Wince try to evade the defense! I can’t wait. Don’t let Rison see this he already has Washington winning Reaction score Reaction score Who do you think will win the game and by how much? You can also post your specific score predictions as well as comments about the game in this thread. I clicked for the commodes by 4, I meant Cowboys by 4. Go Cowboys Reaction score Wentz bounces back, Boys read their hype. Was - 23 Cowboys 20 Naw, this defense keeps it under 20. Boyz by 14. Reaction score Washington only exists these days to give us 2 easy wins in the NFC East. 13-5 the last 18 games and that’s with a lot of down years on our side too. I hope Dan Snyder continues his iron grip on operations. Reaction score This could be a trap game, but I think the defense knows it can get even better. My early prediction Dallas 27 Washington 20 Big trap game. A win though at 3-1 would be a hell of a start with the issues we had game one. Reaction score This game should be cake, the Comms are straight up trash. Dallas 21-10 Not open for further replies.	{"url":"https://cowboyszone.com/threads/who-do-you-think-will-win-the-commanders-vs-cowboys-game-and-by-how-much-week-4.500412/","timestamp":"2024-11-04T13:58:37Z","content_type":"text/html","content_length":"137334","record_id":"<urn:uuid:38708cc8-5ca6-4144-9417-ce8a98928695>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00113.warc.gz"}